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+Flexible Supervised Autonomy for Exploration in
+Subterranean Environments
+Harel Biggie∗
+Computer Science
+University of Colorado Boulder
+Harel.Biggie@colorado.edu
+Eugene R. Rush∗
+Mechanical Engineering
+University of Colorado Boulder
+Eugene.Rush@colorado.edu
+Danny G. Riley
+Computer Science
+University of Colorado Boulder
+Dan.Riley@colorado.edu
+Shakeeb Ahmad
+Aerospace Engineering Sciences
+University of Colorado Boulder
+Shakeeb.Ahmad@colorado.edu
+Michael T. Ohradzansky
+Aerospace Engineering Sciences
+University of Colorado Boulder
+Michael.Ohradzansky@colorado.edu
+Kyle Harlow
+Computer Science
+University of Colorado Boulder
+Kyle.Harlow@colorado.edu
+Michael J. Miles
+Mechanical Engineering
+University of Colorado Boulder
+Mike.Miles@colorado.edu
+Daniel Torres
+Computer Science
+University of Colorado Boulder
+Daniel.TorresDominguez@colorado.edu
+Steve McGuire
+Electrical & Computer Engineering
+University of California Santa Cruz
+steve.mcguire@ucsc.edu
+Eric W. Frew
+Aerospace Engineering Sciences
+University of Colorado Boulder
+Eric.Frew@colorado.edu
+Christoffer Heckman
+Computer Science
+University of Colorado Boulder
+Christoffer.Heckman@colorado.edu
+J. Sean Humbert
+Mechanical Engineering
+University of Colorado Boulder
+Sean.Humbert@colorado.edu
+Abstract
+While the capabilities of autonomous systems have been steadily improving in recent years,
+these systems still struggle to rapidly explore previously unknown environments without the
+aid of GPS-assisted navigation. The DARPA Subterranean (SubT) Challenge aimed to fast
+track the development of autonomous exploration systems by evaluating their performance
+in real-world underground search-and-rescue scenarios. Subterranean environments present a
+plethora of challenges for robotic systems, such as limited communications, complex topology,
+visually-degraded sensing, and harsh terrain. The presented solution enables long-term
+autonomy with minimal human supervision by combining a powerful and independent
+*These authors contributed equally to this work.
+arXiv:2301.00771v1  [cs.RO]  2 Jan 2023
+
+single-agent autonomy stack, with higher level mission management operating over a flexible
+mesh network. The autonomy suite deployed on quadruped and wheeled robots was fully
+independent, freeing the human supervision to loosely supervise the mission and make
+high-impact strategic decisions. We also discuss lessons learned from fielding our system at
+the SubT Final Event, relating to vehicle versatility, system adaptability, and re-configurable
+communications.
+1
+Introduction
+Despite a myriad of developments in sensing, planning, control and state estimation over the last few decades,
+deploying robots in harsh subterranean environments for the purpose of rapid situational awareness presents a
+number of new challenges to robot autonomy. Traditionally, robots rely on a number of complex, interconnected
+sub-processes, such as localization, mapping, and planning, to navigate unknown environments. Maintaining
+accurate state estimates, a process critical to mapping and exploration, is exceptionally challenging in
+subterranean environments. GPS is unavailable for obtaining position estimates and visual-based localization
+methods can be affected by varied lighting conditions and environmental factors such as heavy dust, fog,
+or smoke.
+Subterranean environments, such as mines and caves, are often unstructured and contain
+hazardous obstacles, making navigation with ground vehicles challenging. Additionally, aerial vehicles can
+be exceptionally difficult to deploy in tight constrained underground spaces due to self-induced propeller
+wash. The DARPA Subterranean Challenge (SubT) (DARPA, 2022) aimed to spark new developments in
+the areas of autonomy, perception, mobility, and networking in subterranean environments. In the following
+work, a scalable multi-agent autonomy solution for subterranean exploration developed by the University
+of Colorado’s Team MARBLE for the SubT Challenge is presented, along with critical lessons learned and
+developments made along the way.
+DARPA designed the SubT Challenge to simulate search-and-rescue scenarios in unknown subterranean
+environments, and consisted of three domain-specific circuit events, Tunnel Circuit, Urban Circuit, and Cave
+Circuit, followed by the Final Event, which was a combination of three subterranean domains. Teams were
+challenged with developing robot platforms to deploy in each of the events in search of sets of predefined
+artifacts, such as backpacks or a Bluetooth signal produced by a cell phone. Correct identifications, consisting
+of an artifact classification and location to within a 5m sphere of the ground truth location, resulted in a
+point scored. Placement in the competition was determined by the team which could score the most points
+over a series of one hour deployments. For the Final Event, teams were limited to a single “human supervisor”
+who was able to interact with the systems and visualize any incoming data. Adding to the challenge, teams
+had a limited window of time in which five team members could set up and initialize robots at the entrance
+to the course.
+Team MARBLE’s initial approach to subterranean exploration for the Tunnel and Urban circuit events is
+presented in (Ohradzansky et al., 2021). Initially, a graph-based planning and exploration strategy was
+implemented, the details of which are presented in (Ohradzansky et al., 2020). This solution is suitable
+for tunnel-like mines that have mostly planar corridor-junction structures, because the environment can be
+easily represented by a graph of nodes and edges. A scanning lidar was used to center robots in corridors
+while navigating edge sections as well as avoid obstacles in the environment. However, this approach lacked
+multi-agent coordination, resulting in significant overlap of explored regions by different agents. For the
+Urban and Cave circuit events, a three-dimensional volumetric map representation of the environment was
+generated and used in a frontier-based exploration strategy (Ahmad et al., 2021a; Ahmad et al., 2021b).
+In this approach, the exploration rate of the robot is maximized using a frontier-based (Yamauchi, 1997)
+sampling technique and a fast marching cost-to-go calculation (Sethian, 1999) to select goal poses and plan
+paths to them in three dimensional space. An artificial potential function based obstacle avoidance algorithm
+enables the robot to path follow while avoiding small obstacles in the environment. Our initial approach also
+implemented limited forms of multi-agent coordination in the form of agents sharing goal points and paths.
+
+Other teams developed impressive solutions to the initial Tunnel and Urban Circuit challenges. Team CSIRO, a
+collaboration between the Commonwealth Scientific and Industrial Research Organization (CSIRO), Emesent,
+and Georgia Tech, presents a unique homogeneous sensing solution (Hudson et al., 2021). In this approach,
+heterogeneous teams of robots, including both ground and aerial platforms, share sensor information as a part
+of a decentralized multi-agent SLAM system. Initially, exploration was handled through manual waypoints
+commanded by the human supervisor, but eventually an autonomous exploration algorithm was implemented
+(Williams et al., 2020). A common perception module, the CatPack, is used across all ground vehicles for
+easy reuse of the autonomy stack across different platforms. Similar to Team CSIRO, Team CERBERUS
+also used a heterogeneous team of ground and aerial platforms (Tranzatto et al., 2022a; Papachristos et al.,
+2019; Tranzatto et al., 2022b). In their approach, map information from different agents is fused into an
+optimized global map that is shared back to the agents (Khattak et al., 2020). Similar to the work presented
+in (Ohradzansky et al., 2020), other teams used graph-based planning approaches for global navigation (Dang
+et al., 2019; Dang et al., 2020). Other noteworthy teams and their approaches to autonomous subterranean
+navigation include Team CoSTAR (Santamaria-Navarro et al., 2020; Ebadi et al., 2020; Agha et al., 2021;
+Otsu et al., 2020), CTU-CRAS (Rouˇcek et al., 2020), CMU (Carnegie-Mellon University) (Scherer et al.,
+2021), and NCTU (National Chiao Tung University) (Huang et al., 2019). Additional discussions on the
+challenges, novel developments, and lessons learned from the Tunnel and Urban circuit events are included in
+the following works (Miller et al., 2020; Lajoie et al., 2020). One common theme common to many of these
+approaches is the use of heterogeneous teams of agents with multi-modal sensing solutions. By diversifying
+the sizes, types of locomotion, and sensor modalities of individual robots, teams can be more versatile when
+faced with varied environments, each with a unique set of challenges. This ability to be flexible and adapt to
+the needs of the mission is one of Team MARBLE’s driving philosophies.
+(a)
+(b)
+Figure 1: The fleet is composed of two classes of robotic agents: (a) Clearpath Husky A200, (b) Boston Dynamics
+Spot. Each platform carries a common sensor suite designed for exploration and object detection.
+The format of the challenge necessitated advancements in platform design, robust communication networks,
+intelligent planners, and a balance between autonomy and human decision making for robot fleets. Team
+MARBLE’s solution, which was developed over the course of three different circuit events and showcased
+at the Final Event, led to a third place finish. Specifically, we develop a heterogeneous fleet of autonomous
+robots, each capable of operating independently of human intervention. Our autonomy-first approach employs
+a lightweight graph-based planner that scales to large environments, and has been adapted to reason over
+dynamic environments, such as closing doors and passing agents, as well as take advantage of multi-agent
+coordination. Information sharing between agents is accomplished via a custom mesh networking solution with
+fast reconnect times and configurable message prioritization. The same network provides the human supervisor
+real-time visibility and control of the mission. Our autonomy-first philosophy inspired an autonomous mission
+
+HLSNYA200MESHMERIZE
+IRISS
+000
+BostonDynamig
+200
+ECEmanagement system that frees the human supervisor from agent-level micromanagement and deepens the
+opportunity to strategically accomplish mission goals. In this work we will present each of the components of
+our proposed autonomy system, as well as a detailed performance analysis of our solution and lessons learned
+along the way.
+This paper is organized with the following structure. First, an overview of our system is provided in Section 2.
+The robot platforms developed for the Final Event are described in Section 3, followed by a description of the
+localization system in Section 4, and the artifact detection system in Section 5. The multi-agent components
+of our system include the volumetric mapping pipeline in Section 6 and the graph-based path planning over
+those maps in Section 7. Information transmitted among agents, as well as to and from the human supervisor,
+is mediated by the wireless mesh network communications system described in Section 8, and handled by
+the autonomous mission management system described in Section 9. Finally, we analyze how these systems
+performed in the DARPA SubT Challenge Final Event in Section 10 and discuss lessons learned in Section 11.
+2
+System Overview
+In the following subsections we present a high-level description of Team MARBLE’s approach to the DARPA
+Subterranean Challenge. First, the general concept of operation is described, followed by an overview of each
+of the major components of the developed autonomy solution. A full high level summary of the autonomy
+system can be seen in Figure 2.
+2.1
+Concept of Operations
+Team MARBLE has emphasized development of multi-agent autonomy solutions that are able to operate
+without requiring intervention from a human operator. This aligns with the goals of the DARPA Subterranean
+Challenge where intermittent or unavailable communications with agents from a base station or between agents
+is expected. Therefore, our solution is centered around robust single-agent autonomy, where independent
+robots are able to explore unknown environments and report back to a base station with collected information
+about the environment including map data and detected artifact locations.
+In communication-limited
+environments where information sharing with other agents and the human supervisor may not be available,
+our agents persist and continue to execute the mission.
+While it is important for single, independent agents to be able to explore autonomously, our solution
+incorporates several multi-agent components to improve exploration efficiency. Our fleet is also designed to be
+opportunistic, capitalizing on communication links when they are available to amplify fleet performance. Multi-
+agent coordination is an auxiliary capability that reduces redundant efforts when agents enter communication
+range with one another, and inform each other where they have been and where they plan to go next.
+Our system’s performance can further be improved when communications are available which enables the
+human supervisor to have a holistic perspective of the specific search-and-rescue scenario. This holistic
+perspective empowers the human to make high-level contextual decisions through two types of intervention:
+directing the agent to a specific location by commanding a high-level waypoint, or teleoperating the agent by
+commanding low-level velocity signals. During the 60-minute final event prize run, Team MARBLE’s robot
+fleet was completely autonomous, with the exception of five strategic low-level human supervisor interventions.
+The balance between human input and autonomy is further discussed in Section 9.
+2.2
+Perception
+A modular perception suite, shown in Figure 5b, was designed as the basis for the autonomy stack. The
+primary sensor is the Ouster OS1-64 lidar (Light Detection and Ranging), which provides 3D point clouds for
+mapping, localization, semantic mapping, and obstacle avoidance. A LORD Microstrain 3DM-GX5-15 IMU
+
+2D Lidar
+3D Lidar
+IMU
+RGB Cameras
+Bluetooth
+CO2
+Other Robots
+(MADCAT Instance)
+Localization
+LIO-SAM
+Mapping
+marble mapping
+2D Object Detection
+YOLO
+3D Artifact
+Localization
+Terrain Assessment
+(Husky Only)
+Stair Detector
+(Spot Only)
+Planning
+scan plan
+Path Following
+pure pursuit
+Mission Management
+BOBCAT
+MADCAT
+Comms
+udp mesh
+Base Station
+(Human Supervisor)
+Artifact Reports
+Other Robots
+(MADCAT Instance)
+Maps & Position History
+Velocity Commands
+Input
+Input (Multiagent)
+Output
+Output (Multiagent)
+Software Module
+Human Input
+Figure 2: Overall block diagram showing the high level functionality of the autonomy stack. Inputs are shown in
+green and outputs are shown in red. Software package names are italicized and inputs & outputs which are shared
+between agents are outlined in blue. Terrain assessment and stair detection both add semantic information to the
+map but are only run on the Husky and Spot respectively.
+(Inertial Measurement Unit) is used to measure linear and angular acceleration of the sensor head for use
+in lidar-inertial state estimation. To identify visual artifacts, the systems are equipped with several FLIR
+Blackfly PGE-05S2C-CS cameras and an array of 5W dimmable LEDs for self illumination. The Husky
+platforms were equipped with four cameras facing forward, backward, to the left, and to the right. The Spot
+robots had a similar configuration, save omitting the rear camera due to occlusions caused by the custom-built
+compute and power management system.
+2.3
+Localization
+Localization provides consistent pose information for many downstream autonomy processes including
+volumetric mapping, path planning, artifact detection, and multi-agent coordination. However, ensuring
+reliable localization is difficult in austere underground environments. Because conventional vision-based
+solutions can be unreliable due to dark, feature-poor settings, Team MARBLE utilized lidar-based methods
+and specifically tested and integrated LIO-SAM (Shan et al., 2020), which has fast online loop closures during
+long-duration missions. Several modifications were made to the system to improve localization accuracy and
+reliability, which are further discussed in Section 4. Methods used to align the robots with into a common
+reference frame are also presented in this section.
+2.4
+Exploration
+The exploration algorithm generates safe and traversable paths that lead agents toward unexplored areas
+of previously unseen environments. The developed sampling based path planning algorithm is designed
+to be lightweight, so that it can operate on rapid exploration timescales, regardless of the extent of the
+environment. Computational efficiency is achieved by employing a bifurcated local-global graph for sampling
+unseen frontiers as well as a good enough strategy for final selection. In such time-constrained search and
+rescue settings, even humans will often make rapid decisions rather than dwell for long periods of time to
+make globally optimal ones. The details of the planning algorithm is described in Section 7.
+The planner is also constructed to be flexible, so that additional capabilities could be scaffolded on top of the
+core algorithm. Integration of the planning algorithm with semantic mapping was critical for rough terrain
+and stair traversal, as explained in Section 6. Section 7.4 explains how this planning algorithm re-plans in
+dynamic environments, whether due to doorways that are being opened or closed, or fellow agents that are
+
+passing by. This capability is crucial for robust operation in real-world environments, which cannot assumed
+to be static. Finally, the planner is able to run more efficiently with multiple robots using multi-agent
+coordination is covered in Section 7.3.
+Agents follow paths via a modified pure pursuit controller (Coulter, 1992). The yaw rate command is
+computed by comparing the current agent’s heading against a lookahead point that is a fixed distance along
+the path. Forward speed is regulated based on local proximity to obstacles in the environment and the
+relative heading error to the lookahead point. This results in slower speeds when agents are in cluttered
+environments or experiencing large heading errors. A 2D (RP Lidar) was used for local obstacle avoidance on
+the Husky platform and the Spot platform had built in obstacle avoidance.
+2.5
+Mapping
+Team MARBLE’s mapping framework is based on the open source Octomap package (Hornung et al., 2013),
+and has been customized with additional capabilities including map merging, transmission of difference maps,
+and encoding of semantic information. The core of the mapping framework is a log-odds based probability
+metric for occupied and unoccupied voxels or cells. These cells provide a 3D representation of the environment
+that is later used for navigation. This flexible framework enabled transmission of key environmental features
+such as rough terrain and the location of stairs efficiently through a bandwidth limited communication system.
+The details of our mapping system are provided in Section 6.
+2.6
+Artifact Detection
+The artifact detection system precisely localizes visual artifacts using RGB sensors for visual classification
+and detection and lidar for the depth estimation. Non-visual artifacts such as cellphones and gas are localized
+based on the position of the robot. A weighted median filter fuses together all detections which are sent to
+the human supervisor for final validation as described in Section 5.
+2.7
+Communication Systems
+A mesh networking solution transmits data between robots and back the base station for the human supervisor
+to review. Standard 2.4ghz 802.11 wireless radios based on the ath9k chipset are used for the physical layer.
+The wireless radios are embedded in beacons that can be deployed from the back of the Husky platforms,
+allowing for ad-hoc mesh networks to be established. Meshing technology was provided by Meshmerize (Pandi
+et al., 2019) and a custom UDP based transport layer (udp mesh) was developed. Details of this innovative
+layer can be seen in Section 8.
+2.8
+Multi-Agent Coordination
+Exploring unknown environments with multiple agents can be made more efficient through coordination,
+especially when agents are not within communication range. Sharing information across agents, such as
+explored regions, discovered artifacts, and current behavioral states, allows for more intelligent management
+of multi-agent exploration. The framework called Multi-Agent Data Collaboration for Autonomous Teams
+(MADCAT), provides the multi-agent data sharing capabilities required for the Subterranean Challenge
+mission (Riley and Frew, 2021), including transmission of relevant coordination data and maps, as well as map
+merging functionality and decision making for each agent. Additionally, MADCAT implements Behaviors,
+Objectives and Binary states for Cooperative Autonomous Tasks (BOBCAT), originally presented in (Riley
+and Frew, 2022), for high-level autonomy, decision making, and interfacing with the human supervisor. The
+MADCAT algorithm is discussed in more detail in Section 9.
+
+2.9
+Mission Management
+The human supervisor is able to monitor the fleet’s progression through the unknown subterranean environment
+using a custom GUI operating on a computer at the entrance to the environment (base station). Current
+mission status of all agents in the field (‘Reporting’, ‘Exploring’, ‘Home’, etc.) as well as their location in the
+global map are displayed whenever robots are within communication range of the base station. Additionally,
+the goal point and goal path for each robot is visible, allowing the supervisor to see the intent of each robot.
+The human supervisor is able to take over control of a given agent by either sending manual goal points or
+tele-operating the vehicle using a joystick interface with the base station. Reported artifacts are displayed per
+robot, with the type (survivor, cell phone, backpack, helmet, rope), position, confidence, submission result,
+and an image. The GUI also enables the modification of artifact classes and locations prior to submission to
+the DARPA scoring server. An example of the GUI interface is shown in Figure 3.
+Figure 3: Example of the Human Supervisor’s interface. The custom GUI is on the left, showing a received artifact
+image. The middle is the multi-agent RViz view, with all robots and the complete merged map. The right is a
+third-person follower for each robot (two in this case) with that robot’s original unmerged map.
+3
+Platform Development
+For the final event of the SubT Challenge, Team MARBLE deployed a heterogeneous fleet consisting of
+two Clearpath Husky A200s (H01, H02) and two Boston Dynamics Spots (D01, D02). Examples of each
+platform are shown in Figure 1. The Husky platforms are four-wheeled skid-steer ground vehicles capable of
+carrying heavy payloads, while the Spot quadrupedal “dog” platforms are agile, capable of climbing staircases
+and traversing uneven terrain. The Husky platform is robust and stable, with its generous payload budget
+allowing it to carry six communication beacons. The intended deployment strategy is to first deploy the
+Spot platforms to maximize exploration, and then the Husky platforms to establish the mesh communication
+network.
+For processing power, each Husky is equipped with a 32-core AMD Ryzen CPU equipped with 128GB of
+RAM and 4TB of SSD storage, integrated into a complete platform as shown in Figure 4a. Dual NVIDIA
+GTX 1650 GPUs were used to accelerate object detection inference speed. The primary computer on the
+Spot platforms is an AMD Ryzen 5800U with 64GB of RAM and 2TB of storage which is paired with
+a Jetson Xavier AGX to process the camera streams and perform artifact detection, following a similar
+integration as shown in Figure 4b. Many purpose-built components are common between platforms to reduce
+field maintenance efforts and platform-specific code. Each system is outfitted with a custom power system,
+discussed in Section 3.2, which enables the ability to switch from a wired shore power supply to the onboard
+computer batteries. This leads to more efficient use of the onboard batteries, which are a limiting factor in
+
+interatMorneCameraselet
+2D Nav Goll PubhshPoint+-,*
+nterktMoveCamera
+Image
+合
+AUHONIZED
+backpack:0.70
+Submissions
+wall Time:1600896291.47wallElapsed:400.93the duration of field testing deployments.
+Batteries (4x)
+Shore Power Inlet
+Power
+Management
+Relays
+Platform Control
+EStop Control
+Update Button
+DARPA EStop
+EStop Status
+EStop Button
+Status Display
+Shore Button
+Motion Control
+Main CPU
+Power Rail
+Lidar
+Radios
+Ethernet Switch
+Lighting
+& Camera
+Control
+Area Lighting
+GigE Cameras (4x)
+Beacon
+Deployment
+Host USB
+Mode LEDs
+RPLidar
+Bluetooth
+CO2
+PicoFlexx (2x)
+IMU
+Component
+Custom HW
+Custom SW
+Custom HW & SW
+Power
+GPIO
+Serial
+USB
+USB3
+CAN
+Ethernet
+(a)
+Batteries (2x)
+Shore Power Inlet
+Power
+Management
+EStop Control
+DARPA EStop
+EStop Status
+EStop Button
+Status Display
+Shore Button
+NUC
+Xavier AGX
+Spot Base
+Power Rail
+Lidar
+Radio
+Ethernet Switch
+Lighting
+& Camera
+Control
+Area Lighting
+GigE Cameras (3x)
+Bluetooth
+CO2
+IMU
+Mode LEDs
+Component
+Custom HW
+Custom SW
+Custom HW & SW
+Power
+GPIO
+Serial
+USB
+USB3
+CAN
+Ethernet
+(b)
+Figure 4: Power and signal routing diagrams of customized (a) Husky and (b) Spot platforms.
+3.1
+Communication Beacons
+Underground environments provide limited line-of-sight capabilities for wireless communications. As a
+result, Team MARBLE developed custom communication beacons to complement the custom multi-robot
+coordination solution. This allows for robots to share information with the base station and other robots in
+the field. Each Husky platform is capable of carrying six beacons, each containing a single 2W Doodlelabs
+802.11n radio as seen in Figure 5a. The autonomous beacon deployment mechanism relies on a latching
+solenoid release coupled with a novel passive system to gently lower each beacon to the ground to ensure
+maximum antenna height. Additional design details of the communication beacons can be found in Section
+13.1 of the Appendix.
+(a)
+(b)
+Figure 5: Final robot configurations, with (a) a deployment mechanism loaded with six communication beacons on
+Husky vehicles at Final Event and (b) a modular perception suite installed on both Huskies and Spots.
+
+MARBLE
+HUSKYA2U0TM
+HO1
+LORD SENSING
+icroStrai
+eneerin64-BeamOusterLidar
+LordMicrostrainIMu
+5WLED
+FLIR GigE
+Cameras3.2
+Power and Platform Control Systems
+In order to support each platform’s sensor and processing needs, as well as meet DARPA equipment
+requirements for emergency stop systems, the systems integration efforts relied on several custom hardware
+and software components. Where feasible, these components are shared between the Husky, shown in Figure
+1a, and the Spot platforms, shown in Figure 1b, reducing the development and validation efforts, as well as
+team member operational training requirements.
+Of the custom capabilities developed, the power management subsystem deserves special mention. This
+component implements a hardware-interlocked, ideal diode system to permit downstream electronics to
+source power from either a wall-connected power supply, referred to as shore power, or onboard batteries. By
+switching to shore power, onboard systems can remain powered for development, testing, and analysis while
+the batteries are charged without carrying load. Further, the ideal diode component allows the battery packs
+to load share and charge independently. In contrast to a bus-tied battery system, this ideal diode design
+prevents high-energy charge equalization between packs and allowed each battery pack’s onboard management
+board to function independently. The system also enables live monitoring of current consumption and battery
+voltage, as well as intelligent e-stop management which ensures the robot cannot exit the emergency stopped
+state while connected to shore power.
+Emergency stop requirements dictated that each platform needed the ability to be stopped by a physical
+button, software, and via a DARPA deployed Xbee network. On the Husky control system the emergency
+stop system was integrated directly into the base controller. In contrast, the Spot platform’s emergency stop
+tied into the available API to issue a the robot a “sit” command before terminating power to the motors
+which allowed the robot to be stopped gracefully.
+Several important lessons learned emerged after three years of platform architecture development. We have
+highlighted the most critical lessons below and provide more details about the design and consequences of
+our platform compute systems in the Section 13.2 of the Appendix.
+As part of our field testing campaign, we uncovered an issue where our USB-connected IMU was delayed in
+delivering measurements critical to localization performance. From an integration perspective, our IMU’s
+USB interface was implemented using a standard USB Communications Data Class Abstract Control Model
+(CDC-ACM) interface. Using CDC-ACM for IMU measurements was particularly problematic due to the way
+in which CDC-ACM uses bulk transport. USB has several methods of transferring data from device to host,
+including interrupt, isochronous, and bulk transport. CDC-ACM uses bulk transport, which does not include
+any guarantee for on-time delivery of data. As a consequence, during high CPU load, IMU measurements were
+occasionally delayed and resulted in localization error. In contrast, interrupt and isochronous transports are
+regularly serviced and can deliver on-time data. This problem could be solved in future deployments by either
+replacing the bulk interface with an interrupt interface or by using legacy serial interfaces such as RS-232
+(not typically available on small form factor computing units). However, in practice we found adjusting the
+IMU timing as described in Section 4, was sufficient and did not require engineering new firmware.
+Another critical piece for reliable localization and consequently navigation is sensor synchronization. Synchro-
+nization is fundamentally necessary in order for our onboard sensors to communicate with their respective
+computers, and for those computers onboard individual agents to communicate with each other and the base
+station computer. The technical implementation of our solution is detailed in Section 13.3 of the Appendix.
+4
+Localization
+One of the major challenges in the DARPA Subterranean Challenge is ensuring reliable localization across a
+diverse set of austere environments. Localization is a critical process for an autonomous system, as it provides
+pose information to downstream autonomy processes including volumetric mapping, path planning, artifact
+
+detection, and multi-agent coordination. Section 4.1 details the simultaneous localization and mapping
+solution that was integrated into the autonomy stack, and Section 4.2 describes the process used to align all
+robots to the common DARPA reference frame.
+4.1
+Simultaneous Localization and Mapping
+Simultaneous localization and mapping has relatively mature vision-based solutions (Leutenegger et al., 2015;
+Nobre et al., 2017; Qin et al., 2018), thanks to advances in feature extraction (Cheung and Hamarneh, 2009;
+Bay et al., 2008; Zhan et al., 2018). However, in mission-critical applications such as underground search and
+rescue, visual-inertial solutions are not reliable enough when faced with irregular lighting, specular highlights,
+and feature-poor scenes. Recent work has illuminated the possibility of leveraging thermal-based odometry
+estimation in visually degraded environments (Khattak et al., 2019; Wisth et al., 2021).
+Because underground environments are typically rich in geometric features, lidar-based localization solutions
+are a compelling alternative. Some spaces though, such as a smooth tunnels and corridors, contain relatively
+few longitudinal features, and therefore pose limits to lidar-based perception. Single-echo lidar also struggles
+in austere environments containing fog or smoke, though some recent work has focused on addressing these
+limitations (Shamsudin et al., 2016).
+For the Final Event, Team MARBLE transitioned from Google Cartographer (Hess et al., 2016) to LIO-SAM
+(Shan et al., 2020), since its faster online loop closures during long-duration missions results in greater
+localization accuracy. Extensive testing was conducted in many different environments including parking
+garages, academic buildings, gold mines, and outdoor environments. The fast, lightweight loop closure
+performance can be attributed to performing scan-matching on a local level rather than a global level.
+LIO-SAM additionally performs IMU pre-integration to deskew point clouds, yielding better initialization
+for lidar odometry estimation. Because localization is the foundation to many autonomy modules, it was
+imperative to validate LIO-SAM’s performance onboard the Spot and Husky platforms during large-scale,
+long-duration missions. Some examples of such validation efforts are shown in Section 13.4 of the Appendix.
+Several modifications are made to the system to improve localization accuracy and reliability. First, the
+IMU and lidar sensors are fastened to a 6061 aluminum sensor plate, with a mounting configuration that is
+common between Huskies and Spots. By specifying the relative transform between the two sensors to a high
+precision, the need for extrinsics calibration is reduced. Specifically, the mounting configuration consists of a
+tight-tolerance, precision-ground plate with a flatness tolerance of 0.005”, which greatly improves the roll
+and pitch alignment between the two sensors. By using a high-quality MEMS IMU and such precise sensor
+mounting, the LIO-SAM parameter specifying how much to weight IMU roll, pitch, and yaw measurements
+relative to lidar odometry was increased by a factor of 100. Taken together, these modifications greatly reduce
+accumulated rotation and translation drift, enabling smooth autonomous operation across long missions.
+Secondly, LIO-SAM requires sensor timestamps to be aligned and sensor rates to be consistent. In particular,
+if IMU message rates fluctuate too greatly, the IMU pre-integration factors (Forster et al., 2015) can fail
+and lead to LIO-SAM instability. To reduce sensitivity to fluctuating IMU sensor rates caused by USB
+transmission delays (described in Section 3.2), the IMU timestamp assignment is adjusted when messages are
+not received within 15% of the nominal rate. Additionally, the lidar sensor is synchronized with the onboard
+computer via PTP as discussed in Section 13.3 of the Appendix. These two timing solutions reduce the
+probability of erroneous measurements and greatly improve the stability of LIO-SAM.
+4.2
+Common Reference Frame Alignment
+Accurate multi-robot alignment is a core design decision for the MARBLE localization, mapping, and planning
+systems. Robots are required to share globally aligned map data for planning and navigation. In addition,
+it allows the human supervisor and robots to share global coordinates for artifact locations relative to the
+
+Figure 6: Figure of the MARBLE gate alignment setup including the Leica Total Station (LTS), gate, and an example
+robot. Transforms from the LTS to the robot TRL, from the LTS to the world frame TW L, and the resulting transform
+from the world frame into the TW R
+DARPA-provided world frame. In order to align with the DARPA frame, Apriltags (Malyuta et al., 2019;
+Brommer et al., 2018; Wang and Olson, 2016), retro-reflective targets, and Leica Total Station (LTS) reflectors
+are attached to a gate with relative transforms to the DARPA origin frame. The global frame was assumed to
+be aligned with gravity, but each team was responsible for aligning yaw, and XYZ-translation from their robots
+to the common DARPA frame. In the context of the SubT Challenge, the DARPA frame was purely used to
+align robots into the measured ground truth frame for artifact scoring and map accuracy analysis. However,
+in practice, an accurate initial alignment between robots results in more reliable multi-agent coordination
+and global merging maps.
+In order to align with the common reference frame, MARBLE primarily relied on the LTS reflectors. Based
+on conventional trigonometry and the assumption of needing to maintain less than 5m of error over the course
+of a 1km linear distance, it is determined that an initial alignment target required less than 0.29◦ of error.
+To align the robot, 3 reflective prisms are attached to each robot, and their positions are scanned with an
+LTS. These points VLR are then compared with a ground-truth set VR, determined by the relative locations of
+the prisms to the robots tracking frame via CAD. These two sets of points are used to estimate the transform
+between the LTS recorded positions and the assumed positions by minimizing across the pose X to solve:
+argminX
+�
+||VLR − VRTRL||.
+(1)
+The result is the robot’s position in the LTS frame TRL. An additional calculation is used between scanned
+points of the gate VLG and the provided coordinates VW are used to solve for the gate’s position in the Leica
+frame TW L using the equation:
+argminX
+�
+||VLG − VW TW L||.
+(2)
+Both minimization problems were solved using Horn’s absolute orientation method (Horn, 1987), a closed
+form solution to least squares alignment problems. Given these transforms, the robots position in the world
+frame was calculated by inverting the robot to LTS transform:
+TW R = TW L(TRL)−1.
+(3)
+To further reduce the impact of minor errors in either prism localization or low observability, these transforms
+
+WR
+TRL
+TwLare altered slightly by each robot. The LTS-predicted pitch and roll is substituted with an estimated pitch
+and roll from the lidar-inertial localization system, largely based on the initial measurements of the IMU.
+After these adjustments, yaw estimates had the largest impact on our resulting transforms. Because yaw
+error has the potential to propagate to large translational discrepancies at far distances, it became imperative
+to modify our system. The solution involves increasing the lateral spacing of the prisms mounted on the
+robots, and is described in more detail in Section 13.5 of the Appendix.
+5
+Artifact Detection
+A core component of the SubT Challenge is the detection and localization of objects that could potentially
+indicate human presence. Each artifact needed to be reported within a 5m radius of the ground truth location.
+To achieve this requirement a lidar-inertial based state estimator as described in Section 4 is used. Robots
+are put into a common reference frame based on survey-grade measurements from a Leica Total Station
+(LTS) and objects are projected using the mapping framework described in Section 6. The available sensing
+modalities for various artifacts are discussed in Section 5.1, the visual detection system is described in Section
+5.2, and the non-visual detection system is explained in Section 5.3. The resulting performance of the artifact
+detection system at the Final Event is detailed in Section 10.4.
+5.1
+Sensing Modalities
+Table 1 shows the classes of artifacts present at the final event along with the types of sensing modalities
+capable of detecting each artifact. Each robot in the fleet is equipped with RGB cameras, Bluetooth modules,
+and CO2 sensors which enable the detection of all classes of artifacts using a minimal sensor suite. The
+visual detection system is not trained to detect either the cell phone, due to its small form factor, or the cube
+artifact which was detectable using Bluetooth. The cube artifact had rotating colors which pose significant
+challenges for visual detection methods.
+Artifact Class
+Visual
+Thermal
+Wireless
+CO2
+Survivor
++
+–
+Cell Phone
+–
++
+Backpack
++
+Drill
++
+Fire Extinguisher
++
+Gas
++
+Vent
++
+–
+Helmet
++
+Rope
++
+Cube
+–
++
+Table 1: Sensing modalities, that Team MARBLE utilized (+) and did not utilize (–) for detecting the ten artifact
+classes. Blank entries indicate sensing modalities that are not useful for detecting specific artifact classes.
+
+CO
+25.2
+Visual Detection
+Visual object detection is a well-researched problem in computer vision and state of the art detectors are
+capable of identifying objects in both 2D and 3D. Common 2D detectors are typically based on Convolutional
+Neural Networks (CNN) (Zou et al., 2019), such as region proposal-based networks like Fast R-CNN (Girshick,
+2015). Typically these networks require multiple passes over an image to classify an object and then detect
+where the object is in the image. In contrast, YOLO (Redmon et al., 2016) performs both classification and
+detection in a single regression making it a significantly faster detection: 0.5 FPS for Fast R-CNN and 45
+FPS for YOLO. Object detectors operating in 3D typically use point clouds obtained from a lidar and until
+recently were limited to classification rather than full detection (Maturana and Scherer, 2015; Qi et al., 2017).
+Extensions to 3D classifiers such as Voxelnet (Zhou and Tuzel, 2018) and PointRCNN (Shi et al., 2019) are
+capable of performing object detection on powerful GPUs. These GPUs are impractical from both a size
+and power consumption standpoint for mobile robots. We selected the Yolo V3 (Redmon and Farhadi, 2018)
+model due to the fast and accurate nature of the YOLO (Redmon et al., 2016) family of networks.
+Specifically, for classification and detection, the visual pipeline utilizes a YOLO V3 Tiny (Redmon et al.,
+2016) model with custom trained weights. The model is optimized for Nvidia TensorRT acceleration and
+we infer images at a resolution of 608x608. The Husky platforms are able to perform inference at 60FPS
+on a GTX 1650 based on Nvidia’s Turing architecture with 896 CUDA cores and 112 RT cores. The Spot
+platform uses a Nvidia Jetson Xavier AGX based on the Volta architecture with 512 CUDA Cores and 64
+Tensor cores to perform inference at 40 FPS. These GPUs were chosen to balance performance against size
+and power constraints for on-board compute. The TensorRT YOLO detector outputs a message containing
+the detected artifacts as well as the coordinates of their bounding boxes.
+A systematic procedure targeted at low-light conditions is used to train the model. At each location, data was
+collected using three different brightness levels to minimize the impact of lighting conditions on the model’s
+performance. Specifically, images were taken from past circuit events as well as separate field exercises.
+Images with excessive motion blur were subsequently filtered out and the data was later augmented with
+images that contained false positive defections. The full details of our training procedure can be found in
+Section 13.6 of the Appendix.
+Depth registration is performed using marble mapping as described in Section 6 which is generated by the
+Ouster 64-beam lidar. Utilizing an Octomap based framework allowed us to avoid implementing any additional
+filtering due to the probabilistic nature of the map. Additionally, the Octomap structure aggregates scans
+into the map with temporal memory. This important feature resolves the inconsistency between the 33.2◦
+vertical field of view of the Ouster and the 68◦ vertical field of view of the cameras. At further distances, the
+agent is able to incrementally build out regions near ceilings and floors, overcoming the vertical blind spots of
+the Ouster. Essentially, this temporal memory allows us to decouple the depth measurement from the visual
+artifact detection. The biggest drawback of this approach is the potential for an additional 0.15m of error on
+each detection due to the voxel resolution. However, this error figure still falls within the design constraints
+of localizing an object to within 5m of its desired location.
+After 3D coordinates are obtained via the Artifact Localization node, we run a weighted median filter in
+the world coordinate frame to de-noise the projected location within the Artifact Fusion node in Figure 7.
+Each localized artifact is considered to be part of the same measurement if it is the same class as a previous
+measurement and within 5m of that measurement. We then require five to 10 positive detection events and
+use the median position as the reported position to the human supervisor. The final detection is published
+in a custom ROS message which contains this position as well as a compressed version of a corresponding
+camera image and associated bounding box. The full overview of the artifact system can be seen in Figure 7.
+
+Gig-E Cameras
+TensorRT YOLO
+Artifact
+Localization
+Classification and Detection
+Artifact
+Fusion
+Robot
+Localization/Mapping
+Artifact Message
+Bluetooth
+CO2
+Figure 7: Overview of the artifact detection system. Sensor inputs are shown in red and outputs are shown in green.
+5.3
+Non-Visual Detection
+Cell phone, cube, and gas reports are also fused using a weighted median filter. The Bluetooth and CO2
+detections are simply localized to the position of the robot at the time of detection. Bluetooth detections
+are also grouped together by unique SSIDs and gas detections within 10m of another detection are assumed
+to have originated from the same source. The final positioning of these non-visual artifacts relies on input
+from the human supervisor. Our human supervisor interface was designed to easily allow for movement of
+reported artifacts based on features observed in the map by the human operator. The details regarding the
+accuracy and success rate of these reports can be found in Section 10.4.
+6
+Mapping
+Team MARBLE’s custom mapping package, marble mapping (Riley, 2021) is based on Octomap (Hornung
+et al., 2013) and is used to generate 3D occupancy grid representations of the world. The environment is
+sub-divided into voxels, or cells which are marked as either occupied, free, or unknown using a probabilistic
+log-odds based model operating on sensor returns. The output of marble mapping is a direct input to the path
+planner and also provides depth measurements for visual artifact detection, as well as situational awareness
+for the human supervisor. The Octree (Meagher, 1982) structure of Octomap’s occupancy grids makes
+storing and transmitting maps more efficient than other representations such as point clouds; this efficiency is
+highly desirable when trying to transmit maps over low bandwidth mesh networks. marble mapping extends
+Octomap by enabling map differences for low bandwidth transmission, map merging between multiple robots,
+and the addition of semantic information.
+6.1
+Difference-Based Map Merging
+Despite the efficient encoding of the Octree data structure, regularly transmitting full volumetric maps of the
+explored space is impractical in bandwidth-constrained subterranean environments. Map differences are both
+a natural solution to reduce bandwidth, and have been shown to facilitate efficient data transfers (Sheng
+et al., 2004). In the marble mapping package, modifications to Octomap package were made to generate
+differences between different map sections, or “diff maps” shown in Figure 8. The implementation allows for
+diff maps, or smaller Octree structures, to be created at a predetermined rate, and contains all the mapping
+data for that time interval. The sum of an agent’s diff maps make up its “self” map and the differences can
+be transmitted to other agents. These differences are later merged into the robot’s “merged map” in the the
+map merging process which is shown in Figure 8.
+Merged maps generated from multiple agents are important both for a more complete view of the environment,
+
+Figure 8: Sequential difference maps from top left to bottom right, with the final map on the far right constructed in
+real time for comparison. The diff maps can be merged to fully reconstruct the original map shown in the bottom
+right.
+and they also reduce redundant coverage in coordination strategies (Ko et al., 2003; Simmons et al., 2000;
+Zlot et al., 2002). The marble mapping package enables map merging both for individual agents and on the
+base station which allows agents to intelligently act on the data and provides a holistic view for the human
+supervisor. The system does not re-align maps prior to merging, as it is assumes agents are already in a
+common reference frame as described in Section 4.2. The lack of a re-alignment feature has the potential to
+cause one agent’s map to block pathways in a receiving agent’s map. To mitigate this, each agent prioritizes
+its own map by only appending cells from other maps into “unknown” areas. Areas that have already been
+“seen” by the agent are left untouched which prevents misaligned data from blocking free space. In cases
+where this mitigation procedure is not enough, such as narrow hallways, or a complete loss of localization by
+an agent, “bad” map diffs can be removed by the human supervisor using the base station GUI described in
+Section 9.
+6.2
+Semantic Mapping for Terrain-Aware Navigation
+While the volumetric-based mapping produced by the Octomap framework provides the high-level structure
+of the environment, its resolution, set to a voxel size of 0.15m, is too coarse to capture details needed for
+high fidelity motion planning. In order to augment the existing marble map with terrain information, Team
+MARBLE evaluates the traversability of a given voxel using the normal and curvature values from raw point
+clouds. The planning solution is then able to utilize this semantic information to plan safe paths in Section 7.
+An additional label is attached to each voxel which enables the semantic labeling of staircases for the Spot
+platform.
+Early approaches to evaluating the traversablity of an environment include elevation based maps based on a
+2D lidar (Ye and Borenstein, 2003) but are unable to take advantage of modern 3D sensors. The traversability
+classifier presented here is largely based on the Grid Map framework presented in (Fankhauser and Hutter,
+2016), which evaluates the slope and roughness of point cloud regions to generate a multi-layer surface map
+but only creates a 2D grid rather than a 3D volumetric map. Other fielded approaches in subterranean
+environments include “virtual surfaces” on occupancy maps (Hines et al., 2021) and Conditional-Value-at-
+Risk metrics, such as collision, step size, tip over, and slippage, which are incorporated into a dense 2.5D
+
+(a)
+(b)
+Figure 9: Traversability information (a) of section within the Edgar Experimental Mine in Idaho Springs, CO, USA,
+that contains railroad tracks. Raw traversability values of the lidar point clouds (top) are shown, where white is not
+traversable, and black is traversable. Resulting semantic map (bottom) illustrated non-traversable surfaces such as
+walls in white, traversable surfaces such as the ground in black, and semi-traversable surfaces such as the railroad
+tracks in grey. Note that red voxels do not contain traversability data. An accompanying photo (b) of the section of
+mine is shown for reference.
+gridmap (Fan et al., 2021). These dense methods typically come in the form of high-resolution local maps,
+which enable more precise locomotion over varied terrain. An alternative approach presented in (Kr¨usi
+et al., 2017) computes paths with continuous curvature over raw point clouds. However, by computing
+semantic traversability information, our planning approach only required a low-resolution global map, greatly
+simplifying both mapping and planning systems and allows for sharing of semantic information between
+agents.
+6.2.1
+Traversability Classification & Map Integration
+To estimate the traversability of a voxel, we segment the 3D point cloud produced by the Ouster lidar,
+and evaluate the unit normal vector ˆn and curvature K of each point p at timestep t. All calculations
+are performed with the aid of the pcl package (Rusu and Cousins, 2011) and a traversability value, τp,t, is
+estimated for each point using Equation 4 where ˆk is the gravity-aligned up vector, (1 − |ˆn · ˆk|)3 is a measure
+of the slope of the terrain, and cnorm and ccurv are tunable parameters. The parameter values for the Final
+Event were set to cnorm = 40.0, and ccurv = 4.0, and τp,t ∈ [0, 1].
+τp,t = cnorm(1 − |ˆnp,t · ˆk|)3 + ccurvKp,t ∈ [0, 1]
+(4)
+Traversability is implemented in the Octomap framework using Equation 5 to estimate the traversability, τv,t,
+of a given voxel, v, as a function of the voxel’s occupancy probability, Pocc,v,t. The traversability estimate for
+the voxel is a linear combination its previous traversability estimate, τv,t−1, and new estimate τp,t for the
+points in the voxel. An example of this process is shown in Figure 9.
+τv,t = τv,t−1Pocc,v,t + τp∈V,t(1 − Pocc,v,t) ∈ [0, 1]
+(5)
+
+0.995486
+1.00000
+0.626.2.2
+Stair Classification & Map Integration
+Semantic information on stairs is fused into the mapping framework using the open source StairwayDetection
+(Westfechtel et al., 2018) package and a binary Bayes filter (Thrun et al., 2005). Stair classification of
+point clouds via this approach consists of 4 major steps: (1) pre-analysis, in which the point cloud is
+downsampled and filtered, normal and curvature is estimated for each point, and floor separation is performed;
+(2) segmentation via a region growing algorithm, which segments the point cloud into smooth regions; (3)
+plane extraction, in which the surfaces that make up the riser and tread regions of each stair step are
+extracted; and (4) recognition, where the tread and riser regions are connected and analyzed via a graph to
+determine whether they make up a valid set of stairs.
+(a)
+(b)
+Figure 10: Spot planning up a staircase using the estimated stair voxels in the Octomap shown in blue.
+Stair detections are integrated into the map using a similar mechanism to the log-odds probability which
+determines occupancy in octomap. A binary Bayes filter (Thrun et al., 2005), shown in Equation 6, is used
+to estimate the probability that a given voxel is a part of a staircase. The extracted points from the stairway
+detector are modeled as measurements z where Pstair(n|zstair,t) is the probability that a voxel n is part
+of a staircase. The measurement through time step t is represented by zstair,1:t as shown in Equation 6a.
+Lstair(n|zstair,1:t) as shown in Equation 6b are the corresponding log-odds probabilities which are used for
+fast updates updates to the probabilistic estimate of each voxel. More details of the log-odds formulation are
+provided in (Hornung et al., 2013; Thrun et al., 2005). Our filter is tuned to prioritize true positive detections
+with the following parameters: Pstair(n) = 0.5, Lstair,min = −2.0, Lstair,max = 3.48, Lstair,hit = 4.60,
+Lstair,miss = −0.04.
+Pstair(n|zstair,1:t) =
+�
+1 + 1−Pstair(n|zstair,t)
+Pstair(n|zstair,t)
+1−Pstair(n|zstair,1:t−1)
+Pstair(n|zstair,1:t−1)
+Pstair(n)
+1−Pstair(n)
+�−1
+∈ (0, 1)
+(6a)
+Lstair(n|zstair,1:t) = Lstair(n|zstair,1:t−1) + Lstair(n|zstair,t) ∈ [Lstair,min, Lstair,max] ,
+(6b)
+where Lstair,min ∈ (−∞, 0) , Lstair,max ∈ (0, ∞) ,
+Lstair(n|zstair,t) =
+�
+Lstair,hit > 0 on stairs
+Lstair,miss < 0 on non-stairs
+A raycast operation on the footprint of the vehicle is used to provide a binary signal indicating the robot is
+
+S18on stairs. Additionally, eigenvector decomposition is performed over each cluster of stair voxels to extract a
+straight path along the staircase. These triggers provide waypoints so that the local trajectory follower can
+navigate to the top of the staircase. It’s important to note that since stairs would generally be classified
+as non-traversable, a stair label takes precedence over a traversability label for the Spot platform, which is
+capable of walking up stairs. Additionally, this method requires sufficient lidar scans of the staircase, which is
+generally available when located at the bottom of a staircase but is not when facing the stairs leading down.
+As a result, detecting and navigating a descending staircase is not feasible with the current configuration, but
+could be with a wider field-of-view sensor or programmed forward pitching behavior of the Spot.
+The marble mapping package enables the creation of difference-based Octomaps which allows for efficient
+transmission in underground environments. Furthermore the framework provides semantic and traversability
+information which the planner utilizes to ensure the robot is able to navigate safely. Details of the planner
+are described in Section 7.
+7
+Path Planning
+Team MARBLE’s heterogeneous fleet relies on autonomous path planning onboard each agent to reduce
+the workload of the human supervisor. The path planner running onboard each agent generates safe and
+traversable paths that lead to unexplored areas. Paths are planned on the Octomap-based marble mapping
+framework described in Section 6. Team MARBLE used the same planner on all robots with the only
+difference being the collision-function depending on vehicle’s class. For instance, a wheeled robot cannot
+traverse stairs while a legged robot can. Existing methods discussed in Section 7.1 suffer computational costs
+that make it challenging to scale to large environments. Because the proposed planniner is computationally
+efficient and minimally dependent on tuning gains, it performs well in large-scale environments. Our planner
+makes several significant contributions, such as light on-demand terrain assessment, which is discussed in
+Section 7.2, hierarchical solution-search that also incorporates position history-based multi-agent coordination,
+which is discussed in Section 7.3, and handling of dynamic changes in the environment such as blocked
+passages, which is covered in Section 7.4.
+7.1
+Background
+One of the widely-known methods (Yamauchi, 1997; Ahmad et al., 2021a) for autonomous exploration
+relies on explicitly detecting potential frontiers on an explored map, followed by a path planned toward
+each cluster of frontiers. The method seeded significant developments in the area of autonomous robotic
+exploration since it was first proposed. However, this frontier-based method employs a computationally
+expensive optimization-based approach that plans paths to each frontier cluster, despite the fact that some
+may not be reachable.
+In recent decades, the planning community has witnessed significant advancements in more computationally
+efficient sampling-based approaches for path planning and exploration. One instance of such development
+is an exploration planner that uses Rapidly Exploring Random Trees (RRT) (LaValle, 2006) to sample an
+environment and chooses an optimal path from the set of sampled ones. The method samples the environment
+as a single batch, and therefore is not scalable to large-scale environments. A rectification of this limitation
+is recently proposed by (Dang et al., 2020) where a bifurcation approach is introduced for sampling and
+exploration. This approach implies that the environment is sampled only in the local neighborhood of
+a robot while simultaneously building a sparse graph that scopes the entirety of the explored map. The
+latter is essential to deal with local minima such as dead-ends and also to plan a path back home. Our
+autonomous exploration solution for the SubT Final Event relies on the principle of bifurcation with additional
+contributions in the terrain assessment, solution-search, dynamic obstacle avoidance and coordination.
+The graph-based planners based on sampling and bifurcation approach use high resolution depth images
+
+to compute a 2.5D grid-based elevation map using the technique presented in (Fankhauser et al., 2018).
+This elevation map is further filtered to segment terrain characteristics such as slope, roughness and step
+(Wermelinger et al., 2016). The authors of such graph planners mention the scalability challenges with such
+computationally expensive approaches, which limit their terrain awareness to regions local to the agent. This
+further leads to challenges such as the planned path and the underlying graph being generated with an
+over-optimistic view of the terrain, consequently needing the robot to be backed up if it encounters impassible
+terrain.
+7.2
+Sample-and-Project Strategy
+To rectify the terrain assessment challenges, a sample-and-project approach is followed, similar to the settling-
+based collision-check approach proposed in (Kr¨usi et al., 2017). We perform such checks on an Octomap with
+resolution 0.15m. At this resolution, all of team MARBLE’s robots were at least three voxels wide, providing
+a decent amount of robot footprint to project a robot’s pose on. SubT challenge rules highlight that the
+extremely narrow passages could be around 1m wide, with doorways as narrow as 36 inches. With this in
+mind, Octomap voxel length of 0.15m was small enough to navigate narrow passages and large enough to be
+able to keep up with computational complexity of generating such a map in a large-scale environment. In
+case of a wheeled robot, each voxel in the Octomap is labelled with a roughness value which is obtained using
+high resolution point clouds as described in Section 6.2. However, on Spot legged robots dense roughness
+information is not required because of their onboard terrain assessment. Additionally, for Spots, encode
+semantic information about stairways into the map which overrides the default height parameters of the
+planner. With explicit labels, the planner is able to plan paths over built up staircases despite elevation
+changes the robot would not normally traverse over. More formally, for the legged robots capable of traversing
+stairs, each Octomap voxel maps to a label from the set {‘occupied’, ‘unknown’, ‘free’, ‘stair’}, whereas in
+case of wheeled robots the label set is {‘occupied’, ‘unknown’, ‘free’, ‘rough’}.
+First, the environment is sampled in the local neighborhood of the robot using RRT∗, a variant of RRT
+with optimality considerations. Each tree sample is a robot position parameterized by the robot width
+and length. During sampling, the collision-checks are performed by vertically projecting a query sample to
+find the ground below it. Once the ground is found, the elevation change at the footprint of the sample is
+evaluated if there are enough projections on occupied voxels. In case of a wheeled robot, the average roughness
+information of the footprint voxels is also taken into consideration. For a legged robot, if a threshold amount
+of footprint voxels are labelled as ‘stairs’, the sample is considered collision-free regardless of the elevation or
+roughness check. Expanding an RRT∗ requires checking path segments for collisions instead of isolated robot
+configurations. In order to check such a path segment, a set of robot configurations along the segment is
+checked for traversability. Fig. 11 depicts the terrain assessment process.
+7.3
+Solution Search and Multi-Robot Coordination
+At any replan iteration, a set of potential solutions include all of the local paths sampled using RRT∗ and
+all of the global paths ending at the graph frontiers. The former set of solutions is represented by PL and
+latter solutions belong to the set PG. A set of all local paths that are leading the robot toward areas with
+greater than a defined threshold of volumetric gain and teammate separation are given as PLV and PLS
+respectively. Similarly, a set of all global paths that are leading the robot toward areas with greater than
+a defined threshold of volumetric gain and teammate separation are given as PGV and PGS respectively.
+These sets is highlighted in Figure 12. A typical approach to find an appropriate solution is to form an
+objective function with a combination of exploration objectives such as volumetric gain and exploration
+heading parameterized by the penalty gains. Volumetric gain calculation is, however, a computationally
+expensive operation and limits the amount of frontiers a robot can process in a reasonable amount of time.
+Our approach relies on finding a good enough solution in terms of volumetric gain. To achieve multi-robot
+coordination, the position histories of teammate robots on the network are used. If a path is leading a robot
+to a point such that the minimum distance of the point from the position histories of the teammate robots is
+
+Figure 11: A depiction of sample-and-project strategy for terrain checks on an OctoMap. The figure shows four
+different types of query poses. (a) and (b) depict collision-free samples whereas (c) and (d) are marked under-collision
+or non-traversable.
+more than the mapping range, it is guaranteed that new areas are being explored.
+Following this intuition, the primary objective of Team MARBLE’s solution search method is not to optimize
+for the volumetric gain and the teammate position histories separation, but to accept a solution that has
+a satisfactory amount of volumetric gain and distance from the teammate position histories. The formal
+objective of the path planner is to output a solution that belongs to PLV ∪ PLS, PGV ∪ PGS, PLV or PGV
+in the order of preference. The solution search process makes use of two different objective functions,
+Jα = c0DP (phist, pcand) − c1|θexplore − θcand
+mean|
+− c2|hexplore − hcand
+mean|,
+(7)
+Jβ = −c1|θexplore − θcand
+mean| − c2|hexplore − hcand
+mean|
++ c3GV (pcand(1)) + c4DS(pcand(1), phist
+1
+, ..., phist
+1
+),
+(8)
+where Jα is used to find a candidate path that aligns best with the current exploration heading of the robot
+and Jβ is leveraged to perform a thorough search if required. The sets of points pcand and phist represent a
+list of candidate solutions and the position history of a robot respectively. The exploration heading θexplore
+is calculated by averaging the most recent few points on phist of the robot. The mean heading and mean
+height of a candidate path are denoted by θcand
+mean and hcand
+mean respectively. The function DP accepts two paths
+as arguments and calculates the mean of minimum distance of all points along the first path with the second
+path. Furthermore, the function DS calculates the minimum distance of a candidate path from the position
+histories of all other teammate robots.
+Algorithm 1 provides a deeper insight into the solution search steps. As a first step, a collision-free local
+path is found that best aligns with the direction of travel of the robot. This path is then checked if it has a
+satisfactory amount of volumetric gain and distance from the teammate position histories. If a good-enough
+solution is found at this step, the solution is returned and only a single volumetric gain function call is
+required. Therefore, we save significant computation time during most replan iterations. If a solution is not
+found at this first step then a more thorough search is performed, first through the sampled local paths and
+then through the global paths leading toward graph frontiers. This search is highlighted in Algorithm 1.
+
+Robot Length
+Query Poses
+Local
+Sampling
+(d)
+Region
+(a)
+(c)
+(b)
+X
+X
+max-min
+stair
+inot enough
+max-min
+< thresh
+voxels
+> thresh
+I projections The functions PlanLocally() shown in Algorithm 2, and PlanGlobally() shown in Algorithm 3, are
+responsible for outputting solutions that satisfy both volumetric gain and teammate separation constraints
+if possible, otherwise they output solutions that only satisfy the volumetric gain constraint. In the worst
+case, when neither constraint can be satisfied, paths with maximum teammate separation are returned as a
+contingency.
+This attempt of finding a solution by breaking the potential solution space down into subsets instead of
+having one objective function to optimize over the entire space, helped us avoid extensive gain tuning. During
+testing and final event runs, we found our approach to be scalable for environments of various sizes without a
+need for tuning gains for different environment types. The details of the sampling-based path planner can
+be found in (Ahmad and Humbert, 2022). In this work, a simulation comparison of the proposed planner
+with an existing sampling-based planner (Dang et al., 2020) is presented, highlighting the improvement in
+scalability and computational efficiency.
+Algorithm 1 ScanPlan Solution Search.
+1: pl
+α ← pcand ∈ PL costing minimum Jα
+2: if GV (pl
+α) ≥ vg
+thresh and DS(pl
+α) ≥ sg
+thresh then
+3:
+return pl
+α
+4: end if
+5: pl
+β ← PlanLocally()
+6: if pl
+β is non-empty and DS(pl
+β) ≥ sg
+thresh then
+7:
+return pl
+β
+8: end if
+9: pl ← pl
+β
+10: pg ← PlanGlobally()
+11: if pg is non-empty and DS(pg) ≥ sg
+thresh then
+12:
+return pg
+13: else if pl is empty and pg is empty then
+14:
+return pl
+α
+15: else if pg is empty or
+(pl is non-empty and DS(pl) ≥ DS(pg)) then
+16:
+return pl
+17: else if pl is empty or
+(pg is non-empty and DS(pg) ≥ DS(pl)) then
+18:
+return pg
+19: end if
+Algorithm 2 PlanLocally(). Returns a path in PLV ∩ PLS or PLV in the order of preference.
+1: Jβ
+min ← + inf
+2: pres ← none
+3: success ← false
+4: for pl ∈ PL do
+5:
+if GV (pl) ≥ vg
+thresh and DS(pl) ≥ sd
+thresh and ∼ success then
+6:
+Jβ
+min ← Jβ(pl), pres ← pl, success ← true
+7:
+else if Jβ(pl) ≥ Jβ
+min then
+8:
+continue
+9:
+else if (GV (pl) ≥ vg
+thresh and DS(pl) ≥ sd
+thresh) or (GV (pl) ≥ vg
+thresh and ∼ success) then
+10:
+Jβ
+min ← Jβ(pl), pres ← pl
+11:
+end if
+12: end for
+13: return pres
+
+Algorithm 3 PlanGlobally(). Returns a path in PGV ∩ PGS or PGV in the order of preference.
+1: PGV ← ∅, PGV S ← ∅
+2: for pg ∈ PG do
+3:
+if GV (pg) ≥ vg
+thresh and DS(pg) ≥ sd
+thresh then
+4:
+PGV S ← PGV S ∪ {pg}
+5:
+else if GV (pg) ≥ vg
+thresh then
+6:
+PGV ← PGV ∪ {pg}
+7:
+end if
+8: end for
+9: if PGV S is empty then
+10:
+return path pg ∈ PGV with maximum DS(pg)
+11: end if
+12: pg
+r ← path from PGV S leading to most recent frontier
+13: pg
+c ← path from PGV S leading to closest frontier
+14: if PathLength(pg
+r) ≥ PathLength(pg
+c) then
+15:
+return pg
+c
+16: else
+17:
+return pg
+r
+18: end if
+(a)
+(b)
+(c)
+Figure 12: The figures highlight the planner solution search. The potential solutions for the planner include the paths
+leading toward the leaves of the RRT∗ tree (blue) and the frontiers of the graph (green). (a) The first preference of
+the solution is the path that aligns best with the robot’s exploration heading (bold blue path). In case this path has
+both sufficient volumetric gain and teammate separation, it is returned as a solution. (b) As a second preference, a
+thorough search is performed to find a local or global path that satisfies both constraints. (c) If none of the paths is
+found at both of the steps above then a path that satisfies the volumetric gain constraint is accepted as a potential
+solution.
+7.4
+Dynamic Replanning
+Another challenge faced by the existing graph-based planners is that they rely on building a parallel graph
+representation of the environment. This representation does not naturally reflect changes in the environment,
+such as closed passages which were initially open at the time the graph is built. To handle this exception, the
+graph edges are labeled with a boolean representing its occupancy. During exploration, the planned paths
+are constantly checked for collisions. If a planned path is under-collision, all edges in the local neighborhood
+of the robot are validated for collision and marked accordingly. Moreover, all occupied edges are checked
+for occupancy all the time when the planner finds some idle time which mostly happens when the vehicle
+is following a path. This enables the planner to take into account the cases where an occupied area is free
+again. In the case where the graph search is performed to plan a global path, the occupied edges are ignored.
+
+Exploration Heading
+GraphFrontiers
+TeammatePose
+History
+RRT*Root Node
+Pose HistoryExplorationHeading
+Graph Frontiers
+LOWVOL-GAIN
+TeammatePose
+History
+RRT*Root Node
+Pose History
+LOWTEAMMATE
+SEPARATIONExploration Heading
+GraphFrontiers
+LOWVOL-GAIN
+TeammatePose
+History
+RRT*RootNode
+Pose History8
+Communication Systems
+Effective communication with deployed systems from a fixed human operator is a crucial component of a
+complete robotic exploration system. While robots are capable of independent localization, mapping, and
+artifact detection, the addition of a communication infrastructure is a force multiplier to enable human
+supervisory control, inter-robot coordination, and timely artifact reporting. We developed a mesh network
+system to provide long-reach communications into underground environments which prioritizes reconnection
+times to maximize opportunities for data transfer.
+8.1
+Background
+Previous work has developed several solutions to common problems encountered with deploying mesh networks,
+such as discovery and optimal routing. A wide variety of both closed-source and open-source solutions exist
+that include both hardware and software components. Mesh networking can largely be subdivided into
+three layers: physical, logical, and transport. We will detail several prominent open-source or commercially
+available options for each layer before describing our final solution.
+From a logical layer standpoint, meshing layers lay between the physical transmission of frames over the
+medium and a higher-level protocol such as IP. For mobile robots operating in subterranean environments,
+a responsive mesh layer that minimizes lost link time is a major requirement due to the rapid movement
+of the robots. Further, to reduce integration effort, a mesh layer that operates at layer 2 of an OSI stack
+(for Standardization, 1996) is desirable to allow transparent use of higher-level protocols such as ARP and
+IP. Typically, meshing algorithms such as OLSR (Clausen et al., 2003) and AODV (Perkins et al., 2003)
+select a single best path for routing between nodes which hinders algorithmic performance in dynamic
+environments. A more recent example of a single-path logical meshing layer is Better Approach to Mobile
+Ad-hoc Networking-Advanced (batman-adv) (Seither et al., 2011), an open source implementation of a layer 2
+mesh. In contrast to batman-adv, meshmerize (Pandi et al., 2019) provides multiple paths between nodes to
+ensure a reliable connection while still operating at layer 2; these multiple paths allow for a dramatic decrease
+in reconnect times when mesh topology changes. We relied on meshmerize as our layer 2 meshing solution in
+cooperation with Meshmerize GmBH.
+Only transport layers designed for ROS were considered for ease of integration with the rest of the autonomy
+stack. In a traditional networked ROS architecture, a single computer runs a main node known as the
+rosmaster that coordinates the publish-subscribe mechanisms. When a node wishes to exchange data with
+another node via named topics, the master is consulted to determine the computer to connect to, as in Figure
+13a. A single rosmaster serves as a central directory of nodes and topics; when a subscription to a topic is
+requested, a list of publisher nodes is returned so that point-to-point TCP connections can be made directly
+between publisher and subscriber. These direct TCP connections break down when systems are linked over
+unreliable mesh networks which necessitates the need for an alternative transport mechanism.
+One open source transport layer multimaster fkie (Juan and Cotarelo, 2015) solves the discovery and
+advertisement problems using multi-cast packets and specialized nodes on each machine with an architecture
+shown in Figure 13b. However multimaster fkie does nothing to establish prioritization of data flow. With
+the standard TCP transport provided by ROS, there is no centralized means of monitoring inter-node
+connections to arbitrate data priorities. Prioritization is crucial for monitoring the robot fleet in intermittent
+communication situations. Mission critical data such as artifacts needs to make it through to the human
+supervisor before other auxiliary data such as odometry and maps.
+One alternative to multimaster fkie, Pound1 (Tardioli et al., 2019), is specifically designed for use in unreliable
+mesh networks and implements many of the desired requirements. However, Pound relies on hardcoded
+topic names and fixed addressing information, which limit the flexibility of the system.
+Alternatively,
+1https://github.com/dantard/unizar-pound-ros-pkg
+
+Master
+Node A
+Node B
+Node C
+(a)
+Master α
+Node A
+Node B
+Node C
+Master β
+Node D
+Node E
+Node F
+(b)
+Master α
+Node A
+Node B
+Node C
+udp mesh
+Master β
+Node D
+Node E
+Node F
+udp mesh
+(c)
+Figure 13: Several different types of ROS architectures. Red lines indicate data transfer, black lines indicate directory
+management, and blue lines are data paths that cross network segments. A basic, single-master ROS network node
+graph is shown in (a). A fkie multimaster multi-master ROS network node graph is shown in (b). In contrast to both
+(a) and (b), (c) shows how udp mesh creates a single virtual channel between nodes, shown in green, to implement
+data prioritization.
+nimbro network (Schwarz et al., 2016) implements a similar set of functions with regards to transport over
+wireless networks, but omits prioritization. Crucially, nimbro network still utilizes TCP for reliable inter-robot
+communication, preventing adaptation of core TCP behavior (particularly retransmits) to unreliable mesh
+networks; UDP links are only used for non-guaranteed data delivery.
+8.2
+UDP Mesh
+The main innovation in our system is our transport layer, udp mesh which allows for runtime reconfiguration,
+implements prioritization, and re-implements reliable communication over UDP to allow for more refined
+control over retransmits and fragmentation. Fundamentally, the udp mesh layer uses only unicast and
+broadcast UDP datagrams to implement higher-level services without requiring multicast support.
+In
+principle, multicasting would offer a performance benefit by reducing broadcast traffic. However, in a wireless
+mesh environment, these potential gains are offset by multicast group membership management overhead.
+8.2.1
+Discovery and Address Resolution
+Discovery is the process of identifying nodes that are available for communication. We implement discovery
+through the use of a periodic heartbeat broadcast that advertises the node’s availability and provides name
+resolution information. In concept, this service is similar to the mcast dns service in Linux, where peers
+advertise their naming information to be able to address nodes by hostname instead of layer 2 MAC or layer
+3 IP address. Nodes identified through discovery are added to the list of available nodes for communication as
+well as status reporting. This discovery heartbeat is also used as a lost-communications detector to prevent
+higher-level messages from queueing for unreachable nodes.
+8.2.2
+ROS Message Encapsulation
+In the ROS ecosystem, messages are translated from a message definition language specification into internal
+representations appropriate to the implementing language2. This same language specification is used to
+serialize and deserialize messages; that is, to transform a ROS message into a buffer of bytes suitable for
+transmission over an arbitrary channel. udp mesh implements a generic message passing system such that
+the message to be transmitted is never deserialized, saving a significant amount of processing time in the case
+of complex, large message types such as images. Instead, a generic subscriber is used to acquire the serialized
+bytes for direct use to be transmitted to other nodes. On the receiver side, the transmitted byte stream is
+deserialized to instantiate the message in a format that other ROS ecosystem nodes can readily consume.
+These two functions abstract the transport of arbitrary messages over the udp mesh layer and remove any
+2http://wiki.ros.org/msg
+
+requirement to define a list of acceptable message types.
+8.2.3
+Point to Point Transport
+In the udp mesh system, point-to-point transport is implemented via UDP datagrams. This envelope contains
+provisions for sequence tracking, fragmentation, and message reconstruction. When preparing a message for
+transmission, the byte buffer provided by the ROS encapsulation service is split into chunks that fit inside
+the underlying medium’s maximum transmit unit (MTU). We use the standard 802.11 framing with an MTU
+size of 1500 bytes, out of which 100 bytes are reserved for overhead, leaving 1400 bytes for payload.
+In the implementation of our system, a configurable number of message fragments are permitted to be ‘in
+flight’ at any given time, similar to TCP congestion window control. In order for the next fragment to be
+transmitted, the receiver must send an acknowledgment. During unit testing to determine an appropriate
+value for the number of in-flight fragments permitted, an initial increase yields improved throughput. However,
+after a certain point, throughput decreases as multiple packets are queued for transmission on the medium and
+start to destructively interfere. As a compromise determined via empirical testing, three packets are permitted
+to be in-flight between any two nodes at a time. With this configuration, our transport-layer throughput is
+approximately 20 Mbit/s of payload data, measured using raw images as representative high-density traffic
+over a wired gigabit Ethernet link.
+Retransmits are automatically queued until either an acknowledgment is received or the host is marked offline
+due to non-reception of any heartbeat or acknowledgment messages. Once a host is marked offline, any
+attempts to send messages are discarded. Hosts may become online once again after receipt of a discovery
+message. On the receiver side, the message is kept in a temporary state while the fragments arrive. Should
+message fragments stop arriving, the partial message is purged after a timeout and the host is once again
+marked offline which indicates to higher levels that reliable transport is unavailable. In this case, the higher
+level is BOBCAT, which is discussed in Section 9.
+8.2.4
+Quality of Service
+Quality of Service (QoS) is the notion that some traffic should be prioritized over other traffic for use of a
+limited communications channel, e.g, artifact reports need to arrive before mapping data. Fundamentally,
+TCPROS (the default transport used in ROS v1) is not capable of implementing a QoS scheme where a
+limited channel is shared between different topics (Figure 13c), as every node subscribing to a topic uses an
+individual TCP point-to-point link with no information about other links. This need to prioritize traffic was
+the driving rationale behind the development of the udp mesh layer. As part of the configuration of the layer,
+each topic to be transported includes a priority number. Internally, this priority number is used as a sorting
+key to order message fragments for transmission.
+8.2.5
+Point-to-Multipoint Transport
+Although udp mesh is based around point-to-point message transfer, mission requirements sometimes necessi-
+tate system-wide messaging. For example, broadcast methods are used within the udp mesh layer to manage
+name resolution. To facilitate these type of messages originated at higher levels, a broadcast mechanism is
+provided by the transport layer. For messages that fit within a single MTU, a single, unacknowledged UDP
+broadcast is used to distribute the message. For larger messages, individual links to each node are used to
+send the broadcast as a series of unicast fragments using the same accounting and acknowledgments as the
+point-to-point mechanism.
+
+8.3
+Final Solution
+The final communication solution used meshmerize as the logical layer with udp mesh as the transport layer.
+Both robots and beacons acted as nodes in the mesh with robots carrying 1W radios and beacons carrying
+2W radios. Beacon drops are controlled by the methodology described in Section 9.3. Table 2 shows the
+evolution of our final networking solution from the Tunnel event through the Final Event. Our meshing
+solution, including the meshmerize layer 2 software stack, was implemented on ath9k-compatible 802.11
+hardware, while udp mesh was implemented on high-level compute units. Because of this split and radio
+hardware commonality, all of our radios ran essentially the same firmware image built off of an OpenWRT3
+base. Our beacons only participated in the mesh at layer 2, and as such did not contribute to any broadcast
+traffic associated with udp mesh services. By providing a reliable ROS-compatible mesh networking layer,
+higher-level autonomy and human interface via BOBCAT could be provisioned without knowledge of the
+underlying infrastructure.
+Event
+Physical
+Data Link
+Transport
+Application
+Tunnel
+ath9k
+B.A.T.M.A.N.
+fkie multimaster
+marble multi agent
+Urban
+ath9k
+meshmerize
+fkie multimaster
+marble multi agent
+Final
+ath9k
+meshmerize
+udp mesh
+BOBCAT
+Table 2:
+Evolution of Team MARBLE’s communication system. As a note, ath9k and meshmerize are commercially
+available, B.A.T.M.A.N. and fkie multimaster are open-source software, and upd mesh, marble multi agent, and
+BOBCAT are custom packages developed for the SubT Challenge.
+9
+Mission Management
+While the combination of Team MARBLE’s large scale positioning system, mapping, and planning solutions
+provide a solid foundation for autonomy, higher level cognition and reasoning is required to take full advantage
+of the system. For Team MARBLE, this higher level reasoning consists of a flexible mission management
+solution which keeps the robots on task and allows for higher level instructions from a human supervisor.
+The core of the mission management solution is Behaviors, Objectives and Binary States for Coordinated
+Autonomous Tasks (BOBCAT) (Riley and Frew, 2021). BOBCAT controls the decision-making process for
+each individual agent while a separate process known as Multi-Agent Data Collaboration for Autonomous
+Teams (MADCAT) controls the data sharing and waypoint deconfliction between robots. In this section we
+highlight the design decisions, and algorithm details behind BOBCAT and MADCAT.
+9.1
+BOBCAT
+BOBCAT simplifies the robot and environment states using Monitors such as communication status. The
+Monitors are combined with weighted goals which Objective such as finding artifacts or extending communi-
+cations can be fulfilled. BOBCAT then selects the best Behavior such as exploring or deploying a beacon to
+fulfill and the most important Objectives to execute. A full list of implemented Monitors, Objectives, and
+Behaviors can be seen in Section 13.7 of the Appendix.
+Formally, a BOBCAT is defined by the tuple {x, y, w, M, O, B, πB} where
+• x ∈ X is the system state with state space X.
+• y ∈ Y are the sensor measurements with measurement space Y .
+3http://www.openwrt.org
+
+• w ∈ W = R|O|
++
+is a vector of input weights. These weights are used by the respective Objective
+functions and represent the relative importance of the Objective to the overall mission
+• M is the set of Monitor functions of the form Mi : X × Y → {0, 1} ∀Mi ∈ M. Monitor functions Mi
+return a binary value based on the robot state and measurements.
+• O is the set of Objective functions of the form Oj : W × {0, 1}|M| → {0, Wj} ∀Oj ∈ O. Objective
+functions Oj use the input weight Wj and a logical combination of Monitor outputs to return either
+the input weight or a 0, which indicates the current preference of the objective to be fulfilled.
+• B is the set of Behavior functions of the form Bk : {0, 1}|M| × {0, Wj}|O| → R≥0 × Fk ∀Bk ∈ B.
+Behavior functions Bk sum the outputs of the Objectives associated to that Behavior. Monitor
+outputs may be used to selectively inhibit specific Objective weights during evaluation steps. The
+Behavior function returns a real value that indicates the current utility score of the actions associated
+with that Behavior, and a pointer to an execution function. A Behavior may have a null execution
+function.
+• πB is the policy for selecting the Execution Behavior BE based on each of the Behavior utility scores.
+BOBCAT can be represented graphically as in Figure 14. States and measurements from both the robot
+itself and external agents in a multi-agent scenario feed the various Monitors. This represents what the robot
+“knows”, and provides a binary output to the rest of the system. The Monitor output lines in Figure 14 and
+other figures represent the cases where the Monitor is associated with the respective Objective or Behavior.
+Figure 14: Graphical overview of BOBCAT. Numbers represent binary outputs, output weights, and behavior scores,
+respectively. A full list of monitors, objectives and behaviors is provided in the Appendix in Tables 8, 9, 10 respectively.
+9.2
+MADCAT
+The MADCAT framework depicted in Figure 15 provides the multi-agent data sharing capabilities required
+for the mission. The framework includes transmission of relevant coordination data and maps, as well as
+map merging functionality and decision making for each agent. MADCAT uses BOBCAT to accomplish the
+high-level mission management for individual agents with additional higher-level direction provided by the
+human supervisor.
+
+Monitor 1
+-0/1
+Objective 1
+.0
+Monitor 2
+0/ 1
+Behavior 1
+3.0
+Objective 2
+2.0-
+States and
+Measurements
+(Internal and
+Monitor 3
+0/1
+Behavior 2
+3.5
+External)
+Objective 3
+0.5-
+Monitor 4
+0/1
+BehaviorP
+1.0
+Objective O
+1.0
+Monitor N
+-0 / 1-
+Binary Output
+OutputWeights
+BehaviorScoresFigure 15: Overview of the Multi-Agent Data Collaboration for Autonomous Teams (MADCAT) framework.
+9.2.1
+Messages
+MADCAT sends most messages by broadcast with no acknowledgement required, and therefore does not
+require the sender to needlessly wait. This allows any agent who receives the message to act accordingly
+without a requirement to respond. This is helpful in the event the sender leaves communications range shortly
+after the broadcast. An exception to this policy is made for high bandwidth data such as maps and images,
+because the receiver can not act on incomplete data. Bandwidth is not strictly managed, but instead uses a
+‘best-available’ strategy consistent with the prioritizations assigned to differing message classes, e.g. telemetry,
+supervisor commands, maps, and FPV.
+An agent’s pertinent local messages are concatenated into a single message in order to limit the number of
+messages broadcast over the communications channels. Messages are re-built and broadcast every second.
+Only the most recent message is needed for time-varying data such as odometry or the current goalpoint and
+any older messages are discareded. Other data such as artifact reports and relay locations are appended to a
+growing list, so any message a remote agent received has all of this type of data. Larger data that could
+grow to become impractical to transmit repeatedly, such as maps and images, use a point-to-point handshake
+transmission. Messages are deconflicted using sequence numbers, to allow agents to share the messages of
+other agents but ensure only the latest data is used by the receiver.
+Each agent’s broadcast message contains not only its own local data, but that of any neighbor agents as well.
+This allows downstream agents who can communicate with agent A but not agent B to still receive relatively
+current information from agent B.
+9.2.2
+Artifact Report Management
+Agents keep track of both their own detected artifacts using the procedure described in Section 5 as well any
+artifacts they have received from other agents. The framework aggregates all of the artifact reports and the
+
+Remote Agents
+MapDiffs
+Odometry
+MissionElements
+SupervisorInputs
+Goal
+Mapping
+Remote Message Aggregation
+Message Deconfliction
+Map Diffs
+Monitors
+Objectives
+Behaviors
+Odometry
+MissionElements
+Local Message
+Output Aggregation
+Aggregation
+SupervisorInputs
+Goal Selection
+GoalArray
+Local ControlArtifact monitor determines if the agent needs to return to communications to report the new information
+to the base station. The base station further parses these messages for display, selection, and transmission
+to the scoring system. More details of this display can be found in Section 9.5. Images are sent using a
+point-to-point request system over a low priority channel to reduce bandwidth requirements.
+The BOBCAT Artifact monitor which is triggered by 3 unreported artifacts or 5 minutes of exploration with
+a pending artifact, ultimately determines whether the robot should return to communications for unreported
+artifacts. The raw artifact reports are always used in this determination, but transmission of images is
+configurable. By default, and as configured during the Final Event Prize Run, artifact images not received by
+the base station are considered unreported artifacts, and will force the robot to return to communications
+until they are fully transmitted.
+9.3
+Beacon Deployment
+The framework is responsible for identifying locations to deploy communications relays to extend the
+communications reach into the environment. It uses a combination of communications status, distance and
+turn detection to identify potential locations. The human supervisor is also able to command drops based on
+a robot’s location on the map.
+9.4
+Goal Selection
+Some behaviors, particularly Explore, require a goal selection step once that behavior has been chosen to
+execute. The goals either come from the global planner described in Section 7 or from human supervisor
+input. If two agents goals are found to be conflicting, BOBCAT requests a new goal from the planner which
+provides a path to the next closet goal point.
+9.5
+Human Supervisor Interface
+The human supervisor interacts with BOBCAT using a custom GUI shown in Figure 16. This interface
+allows the human supervisor to set a goal point for the robot using an Interactive Marker. MADCAT then
+passes this goal to the robot through the communications network if a connection is available. The human
+supervisor can also remotely control robots using an Xbox controller when communication systems allow.
+Figure 16: Example of the human supervisor interface showing the end of Team MARBLE’s final run
+First-person view (FPV) allows the human supervisor to see the environment from the robot’s perspective in
+semi-real-time, instead of just through the map representation and infrequent artifact images. In addition to
+increased situational awareness it allows the human supervisor to identify visual artifacts that may be missed
+
+EilePanelsHelp
+DARPA
+Arifacts Beacons Reset
+HO1
+Fused Artifacts
+HO2
+ETO
+Reset
+31 fps 
+个stop
+H03
+Raw Robot Artifacts
+4.14m
+H03
+Reset Left-click: Rotate. Middle-Click: Move X/Y, Right-click: Move Z, shift: More c 31 fps
+DO2
+Backpaicl
+ Eile Panels Help
+Backpack
+Reset Left-click: Rotate. Middle-Click: Move X/Y. Right-click: Move Z. shift: More c 31 fps
+Submissions
+Type
+Position
+Notes
+Response
+Drill
+21.83, 2.63,-1.73
++1 points
+Backpack
+27.23.5.18.-1.28
++1points
+New Repot
+Transform Transportcustom Arifact
+End Mission
+TIme
+DS Time:1632420024.41ROS Elapsed:4940.81by the on-board artifact detection system, or identify them more rapidly. Finally, FPV helps the human
+supervisor during manual teleoperation in the event it is needed to direct the robot.
+Flexibility of our udp mesh communication architecture made it possible to rapidly adapt to mission specific
+constraints. During the Final Event, we observed extra bandwidth in the communication system and decided
+to add FPV to our Spot robots to enhance their exploration potential. Compressed images from the Spot
+forward facing cameras were transmitted as 1 Hz low-priority messages. On the base station computer, these
+images were displayed live in RViz, and saved locally so the human supervisor can review that at any time.
+The additional scoring potential of FPV is highlighted in Section 10.4.
+Several features of the human supervisor GUI were designed to help reduce operator workload. First, another
+artifact fusion process runs on the Base Station computer, to aid the operator in tracking artifacts reported
+by multiple vehicles. If reports of the same type are within 3m of prior reports, they are fused to the mean
+position. Redundant artifact reports already been seen by another robot appear in a light gray color. In
+contrast, new reports flash with large white text to bring attention to the human supervisor. When submitting
+artifact reports, the human supervisor can select from individual or fused reports. If an artifact is successfully
+scored, the submission is locked out to prevent re-submitting. If it does not result in a score, the operator
+can utilize additional map, trajectory, and FPV information to improve the estimated position. New reports
+can be submitted by shifting fused artifacts icons in the map or specifying a manual position.
+10
+Final Event Results
+The SubT Final Event Prize Run on September 23, 2021 provided Team MARBLE an excellent opportunity
+to evaluate the performance of our complete supervised autonomy solution, and we share our results in this
+section. First, an overview is provided in Section 10.1, which includes an outline of the mission objectives,
+a description of the previously unknown course in Section, as well as a high-level summary of the results.
+Localization and mapping results are presented in Section 10.2. Further analysis of the planner and resulting
+exploration effort is described in Section 10.3. Artifact detection results are thoroughly analyzed in Section
+10.4. The communication environment was friendlier than expected, and in Section 10.5, we discuss how we
+capitalized on that opportunity. Mission management results are detailed in Section 10.6, including the five
+instances where the human supervisor manual intervened, as well as the seven artifacts that were scored via
+FPV imagery. Together, this section elucidates how our systems worked together to score 18 artifacts, while
+also discussing the areas that limited even higher performance. Data from Team MARBLE’s deployment
+during the Final Event Prize Run is publicly available and discussed in Section 10.7.
+10.1
+Overview
+The mission objectives were to accurately report as many of the 40 artifacts in the course as possible during
+the mission. There are three hard constraints: the mission is 60 minutes long, there are a total of 40
+attempts to report artifacts, and only one human, the human supervisor, is permitted to supervise the mission,
+manipulate robotic agents, and submit artifact reports.
+The final course was custom constructed as illustrated in Figure 17, which breaks the course out into distinct
+tunnel, urban, and cave environments. The course contains numerous hazards and challenges: rough terrain,
+railroad tracks, slippery surfaces, ramps, stairs, large drop-offs, rocky cliffs, narrow hallways, low-to-the-
+ground corridors, wide-open caverns, fog, standing water, dynamic obstacles, trap doors, and a degraded
+communications environment. These challenges are discussed further in Section 10.3.
+Here, we provide a brief high-level summary of the artifacts scored and the extent of the environment explored.
+Team MARBLE scored 18 of the 40 artifacts and explored roughly half of the environment. For reference,
+the top-scoring team scored 23 artifacts, and the performance for all teams is listed in Section 13.9 of the
+
+Appendix. The location and class of all 40 artifacts can be visualized in the context of the course map shown
+in Figure 17. This map also indicates which regions of the course that Team MARBLE explored as well
+as the 18 scored artifacts. The artifacts that Team MARBLE scored are also listed in Table 3, ordered
+chronologically from mission start to mission end. Each of the 18 artifacts in Table 3 correspond by ID to the
+scored artifacts in Figure 17. Further analysis of artifact detection results are detailed in Section 10.4.
+Staging Area
+Explored Cave
+Unexplored Cave
+Explored Tunnel
+Unexplored Tunnel
+Explored Urban
+Unexplored Urban
+Backpack
+Cell Phone
+Cube
+Drill
+Fire Extinguisher
+Gas
+Helmet
+Rope
+Survivor
+Vent
+Artifact Type
+Scored
+Not Scored
+Limestone
+Pillars
+0
+25
+50
+meters
+L51
+L53
+L31
+L32
+L59
+L55
+L58
+L38
+L34
+L36
+L40
+L26
+L24
+L22
+L47
+L08
+L11
+L67
+L19L45
+L17
+L21
+L50
+L44
+L68
+L72
+L74
+L71
+L77
+L78
+L79
+L80
+L16
+L15
+L13
+L05
+L02
+L42
+L64
+L62
+Entrance
+Figure 17: Course map of the 60-minute Final Event Prize Run designed by DARPA, highlighting the 18 of 40 artifacts
+scored by Team MARBLE, along with the areas of the tunnel, urban, and cave sections were explored by agents.
+Score
+Time [mm:ss]
+Type
+ID
+Error [m]
+1
+01:08
+Drill
+L51
+0.57
+2
+01:23
+Backpack
+L53
+2.23
+3
+06:23
+Rope
+L55
+0.84
+4
+12:03
+Survivor
+L26
+0.62
+5
+16:35
+Survivor
+L32
+1.40
+6
+17:23
+Gas
+L08
+1.80
+7
+28:08
+Fire Extinguisher
+L31
+1.31
+8
+35:51
+Drill
+L34
+1.43
+9
+36:53
+Fire Extinguisher
+L38
+2.82
+10
+37:08
+Cube
+L36
+3.94
+11
+37:58
+Backpack
+L40
+1.40
+12
+38:47
+Rope
+L67
+2.87
+13
+47:53
+Cube
+L11
+1.83
+14
+50:33
+Cell Phone
+L22
+4.06
+15
+50:45
+Cell Phone
+L47
+4.00
+16
+51:48
+Cell Phone
+L59
+2.15
+17
+52:45
+Gas
+L24
+2.55
+18
+56:33
+Helmet
+L58
+1.74
+Table 3:
+List of all artifacts scored by Team MARBLE during the 60-minute Final Event Prize Run, along with
+the corresponding mission time when artifacts were reported and scored, artifact type, unique DARPA-assigned ID
+numbers, and Euclidean distance error between the reported and ground truth location of the artifact.
+
+10.2
+Localization & Mapping
+A secondary objective of the 60-minute Final Event Prize Run was to rapidly map the environment and
+transmit the real-time map back to DARPA every ten seconds. The map takes the form of a point cloud, or
+a collection of three-dimensional points that represent occupied space in the environment, e.g. floors, walls,
+ceilings. Figure 18 provides a comparison of the DARPA-generated ground truth map in black against the
+final map submitted by Team MARBLE, split into inliers in green and outliers in red.
+Inliers
+Outliers
+Ground Truth
+25
+0
+50
+meters
+Figure 18: Final point cloud map submitted by Team MARBLE to DARPA staff during the Final Event Prize Run.
+DARPA has generously collected, processed, and shared this map data with participating teams (Schang
+et al., 2021). Inliers are defined as points within 1m of ground truth map points, and outliers are defined
+as points outside 1m. Map coverage is a metric representing the ratio of the environment explored, and is
+defined as
+map coverage = ground truth points within 1m of an inlier point
+total ground truth points
+.
+(9)
+Map error or deviation is a metric representing the ratio of the submitted map that is inaccurate relative to
+the ground truth map, and is defined as
+map deviation =
+outlier points
+total submitted points.
+(10)
+Figure 19 shows that map coverage steadily increases throughout the mission, with some periods of rapid
+exploration, and by the end of the mission, nearly 50% of the environment has been mapped. Map error on
+the other hand, increases modestly throughout the mission due to localization drift. However, it increases
+significantly due to a localization failure on D01, which is discussed further in Section 10.3.4. This failure
+generated erroneous sections of map which are mistakenly appear as a long winding corridor in Figure 18.
+
+File  Panels  Help
+Reset
+21 fp0
+10
+20
+30
+40
+50
+60
+Mission Time [minutes]
+0
+5
+10
+15
+20
+25
+30
+35
+40
+45
+50
+Map Statistic [%]
+0
+2
+4
+6
+8
+10
+12
+14
+16
+18
+Cumulative Score
+Map Coverage [%]
+Map Error [%]
+Cumulative Score
+Figure 19: Map coverage, error, and cumulative score throughout Team MARBLE’s deployment during the DARPA
+SubT Final Event Prize Run.
+(a)
+(b)
+(c)
+(d)
+Figure 20: The final maps from (a) D01, (b) H01, (c) D02, and (d) H02.
+10.3
+Planning
+During the entire 60-minute mission and across a diversity of environments, none of the robots were teleoperated
+due to a planner failure or to improve the volumetric gain, which is seen as a successful demonstration of the
+planning reliability and flexibility. The human supervisor only intervened to complement autonomy with
+
+human-level cognition and intelligence, as discussed further in Section 10.6.
+As an overview, Figure 21 presents three scenarios from the Final Event Prize Run, which demonstrate
+negative obstacle avoidance, multi-agent coordination based on teammate position histories, and teleoperation
+initiation by the human supervisor. Below, Section 10.3.1 highlights the exploration performance, Section
+10.3.2 shares examples of agents avoiding treacherous terrain, and Section 10.3.3 demonstrates the planner
+adapting to dynamic changes in the environment. Section 10.3.4 discusses system limitations that prevented
+the agents from exploring the entire course.
+Figure 21: The figure shows three snapshots from the SubT Final Event Prize Round run. The blue lines in the
+figure represents the locally sampled RRT∗ tree and the green lines represent the global graph. (a) The robot can be
+seen to avoid sampling over the negative obstacles i.e., the edge of the subway platform, owing to the settling-based
+collision-checks. (b) The agent planned away from the position history of a teammate robot that was launched before
+it, demonstrating effective multi-agent coordination. (c) This snapshot shows an instance where teleoperation was
+initialized on one of the robots. The robot can be seen to plan a path that had sufficient volumetric gain leading the
+robot toward a frontier in the urban area that had not been seen by any other robots. However, the human supervisor
+decided to teleoperate the robot through the initial section of the tunnel area and then let it autonomously explore
+the tunnel.
+10.3.1
+Exploration
+The volume of unseen area explored by each agent across the mission is illustrated in Figure 22. Unlike the
+statistics of the global map submitted to DARPA in Figure 18, Figure 22 presents agent-specific exploration
+information stored onboard each robot. D02 was the largest contributor to overall exploration, partly because
+it was launched first and had the most time to explore. D01 played a complementary role by exploring most
+of the tunnel environment, which remained unexplored by D02. Because the subset of the environment safely
+traversable by the wheel robots was relatively more constrained, it naturally led to less exploration from the
+Huskies across the mission.
+Besides showing the volume explored by each of the robots, Figure 22 also shows time periods during which
+the default planner exploration behavior was paused for higher-level mission management and teleoperation
+commands. These interjections by the autonomous mission management system were primarily triggered
+when agents approached each other, and resulted in lower-priority agents pausing and higher-priority agents
+resuming their task. During these encounters, agents appear as fast-approaching dynamic objects, and this
+simple procedure was employed rather than incorporating reactive obstacle avoidance into the planner.
+10.3.2
+Treacherous Terrain
+The path planning solution successfully kept each agent safe from collision throughout the entire mission. The
+Spot robots, which explored more challenging features in the environment, fully avoided negative obstacles,
+such as shear drop-offs and rocky slopes, and traversed up and down stairs.
+
+(a)
+(b)
+(c)
+Teammate
+Teammate Position
+Position History
+Histories
+Subway
+Station
+PlannedPath
+0.00000
+Robot
+0.00000
+Position
+.000000
+DO
+Planned Path
+Negative
+Obstacle
+Robot Position
+Robot Position
+Platform0
+10
+20
+30
+40
+50
+60
+Time Elapsed (min)
+0
+500
+1000
+1500
+2000
+2500
+Explored Volume (
+m 3)
+D01
+D02
+H01
+H02
+Go-Home
+Planner-Off
+Figure 22: Volumetric gain explored by each robot across the 60-minute Final Event Prize Run. Planner-Off, denoted
+by blue lines, represents instances when the planner was paused to allow the autonomous mission-management system
+to take over when robots are in close proximity, as well as the several interventions when the human supervisor
+manually teleoperated robots. Go-Home, denoted by black lines, represents instances when the planner began returning
+home to reconnect to the network and report new artifact reports and map data.
+The Spot robots successfully traversed up and down the small set of stairs leading up to the subway platform,
+as shown in Figure 23. However, when stairs where first encountered from above, agents did not plan down
+them due to the limited ±16.5◦ vertical field of view of onboard Ouster OS-1-64 lidar sensors, as shown in
+Figure 24a. In addition, the Spots can only safely walk down stairs backwards, and therefore additional logic
+would be be required to autonomously traverse those stairs.
+Both Spot robots thoroughly explored the subway platform, approaching the edge, but never stepping and
+falling over, as demonstrated by D02 in Figure 24. Additionally, the same Spot robot, D02, explored the
+entire cavern autonomously without entering treacherous terrain, as shown in Figure 25.
+(a)
+(b)
+Figure 23: Instance of (a) D01 planning up stairs with the green edges of the graph, pink planned path, and associated
+semantic map with blue voxels representing stairs, along with (b) FPV imagery.
+10.3.3
+Dynamic Environment
+The planner has the ability to adapt to dynamic environments, such as closing or opening of doors, falling
+rubble, as well as other situations also lead to dynamic changes in the map, including other nearby mobile
+agents, and localization and mapping error.
+During the Final Event Prize Run, D01 traveled through a side branch of the urban environment, triggering
+a trap door. Figure 26 shows the planner adapting to the dynamic environment by re-assigning previously
+
++Entry(a)
+(b)
+Figure 24: Instance of (a) D02 thoroughly exploring the subway platform without planning over the edge, with green
+edges of the graph, pink planned path, and associated map, along with (b) FPV imagery.
+(a)
+(b)
+Figure 25: Instance of (a) D02 autonomously exploring the entire cavern, traversing the safer surfaces and avoiding
+treacherous areas, such as the one shown in (b).
+traversable edges as untraversable.
+In addition, there were several instances of temporary localization and mapping error, which caused erroneous
+new map data to change previously traversable edges of the planning graph to untraversable. In each case,
+the planner adapted to the new scenario, and continuously operated throughout the localization drift and
+loop closure correction. An example of this is included in Section 13.10 of the Appendix.
+When agents pass each other, they appear as fast-moving dynamic obstacles, and cannot re-plan around one
+another fast enough. Therefore, an agent-based prioritization scheme prevents both collision and deadlock, by
+enforcing one agent to wait while the other passes. Examples can be found in Section 13.10 of the Appendix.
+10.3.4
+Limitations
+However, several limitations did prevent agents from exploring roughly half of the course. In total, there
+were three types of bottlenecks: constrained corridors, slippery surfaces, and downward sets of stairs, each
+exposing unique limitations within the autonomy system. None of these are limitations of the planner itself,
+but rather limitations of mapping, mobility, and perception.
+The planner did not plan through all constrained spaces because the selected planning parameters for agent
+width and height did not allow the graph to propagate through especially short and narrow spaces. These
+parameters were intentionally chosen to be conservative to prevent the agent from moving along an unsafe
+trajectory. Utilizing a higher-resolution local map and planner could result in a more agile robot that could
+safely traverse those spaces. Additionally, autonomously transitioning into a crouching gait could improve the
+Spots ability to traverse spaces with low ceilings. Examples are included in Section 13.11 of the Appendix.
+
+Subr(a)
+(b)
+(c)
+(d)
+Figure 26: Early in the mission, D01 walked by the corridor with the trap door, as shown by (b) the left camera (1:47).
+The agent later to returns to the corridor, (a) walks under the trap door (10:34), and soon after sees it has closed, as
+shown by (d) the right camera (11:01). After seeing the trap door close, (c) the updated map and graph show (red)
+edges as untraversable (11:11). Had the agent moved closer to the trap door, and fully been within mapping range, all
+edges would have updated as untraversable.
+The second limitation is slippery surfaces, and led to D01 slipping and falling in the cave section. Some of
+these rocky surfaces were intentionally designed to be slippery, and plenty of humans walking through the
+course after the event also slipped and fell. After D01 fell, it also experienced a localization failure. Methods
+to recovery the system from an event such as this one would involve implementing a fall detection algorithm,
+as well as autonomous self-righting and localization reset logic.
+10.4
+Artifact Detection
+In this section, we present performance results of the artifact detection and reporting system. During the
+60-minute Final Event Prize Run, Team MARBLE scored a total of 18 artifacts out of the 40 artifacts in
+the environment. Figure 27 presents a flow diagram that summarizes how our team scored 18 artifacts and
+the limitations that resulted in the remaining 22 from being scored. Of all 40 artifacts, our agents explored
+enough of the environment that they were in the vicinity of 25 artifacts, leaving 15 unexplored due to mobility
+challenges discussed in Section 10.3. Team MARBLE reported 19 of the 25 artifacts that were explored,
+and successfully scored 18 of those 19 reported. A map of the area explored and scored artifacts is shown
+in Figure 28. Details of these 25 explored artifacts, of which 18 were scored, one was missed, and six were
+unreported, are shared in Section 13.13 of the Appendix.
+Of the 18 artifacts that Team MARBLE successfully scored, 11 were scored by autonomous robot reports, two
+were scored by the human supervisor modifying the position of autonomous reports, and five were scored by
+the human supervisor via robot FPV imagery. One artifact was reported but did not score due to localization
+error in excess of 5m. There were six artifacts that agents saw, but did not report due to errors in the
+autonomous artifact detection system. The human supervisor received information regarding three of these
+six artifacts but missed them due to high workload demands during the mission. The other three artifacts
+were located in areas that prevented agents from communicating back to the human supervisor.
+10.4.1
+Visual Detection
+The focus of this section is to quantify the performance of the visual artifact detection system. Of the 18
+artifacts that Team MARBLE scored, 11 of them were visual artifacts, as shown by Table 4. First, we focus
+on the six artifacts (L51, L53, L26, L34, L40, L67) that were successfully reported by the agents’ autonomous
+artifact detection systems in the course. All six artifacts were accurately localized to within 5m. In fact, the
+largest error for a visual artifact was 2.87m (L67). This eliminated the need for the human supervisor to
+spend time trying to correctly localize artifacts.
+The autonomous artifact detection system filters raw frame-to-frame detections onboard the agent, with the
+
+Figure 27: Flow diagram illustrating how Team MARBLE scored 18 of the 40 artifacts in the course, and the
+limitations preventing the remaining 22 artifacts from being scored.
+aim to reduce the number of false positive and redundant artifact reports. Because the human supervisor has
+limited bandwidth, unnecessary distractions detract from the completing other mission-related tasks. In the
+process of scoring these six visual artifacts, the human supervisor had to process 21 artifact reports from the
+automated artifact detection systems onboard agents. Of these, 11 were true positives, six were approved by
+the human supervisor and successfully reported, and five were ignored because they were redundant reports
+that were previously scored. The other 10 reports were also ignored by the human supervisor because they
+were false positives.
+In total, there were only four false reports that the human supervisor submitted. One was cause by human
+error, the other three were caused by erroneous CO2 reports, as detailed in Section 13.16 of the Appendix.
+Agents in the course failed to autonomously report the other five artifacts (L55, L32, L31, L38, L58), but did
+transmit FPV imagery back to the human supervisor, who manually reported and scored them. The fact
+that the autonomous artifact detection system did not detect five of the 11 visual artifacts it saw, indicates
+that certain reliability limitations exist. Team MARBLE acknowledged this limitation and relied on the the
+human supervisor and FPV system to fill in that void, which is further in Section 10.6.
+10.4.2
+Non-Visual Detection
+Team MARBLE scored seven non-visual artifacts, i.e. cell phone, cube, and gas, but as shown in Table
+5, required submitting more reports due to difficulty around accurately localizing the source. The main
+limitation is that the detection scheme relies on threshold-based logic for RF and CO2 levels, and when
+triggered, simply reports the current location of the agent. The thresholds were intentionally set low to
+increase the probability of detection when agents pass by the vicinity of these non-visual artifacts. Of the
+seven non-visual artifacts scored, five of them (L08, L36, L22, L47, L24) were scored via the autonomous
+robot reports, with an average error of 3.27m. The remaining two artifacts (L11, L59) were scored by the
+human supervisor manually adjusting the reported artifact locations.
+10.5
+Communications
+The performance of the communication system was evaluated in the final run using both qualitative and
+quantitative measures. Subjectively, the human operator was able to employ live FPV video from the robots,
+a capability that directly contributed to team’s third place finish. The robots were in communication with
+
+Robot Score:11
+ReportedArtifacts:19
+ScoredArtifacts:18
+ExploredArtifacts:25
+Hybrid Score:2
+AllArtifacts:40
+Human Score:5
+NotScoredArtifacts:1
+Localization Error:1
+Unreported Artifacts:6
+Robot Detection Error:6
+HumanMissed:3
+NoCommunications:3
+UnexploredArtifacts:15
+Limited Mobility: 15Figure 28: Locations of ground truth artifacts in white, reports that scored in green, and reports that did not score in
+red, overlaid on ground truth map of the course.
+Artifact Type
+Scored
+Not Scored
+Unreported
+Unexplored
+Total
+Survivor
+2
+0
+0
+1
+3
+Cell Phone
+3
+0
+0
+1
+4
+Backpack
+2
+0
+0
+3
+5
+Drill
+2
+0
+0
+2
+4
+Fire Extinguisher
+2
+0
+1
+1
+4
+Gas
+2
+0
+1
+0
+3
+Vent
+0
+0
+3
+1
+4
+Helmet
+1
+0
+1
+3
+5
+Rope
+2
+0
+0
+3
+5
+Cube
+2
+1
+0
+0
+3
+Total
+18
+1
+6
+15
+40
+Table 4:
+Artifact statistics for Team MARBLE during the Final Event Prize Run. A total of 18 artifacts were scored.
+Only one artifact was reported but not scored, which was due to localization error greater than 5m. Agents were in
+the vicinity of six artifacts, but they went unreported due to autonomous artifact detection failure and in some cases,
+also missed by the human supervisor due to excess workload. The remaining 15 unexplored artifacts were never seen
+by agents because they were located in parts of the course that were never reached.
+the base station over the majority of the explored regions of the course, as shown by the blue-green hues in
+Figure 29a, allowing the human supervisor to monitor and intervene as needed. Overall, 125.2 MB of data
+was transferred through the communications network, including all map segments, telemetry, artifact reports,
+and other data products. Of that, FPV video comprised 51.6 MB. The latency of the mesh networking
+solution was evaluated using inter-message arrival times of a heartbeat message sent from a long-ranging
+robot to the base station. This mission management message, transmitted regularly from the robot, was
+
+Attempt Type
+Scored
+Missed
+False
+Total
+Survivor
+2
+0
+0
+2
+Cell Phone
+3
+5
+0
+8
+Backpack
+2
+0
+1
+3
+Drill
+2
+0
+0
+2
+Fire Extinguisher
+2
+0
+1
+3
+Gas
+2
+1
+3
+6
+Vent
+0
+0
+0
+0
+Helmet
+1
+0
+0
+1
+Rope
+2
+0
+0
+2
+Cube
+2
+5
+0
+7
+Total
+18
+11
+5
+34
+Table 5: Artifact report statistics for Team MARBLE during the Final Event Prize Run. A total of 34 reports, or
+attempts, were made throughout the mission. A total of 18 attempts resulted in scores, 11 attempts were misses and
+did not result in a score due to localization error greater than 5m, and five attempts were false attempts in that they
+were false positives and no artifact of that class was in the vicinity.
+part of our protocol scheme and is analyzed as a message of convenience. A distribution of the inter-message
+arrival times of these heartbeat messages from D01 to the base station is shown in Figure 29b.
+Since these messages originate from the robot at 1 Hz, an ideal system would observe all inter-message arrival
+times to be one second in duration. As our mission management system on the robot does not run with hard
+realtime constraints, the 1 Hz publish rate is an estimate that includes noise due to process load, etc. On the
+base station, there was an approximately normal (N(1.00, 0.04)) distribution of arrival times. Since messages
+may experience delays, the immediately following message may exhibit an inter-message time of less than one
+second, leading to the symmetry apparent in Figure 29b. Our key observation of this plot is that the bulk
+of messages arrive within five percent of their expected times across a distance of hundreds of meters and
+multiple mesh hops, validating the performance of the entire mesh networking solution.
+(a)
+0.85
+0.9
+0.95
+1
+1.05
+1.1
+1.15
+Time (s)
+(b)
+Figure 29: Performance results of the communication systems, including (a) the map of the Team MARBLE’s
+deployment during the Final Event Prize Run, overlaid with locations of robot connection (C) to the network in
+blue-green, and locations of robot disconnection (D) from the network in red-magenta, as well as (b) the distribution
+of inter-message arrival times for D01 with a nominal publishing rate of 1 Hz, overlaid with N(1.00, 0.04). The
+highlighted red region represents the first σ value which contains 68% of the message times.
+
+C
+D
+D01
+D02
+H01
+H0210.6
+Mission Management
+Overall, the mission management system was able to keep the robots on task with minimal human supervisor
+intervention. However, when needed, the interventions were crucial towards both the exploration capabilities
+of the system and the final event performance. Figure 30 presents a detailed timeline of the four agents in
+the field, as well as the human supervisor. The four robot launches and five robot interventions were the only
+times when the human supervisor used teleoperation. There were only two other types of instructions agents
+received from the human supervisor. One was commanding H02 to drop two communication beacons. The
+other was commanding D01 to return home three times, each occurring while the human supervisor was also
+teloperating D01 through the fog.
+Figure 30: Mission management timeline for all robots and human supervisor during the 60-minute Final Event Prize
+Run. When not manually teleoperating a robot, the human supervisor was monitoring the mission, which includes
+watching live FPV streams from D01 and D02, reviewing incoming artifact reports from agents, reporting artifacts to
+DARPA, and in the last five minutes of the mission, checking archived FPV images from the Spots for previously
+missed artifacts.
+10.6.1
+Robot Launches
+The human supervisor was under immense pressure to optimally balance many competing tasks during the
+60-minute Final Event Prize Run. With such limited time, the primary objective at the beginning of the
+mission is to launch robots into the environment as fast as possible. Through extensive practice and full-scale
+comprehensive field deployments, discussed further in Section 11.2, Team MARBLE launched all four robots,
+with a mean launch time of 41 seconds, as shown in Table 6.
+Two agents experienced failures late in the mission, reducing overall fleet utilization rate from 92% to 73%.
+H02 experienced a hardware failure (36:48), and post-event inspection revealed better vibration isolation
+of the computing system would reduce the likelihood of such a failure in the future. D01 experienced a
+mobility failure (39:46), in which the Spot slipped on a slick rock and fell over. The agent then experienced
+a localization instability due to the large induced velocity. To recover from such an incident in the future,
+Team MARBLE could implement an autonomous self-righting maneuver and localization reset logic.
+Despite the fact that some artifacts were not detected and reported by the autonomous board artifact
+detection system, the human supervisor filled in the void. The human supervisor reported and scored five
+artifacts that were seen via robot FPV imagery, but not autonomously detected. Of these, the human
+supervisor saw four (L55, L32, L31, L38) from live FPV streams, while one (L58) was found while reviewing
+
+aunch
+narrow
+D02
+stairs
+cave
+Launch
+D01
+fog
+mobility failure
+Launch
+H02
+entrance
+hardware failure
+Launch
+H01
+entrance
+HS
+0
+5
+10
+15
+20
+25
+30
+35
+40
+45
+50
+55
+60
+Mission Time
+[minutes]
+autonomous
+manual teleoperation
+not in operation
+mission monitoring
+operationLaunch
+Agent
+Duration
+Teleoperation Window
+Course Entry
+[s]
+[mm:ss - mm:ss]
+[mm:ss]
+RL1
+D02
+75
+-00:37 - 00:38
+00:02
+RL2
+D01
+55
+01:29 - 02:24
+01:44
+RL3
+H02
+35
+07:15 - 07:50
+07:25
+RL4
+H01
+50
+10:57 - 11:32
+11:08
+Mean
+41
+Table 6: List of the four robot launch (RL) sequences executed by the human supervisor during the 60-minute Final
+Event Prize Run. Extensive process streamlining and repeated practice deployments resulted in quick and repeatable
+launch sequences. The course entry column represents the mission time at which the agent crossed into the course.
+archived FPV images near the end of the mission. More details of these artifact reports are presented in
+Section 13.17 of the Appendix.
+10.6.2
+Robot Interventions
+In total, the human supervisor intervened in the autonomous fleet five times during the 60-minute mission.
+Table 7 presents the duration of each intervention, with a mean length of 217 seconds, as well as the reason
+for intervening. Each of these interventions was in the form of manual teleoperation, in which FPV streams
+were available at the base station, and velocity commands from the human supervisor was transmitted to the
+remote agent in the field. Of the five manual teleoperation interventions, only two had significant impact on
+the mission. The first (RI1) was navigating D02 through the narrow cave corridor early in the mission (3:02
+- 7:02), which led the agent to the small cavern and two artifact scores. The second (RI2) was navigating
+D01 through fog in the tunnel section midway through the mission (21:54 - 31:52), leading to six artifact
+scores. Figure 31 provides a side-by-side comparison of the full-resolution FPV imagery processed onboard
+by the autonomous artifact detection system and the compressed FPV imagery transmitted to the human
+supervisor.
+Two interventions (RI3, RI5) commanding agents back into the course, was out of a shear abundance of
+caution. The planning algorithm is configured so that the staging area is treated as explored, so agents
+should not attempt to explore it. One intervention failed (RI4), in which the human supervisor attempted to
+teleoperate D02 down the stairs by the subway platform. This attempt failed because communication to and
+from the robot was intermittent.
+(a)
+(b)
+Figure 31: Imagery during robot intervention 2 (RI2). when human supervisor was pushing D01 through the foggy
+area in the tunnel environment, eventually leading to six additional points. Shown is a comparison of (a) full-resolution
+FPV imagery onboard the agent and (b) compressed low-resolution FPV imagery transmitted to the human supervisor.
+
+The main takeaways from these results is that our agents are highly autonomous, leaving the human supervisor
+to focus on mission monitoring and targeting strategic, high-value intervention opportunities. The mission
+management system enabled convenient transition between autonomy and manual operation, while the
+communication system enabled visibility and control over the agents in the field.
+Intervention
+Agent
+Duration
+Window
+Goal
+Success
+Points
+[s]
+[mm:ss]
+RI1
+D02
+240
+03:02 - 07:02
+Enter narrow cave corridor
++
+6
+RI2
+D01
+598
+21:54 - 31:52
+Enter foggy tunnel area
++
+2
+RI3
+H02
+24
+35:04 - 35:28
+Avoid course exit
++
+0
+RI4
+D02
+200
+39:24 - 42:44
+Walk down stairs
+–
+0
+RI5
+H01
+21
+49:40 - 50:01
+Avoid course exit
++
+0
+Mean
+217
+1.6
+Table 7: List of the five robot interventions (RI) executed by the human supervisor during the 60-minute Final Event
+Prize Run. Our concept of operations relies on autonomous multi-agent exploration, and does not necessitate manual
+waypoints or teleoperation from the human supervisor. Therefore, agents were completely autonomous, except for
+the human supervisor input during these five instances of teleoperation. The interventions goals varied, but were
+all specific scenarios where human intervention would augment autonomous agent capabilities in a mission-relevant
+manner.
+10.7
+Open-Source Data
+During the final run at the final event, our team collected a significant amount of data related to autonomous
+subterranean exploration in the form of ROS “rosbags”. Our datasets are split up by agent, and each set
+contains a rosbag of the system inputs, mostly consisting of raw sensor data, and another rosbag of the
+outputs used for visualization and performance monitoring. The complete collection of data recorded at the
+final event can be found at https://arpg.github.io/marble.
+11
+Lessons Learned
+Underground exploration of previously unknown environments, especially in a time and resource-constrained
+search-and-rescue context, requires a highly adaptable human-robot team. The lessons learned presented in
+the following sections enhance our proposed system’s flexibility across mobility, communications, human-robot
+teaming, and multi-agent coordination.
+11.1
+Platform Mobility
+Systems with heterogeneous platforms allow for specialization by each platform for both specific environments,
+and roles which benefit the entire mission. Specifically, in Team MARBLE’s case, the addition of Spot
+platforms into an exploration role enabled rapid multi-story expeditions.
+This capability was further
+augmented with the utility of higher-payload, wheeled Huskies, carrying communication beacons. When
+deployed, these beacons allowed the Spot platforms to report artifacts without the need to traverse “home,”
+leading to much more efficient exploration. No single, platform is as performant as the combination of
+platforms operating in different roles based on each one of their strengths.
+11.2
+Testing and Validation
+Significant testing and validation was conducted for every element of the Team MARBLE system. These
+tests taken over a variety of environments, shown in Figure 32, reduced mission-to-mission variability, and
+
+increased system-wide adaptability. Through these tests, Team MARBLE locked in well tested solutions with
+minimal unexplained errors. This reduced changes on the system as we approached the final competition,
+given that new solutions had to be verified against similarly stringent tests. To our knowledge our team made
+the fewest hardware and software based adjustments from the preliminary runs to the final run to achieve our
+score, instead relying on our testing having eliminated all significant errors outside of a so-called ”Poisson
+distributed fatal error.” These types of errors were mission ending to any individual robot but difficult to
+predict or adapt to in a meaningful sense. These can be seen in our final competition run where D01 fell over
+on rough steep terrain, and where H02 suffered a hardware error.
+A secondary goal of these full-scale deployments was to reduce human-based variability in performance.
+They helped both the human supervisor and pit crew prepare for the stressors of operating and responding
+quickly to complex interactions between robots, the environment, and potential failure modes. At no point
+was a single test considered sufficient for validating a solution as sufficient; instead all solutions released
+demonstrated repeatability across the same and varied environments. For reference, all full scale deployments
+are tabulated in Section 13.18 of the Appendix.
+(a)
+(b)
+(c)
+Figure 32: Team MARBLE conducted comprehensive field deployments at various sites including (a) the Edgar
+Experimental Mine operated by Colorado School of Mines, located in Idaho Springs, CO. which tested terrain, distance
+travelled, and communications; (b) the Engineering Center complex also located on University of Colorado Boulder
+Main Campus which tested multi-story navigation and repeated features from urban environments; and (c) the Folsom
+Parking Garage located on University of Colorado Boulder Main Campus in Boulder, CO. which tested planning in
+open spaces and vertical localization across multiple stories.
+11.3
+System Adaptability
+The challenges posed by operating a system in an unknown environment necessitate a high level of system
+adaptability. Predicting every capability that a system will need for a given mission, such as search and
+rescue, is impractical. Instead, having a highly flexible system capable of adapting to unknown situations
+is crucial. Having already addressed the mobility considerations in Section 11.1, we also found software
+adaptability key to our success. For instance, the flexible communication network enabled the ability to
+pass on FPV to the human supervisor with a simple configuration change. While our system was designed
+with autonomy in mind and the capability was not previously planned for, the adaptation proved invaluable.
+Minimal human supervision directly impacted the final score and exploration capabilities of the system as a
+whole.
+The notion of adaptability extends to other software architecture as well. The artifact detection system was
+adjusted during the final competition to include SSID information for Bluetooth artifacts. These artifacts
+could then be more accurately merged between agents and by the human supervisor, despite inaccurate
+positioning from wireless signals.
+In some cases, adaptability is explicitly accounted for in our design, most notably in the mission management
+system. BOBCAT has many parameters which allowed the human supervisor to adjust exploration activity,
+including how long a robot can explore before reporting detected artifacts, whether a robot should find
+multiple artifacts before reporting, and how the system should adjust to time constraints in the mission.
+
+STARTHERE11.4
+Autonomy and Human Robot Interaction
+Team MARBLE emphasized autonomous system design as our primary goal. Since only one person, the
+human supervisor, was permitted to interact with agents while they were deployed, having robots controlling
+their own paths and decision making was key to reducing the cognitive load required to manage mission
+objectives. Despite the focus on autonomy, the human supervisor is still necessary to maximize system
+performance. As Team MARBLE approached the final competition, we found targeted areas where direct
+human supervisor control improved results. This influenced decisions regarding the number of deployed
+platforms, how artifacts were passed from agents to the human supervisor, and how we recovered from
+anomalous robot behavior.
+To maximize the human supervisor’s ability to track the data streams present across the fleet, only four
+robots were deployed. While a larger fleet might have enabled a rapid exploration of the environment, it
+could reduce the human supervisor’s ability to meaningfully address problems arising from any specific robot.
+The artifact detection system was designed to filter against false positives before passing information to the
+human supervisor. Artifacts had to consistently detected in multiple frames, and be a sufficient distance from
+existing artifact estimates. Where possible, artifacts were returned with additional information including
+SSID for Bluetooth artifacts and images for visual artifacts. This data allowed the human supervisor to sort
+remaining false positives quickly without being distracting from the core mission objectives.
+Specifically, the human supervisor spent most of their operational load solving problems the robots were
+incapable of correcting through their autonomy stack. For example, during the Final Event Prize Run, the
+human supervisor used the first person vision to navigate through dense fog, which the robot was incapable
+of planning through autonomously. After this intervention, the human supervisor allowed the robot to return
+to autonomous operations, where it explored several new areas, and reported a total of six new artifacts.
+The human supervisor was able to opportunistically intervene like this, only because the other agents were
+operating autonomously without supervision.
+11.5
+Retrospective
+It’s important, after a three-year effort of this size and scope, to examine some of the bigger picture questions.
+What would we do differently next time? What did we wish we knew at the start, that we know now? The
+answer to the first two questions is that we would have spent more time at the onset to scope out short-term
+and long-term development goals. We excelled at the Tunnel Event, placing fourth amongst a large group
+of competitors. This was mostly due to meeting well-scoped short-term goals within the one year cycle.
+However, during the six-month development period for the Urban Event, we focused on long-term goals that
+we ultimately only partially validated, leading to a disappointing performance. In hindsight, we should have
+focused on making our already capable platforms more capable, rather than spreading our resources thin
+across many thrusts. However, what our team excelled at after this experience, was pivoting and adopting two
+new strategies. First, we leaned into student-led project management, which led to greater team coordination,
+a key component of rapid and effective system development. Secondly, we focused on long-term development
+goals, and given a year and a half, had the time to adequately develop, test, and validate each subsystem,
+each fully autonomous robot, as well as our entire fleet in numerous search and rescue missions. Together,
+these two powerful changes allowed a small, lean team, produce a highly functional autonomy solution that
+could perform under pressure.
+12
+Conclusions
+In this paper, we showcase our flexible autonomy solution for exploring unknown subterranean environments.
+The highly performant autonomy solution directly lead to a third place finish at the DARPA SubT Final
+
+Event which was focused on search and rescue. Moreover, the specific innovations presented in graph planning,
+flexible communications, and mission management are directly applicable to other multi-robot teaming
+applications under limited human supervision.
+Specifically, the combination of legged and wheeled robots allowed for heterogeneous teaming enabling both
+rapid exploration and a robust communication network. The deployed mesh network which is described
+in Section 8 enables flexible configuration and prioritization of data sent both to other robots, and a
+human supervisor for review. A powerful graph planning framework, as described in Section 7, paired with
+semantically encoded Octomaps enabled safe, rapid exploration. The final system was able to explore a
+variety of underground environments, including gold mines and subway stations, with minimal human input.
+Human input was reserved for specific situations where higher-level reasoning had the potential to improve
+the mission. The flexible mission management system described in Section 9 enables safe transitions between
+human input and the underlying autonomy system. One of the most important lessons learned from the
+developed system is that focusing on autonomy is core for human robot teaming. Robots need to be able to
+make decisions on their own, enabling the limited human resource to only act in critical situations.
+
+Acknowledgments
+This work was supported through the DARPA Subterranean Challenge, cooperative agreement number
+HR0011-18-2-0043. A special thanks to Andrew Beathard, Nikolaas Bender, Cesar Galan, Nicole Gunderson,
+Davis Landry, Greg Lund, Cole Radetich, Ben Rautio, Zoe Turin for assisting with the design, testing, and
+deployments of our platforms and algorithms. Thank you to the Colorado School of Mines and the Edgar
+Experimental Mine for allowing us to conduct mock deployments in the mine. Special thanks also to Simon
+Wunderlich and his team at Meshmerize GmBH for their mesh networking support. Finally, thank you to all
+of the DARPA staff who have planned and executed absolutely incredible Subterranean Challenge system
+track circuit events.
+
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+
+13
+Appendix
+13.1
+Communication Beacon Design
+Each beacon consists of two 3D printed nylon-carbon fiber infused internal brackets serving as structural
+support components. Figure 33a shows the top of each beacon that contains a charging port, power button
+and LED power status indicator between the antennas. The two stabilizers seen towards the front provide
+longitudinal stability for the beacon once deployed on the ground. These complement the steel counterweights
+on the rear of the beacon. The deployment mechanism shown in Figure 33b uses solenoids to hold the beacons
+in place. When the solenoid is released, the beacon falls at a consistent rate using a constant force spring.
+(a)
+(b)
+Figure 33: On the left (a) a perspective view of the beacon design as well as (b) a side view of the beacon attached to
+the deployment mechanism.
+13.2
+Platform Compute Design
+When designing our system, we sought to balance the ease of development with a single, monolithic compute
+unit versus potential integration challenges of a distributed system.
+Early on, we decided that, given
+uncertain compute requirements, we should attempt to pack as much compute as possible into the Husky
+platforms. This decision had a wide array of collateral consequences, including power system requirements,
+cooling requirements, and mechanical considerations. Further, the decision to not utilize an industrial-style
+motherboard, but rather a consumer gaming motherboard, created integration difficulties that might have
+otherwise been avoided. For example, the power requirements of our Ryzen Threadripper-based system were
+roughly 425W at peak consumption of both CPU and GPUs, at the limit of potential commercially available
+DC/DC ATX power supplies. Using discrete GPUs mounted in PCI Express slots presented a mechanical
+challenge, particularly for shock and vibration mounting, which was discovered as a failure mode late in the
+design process. In contrast, the distributed architecture developed for our Spot platforms utilized significantly
+less power and space while delivering similar performance. Future system designs are more likely to follow a
+distributed approach, rather than a monolithic approach, to ease mechanical and electrical integration efforts
+at the expense of only minimal added software effort.
+At a deeper level of inter-robot module communications, we underestimated the challenges of differing
+ground potentials. After shorting out several serial links between components and having unreliable USB
+communication, we realized that several ground loops were responsible. By adding serial optocouplers and
+
+Solenoid Catch
+Antennas
+Spring
+Catch
+PowerButton
+Counterweight
+StabilizerSolenoid Catch
+ConstantForce
+Spring
+Solenoid
+C-Channel
+Stabilizer
+Counterweightconverting to Ethernet-based platform control, these ground loops were eliminated, resulting in highly reliable
+platforms. Future development would rely exclusively on differential signalling such as controller-area network
+or Ethernet for inter-module communication.
+13.3
+Sensor Synchronization Design
+To effectively share sensor data between robots, sensor and system timing has to be considered. Our lidar
+solution could utilize IEEE 1588v2 timing (also known as the Precision Time Protocol v2 or PTP), but
+our wireless mesh networking solution could not support IEEE-1588v2. Therefore, we implement Network
+Time Protocol (NTP) between robots and PTP within the same robot. On startup, each robot attempts to
+synchronize with the Base Station using NTP over the mesh network. This synchronization step is critical for
+multi-robot operations and coordination to provide a consistent time basis across all nodes. In testing, the
+relative time drift (a few milliseconds) over the course of a run (1 hour) was not significant enough to cause
+problems. If the Base Station is unreachable (say for single-robot testing), the robot falls back to its own
+battery-backed realtime clock as a time source. In either case, after attempted synchronization, no further
+attempts are made to match times with any other robot or the Base Station. In testing, we observed that
+clock slews resulting from attempted time synchronizations as robots entered and left communications range
+had a negative impact on localization performance.
+In the Threadripper monolithic architecture used on the Husky platforms, PTP on the secondary Ethernet
+interface functioned perfectly, allowing the Threadripper to become the grandmaster and the lidar to follow
+along, as shown in Figure 34a. However, in the distributed Xavier+NUC architecture (shown in Figure 34b),
+designing an effective PTP interface encountered significant challenges.
+Threadripper
+Base Station
+Lidar
+PTP Grandmaster
+NTP
+PTP
+(a)
+NUC
+Base Station
+Xavier AGX
+Lidar
+PTP Grandmaster
+NTP
+PTP
+(b)
+Figure 34: Block diagrams showing Precision Time Protocol (PTP) distribution between system components on the
+Threadripper (a) and Xavier+NUC (b).
+Fundamentally, PTP requires hardware support in order to function by performing sensitive timing operations
+as close to transport medium as possible. The network hardware options on the Spot included a Realtek
+r8125, an Nvidia platform SoC module, and a quad-port Intel i210 card. Realtek r8125 support for PTP was
+not functional, as verified by the phc ctl utility; this relatively new chipset relies on an out-of-tree kernel
+driver for Linux at the time of our development. The NVidia platform module appeared to support PTP
+when interrogated by phc ctl, but on further investigation through network traffic inspection, the platform
+module was not inserting the correct information into outbound Ethernet traffic. Our final solution is based
+on using a spare port on the Intel i210 card, which had robust, verifiable PTP support. As the NUC lacks a
+port with viable PTP support, we fall back on NTP as a synchronization method, relying on the Xavier as
+the robot’s grandmaster time source.
+As an aid to the community, we offer the following verification steps to assist in debugging PTP issues. First,
+verify that there is hardware support via ethtool -T <iface> to verify kernel-level hardware PTP support.
+Second, use phc ctl <iface> cmp to verify that the Ethernet hardware clock is synchronized to the Linux
+system clock. Finally, ensure that the hardware timestamps encoded in the PTP network traffic match
+
+the local system by capturing network traffic from a PTP-enabled grandmaster Ethernet interface. These
+steps can verify that the PTP software stack is broadcasting the system time via the Ethernet hardware to
+downstream consumers.
+13.4
+Large-Scale Localization Validation
+Validation testing of LIO-SAM onboard Spot and Husky platforms was imperative for ensuring sufficient
+accuracy, speed, and stability for long-duration and large-scale missions. Some examples include an outdoor
+test at CU South Campus, as shown in Figure 35, and as well as test that begins at the CU Engineering
+Center building and treks across campus to the bottom of a three-story underground parking garage, as
+shown in Figure 36.
+(a)
+Imagery: @2022 Maxar Technologies, U.S. Geological Survey, USDA/FPAC/GEO, Map data @2022
+0
+100
+200
+meters
+(b)
+Figure 35: On the left (a) photo taken at CU South Campus of a Husky robot during a large-scale, long-duration
+localization test. On the right (b), is an LIO-SAM point cloud map, denoted by small black dots, and LIO-SAM robot
+trajectory denoted by larger dots colored by elevation, overlaid with Google Maps satellite imagery of the area. The
+Husky was manually controlled, beginning in the CU South parking lot, continuing along a dirt path and up a hill,
+looping back, and ending back at the parking lot.
+13.5
+Common Reference Frame Alignment Optimization
+To further reduce the yaw error of the common reference frame alignment, the lateral spacing of the robot
+prisms was increased.
+In our initial testing, we found that the roughly 120mm plates attached to the robots along with the prisms
+1.5mm centering error lead to consistent variance in the resulting yaw of approximately 0.7◦. This aligned
+with a calculated value of arcsin(1.5/120) = 0.72◦. In order to reduce the impact of this error, a mechanical
+bar holding two of the prisms at a distance of 655mm, is added to each robot. The resulting angle error
+after adding the bar was arcsin(1.5/655) = 0.13◦. This was consistent with external testing. For a rough
+comparison see Figure 37a, where a test was conducted using a mock robot plate and the prism separating
+bar. The difference from the average of each test shows a higher precision for the tests conducted with the
+prism bar in place. An example setup for these test is shown in Figure 37b.
+
+Imagery @2022 CNES / Airbus, Maxar Technologies, U.S. Geological Survey, USDA/FPAC/GEO, Map data @2022
+Figure 36: Overlay of LIO-SAM point cloud map, denoted by small black dots, and LIO-SAM robot trajectory
+denoted by larger dots colored by elevation, with Google Maps during a large-scale, long-duration localization test at
+CU Main Campus. This test was conducted on a Spot robot, that was manually controlled, beginning in the CU
+Engineering Center building on the bottom left corner of the map, continuing across campus, ending at the bottom of
+the three-story underground Folsom Parking Garage on the top right corner of the map.
+Plate
+Bar
+Prism Base
+-0.15
+-0.1
+-0.05
+0
+0.05
+0.1
+Yaw (deg)
+(a)
+(b)
+Figure 37: A comparison of transforms generated by LTS using the standard robot sensor plate, and after attaching
+prisms to a bar placed on the plate instead (a). The difference from the average, and precision of the bar is higher
+than without the bar. Setup for a prism test with a mock gate highlighted in red (b). The mock gate was designed to
+have the same dimension as a robot sensor plate
+13.6
+Artifact Detection Training Procedure
+A systematic procedure targeted at low-light conditions is used to train the model. At each location, data was
+collected using three different brightness levels to minimize the impact of lighting conditions on the model’s
+performance. Specifically, images were taken from past circuit events as well as separate field exercises.
+Remote data collection sessions took place inside the dark and rocky Edgar Experimental Mine in Idaho
+Springs, Colorado. Local data collection took place on University of Colorado Boulder campus, primarily
+within the outdoor courtyard of our Engineering Center building, and during evening hours when there was
+
+0
+50
+100
+meters0
+50
+100
+L
+meters0
+50
+100
+meters0
+50
+100
+L
+metersno natural illumination. The data was collected with three onboard illumination levels: 0%, 50%, and 100%.
+The cameras, FLIR Blackfly PGE-05S2C-CS GigEVision cameras, were mounted in cardinal directions on
+the robots as shown in Figure 5b.
+Photos in which the artifacts suffered excessive motion blur and occlusions, determined by the ability of the
+human reviewer to detect the artifact, were removed from the data set. After the AD pipeline was trained on
+this initial data collection effort, we found that it did not generalize well to new environments. Therefore, we
+augmented the dataset with additional imagery collected from a greater diversity of backgrounds, including a
+nearby university loading dock. The datasets used for training are summarized in the Appendix. The dataset
+was later augmented with data from areas where false positives were frequently identified in order to reduce
+the identification of these false positives.
+13.7
+BOBCAT Components
+BOBCAT calculates Objective weights and Behavior scores to select an Execution Behavior whenever one
+or more Monitor outputs or Objective input weights change. Behavior execution functions may be either
+blocking or non-blocking. They should be non-blocking to the maximum extent possible, to increase reactivity
+and allow Behaviors to change at any time. Behaviors that need to block may be required for certain actions
+that must be completed before the robot can do something different. If a Behavior is blocking, BOBCAT
+will delay evaluation until the actions have completed. Tables 8, 9, and 10 provide an exhaustive list of the
+Monitors, Objective, and Behaviors, respectively.
+Monitor
+Criteria
+ExploreToGoal
+Received command to explore to a specific goalpoint, either submitted by the
+human supervisor or generated by a node other than the global planner.
+iExplore
+iGoToGoal
+iStop
+iDeployBeacon
+iGoHome
+The associated input command has been sent by the human supervisor from the GUI
+or joystick to execute a specific behavior.
+NearbyRobot
+This and any other robot’s paths are within 2m of each other.
+Beacon
+A communications beacon is available and other criteria has been met to deploy it.
+ReverseDrop
+A communications beacon is available and communications have been lost with the
+base station for 10 seconds.
+Comms
+Any message has been received from the base station in the last 3 seconds.
+Artifact
+There are at least 3 unreported artifacts, or it has been at least 5 minutes since the
+first unreported artifact was detected.
+Table 8: MARBLE Monitors, with a description of robot state requirements for an output of 1.
+13.8
+Artifact Detection Training Data
+Table 11 presents a summary of the training data Team MARBLE collected and annotated. Our main data
+collection effort was conducted in the Engineering Center Courtyard at University of Colorado Boulder,
+during late evening hours when there was no natural illumination. This location was chosen because we
+believed it be more representative of a subterranean environment, with the large amount of concrete and low
+illumination levels. Another smaller dataset was collected at the Edgar Mine in order to introduce data from
+tunnel-level environments. Real-world imagery from the Tunnel Event at the NIOSH Mine and the Urban
+Event at the Satsop Nuclear Power Plant was incorporated as well. To round out training, a final data set at
+
+Objective
+Evaluation Function
+Description
+FindArtifacts
+iwFindArtifacts
+Find, identify and localize artifacts. Always active.
+Input
+iwInput ∗ OR(Input Monitors)
+Allow supervisor to override autonomy when
+necessary.
+BeSafe
+iwBeSafe ∗ NearbyRobot
+Safety of the robot, particularly collision with
+other robots.
+ExtendComms
+iwExtendComms ∗
+(Beacon || ReverseDrop)
+Extend communications as far into the
+environment as possible to reduce any delay in
+reports and minimize robot travel back and forth.
+MaintainComms
+iwMaintainComms ∗ !Comms
+Communicate with the base station, either by
+staying in or returning to communications.
+ReportArtifacts
+iwReportArtifacts ∗ Artifact
+Report artifact types and locations to base station.
+Table 9: MARBLE Objectives. Evaluation functions calculate the weight for each objective. Components prefixed by
+“iw” represent the input weight of the objective, while monitor names represent the binary output of the monitor.
+Behavior
+Evaluation Function
+Actions
+Explore
+owFindArtifacts ∗ !ExploreToGoal +
+owInput ∗ iExplore
+Request global planner to plan to
+unexplored areas.
+GoToGoal
+owFindArtifacts ∗ ExploreToGoal +
+owInput ∗ iGoToGoal
+Request global planner to plan a path to a
+specific goalpoint.
+Stop
+owBeSafe + owInput ∗ iStop
+Controller stops using plan generated by
+global planner. Causes robot to stop
+autonomous movement, but will still
+accept manual movement by human
+supervisor.
+DeployBeacon
+owExtendComms ∗ !ReverseDrop +
+owInput ∗ iDeployBeacon
+Initiate beacon deployment maneuver,
+which positions robot, stops, and deploys a
+beacon.
+GoHome
+owReportArtifacts +
+owMaintainComms+owExtendComms∗
+ReverseDrop + owInput ∗ iGoHome
+Request global planner to plan a path to
+the starting point.
+Table 10: MARBLE Behaviors. Evaluation functions calculate the score for each behavior for use by the policy πB.
+Components prefixed by “ow” represent the output weight of the objective, while monitor names represent the binary
+output of the monitor.
+
+the loading dock at the Engineering Center was added.
+Dataset Name
+FPS
+Images
+Labels
+Courtyard
+10
+54523
+4837
+8299
+2682
+3190
+6219
+3928
+3654
+32809
+Edgar Mine
+10
+3603
+254
+159
+148
+142
+122
+187
+84
+1096
+NIOSH Mine
+1
+13437
+118
+45
+106
+173
+0
+0
+0
+442
+Satsop Nuclear
+15
+33953
+0
+630
+0
+0
+302
+0
+0
+932
+Loading Dock
+10
+6923
+184
+234
+143
+143
+76
+329
+116
+1225
+Total
+112439
+5393
+9367
+3079
+3648
+6719
+4444
+3854
+36504
+Table 11: List of data sets utilized to train the artifact detection system. Also listed is the frames per second (FPS) of
+the imagery, the total number of images, the number of labels for each artifact, and the total number of artifact labels.
+At the Final Event, our artifact detection system did not correctly detect any vents. It also falsely detected
+many white walls as vents. One reason for this poor performance is likely because the vent we trained on was
+different than the vent at the Final Event. To support the vent, white sides were added, making it appear as
+a box, as shown in Figure 38.
+Figure 38: Team MARBLE trained on a vent with white-walled sides.
+13.9
+Competition Results
+Table 12 lists the eight teams that competed in the DARPA Subterranean Challenge Final Event, and the
+number of artifacts each team found during the 60-minute Prize Run, out of a maximum possible score of 40.
+Score
+Team
+Funding
+23
+CERBERUS
+DARPA
+23
+CSIRO Data61
+DARPA
+18
+MARBLE
+DARPA
+17
+Explorer
+DARPA
+13
+CoSTAR
+DARPA
+7
+CTU-CRAS-NORLAB
+DARPA
+2
+Coordinated Robotics
+Self
+2
+Robotika
+Self
+Table 12:
+List of final scores of all teams that participated in the 60-minute Final Event Prize Run.
+13.10
+Planning in a Dynamic Environment
+The planning algorithm presented in this paper has the capability of adapting to dynamic environments,
+i.e. closing or opening of doors, falling rubble, etc. Other situations also lead to dynamic changes in the
+
+map, including other nearly mobile agents, as well as localization and mapping error. The paper presents
+the planner response to a trap door, and here in the Appendix, additionally present examples of planner
+responses to robot-robot encounters as well as localization and mapping error.
+Robots often came withing close proximity of other robots during the mission. Some examples of this
+are shown in Figure 39. Rather than resolve these interactions through the planner, which operates on a
+slower timescale that fast-approaching robots, the autonomous mission management system specifies the
+higher-priority agent continue while the lower-priority agent wait. The planner marks previously traversable
+edges as untraversable, and later updates them as traversable again once the other agents leaves the vicinity.
+(a)
+(b)
+(c)
+(d)
+Figure 39: Robot-robot interactions
+Figure 40 presents an instance when the planner on D01 adapted to localization and mapping drift. The
+erroneous new map data caused many of the previously traversable edges of the graph to change to
+untraversable. The planner adapted to the new scenario, helping the agent return to the main corridor,
+at which point a loop closure occurred, correcting the agent’s pose. The planner operated continuously
+throughout this localization drift and loop closure correction.
+13.11
+Planning in Constrained Spaces
+The planner parameters were configured such that the agents would operate safely and not attempt to traverse
+highly confined spaces. One limitation to this approach, is that agents could not autonomously plan and
+traverse some areas, such as the utility corridor with low ceilings and narrow cave section, as shown in Figures
+41 and 42 respectively. The human supervisor was able to teleoperate the Spot through the cave section, and
+it autonomously traversed it when later exiting the cavern.
+13.12
+Artifact Reports
+All true positive 43 and false positive artifact detections can be seen in Figure 44.
+13.13
+Twenty-Five Explored Artifacts
+This section focuses on all 25 artifacts that Team MARBLE agents were in the vicinity of during the
+Final Event Prize Run. To summarize, 18 of these artifacts were scored, one was not scored, and six were
+unreported. These three categories of artifacts are discussed in Section 13.13.1, Section 13.14, and Section
+13.15, respectively.
+Each artifact has at least one attempt associated with it. The eighteen scored artifacts all end with a scored
+attempt (SA), and some will have multiple missed attempts (MA) before reaching a scored attempt. The
+one missed artifact was not scored, so it only has missed attempts associated with it. The six unreported
+artifacts have no attempts associated with them.
+
+(a)
+(b)
+(c)
+(d)
+(e)
+(f)
+Figure 40: Instance of D01 experiencing localization drift and erroneous mapping, causing planner to mark traversable
+(green) edges in the graph as no longer traversable (red). The progression of events is first (a) no localization error
+(13:03), (b) initial localization and mapping error (17:12), (c, d) continued localization and mapping error (13:53,
+17:16), (e) loop closure correcting the robot pose (17:34), teleporting the agent away from its planned path (pink),
+and (f) continuing on the mission (18:23).
+13.13.1
+Eighteen Scored Artifacts
+L51 Drill (SA1): This drill was the first artifact scored, within 1 minute and 8 seconds of the mission start.
+D02 reported and scored the drill as it passed through the first junction of the course, splitting it into tunnel,
+urban, and cave corridors (SA1). The HS also saw the drill via D02 live FPV view and would have attempted
+to score it had D02 not automatically reported it.
+L53 Backpack (SA2): D02 quickly continued into the cave corridor, reporting and scoring the backpack
+(SA2). If D02 did not autonomously score the backpack, the HS may have scored it via D02 live FPV. Had
+that failed too, H02 and H01 accurately reported the backpack later in the mission and would have scored it.
+
+(a)
+(b)
+Figure 41: Mobility limitations.
+(a)
+(b)
+Figure 42: Mobility limitations.
+L55 Rope (SA3): Because D02 had difficulty traversing the narrow cave corridor, the HS manually
+teleoperated the agent through this section of the course, and in the process, saw the rope via D02 live FPV
+and reported it (SA3). D02 did not automatically detect the rope, likely due to poor lighting conditions.
+L26 Survivor (SA4): While D02 immediately explored the cave section, D01 began exploring the urban
+section and autonomously reported the survivor and scored it (SA4). Later in the mission, H01 accurately
+reported L26 and would have scored had D01 missed it at the beginning of the mission.
+L32 Survivor (SA5): The HS saw the survivor through D02 live FPV stream, submitted a manual report
+and scored the artifact (SA5). Later in the mission, accurate autonomous reports from D01 and H01 would
+have scored the artifact, had the HS had not already scored it.
+L08 Gas (SA6): D01 autonomously reported and scored the gas artifact as it traversed the urban environment
+(SA6).
+L31 Fire Extinguisher (SA7): During the middle of the mission, the HS teleoperated D01 through a
+foggy section of the tunnel environment. In this process, the HS saw the fire extinguisher through D01 live
+FPV stream, and scored the artifact via manual report (SA7). After this manual intervention, D01 went onto
+help score five more artifacts, L34, L38, L36, L40, and L67. The L31 fire extinguisher was also seen at the
+end of the mission when the HS was reviewing the D02 archived FPV images.
+L34 Drill (SA8): After the HS teleoperated D01 through the fog, D01 autonomously reported and scored
+the drill (SA8). The HS also saw the drill through D01 live FPV stream, and would have reported the artifact
+manually had D01 not already scored it.
+
+(a)
+(b)
+(c)
+(d)
+(e)
+(f)
+(g)
+(h)
+(i)
+(j)
+(k)
+Figure 43: All true positive reports of visual artifacts from autonomous artifact detection systems onboard remote
+agents D02 (a, b), D01 (c, d, e, f, g), H02 (h), and H01 (i, j, k), in order from mission start to mission end.
+L38 Fire Extinguisher (SA9): After teleoperating D01 through the fog, the HS saw the fire extinguisher
+though D01 live FPV stream and manually scored the artifact (SA9).
+L36 Cube (SA10): After the HS teleoperated D01 through the fog, D01 autonomously reported and scored
+the cube (SA10).
+L40 Backpack (SA11): After the HS teleoperated D01 through the fog, D01 autonomously reported and
+scored the backpack (SA11).
+L67 Rope (SA12): After the HS teleoperated D01 through the fog, D01 autonomously reported and scored
+the rope. The HS also saw the artifact though D01 live FPV stream, and would have manually reported the
+artifact had D01 not already scored it (SA12).
+L11 Cube (MA5, MA6, SA13): D01 and D02 both autonomously reported the cube, but were both
+missed attempts (MA5, MA6), with corresponding errors of 10.73m and 9.42m. The HS then used the
+location of the missed attempts to manually submit an adjusted location, scoring the cube (SA13) with an
+error of 1.83m.
+L22 Cell Phone (MA2, MA4, SA14): D01 autonomously reported the cell phone early in the mission,
+but was a missed attempt (MA2) with an error of 19.47m. H02 also autonomously reported the cell phone,
+but the report was a missed attempt (MA4) with an error of 7.77m. Near the end of the mission, H01
+accurately localized the cell phone and scored the artifact (SA14), with an error of 4.06m.
+
+drill:0.59backpack:0.90baakpack.6.55backpack:0.71survivor:0.81drill:0.94
+国rope:0.93backpack:0.99survivor:0.76backpack:0.93survivor:1.00(a)
+(b)
+(c)
+(d)
+(e)
+(f)
+(g)
+(h)
+(i)
+(j)
+Figure 44: All false positive reports of visual artifacts from autonomous artifact detection systems onboard remote
+agents D02 (a, b, c, d), D01 (e, f, g, h), H02 (i), and H01 (j), in order from mission start to mission end.
+L47 Cell Phone (SA15): D02 autonomously reported the cell phone along the subway platform and scored
+it (SA15).
+L59 Cell Phone (MA1, MA3, MA10, SA16): Located in the left branch of the cave section, the cell
+phone was autonomously reported by D02, but was a missed attempt (MA1) with an error of 13.69m. The HS
+then manually submitted the cell phone with an adjusted location, but this was also a missed attempt (MA3)
+with an error of 7.74m. Later in the mission, a third missed attempt occurred (MA10), as D01 autonomously
+reported the cell phone with an error of 9.97m. Immediately after, the HS used the reported locations from
+D02 and D01 to submit another manual report with an adjusted location, and scored the artifact (SA16),
+with an error of 2.15m.
+L24 Gas (MA11, SA17): The gas was autonomously reported by H01, but with an error of 5.38m, resulted
+in a missed attempt (MA11). Soon after, H02 traversed the same space and autonomously reported the gas
+independently of H01. This report better estimated the position of the artifact, with an error of 2.55m, and
+scored (SA17) the gas artifact.
+L58 Helmet (SA18): The HS reviewed the D02 archived FPV images near the end of the mission, and saw
+the helmet in the cavern. The report was accurate to 1.74m and scored Team MARBLES 18th and final
+point of the mission (SA18). This point was made possible by the HS teleoperation through the narrow cave
+corridor.
+
+backpack:0.69helmet:0.91backpack:0.84vent:0.53vevent:0.73vent:0.83vent:0.82backpack:0.81backpack:013.14
+One Missed Artifact
+L64 Cube (MA7, MA8, MA9): This cube artifact was located atop a steep slope found along the main
+corridor of the cave section. D01 autonomously reported the cube but was a missed attempt with an error
+of 8.24m (MA7). Based on that experience, the HS manually submitted two reports at adjacent locations,
+both of which were missed attempts (MA8, MA9), with errors of 12.23m and 20.15m. This artifact was never
+scored during the mission. Figure 45b shows the failed score attempts circled in red, and the actual position
+circled in green.
+(a)
+(b)
+Figure 45: Helmet FPV image transmitted to human supervisor, but missed due to workload (a). L64 Cube location,
+circled in green, and incorrect submissions, circled in red (b).
+13.15
+Six Unreported Artifacts
+L02 Vent L02 vent was not detected by the on-board artifact detection system, but was available in the
+human supervisor’s FPV feed. At approximately 20 minutes into the mission, D01 (Spot) was returning home
+due to a localization error noticed by the human supervisor. While it was returning, the Supervisor turned
+attention to other robots, and did not scan the FPV feed for some time. The robot stopped for 33 seconds
+and transmitted a series of images similar to Figure 46. Then the robot moved forward and was stuck on the
+corner underneath the vent, due to the localization error, and this is when the Supervisor returned attention
+to the robot. The FPV images were stored for later review by the Supervisor, but due to workload, these
+images were never reviewed during the mission, and thus the artifact remained unscored.
+L05 Vent Approximately 7 minutes prior to the end of the mission, D02 reported a vent and transmitted
+the image in Figure 47 to the base station. Unfortunately, due to workload and poor notification design
+in the GUI, the human supervisor never noticed the report, and thus it was not submitted. Although the
+detection system identified the bucket as a vent, the Supervisor could easily see the actual vent above it and
+the reported position was 1.24m from the actual position.
+L62 Helmet L62 helmet located in the cave section near the tunnel intersection was seen via FPV, as in
+Figure 45a but not detected by the robot. This was transmitted by D01 and also available for review by the
+human supervisor, but due to workload the images were not reviewed during the mission.
+L13 Gas Approximately 10 minutes prior to the end of the mission, D02 passed approximately within 1m of
+this gas artifact, but provided no reports. Further analysis shows the only gas detection D02 had was a false
+report in an area with no CO2 nearby. The only other robot to go near this gas was D01, but it only went
+near a doorway leading to the area where the gas was located, and had already reported and scored another
+gas artifact near that location, so if it did detect L13 it would have assumed it was the same as the previous
+artifact.
+
+Figure 46: Vent seen by D01 while stopped and trans-
+mitted to human supervisor, but missed due to human
+supervisor workload.
+Figure 47: Vent detected and reported by D02 late in
+mission but missed by the human supervisor due to work-
+load.
+L21 Vent This vent was in the subway platform area of the environment. Both Spot robots viewed the
+vent with various cameras, but never detected it, likely due to the white wall background. Additionally,
+communications to the base station were limited in this area, and none of the FPV images relayed to the
+human supervisor had the vent in view. Interestingly, D01 did provide a false report of a vent in this area, and
+transmitted an image seen in Figure 48, which the human supervisor discarded as a false report. According to
+the truth data, the location reported was 3.93m from the actual vent, and so would have scored if submitted.
+However, visual analysis indicates the vent position in the ground truth file appears 2m off, which would not
+have scored.
+L42 Fire Extinguisher This fire extinguisher was seen only with the right-facing camera on D01 for only a
+few frames, as seen in Figure 49, which was not enough to trigger a detection and report. The robot was
+outside of communications, so even if the front camera had seen it, it would not have been available for the
+human supervisor.
+Figure 48: False vent reported close to actual vent by
+D01.
+Figure 49: Missed fire extinguisher only seen for a few
+frames by the right camera of D01.
+
+vent:0.53VeHH13.16
+Two False Artifacts
+This section focuses on two false artifacts that Team MARBLE reported, but were actually the result of false
+detections. Each false artifact has at least one false attempt (FA) associated with it. These false attempts
+are listed in Table 13.
+Gas (FA1, FA2, FA5): Gas was autonomously reported by H02 early in the mission (FA1). The HS
+modified the location and manually reported again (FA2), but did not score. Later in the mission, D02
+reported gas again in a very similar location as D01. This increased the confidence in the HS that gas was in
+the area, so the HS modified the location of D02 report, but this too did not score (FA4). It remains unknown
+why both agents detected elevated levels of CO2, but it is likely that a source in that vicinity existed, even if
+it was not an gas artifact.
+Fire Extinguisher (FA3): Due to an unknown cause, the HS inadvertantly submitted the same report
+twice, which was a fire extinguisher at xyz-coordinates of (20.60, 20.59, -2.64). Because the fire extinguisher
+was scored by the last report, this report did not score, but simply wasted a report.
+Backpack (FA4): The image was not clear, but the HS attempted to report it, and it did not score.
+13.17
+Tabulated Reports
+All reports that Team MARBLE submitted to DARPA are listed in Table 13.
+13.18
+Field Deployments
+Table 14 lists the dates and locations of all full-scale deployments, within the context of the events at the
+DARPA SubT Challenge.
+
+Report
+ID
+Type
+Error
+Score
+Time
+Scorer
+Assister
+[m]
+[mm:ss]
+SA1
+L51
+0.57
+1
+01:08
+D02
+SA2
+L53
+2.23
+2
+01:23
+D02
+SA3
+L55
+0.84
+3
+06:23
+HS†
+D02 (live FPV stream)
+SA4
+L26
+0.62
+4
+12:03
+D01
+MA1
+L59
+13.69
+14:09
+D02
+MA2
+L22
+19.47
+14:13
+D01
+MA3
+L59
+7.74
+14:57
+HS
+D02 (MA1)
+SA5
+L32
+1.40
+5
+16:35
+HS
+D02 (live FPV stream)
+SA6
+L08
+1.80
+6
+17:23
+D01
+FA1
+—
+—
+17:33
+H02
+FA2
+—
+—
+17:59
+HS
+H02 (FA1)
+MA4
+L22
+7.77
+18:38
+H02
+MA5
+L11
+10.73
+19:49
+D01
+SA7
+L31
+1.31
+7
+28:08
+HS†
+D01 (live FPV stream)
+FA3
+—
+—
+33:38
+HS
+SA8
+L34
+1.43
+8
+35:51
+D01†
+SA9
+L38
+2.82
+9
+36:53
+HS†
+D01 (live FPV stream)
+SA10
+L36
+3.94
+10
+37:08
+D01†
+SA11
+L40
+1.40
+11
+37:58
+D01†
+SA12
+L67
+2.87
+12
+38:47
+D01†
+FA4
+—
+—
+45:18
+D02
+FA5
+—
+—
+46:44
+HS
+H02 (FA1), HS (FA2) & D02
+MA6
+L11
+9.42
+47:31
+D02
+SA13
+L11
+1.83
+13
+47:53
+HS
+D01 (MA5), D02 (MA6)
+MA7
+L64
+8.24
+48:12
+D01
+MA8
+L64
+12.23
+48:42
+HS
+D01 (MA7)
+MA9
+L64
+20.15
+49:09
+HS
+D01 (MA7) & HS (MA8)
+SA14
+L22
+4.06
+14
+50:33
+H01
+D01 (MA2), H02 (MA4), D02
+SA15
+L47
+4.00
+15
+50:45
+D02
+MA10
+L59
+9.07
+50:57
+D01
+SA16
+L59
+3.15
+16
+51:48
+HS
+D02 (MA1), HS (MA3), D01 (MA10)
+MA11
+L24
+5.38
+52:20
+H01
+SA17
+L24
+2.55
+17
+52:45
+H02
+SA18
+L58
+1.74
+18
+56:33
+HS†
+D02 (archived FPV images)
+Table 13: List of all artifact reports submitted by Team MARBLE during the 60-minute Final Event Prize Run.
+Bolded entries represent reports that resulted in a score. There are three types of reports: a Scored Attempt (SA), a
+Missed Attempt (MA) due to error exceeding 5m, and a False Attempt (FA) due to a false positive detection. Listed
+next are artifact ID, artifact type, error, cumulative score, and time since mission start. The scorer is the agent that
+submitted the report and scored (or attempted to score), and the assister is the agent(s) that provided information
+that aided the scorer in scoring (or attempting to score). The reporting of these artifacts was completely autonomous,
+save artifacts scored by the HS as well as those with a (†), denoting artifacts that were seen as a result of the HS
+temporarily teleoperating the agent into new areas of the course.
+
+co
+2CO
+2Date
+Deployment
+Environment
+Onsite
+Location
+Apr 7, 2019
+STIX Event
+Edgar Experimental Mine
+Idaho Springs, CO
+Jul 29, 2019
+Pre-Tunnel 1
+Edgar Experimental Mine
+Idaho Springs, CO
+Aug 1, 2019
+Pre-Tunnel 2
+Edgar Experimental Mine
+Idaho Springs, CO
+Aug 6, 2019
+Pre-Tunnel 3
+Edgar Experimental Mine
+Idaho Springs, CO
+Aug 9, 2019
+Pre-Tunnel 4
+Edgar Experimental Mine
+Idaho Springs, CO
+Aug 12, 2019
+Pre-Tunnel 5
+Edgar Experimental Mine
+Idaho Springs, CO
+Aug 17, 2019
+Tunnel Event
+NIOSH Exp. & Safety Research Mines
+Pittsburgh, PA
+Feb 4, 2020
+Pre-Urban 1
+Geotech Warehouse
+Denver, CO
+Feb 8, 2020
+Pre-Urban 2
+Geotech Warehouse
+Denver, CO
+Feb 12, 2020
+Pre-Urban 3
+Geotech Warehouse
+Denver, CO
+Feb 21, 2020
+Urban Event
+Satsop Nuclear Power Plant
+Elma, WA
+Aug 4, 2020
+Pre-Cave 1
+Edgar Experimental Mine
+Idaho Springs, CO
+Sep 17, 2020
+Pre-Cave 2
+Eng. Center (L1)
+X
+Boulder, CO
+Sep 19, 2020
+Pre-Cave 3
+Eng. Center (L1)
+X
+Boulder, CO
+Sep 21, 2020
+Cave Event*
+Edgar Experimental Mine
+Idaho Springs, CO
+Jul 13, 2021
+Pre-Final 1
+Folsom Parking Garage
+X
+Boulder, CO
+Jul 14, 2021
+Pre-Final 2
+Folsom Parking Garage
+X
+Boulder, CO
+Jul 15, 2021
+Pre-Final 3
+Edgar Experimental Mine
+Idaho Springs, CO
+Aug 13, 2021
+Pre-Final 4
+Folsom Parking Garage
+X
+Boulder, CO
+Aug 17, 2021
+Pre-Final 5
+Eng. Center (LL & Courtyard)
+X
+Boulder, CO
+Aug 19, 2021
+Pre-Final 6
+Sust., Energy, and Env. Community
+X
+Boulder, CO
+Aug 24, 2021
+Pre-Final 7
+Eng. Center (LL & Courtyard)
+X
+Boulder, CO
+Aug 26, 2021
+Pre-Final 8
+Edgar Experimental Mine
+Idaho Springs, CO
+Sep 1, 2021
+Pre-Final 9
+Edgar Experimental Mine
+Idaho Springs, CO
+Sep 8, 2021
+Pre-Final 10
+Eng. Center (LL & Courtyard)
+X
+Boulder, CO
+Sep 10, 2021
+Pre-Final 11
+Eng. Center (LL & Courtyard)
+Boulder, CO
+Sep 12, 2021
+Pre-Final 12
+Eng. Center (L2) + Rustandy
+X
+Boulder, CO
+Sep 14, 2021
+Pre-Final 13
+Eng. Center (L2) + Rustandy
+X
+Boulder, CO
+Sep 21, 2021
+Final Event
+Louisville, Megacavern
+Louisville, KY
+Table 14:
+List of all full-scale field deployments. For the Final Event, Team MARBLE gave greater resources to
+system performance validation and human-robot teaming practice. This realized itself as more frequent and diverse
+field deployments. For the circuit events, just two to three weeks were spent on field deployments, whereas for the
+Final Event, the team devoted two months. Instead of practicing in just one or two environments, the team was
+asked to perform in five unique environments. Selecting locations that were ”onsite” University of Colorado Boulder
+campus enabled the team to be more nimble and operationally efficient. The (*) denotes that the Cave Event was a
+self-managed mock event.
+
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+page_content='edu  Michael T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Ohradzansky  Aerospace Engineering Sciences  University of Colorado Boulder  Michael.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='Ohradzansky@colorado.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='edu  Kyle Harlow  Computer Science  University of Colorado Boulder  Kyle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='Harlow@colorado.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='edu  Michael J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Miles  Mechanical Engineering  University of Colorado Boulder  Mike.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='Miles@colorado.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='edu  Daniel Torres  Computer Science  University of Colorado Boulder  Daniel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='TorresDominguez@colorado.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='edu  Steve McGuire  Electrical & Computer Engineering  University of California Santa Cruz  steve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='mcguire@ucsc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='edu  Eric W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Frew  Aerospace Engineering Sciences  University of Colorado Boulder  Eric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='Frew@colorado.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='edu  Christoffer Heckman  Computer Science  University of Colorado Boulder  Christoffer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='Heckman@colorado.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='edu  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Sean Humbert  Mechanical Engineering  University of Colorado Boulder  Sean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='Humbert@colorado.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='edu  Abstract  While the capabilities of autonomous systems have been steadily improving in recent years,  these systems still struggle to rapidly explore previously unknown environments without the  aid of GPS-assisted navigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The DARPA Subterranean (SubT) Challenge aimed to fast  track the development of autonomous exploration systems by evaluating their performance  in real-world underground search-and-rescue scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Subterranean environments present a  plethora of challenges for robotic systems, such as limited communications, complex topology,  visually-degraded sensing, and harsh terrain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The presented solution enables long-term  autonomy with minimal human supervision by combining a powerful and independent  These authors contributed equally to this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='00771v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='RO]  2 Jan 2023  single-agent autonomy stack, with higher level mission management operating over a flexible  mesh network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The autonomy suite deployed on quadruped and wheeled robots was fully  independent, freeing the human supervision to loosely supervise the mission and make  high-impact strategic decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' We also discuss lessons learned from fielding our system at  the SubT Final Event, relating to vehicle versatility, system adaptability, and re-configurable  communications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  1  Introduction  Despite a myriad of developments in sensing, planning, control and state estimation over the last few decades,  deploying robots in harsh subterranean environments for the purpose of rapid situational awareness presents a  number of new challenges to robot autonomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Traditionally, robots rely on a number of complex, interconnected  sub-processes, such as localization, mapping, and planning, to navigate unknown environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Maintaining  accurate state estimates, a process critical to mapping and exploration, is exceptionally challenging in  subterranean environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' GPS is unavailable for obtaining position estimates and visual-based localization  methods can be affected by varied lighting conditions and environmental factors such as heavy dust, fog,  or smoke.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  Subterranean environments, such as mines and caves, are often unstructured and contain  hazardous obstacles, making navigation with ground vehicles challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Additionally, aerial vehicles can  be exceptionally difficult to deploy in tight constrained underground spaces due to self-induced propeller  wash.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The DARPA Subterranean Challenge (SubT) (DARPA, 2022) aimed to spark new developments in  the areas of autonomy, perception, mobility, and networking in subterranean environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' In the following  work, a scalable multi-agent autonomy solution for subterranean exploration developed by the University  of Colorado’s Team MARBLE for the SubT Challenge is presented, along with critical lessons learned and  developments made along the way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  DARPA designed the SubT Challenge to simulate search-and-rescue scenarios in unknown subterranean  environments, and consisted of three domain-specific circuit events, Tunnel Circuit, Urban Circuit, and Cave  Circuit, followed by the Final Event, which was a combination of three subterranean domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Teams were  challenged with developing robot platforms to deploy in each of the events in search of sets of predefined  artifacts, such as backpacks or a Bluetooth signal produced by a cell phone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Correct identifications, consisting  of an artifact classification and location to within a 5m sphere of the ground truth location, resulted in a  point scored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Placement in the competition was determined by the team which could score the most points  over a series of one hour deployments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' For the Final Event, teams were limited to a single “human supervisor”  who was able to interact with the systems and visualize any incoming data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Adding to the challenge, teams  had a limited window of time in which five team members could set up and initialize robots at the entrance  to the course.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  Team MARBLE’s initial approach to subterranean exploration for the Tunnel and Urban circuit events is  presented in (Ohradzansky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Initially, a graph-based planning and exploration strategy was  implemented, the details of which are presented in (Ohradzansky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' This solution is suitable  for tunnel-like mines that have mostly planar corridor-junction structures, because the environment can be  easily represented by a graph of nodes and edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' A scanning lidar was used to center robots in corridors  while navigating edge sections as well as avoid obstacles in the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' However, this approach lacked  multi-agent coordination, resulting in significant overlap of explored regions by different agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' For the  Urban and Cave circuit events, a three-dimensional volumetric map representation of the environment was  generated and used in a frontier-based exploration strategy (Ahmad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=', 2021a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Ahmad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=', 2021b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  In this approach, the exploration rate of the robot is maximized using a frontier-based (Yamauchi, 1997)  sampling technique and a fast marching cost-to-go calculation (Sethian, 1999) to select goal poses and plan  paths to them in three dimensional space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' An artificial potential function based obstacle avoidance algorithm  enables the robot to path follow while avoiding small obstacles in the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Our initial approach also  implemented limited forms of multi-agent coordination in the form of agents sharing goal points and paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  Other teams developed impressive solutions to the initial Tunnel and Urban Circuit challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Team CSIRO, a  collaboration between the Commonwealth Scientific and Industrial Research Organization (CSIRO), Emesent,  and Georgia Tech, presents a unique homogeneous sensing solution (Hudson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' In this approach,  heterogeneous teams of robots, including both ground and aerial platforms, share sensor information as a part  of a decentralized multi-agent SLAM system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Initially, exploration was handled through manual waypoints  commanded by the human supervisor, but eventually an autonomous exploration algorithm was implemented  (Williams et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' A common perception module, the CatPack, is used across all ground vehicles for  easy reuse of the autonomy stack across different platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Similar to Team CSIRO, Team CERBERUS  also used a heterogeneous team of ground and aerial platforms (Tranzatto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=', 2022a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Papachristos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=',  2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Tranzatto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=', 2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' In their approach, map information from different agents is fused into an  optimized global map that is shared back to the agents (Khattak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Similar to the work presented  in (Ohradzansky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=', 2020), other teams used graph-based planning approaches for global navigation (Dang  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Dang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Other noteworthy teams and their approaches to autonomous subterranean  navigation include Team CoSTAR (Santamaria-Navarro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Ebadi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Agha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  Otsu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=', 2020), CTU-CRAS (Rouˇcek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=', 2020), CMU (Carnegie-Mellon University) (Scherer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=',  2021), and NCTU (National Chiao Tung University) (Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Additional discussions on the  challenges, novel developments, and lessons learned from the Tunnel and Urban circuit events are included in  the following works (Miller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Lajoie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' One common theme common to many of these  approaches is the use of heterogeneous teams of agents with multi-modal sensing solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' By diversifying  the sizes, types of locomotion, and sensor modalities of individual robots, teams can be more versatile when  faced with varied environments, each with a unique set of challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' This ability to be flexible and adapt to  the needs of the mission is one of Team MARBLE’s driving philosophies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  (a)  (b)  Figure 1: The fleet is composed of two classes of robotic agents: (a) Clearpath Husky A200, (b) Boston Dynamics  Spot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Each platform carries a common sensor suite designed for exploration and object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  The format of the challenge necessitated advancements in platform design, robust communication networks,  intelligent planners, and a balance between autonomy and human decision making for robot fleets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Team  MARBLE’s solution, which was developed over the course of three different circuit events and showcased  at the Final Event, led to a third place finish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Specifically, we develop a heterogeneous fleet of autonomous  robots, each capable of operating independently of human intervention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Our autonomy-first approach employs  a lightweight graph-based planner that scales to large environments, and has been adapted to reason over  dynamic environments, such as closing doors and passing agents, as well as take advantage of multi-agent  coordination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Information sharing between agents is accomplished via a custom mesh networking solution with  fast reconnect times and configurable message prioritization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The same network provides the human supervisor  real-time visibility and control of the mission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Our autonomy-first philosophy inspired an autonomous mission  HLSNYA200MESHMERIZE  IRISS  000  BostonDynamig  200  ECEmanagement system that frees the human supervisor from agent-level micromanagement and deepens the  opportunity to strategically accomplish mission goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' In this work we will present each of the components of  our proposed autonomy system, as well as a detailed performance analysis of our solution and lessons learned  along the way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  This paper is organized with the following structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' First, an overview of our system is provided in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  The robot platforms developed for the Final Event are described in Section 3, followed by a description of the  localization system in Section 4, and the artifact detection system in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The multi-agent components  of our system include the volumetric mapping pipeline in Section 6 and the graph-based path planning over  those maps in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Information transmitted among agents, as well as to and from the human supervisor,  is mediated by the wireless mesh network communications system described in Section 8, and handled by  the autonomous mission management system described in Section 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Finally, we analyze how these systems  performed in the DARPA SubT Challenge Final Event in Section 10 and discuss lessons learned in Section 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  2  System Overview  In the following subsections we present a high-level description of Team MARBLE’s approach to the DARPA  Subterranean Challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' First, the general concept of operation is described, followed by an overview of each  of the major components of the developed autonomy solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' A full high level summary of the autonomy  system can be seen in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='1  Concept of Operations  Team MARBLE has emphasized development of multi-agent autonomy solutions that are able to operate  without requiring intervention from a human operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' This aligns with the goals of the DARPA Subterranean  Challenge where intermittent or unavailable communications with agents from a base station or between agents  is expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Therefore, our solution is centered around robust single-agent autonomy, where independent  robots are able to explore unknown environments and report back to a base station with collected information  about the environment including map data and detected artifact locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  In communication-limited  environments where information sharing with other agents and the human supervisor may not be available,  our agents persist and continue to execute the mission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  While it is important for single, independent agents to be able to explore autonomously, our solution  incorporates several multi-agent components to improve exploration efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Our fleet is also designed to be  opportunistic, capitalizing on communication links when they are available to amplify fleet performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Multi-  agent coordination is an auxiliary capability that reduces redundant efforts when agents enter communication  range with one another, and inform each other where they have been and where they plan to go next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  Our system’s performance can further be improved when communications are available which enables the  human supervisor to have a holistic perspective of the specific search-and-rescue scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' This holistic  perspective empowers the human to make high-level contextual decisions through two types of intervention:  directing the agent to a specific location by commanding a high-level waypoint, or teleoperating the agent by  commanding low-level velocity signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' During the 60-minute final event prize run, Team MARBLE’s robot  fleet was completely autonomous, with the exception of five strategic low-level human supervisor interventions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  The balance between human input and autonomy is further discussed in Section 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='2  Perception  A modular perception suite, shown in Figure 5b, was designed as the basis for the autonomy stack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The  primary sensor is the Ouster OS1-64 lidar (Light Detection and Ranging), which provides 3D point clouds for  mapping, localization, semantic mapping, and obstacle avoidance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' A LORD Microstrain 3DM-GX5-15 IMU  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='2D Lidar  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='3D Lidar  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='IMU  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='RGB Cameras  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='Bluetooth  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='CO2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='Other Robots  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='(MADCAT Instance)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='Localization  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='LIO-SAM  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='Mapping  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='marble mapping  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='2D Object Detection  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='YOLO  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='3D Artifact  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='Localization  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='Terrain Assessment  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='(Husky Only)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='Stair Detector  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='(Spot Only)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='Planning  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='scan plan  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='Path Following  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='pure pursuit  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='Mission Management  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='BOBCAT  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='MADCAT  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='Comms  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='udp mesh  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='Base Station  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='(Human Supervisor)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='Artifact Reports  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='Other Robots  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='(MADCAT Instance)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='Maps & Position History  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='Velocity Commands  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='Input  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='Input (Multiagent)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='Output  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='Output (Multiagent)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='Software Module  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='Human Input  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='Figure 2: Overall block diagram showing the high level functionality of the autonomy stack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Inputs are shown in  green and outputs are shown in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Software package names are italicized and inputs & outputs which are shared  between agents are outlined in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Terrain assessment and stair detection both add semantic information to the  map but are only run on the Husky and Spot respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  (Inertial Measurement Unit) is used to measure linear and angular acceleration of the sensor head for use  in lidar-inertial state estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' To identify visual artifacts, the systems are equipped with several FLIR  Blackfly PGE-05S2C-CS cameras and an array of 5W dimmable LEDs for self illumination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The Husky  platforms were equipped with four cameras facing forward, backward, to the left, and to the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The Spot  robots had a similar configuration, save omitting the rear camera due to occlusions caused by the custom-built  compute and power management system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='3  Localization  Localization provides consistent pose information for many downstream autonomy processes including  volumetric mapping, path planning, artifact detection, and multi-agent coordination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' However, ensuring  reliable localization is difficult in austere underground environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Because conventional vision-based  solutions can be unreliable due to dark, feature-poor settings, Team MARBLE utilized lidar-based methods  and specifically tested and integrated LIO-SAM (Shan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=', 2020), which has fast online loop closures during  long-duration missions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Several modifications were made to the system to improve localization accuracy and  reliability, which are further discussed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Methods used to align the robots with into a common  reference frame are also presented in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='4  Exploration  The exploration algorithm generates safe and traversable paths that lead agents toward unexplored areas  of previously unseen environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The developed sampling based path planning algorithm is designed  to be lightweight, so that it can operate on rapid exploration timescales, regardless of the extent of the  environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Computational efficiency is achieved by employing a bifurcated local-global graph for sampling  unseen frontiers as well as a good enough strategy for final selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' In such time-constrained search and  rescue settings, even humans will often make rapid decisions rather than dwell for long periods of time to  make globally optimal ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The details of the planning algorithm is described in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  The planner is also constructed to be flexible, so that additional capabilities could be scaffolded on top of the  core algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Integration of the planning algorithm with semantic mapping was critical for rough terrain  and stair traversal, as explained in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='4 explains how this planning algorithm re-plans in  dynamic environments, whether due to doorways that are being opened or closed, or fellow agents that are  passing by.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' This capability is crucial for robust operation in real-world environments, which cannot assumed  to be static.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Finally, the planner is able to run more efficiently with multiple robots using multi-agent  coordination is covered in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  Agents follow paths via a modified pure pursuit controller (Coulter, 1992).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The yaw rate command is  computed by comparing the current agent’s heading against a lookahead point that is a fixed distance along  the path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Forward speed is regulated based on local proximity to obstacles in the environment and the  relative heading error to the lookahead point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' This results in slower speeds when agents are in cluttered  environments or experiencing large heading errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' A 2D (RP Lidar) was used for local obstacle avoidance on  the Husky platform and the Spot platform had built in obstacle avoidance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='5  Mapping  Team MARBLE’s mapping framework is based on the open source Octomap package (Hornung et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=', 2013),  and has been customized with additional capabilities including map merging, transmission of difference maps,  and encoding of semantic information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The core of the mapping framework is a log-odds based probability  metric for occupied and unoccupied voxels or cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' These cells provide a 3D representation of the environment  that is later used for navigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' This flexible framework enabled transmission of key environmental features  such as rough terrain and the location of stairs efficiently through a bandwidth limited communication system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  The details of our mapping system are provided in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='6  Artifact Detection  The artifact detection system precisely localizes visual artifacts using RGB sensors for visual classification  and detection and lidar for the depth estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Non-visual artifacts such as cellphones and gas are localized  based on the position of the robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' A weighted median filter fuses together all detections which are sent to  the human supervisor for final validation as described in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='7  Communication Systems  A mesh networking solution transmits data between robots and back the base station for the human supervisor  to review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Standard 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='4ghz 802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='11 wireless radios based on the ath9k chipset are used for the physical layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  The wireless radios are embedded in beacons that can be deployed from the back of the Husky platforms,  allowing for ad-hoc mesh networks to be established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Meshing technology was provided by Meshmerize (Pandi  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=', 2019) and a custom UDP based transport layer (udp mesh) was developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Details of this innovative  layer can be seen in Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='8  Multi-Agent Coordination  Exploring unknown environments with multiple agents can be made more efficient through coordination,  especially when agents are not within communication range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Sharing information across agents, such as  explored regions, discovered artifacts, and current behavioral states, allows for more intelligent management  of multi-agent exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The framework called Multi-Agent Data Collaboration for Autonomous Teams  (MADCAT), provides the multi-agent data sharing capabilities required for the Subterranean Challenge  mission (Riley and Frew, 2021), including transmission of relevant coordination data and maps, as well as map  merging functionality and decision making for each agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Additionally, MADCAT implements Behaviors,  Objectives and Binary states for Cooperative Autonomous Tasks (BOBCAT), originally presented in (Riley  and Frew, 2022), for high-level autonomy, decision making, and interfacing with the human supervisor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The  MADCAT algorithm is discussed in more detail in Section 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='9  Mission Management  The human supervisor is able to monitor the fleet’s progression through the unknown subterranean environment  using a custom GUI operating on a computer at the entrance to the environment (base station).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Current  mission status of all agents in the field (‘Reporting’, ‘Exploring’, ‘Home’, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=') as well as their location in the  global map are displayed whenever robots are within communication range of the base station.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Additionally,  the goal point and goal path for each robot is visible, allowing the supervisor to see the intent of each robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  The human supervisor is able to take over control of a given agent by either sending manual goal points or  tele-operating the vehicle using a joystick interface with the base station.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Reported artifacts are displayed per  robot, with the type (survivor, cell phone, backpack, helmet, rope), position, confidence, submission result,  and an image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The GUI also enables the modification of artifact classes and locations prior to submission to  the DARPA scoring server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' An example of the GUI interface is shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  Figure 3: Example of the Human Supervisor’s interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The custom GUI is on the left, showing a received artifact  image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The middle is the multi-agent RViz view, with all robots and the complete merged map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The right is a  third-person follower for each robot (two in this case) with that robot’s original unmerged map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  3  Platform Development  For the final event of the SubT Challenge, Team MARBLE deployed a heterogeneous fleet consisting of  two Clearpath Husky A200s (H01, H02) and two Boston Dynamics Spots (D01, D02).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Examples of each  platform are shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The Husky platforms are four-wheeled skid-steer ground vehicles capable of  carrying heavy payloads, while the Spot quadrupedal “dog” platforms are agile, capable of climbing staircases  and traversing uneven terrain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The Husky platform is robust and stable, with its generous payload budget  allowing it to carry six communication beacons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The intended deployment strategy is to first deploy the  Spot platforms to maximize exploration, and then the Husky platforms to establish the mesh communication  network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  For processing power, each Husky is equipped with a 32-core AMD Ryzen CPU equipped with 128GB of  RAM and 4TB of SSD storage, integrated into a complete platform as shown in Figure 4a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Dual NVIDIA  GTX 1650 GPUs were used to accelerate object detection inference speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The primary computer on the  Spot platforms is an AMD Ryzen 5800U with 64GB of RAM and 2TB of storage which is paired with  a Jetson Xavier AGX to process the camera streams and perform artifact detection, following a similar  integration as shown in Figure 4b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Many purpose-built components are common between platforms to reduce  field maintenance efforts and platform-specific code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Each system is outfitted with a custom power system,  discussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='2, which enables the ability to switch from a wired shore power supply to the onboard  computer batteries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' This leads to more efficient use of the onboard batteries, which are a limiting factor in  interatMorneCameraselet  2D Nav Goll PubhshPoint+-,*  nterktMoveCamera  Image  合  AUHONIZED  backpack:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='70  Submissions  wall Time:1600896291.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='47wallElapsed:400.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='93the duration of field testing deployments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='Batteries (4x)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='Shore Power Inlet  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='Power  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='Management  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='Relays  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='Platform Control  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='EStop Control  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='Update Button  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='DARPA EStop  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='EStop Status  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='EStop Button  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='Status Display  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='Shore Button  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='Motion Control  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='Main CPU  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='Power Rail  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='Lidar  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='Radios  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='Ethernet Switch  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='Lighting  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='& Camera  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='Control  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='Area Lighting  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='GigE Cameras (4x)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='Beacon  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='Deployment  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='Host USB  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='Mode LEDs  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='RPLidar  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='Bluetooth  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='CO2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='PicoFlexx (2x)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='IMU  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='Component  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='Custom HW  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='Custom SW  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='Custom HW & SW  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='Power  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='GPIO  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='Serial  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='USB  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
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+page_content='CAN  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
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+page_content='(a)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='Batteries (2x)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='Shore Power Inlet  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='Power  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='Management  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='EStop Control  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='DARPA EStop  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='EStop Status  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
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+page_content='Status Display  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='Shore Button  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='NUC  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='Xavier AGX  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='Spot Base  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='Power Rail  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='Lidar  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='Radio  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='Ethernet Switch  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='Lighting  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='& Camera  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='Control  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='Area Lighting  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='GigE Cameras (3x)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='Bluetooth  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='CO2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='IMU  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='Mode LEDs  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='Component  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='Custom HW  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='Custom SW  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='Custom HW & SW  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='Power  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='GPIO  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='Serial  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='USB  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='USB3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='CAN  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='Ethernet  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='(b)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='Figure 4: Power and signal routing diagrams of customized (a) Husky and (b) Spot platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='1  Communication Beacons  Underground environments provide limited line-of-sight capabilities for wireless communications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' As a  result, Team MARBLE developed custom communication beacons to complement the custom multi-robot  coordination solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' This allows for robots to share information with the base station and other robots in  the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Each Husky platform is capable of carrying six beacons, each containing a single 2W Doodlelabs  802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='11n radio as seen in Figure 5a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The autonomous beacon deployment mechanism relies on a latching  solenoid release coupled with a novel passive system to gently lower each beacon to the ground to ensure  maximum antenna height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Additional design details of the communication beacons can be found in Section  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='1 of the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  (a)  (b)  Figure 5: Final robot configurations, with (a) a deployment mechanism loaded with six communication beacons on  Husky vehicles at Final Event and (b) a modular perception suite installed on both Huskies and Spots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  MARBLE  HUSKYA2U0TM  HO1  LORD SENSING  icroStrai  eneerin64-BeamOusterLidar  LordMicrostrainIMu  5WLED  FLIR GigE  Cameras3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='2  Power and Platform Control Systems  In order to support each platform’s sensor and processing needs, as well as meet DARPA equipment  requirements for emergency stop systems, the systems integration efforts relied on several custom hardware  and software components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Where feasible, these components are shared between the Husky, shown in Figure  1a, and the Spot platforms, shown in Figure 1b, reducing the development and validation efforts, as well as  team member operational training requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  Of the custom capabilities developed, the power management subsystem deserves special mention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' This  component implements a hardware-interlocked, ideal diode system to permit downstream electronics to  source power from either a wall-connected power supply, referred to as shore power, or onboard batteries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' By  switching to shore power, onboard systems can remain powered for development, testing, and analysis while  the batteries are charged without carrying load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Further, the ideal diode component allows the battery packs  to load share and charge independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' In contrast to a bus-tied battery system, this ideal diode design  prevents high-energy charge equalization between packs and allowed each battery pack’s onboard management  board to function independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The system also enables live monitoring of current consumption and battery  voltage, as well as intelligent e-stop management which ensures the robot cannot exit the emergency stopped  state while connected to shore power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  Emergency stop requirements dictated that each platform needed the ability to be stopped by a physical  button, software, and via a DARPA deployed Xbee network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' On the Husky control system the emergency  stop system was integrated directly into the base controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' In contrast, the Spot platform’s emergency stop  tied into the available API to issue a the robot a “sit” command before terminating power to the motors  which allowed the robot to be stopped gracefully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  Several important lessons learned emerged after three years of platform architecture development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' We have  highlighted the most critical lessons below and provide more details about the design and consequences of  our platform compute systems in the Section 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='2 of the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  As part of our field testing campaign, we uncovered an issue where our USB-connected IMU was delayed in  delivering measurements critical to localization performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' From an integration perspective, our IMU’s  USB interface was implemented using a standard USB Communications Data Class Abstract Control Model  (CDC-ACM) interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Using CDC-ACM for IMU measurements was particularly problematic due to the way  in which CDC-ACM uses bulk transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' USB has several methods of transferring data from device to host,  including interrupt, isochronous, and bulk transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' CDC-ACM uses bulk transport, which does not include  any guarantee for on-time delivery of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' As a consequence, during high CPU load, IMU measurements were  occasionally delayed and resulted in localization error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' In contrast, interrupt and isochronous transports are  regularly serviced and can deliver on-time data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' This problem could be solved in future deployments by either  replacing the bulk interface with an interrupt interface or by using legacy serial interfaces such as RS-232  (not typically available on small form factor computing units).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' However, in practice we found adjusting the  IMU timing as described in Section 4, was sufficient and did not require engineering new firmware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  Another critical piece for reliable localization and consequently navigation is sensor synchronization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Synchro-  nization is fundamentally necessary in order for our onboard sensors to communicate with their respective  computers, and for those computers onboard individual agents to communicate with each other and the base  station computer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The technical implementation of our solution is detailed in Section 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='3 of the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  4  Localization  One of the major challenges in the DARPA Subterranean Challenge is ensuring reliable localization across a  diverse set of austere environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Localization is a critical process for an autonomous system, as it provides  pose information to downstream autonomy processes including volumetric mapping, path planning, artifact  detection, and multi-agent coordination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='1 details the simultaneous localization and mapping  solution that was integrated into the autonomy stack, and Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='2 describes the process used to align all  robots to the common DARPA reference frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='1  Simultaneous Localization and Mapping  Simultaneous localization and mapping has relatively mature vision-based solutions (Leutenegger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  Nobre et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Qin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=', 2018), thanks to advances in feature extraction (Cheung and Hamarneh, 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  Bay et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=', 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Zhan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' However, in mission-critical applications such as underground search and  rescue, visual-inertial solutions are not reliable enough when faced with irregular lighting, specular highlights,  and feature-poor scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Recent work has illuminated the possibility of leveraging thermal-based odometry  estimation in visually degraded environments (Khattak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Wisth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  Because underground environments are typically rich in geometric features, lidar-based localization solutions  are a compelling alternative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Some spaces though, such as a smooth tunnels and corridors, contain relatively  few longitudinal features, and therefore pose limits to lidar-based perception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Single-echo lidar also struggles  in austere environments containing fog or smoke, though some recent work has focused on addressing these  limitations (Shamsudin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  For the Final Event, Team MARBLE transitioned from Google Cartographer (Hess et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=', 2016) to LIO-SAM  (Shan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=', 2020), since its faster online loop closures during long-duration missions results in greater  localization accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Extensive testing was conducted in many different environments including parking  garages, academic buildings, gold mines, and outdoor environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The fast, lightweight loop closure  performance can be attributed to performing scan-matching on a local level rather than a global level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  LIO-SAM additionally performs IMU pre-integration to deskew point clouds, yielding better initialization  for lidar odometry estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Because localization is the foundation to many autonomy modules, it was  imperative to validate LIO-SAM’s performance onboard the Spot and Husky platforms during large-scale,  long-duration missions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Some examples of such validation efforts are shown in Section 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='4 of the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  Several modifications are made to the system to improve localization accuracy and reliability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' First, the  IMU and lidar sensors are fastened to a 6061 aluminum sensor plate, with a mounting configuration that is  common between Huskies and Spots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' By specifying the relative transform between the two sensors to a high  precision, the need for extrinsics calibration is reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Specifically, the mounting configuration consists of a  tight-tolerance, precision-ground plate with a flatness tolerance of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='005”, which greatly improves the roll  and pitch alignment between the two sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' By using a high-quality MEMS IMU and such precise sensor  mounting, the LIO-SAM parameter specifying how much to weight IMU roll, pitch, and yaw measurements  relative to lidar odometry was increased by a factor of 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Taken together, these modifications greatly reduce  accumulated rotation and translation drift, enabling smooth autonomous operation across long missions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  Secondly, LIO-SAM requires sensor timestamps to be aligned and sensor rates to be consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' In particular,  if IMU message rates fluctuate too greatly, the IMU pre-integration factors (Forster et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=', 2015) can fail  and lead to LIO-SAM instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' To reduce sensitivity to fluctuating IMU sensor rates caused by USB  transmission delays (described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='2), the IMU timestamp assignment is adjusted when messages are  not received within 15% of the nominal rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Additionally, the lidar sensor is synchronized with the onboard  computer via PTP as discussed in Section 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='3 of the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' These two timing solutions reduce the  probability of erroneous measurements and greatly improve the stability of LIO-SAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='2  Common Reference Frame Alignment  Accurate multi-robot alignment is a core design decision for the MARBLE localization, mapping, and planning  systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Robots are required to share globally aligned map data for planning and navigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' In addition,  it allows the human supervisor and robots to share global coordinates for artifact locations relative to the  Figure 6: Figure of the MARBLE gate alignment setup including the Leica Total Station (LTS), gate, and an example  robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Transforms from the LTS to the robot TRL, from the LTS to the world frame TW L, and the resulting transform  from the world frame into the TW R  DARPA-provided world frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' In order to align with the DARPA frame, Apriltags (Malyuta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  Brommer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Wang and Olson, 2016), retro-reflective targets, and Leica Total Station (LTS) reflectors  are attached to a gate with relative transforms to the DARPA origin frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The global frame was assumed to  be aligned with gravity, but each team was responsible for aligning yaw, and XYZ-translation from their robots  to the common DARPA frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' In the context of the SubT Challenge, the DARPA frame was purely used to  align robots into the measured ground truth frame for artifact scoring and map accuracy analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' However,  in practice, an accurate initial alignment between robots results in more reliable multi-agent coordination  and global merging maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  In order to align with the common reference frame, MARBLE primarily relied on the LTS reflectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Based  on conventional trigonometry and the assumption of needing to maintain less than 5m of error over the course  of a 1km linear distance, it is determined that an initial alignment target required less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='29◦ of error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  To align the robot, 3 reflective prisms are attached to each robot, and their positions are scanned with an  LTS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' These points VLR are then compared with a ground-truth set VR, determined by the relative locations of  the prisms to the robots tracking frame via CAD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' These two sets of points are used to estimate the transform  between the LTS recorded positions and the assumed positions by minimizing across the pose X to solve:  argminX  �  ||VLR − VRTRL||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  (1)  The result is the robot’s position in the LTS frame TRL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' An additional calculation is used between scanned  points of the gate VLG and the provided coordinates VW are used to solve for the gate’s position in the Leica  frame TW L using the equation:  argminX  �  ||VLG − VW TW L||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  (2)  Both minimization problems were solved using Horn’s absolute orientation method (Horn, 1987), a closed  form solution to least squares alignment problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Given these transforms, the robots position in the world  frame was calculated by inverting the robot to LTS transform:  TW R = TW L(TRL)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  (3)  To further reduce the impact of minor errors in either prism localization or low observability, these transforms  WR  TRL  TwLare altered slightly by each robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The LTS-predicted pitch and roll is substituted with an estimated pitch  and roll from the lidar-inertial localization system, largely based on the initial measurements of the IMU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  After these adjustments, yaw estimates had the largest impact on our resulting transforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Because yaw  error has the potential to propagate to large translational discrepancies at far distances, it became imperative  to modify our system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The solution involves increasing the lateral spacing of the prisms mounted on the  robots, and is described in more detail in Section 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='5 of the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  5  Artifact Detection  A core component of the SubT Challenge is the detection and localization of objects that could potentially  indicate human presence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Each artifact needed to be reported within a 5m radius of the ground truth location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  To achieve this requirement a lidar-inertial based state estimator as described in Section 4 is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Robots  are put into a common reference frame based on survey-grade measurements from a Leica Total Station  (LTS) and objects are projected using the mapping framework described in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The available sensing  modalities for various artifacts are discussed in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='1, the visual detection system is described in Section  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='2, and the non-visual detection system is explained in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The resulting performance of the artifact  detection system at the Final Event is detailed in Section 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='1  Sensing Modalities  Table 1 shows the classes of artifacts present at the final event along with the types of sensing modalities  capable of detecting each artifact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Each robot in the fleet is equipped with RGB cameras, Bluetooth modules,  and CO2 sensors which enable the detection of all classes of artifacts using a minimal sensor suite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The  visual detection system is not trained to detect either the cell phone, due to its small form factor, or the cube  artifact which was detectable using Bluetooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The cube artifact had rotating colors which pose significant  challenges for visual detection methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  Artifact Class  Visual  Thermal  Wireless  CO2  Survivor  +  –  Cell Phone  –  +  Backpack  +  Drill  +  Fire Extinguisher  +  Gas  +  Vent  +  –  Helmet  +  Rope  +  Cube  –  +  Table 1: Sensing modalities, that Team MARBLE utilized (+) and did not utilize (–) for detecting the ten artifact  classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Blank entries indicate sensing modalities that are not useful for detecting specific artifact classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  CO  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='2  Visual Detection  Visual object detection is a well-researched problem in computer vision and state of the art detectors are  capable of identifying objects in both 2D and 3D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Common 2D detectors are typically based on Convolutional  Neural Networks (CNN) (Zou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=', 2019), such as region proposal-based networks like Fast R-CNN (Girshick,  2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Typically these networks require multiple passes over an image to classify an object and then detect  where the object is in the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' In contrast, YOLO (Redmon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=', 2016) performs both classification and  detection in a single regression making it a significantly faster detection: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='5 FPS for Fast R-CNN and 45  FPS for YOLO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Object detectors operating in 3D typically use point clouds obtained from a lidar and until  recently were limited to classification rather than full detection (Maturana and Scherer, 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Qi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  Extensions to 3D classifiers such as Voxelnet (Zhou and Tuzel, 2018) and PointRCNN (Shi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=', 2019) are  capable of performing object detection on powerful GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' These GPUs are impractical from both a size  and power consumption standpoint for mobile robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' We selected the Yolo V3 (Redmon and Farhadi, 2018)  model due to the fast and accurate nature of the YOLO (Redmon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=', 2016) family of networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  Specifically, for classification and detection, the visual pipeline utilizes a YOLO V3 Tiny (Redmon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=',  2016) model with custom trained weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The model is optimized for Nvidia TensorRT acceleration and  we infer images at a resolution of 608x608.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The Husky platforms are able to perform inference at 60FPS  on a GTX 1650 based on Nvidia’s Turing architecture with 896 CUDA cores and 112 RT cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The Spot  platform uses a Nvidia Jetson Xavier AGX based on the Volta architecture with 512 CUDA Cores and 64  Tensor cores to perform inference at 40 FPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' These GPUs were chosen to balance performance against size  and power constraints for on-board compute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The TensorRT YOLO detector outputs a message containing  the detected artifacts as well as the coordinates of their bounding boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  A systematic procedure targeted at low-light conditions is used to train the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' At each location, data was  collected using three different brightness levels to minimize the impact of lighting conditions on the model’s  performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Specifically, images were taken from past circuit events as well as separate field exercises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  Images with excessive motion blur were subsequently filtered out and the data was later augmented with  images that contained false positive defections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The full details of our training procedure can be found in  Section 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='6 of the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  Depth registration is performed using marble mapping as described in Section 6 which is generated by the  Ouster 64-beam lidar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Utilizing an Octomap based framework allowed us to avoid implementing any additional  filtering due to the probabilistic nature of the map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Additionally, the Octomap structure aggregates scans  into the map with temporal memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' This important feature resolves the inconsistency between the 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='2◦  vertical field of view of the Ouster and the 68◦ vertical field of view of the cameras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' At further distances, the  agent is able to incrementally build out regions near ceilings and floors, overcoming the vertical blind spots of  the Ouster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Essentially, this temporal memory allows us to decouple the depth measurement from the visual  artifact detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The biggest drawback of this approach is the potential for an additional 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='15m of error on  each detection due to the voxel resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' However, this error figure still falls within the design constraints  of localizing an object to within 5m of its desired location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  After 3D coordinates are obtained via the Artifact Localization node, we run a weighted median filter in  the world coordinate frame to de-noise the projected location within the Artifact Fusion node in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  Each localized artifact is considered to be part of the same measurement if it is the same class as a previous  measurement and within 5m of that measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' We then require five to 10 positive detection events and  use the median position as the reported position to the human supervisor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The final detection is published  in a custom ROS message which contains this position as well as a compressed version of a corresponding  camera image and associated bounding box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The full overview of the artifact system can be seen in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  Gig-E Cameras  TensorRT YOLO  Artifact  Localization  Classification and Detection  Artifact  Fusion  Robot  Localization/Mapping  Artifact Message  Bluetooth  CO2  Figure 7: Overview of the artifact detection system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Sensor inputs are shown in red and outputs are shown in green.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='3  Non-Visual Detection  Cell phone, cube, and gas reports are also fused using a weighted median filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The Bluetooth and CO2  detections are simply localized to the position of the robot at the time of detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Bluetooth detections  are also grouped together by unique SSIDs and gas detections within 10m of another detection are assumed  to have originated from the same source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The final positioning of these non-visual artifacts relies on input  from the human supervisor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Our human supervisor interface was designed to easily allow for movement of  reported artifacts based on features observed in the map by the human operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The details regarding the  accuracy and success rate of these reports can be found in Section 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  6  Mapping  Team MARBLE’s custom mapping package, marble mapping (Riley, 2021) is based on Octomap (Hornung  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=', 2013) and is used to generate 3D occupancy grid representations of the world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The environment is  sub-divided into voxels, or cells which are marked as either occupied, free, or unknown using a probabilistic  log-odds based model operating on sensor returns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The output of marble mapping is a direct input to the path  planner and also provides depth measurements for visual artifact detection, as well as situational awareness  for the human supervisor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The Octree (Meagher, 1982) structure of Octomap’s occupancy grids makes  storing and transmitting maps more efficient than other representations such as point clouds;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' this efficiency is  highly desirable when trying to transmit maps over low bandwidth mesh networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' marble mapping extends  Octomap by enabling map differences for low bandwidth transmission, map merging between multiple robots,  and the addition of semantic information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='1  Difference-Based Map Merging  Despite the efficient encoding of the Octree data structure, regularly transmitting full volumetric maps of the  explored space is impractical in bandwidth-constrained subterranean environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Map differences are both  a natural solution to reduce bandwidth, and have been shown to facilitate efficient data transfers (Sheng  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=', 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' In the marble mapping package, modifications to Octomap package were made to generate  differences between different map sections, or “diff maps” shown in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The implementation allows for  diff maps, or smaller Octree structures, to be created at a predetermined rate, and contains all the mapping  data for that time interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The sum of an agent’s diff maps make up its “self” map and the differences can  be transmitted to other agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' These differences are later merged into the robot’s “merged map” in the the  map merging process which is shown in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  Merged maps generated from multiple agents are important both for a more complete view of the environment,  Figure 8: Sequential difference maps from top left to bottom right, with the final map on the far right constructed in  real time for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The diff maps can be merged to fully reconstruct the original map shown in the bottom  right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  and they also reduce redundant coverage in coordination strategies (Ko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=', 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Simmons et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=', 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  Zlot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=', 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The marble mapping package enables map merging both for individual agents and on the  base station which allows agents to intelligently act on the data and provides a holistic view for the human  supervisor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The system does not re-align maps prior to merging, as it is assumes agents are already in a  common reference frame as described in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The lack of a re-alignment feature has the potential to  cause one agent’s map to block pathways in a receiving agent’s map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' To mitigate this, each agent prioritizes  its own map by only appending cells from other maps into “unknown” areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Areas that have already been  “seen” by the agent are left untouched which prevents misaligned data from blocking free space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' In cases  where this mitigation procedure is not enough, such as narrow hallways, or a complete loss of localization by  an agent, “bad” map diffs can be removed by the human supervisor using the base station GUI described in  Section 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='2  Semantic Mapping for Terrain-Aware Navigation  While the volumetric-based mapping produced by the Octomap framework provides the high-level structure  of the environment, its resolution, set to a voxel size of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='15m, is too coarse to capture details needed for  high fidelity motion planning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' In order to augment the existing marble map with terrain information, Team  MARBLE evaluates the traversability of a given voxel using the normal and curvature values from raw point  clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The planning solution is then able to utilize this semantic information to plan safe paths in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  An additional label is attached to each voxel which enables the semantic labeling of staircases for the Spot  platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  Early approaches to evaluating the traversablity of an environment include elevation based maps based on a  2D lidar (Ye and Borenstein, 2003) but are unable to take advantage of modern 3D sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The traversability  classifier presented here is largely based on the Grid Map framework presented in (Fankhauser and Hutter,  2016), which evaluates the slope and roughness of point cloud regions to generate a multi-layer surface map  but only creates a 2D grid rather than a 3D volumetric map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Other fielded approaches in subterranean  environments include “virtual surfaces” on occupancy maps (Hines et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=', 2021) and Conditional-Value-at-  Risk metrics, such as collision, step size, tip over, and slippage, which are incorporated into a dense 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='5D  (a)  (b)  Figure 9: Traversability information (a) of section within the Edgar Experimental Mine in Idaho Springs, CO, USA,  that contains railroad tracks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Raw traversability values of the lidar point clouds (top) are shown, where white is not  traversable, and black is traversable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Resulting semantic map (bottom) illustrated non-traversable surfaces such as  walls in white, traversable surfaces such as the ground in black, and semi-traversable surfaces such as the railroad  tracks in grey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Note that red voxels do not contain traversability data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' An accompanying photo (b) of the section of  mine is shown for reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  gridmap (Fan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' These dense methods typically come in the form of high-resolution local maps,  which enable more precise locomotion over varied terrain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' An alternative approach presented in (Kr¨usi  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=', 2017) computes paths with continuous curvature over raw point clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' However, by computing  semantic traversability information, our planning approach only required a low-resolution global map, greatly  simplifying both mapping and planning systems and allows for sharing of semantic information between  agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='1  Traversability Classification & Map Integration  To estimate the traversability of a voxel, we segment the 3D point cloud produced by the Ouster lidar,  and evaluate the unit normal vector ˆn and curvature K of each point p at timestep t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' All calculations  are performed with the aid of the pcl package (Rusu and Cousins, 2011) and a traversability value, τp,t, is  estimated for each point using Equation 4 where ˆk is the gravity-aligned up vector, (1 − |ˆn · ˆk|)3 is a measure  of the slope of the terrain, and cnorm and ccurv are tunable parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The parameter values for the Final  Event were set to cnorm = 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='0, and ccurv = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='0, and τp,t ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  τp,t = cnorm(1 − |ˆnp,t · ˆk|)3 + ccurvKp,t ∈ [0, 1]  (4)  Traversability is implemented in the Octomap framework using Equation 5 to estimate the traversability, τv,t,  of a given voxel, v, as a function of the voxel’s occupancy probability, Pocc,v,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The traversability estimate for  the voxel is a linear combination its previous traversability estimate, τv,t−1, and new estimate τp,t for the  points in the voxel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' An example of this process is shown in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  τv,t = τv,t−1Pocc,v,t + τp∈V,t(1 − Pocc,v,t) ∈ [0, 1]  (5)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='995486  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='00000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='626.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='2  Stair Classification & Map Integration  Semantic information on stairs is fused into the mapping framework using the open source StairwayDetection  (Westfechtel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=', 2018) package and a binary Bayes filter (Thrun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=', 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Stair classification of  point clouds via this approach consists of 4 major steps: (1) pre-analysis, in which the point cloud is  downsampled and filtered, normal and curvature is estimated for each point, and floor separation is performed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  (2) segmentation via a region growing algorithm, which segments the point cloud into smooth regions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' (3)  plane extraction, in which the surfaces that make up the riser and tread regions of each stair step are  extracted;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' and (4) recognition, where the tread and riser regions are connected and analyzed via a graph to  determine whether they make up a valid set of stairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  (a)  (b)  Figure 10: Spot planning up a staircase using the estimated stair voxels in the Octomap shown in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  Stair detections are integrated into the map using a similar mechanism to the log-odds probability which  determines occupancy in octomap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' A binary Bayes filter (Thrun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=', 2005), shown in Equation 6, is used  to estimate the probability that a given voxel is a part of a staircase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The extracted points from the stairway  detector are modeled as measurements z where Pstair(n|zstair,t) is the probability that a voxel n is part  of a staircase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The measurement through time step t is represented by zstair,1:t as shown in Equation 6a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  Lstair(n|zstair,1:t) as shown in Equation 6b are the corresponding log-odds probabilities which are used for  fast updates updates to the probabilistic estimate of each voxel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' More details of the log-odds formulation are  provided in (Hornung et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Thrun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=', 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Our filter is tuned to prioritize true positive detections  with the following parameters: Pstair(n) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='5, Lstair,min = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='0, Lstair,max = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='48, Lstair,hit = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='60,  Lstair,miss = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  Pstair(n|zstair,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='1:t) =  �  1 + 1−Pstair(n|zstair,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='t)  Pstair(n|zstair,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='t)  1−Pstair(n|zstair,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='1:t−1)  Pstair(n|zstair,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='1:t−1)  Pstair(n)  1−Pstair(n)  �−1  ∈ (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' 1)  (6a)  Lstair(n|zstair,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='1:t) = Lstair(n|zstair,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='1:t−1) + Lstair(n|zstair,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='t) ∈ [Lstair,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='min,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Lstair,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='max] ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  (6b)  where Lstair,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='min ∈ (−∞,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' 0) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Lstair,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='max ∈ (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' ∞) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  Lstair(n|zstair,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='t) =  �  Lstair,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='hit > 0 on stairs  Lstair,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='miss < 0 on non-stairs  A raycast operation on the footprint of the vehicle is used to provide a binary signal indicating the robot is  S18on stairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Additionally, eigenvector decomposition is performed over each cluster of stair voxels to extract a  straight path along the staircase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' These triggers provide waypoints so that the local trajectory follower can  navigate to the top of the staircase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' It’s important to note that since stairs would generally be classified  as non-traversable, a stair label takes precedence over a traversability label for the Spot platform, which is  capable of walking up stairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Additionally, this method requires sufficient lidar scans of the staircase, which is  generally available when located at the bottom of a staircase but is not when facing the stairs leading down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  As a result, detecting and navigating a descending staircase is not feasible with the current configuration, but  could be with a wider field-of-view sensor or programmed forward pitching behavior of the Spot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  The marble mapping package enables the creation of difference-based Octomaps which allows for efficient  transmission in underground environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Furthermore the framework provides semantic and traversability  information which the planner utilizes to ensure the robot is able to navigate safely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Details of the planner  are described in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  7  Path Planning  Team MARBLE’s heterogeneous fleet relies on autonomous path planning onboard each agent to reduce  the workload of the human supervisor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The path planner running onboard each agent generates safe and  traversable paths that lead to unexplored areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Paths are planned on the Octomap-based marble mapping  framework described in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Team MARBLE used the same planner on all robots with the only  difference being the collision-function depending on vehicle’s class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' For instance, a wheeled robot cannot  traverse stairs while a legged robot can.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Existing methods discussed in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='1 suffer computational costs  that make it challenging to scale to large environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Because the proposed planniner is computationally  efficient and minimally dependent on tuning gains, it performs well in large-scale environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Our planner  makes several significant contributions, such as light on-demand terrain assessment, which is discussed in  Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='2, hierarchical solution-search that also incorporates position history-based multi-agent coordination,  which is discussed in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='3, and handling of dynamic changes in the environment such as blocked  passages, which is covered in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='1  Background  One of the widely-known methods (Yamauchi, 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Ahmad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=', 2021a) for autonomous exploration  relies on explicitly detecting potential frontiers on an explored map, followed by a path planned toward  each cluster of frontiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The method seeded significant developments in the area of autonomous robotic  exploration since it was first proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' However, this frontier-based method employs a computationally  expensive optimization-based approach that plans paths to each frontier cluster, despite the fact that some  may not be reachable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  In recent decades, the planning community has witnessed significant advancements in more computationally  efficient sampling-based approaches for path planning and exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' One instance of such development  is an exploration planner that uses Rapidly Exploring Random Trees (RRT) (LaValle, 2006) to sample an  environment and chooses an optimal path from the set of sampled ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The method samples the environment  as a single batch, and therefore is not scalable to large-scale environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' A rectification of this limitation  is recently proposed by (Dang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=', 2020) where a bifurcation approach is introduced for sampling and  exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' This approach implies that the environment is sampled only in the local neighborhood of  a robot while simultaneously building a sparse graph that scopes the entirety of the explored map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The  latter is essential to deal with local minima such as dead-ends and also to plan a path back home.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Our  autonomous exploration solution for the SubT Final Event relies on the principle of bifurcation with additional  contributions in the terrain assessment, solution-search, dynamic obstacle avoidance and coordination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  The graph-based planners based on sampling and bifurcation approach use high resolution depth images  to compute a 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='5D grid-based elevation map using the technique presented in (Fankhauser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  This elevation map is further filtered to segment terrain characteristics such as slope, roughness and step  (Wermelinger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The authors of such graph planners mention the scalability challenges with such  computationally expensive approaches, which limit their terrain awareness to regions local to the agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' This  further leads to challenges such as the planned path and the underlying graph being generated with an  over-optimistic view of the terrain, consequently needing the robot to be backed up if it encounters impassible  terrain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='2  Sample-and-Project Strategy  To rectify the terrain assessment challenges, a sample-and-project approach is followed, similar to the settling-  based collision-check approach proposed in (Kr¨usi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' We perform such checks on an Octomap with  resolution 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='15m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' At this resolution, all of team MARBLE’s robots were at least three voxels wide, providing  a decent amount of robot footprint to project a robot’s pose on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' SubT challenge rules highlight that the  extremely narrow passages could be around 1m wide, with doorways as narrow as 36 inches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' With this in  mind, Octomap voxel length of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='15m was small enough to navigate narrow passages and large enough to be  able to keep up with computational complexity of generating such a map in a large-scale environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' In  case of a wheeled robot, each voxel in the Octomap is labelled with a roughness value which is obtained using  high resolution point clouds as described in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' However, on Spot legged robots dense roughness  information is not required because of their onboard terrain assessment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Additionally, for Spots, encode  semantic information about stairways into the map which overrides the default height parameters of the  planner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' With explicit labels, the planner is able to plan paths over built up staircases despite elevation  changes the robot would not normally traverse over.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' More formally, for the legged robots capable of traversing  stairs, each Octomap voxel maps to a label from the set {‘occupied’, ‘unknown’, ‘free’, ‘stair’}, whereas in  case of wheeled robots the label set is {‘occupied’, ‘unknown’, ‘free’, ‘rough’}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  First, the environment is sampled in the local neighborhood of the robot using RRT∗, a variant of RRT  with optimality considerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Each tree sample is a robot position parameterized by the robot width  and length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' During sampling, the collision-checks are performed by vertically projecting a query sample to  find the ground below it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Once the ground is found, the elevation change at the footprint of the sample is  evaluated if there are enough projections on occupied voxels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' In case of a wheeled robot, the average roughness  information of the footprint voxels is also taken into consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' For a legged robot, if a threshold amount  of footprint voxels are labelled as ‘stairs’, the sample is considered collision-free regardless of the elevation or  roughness check.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Expanding an RRT∗ requires checking path segments for collisions instead of isolated robot  configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' In order to check such a path segment, a set of robot configurations along the segment is  checked for traversability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' 11 depicts the terrain assessment process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='3  Solution Search and Multi-Robot Coordination  At any replan iteration, a set of potential solutions include all of the local paths sampled using RRT∗ and  all of the global paths ending at the graph frontiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The former set of solutions is represented by PL and  latter solutions belong to the set PG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' A set of all local paths that are leading the robot toward areas with  greater than a defined threshold of volumetric gain and teammate separation are given as PLV and PLS  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Similarly, a set of all global paths that are leading the robot toward areas with greater than  a defined threshold of volumetric gain and teammate separation are given as PGV and PGS respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  These sets is highlighted in Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' A typical approach to find an appropriate solution is to form an  objective function with a combination of exploration objectives such as volumetric gain and exploration  heading parameterized by the penalty gains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Volumetric gain calculation is, however, a computationally  expensive operation and limits the amount of frontiers a robot can process in a reasonable amount of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  Our approach relies on finding a good enough solution in terms of volumetric gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' To achieve multi-robot  coordination, the position histories of teammate robots on the network are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' If a path is leading a robot  to a point such that the minimum distance of the point from the position histories of the teammate robots is  Figure 11: A depiction of sample-and-project strategy for terrain checks on an OctoMap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The figure shows four  different types of query poses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' (a) and (b) depict collision-free samples whereas (c) and (d) are marked under-collision  or non-traversable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  more than the mapping range, it is guaranteed that new areas are being explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  Following this intuition, the primary objective of Team MARBLE’s solution search method is not to optimize  for the volumetric gain and the teammate position histories separation, but to accept a solution that has  a satisfactory amount of volumetric gain and distance from the teammate position histories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The formal  objective of the path planner is to output a solution that belongs to PLV ∪ PLS, PGV ∪ PGS, PLV or PGV  in the order of preference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The solution search process makes use of two different objective functions,  Jα = c0DP (phist, pcand) − c1|θexplore − θcand  mean|  − c2|hexplore − hcand  mean|,  (7)  Jβ = −c1|θexplore − θcand  mean| − c2|hexplore − hcand  mean|  + c3GV (pcand(1)) + c4DS(pcand(1), phist  1  , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=', phist  1  ),  (8)  where Jα is used to find a candidate path that aligns best with the current exploration heading of the robot  and Jβ is leveraged to perform a thorough search if required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The sets of points pcand and phist represent a  list of candidate solutions and the position history of a robot respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The exploration heading θexplore  is calculated by averaging the most recent few points on phist of the robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The mean heading and mean  height of a candidate path are denoted by θcand  mean and hcand  mean respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The function DP accepts two paths  as arguments and calculates the mean of minimum distance of all points along the first path with the second  path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Furthermore, the function DS calculates the minimum distance of a candidate path from the position  histories of all other teammate robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  Algorithm 1 provides a deeper insight into the solution search steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' As a first step, a collision-free local  path is found that best aligns with the direction of travel of the robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' This path is then checked if it has a  satisfactory amount of volumetric gain and distance from the teammate position histories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' If a good-enough  solution is found at this step, the solution is returned and only a single volumetric gain function call is  required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Therefore, we save significant computation time during most replan iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' If a solution is not  found at this first step then a more thorough search is performed, first through the sampled local paths and  then through the global paths leading toward graph frontiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' This search is highlighted in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  Robot Length  Query Poses  Local  Sampling  (d)  Region  (a)  (c)  (b)  X  X  max-min  stair  inot enough  max-min  < thresh  voxels  > thresh  I projections The functions PlanLocally() shown in Algorithm 2, and PlanGlobally() shown in Algorithm 3, are  responsible for outputting solutions that satisfy both volumetric gain and teammate separation constraints  if possible, otherwise they output solutions that only satisfy the volumetric gain constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' In the worst  case, when neither constraint can be satisfied, paths with maximum teammate separation are returned as a  contingency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  This attempt of finding a solution by breaking the potential solution space down into subsets instead of  having one objective function to optimize over the entire space, helped us avoid extensive gain tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' During  testing and final event runs, we found our approach to be scalable for environments of various sizes without a  need for tuning gains for different environment types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The details of the sampling-based path planner can  be found in (Ahmad and Humbert, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' In this work, a simulation comparison of the proposed planner  with an existing sampling-based planner (Dang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=', 2020) is presented, highlighting the improvement in  scalability and computational efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  Algorithm 1 ScanPlan Solution Search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='1: pl  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='α ← pcand ∈ PL costing minimum Jα  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='2: if GV (pl  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='α) ≥ vg  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='thresh and DS(pl  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='α) ≥ sg  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='thresh then  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='3:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='return pl  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='α  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='4: end if  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='5: pl  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='β ← PlanLocally()  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='6: if pl  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='β is non-empty and DS(pl  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='β) ≥ sg  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='thresh then  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='7:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='return pl  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='β  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='8: end if  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='9: pl ← pl  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='β  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='10: pg ← PlanGlobally()  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='11: if pg is non-empty and DS(pg) ≥ sg  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='thresh then  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='12:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='return pg  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='13: else if pl is empty and pg is empty then  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='14:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='return pl  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='α  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='15: else if pg is empty or  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='(pl is non-empty and DS(pl) ≥ DS(pg)) then  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='16:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='return pl  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='17: else if pl is empty or  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='(pg is non-empty and DS(pg) ≥ DS(pl)) then  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='18:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='return pg  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='19: end if  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='Algorithm 2 PlanLocally().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Returns a path in PLV ∩ PLS or PLV in the order of preference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  1: Jβ  min ← + inf  2: pres ← none  3: success ← false  4: for pl ∈ PL do  5:  if GV (pl) ≥ vg  thresh and DS(pl) ≥ sd  thresh and ∼ success then  6:  Jβ  min ← Jβ(pl), pres ← pl, success ← true  7:  else if Jβ(pl) ≥ Jβ  min then  8:  continue  9:  else if (GV (pl) ≥ vg  thresh and DS(pl) ≥ sd  thresh) or (GV (pl) ≥ vg  thresh and ∼ success) then  10:  Jβ  min ← Jβ(pl), pres ← pl  11:  end if  12: end for  13: return pres  Algorithm 3 PlanGlobally().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Returns a path in PGV ∩ PGS or PGV in the order of preference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  1: PGV ← ∅,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' PGV S ← ∅  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='2: for pg ∈ PG do  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='3:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='if GV (pg) ≥ vg  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='thresh and DS(pg) ≥ sd  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='thresh then  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='4:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='PGV S ← PGV S ∪ {pg}  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='5:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='else if GV (pg) ≥ vg  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='thresh then  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='6:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='PGV ← PGV ∪ {pg}  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='7:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='end if  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='8: end for  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='9: if PGV S is empty then  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='10:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='return path pg ∈ PGV with maximum DS(pg)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='11: end if  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='12: pg  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='r ← path from PGV S leading to most recent frontier  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='13: pg  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='c ← path from PGV S leading to closest frontier  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='14: if PathLength(pg  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='r) ≥ PathLength(pg  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='c) then  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='15:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='return pg  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='c  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='16: else  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='17:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='return pg  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='r  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='18: end if  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='(a)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='(b)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='(c)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='Figure 12: The figures highlight the planner solution search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The potential solutions for the planner include the paths  leading toward the leaves of the RRT∗ tree (blue) and the frontiers of the graph (green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' (a) The first preference of  the solution is the path that aligns best with the robot’s exploration heading (bold blue path).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' In case this path has  both sufficient volumetric gain and teammate separation, it is returned as a solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' (b) As a second preference, a  thorough search is performed to find a local or global path that satisfies both constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' (c) If none of the paths is  found at both of the steps above then a path that satisfies the volumetric gain constraint is accepted as a potential  solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='4  Dynamic Replanning  Another challenge faced by the existing graph-based planners is that they rely on building a parallel graph  representation of the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' This representation does not naturally reflect changes in the environment,  such as closed passages which were initially open at the time the graph is built.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' To handle this exception, the  graph edges are labeled with a boolean representing its occupancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' During exploration, the planned paths  are constantly checked for collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' If a planned path is under-collision, all edges in the local neighborhood  of the robot are validated for collision and marked accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Moreover, all occupied edges are checked  for occupancy all the time when the planner finds some idle time which mostly happens when the vehicle  is following a path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' This enables the planner to take into account the cases where an occupied area is free  again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' In the case where the graph search is performed to plan a global path, the occupied edges are ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  Exploration Heading  GraphFrontiers  TeammatePose  History  RRT*Root Node  Pose HistoryExplorationHeading  Graph Frontiers  LOWVOL-GAIN  TeammatePose  History  RRT*Root Node  Pose History  LOWTEAMMATE  SEPARATIONExploration Heading  GraphFrontiers  LOWVOL-GAIN  TeammatePose  History  RRT*RootNode  Pose History8  Communication Systems  Effective communication with deployed systems from a fixed human operator is a crucial component of a  complete robotic exploration system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' While robots are capable of independent localization, mapping, and  artifact detection, the addition of a communication infrastructure is a force multiplier to enable human  supervisory control, inter-robot coordination, and timely artifact reporting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' We developed a mesh network  system to provide long-reach communications into underground environments which prioritizes reconnection  times to maximize opportunities for data transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='1  Background  Previous work has developed several solutions to common problems encountered with deploying mesh networks,  such as discovery and optimal routing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' A wide variety of both closed-source and open-source solutions exist  that include both hardware and software components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Mesh networking can largely be subdivided into  three layers: physical, logical, and transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' We will detail several prominent open-source or commercially  available options for each layer before describing our final solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  From a logical layer standpoint, meshing layers lay between the physical transmission of frames over the  medium and a higher-level protocol such as IP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' For mobile robots operating in subterranean environments,  a responsive mesh layer that minimizes lost link time is a major requirement due to the rapid movement  of the robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Further, to reduce integration effort, a mesh layer that operates at layer 2 of an OSI stack  (for Standardization, 1996) is desirable to allow transparent use of higher-level protocols such as ARP and  IP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Typically, meshing algorithms such as OLSR (Clausen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=', 2003) and AODV (Perkins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=', 2003)  select a single best path for routing between nodes which hinders algorithmic performance in dynamic  environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' A more recent example of a single-path logical meshing layer is Better Approach to Mobile  Ad-hoc Networking-Advanced (batman-adv) (Seither et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=', 2011), an open source implementation of a layer 2  mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' In contrast to batman-adv, meshmerize (Pandi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=', 2019) provides multiple paths between nodes to  ensure a reliable connection while still operating at layer 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' these multiple paths allow for a dramatic decrease  in reconnect times when mesh topology changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' We relied on meshmerize as our layer 2 meshing solution in  cooperation with Meshmerize GmBH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  Only transport layers designed for ROS were considered for ease of integration with the rest of the autonomy  stack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' In a traditional networked ROS architecture, a single computer runs a main node known as the  rosmaster that coordinates the publish-subscribe mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' When a node wishes to exchange data with  another node via named topics, the master is consulted to determine the computer to connect to, as in Figure  13a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' A single rosmaster serves as a central directory of nodes and topics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' when a subscription to a topic is  requested, a list of publisher nodes is returned so that point-to-point TCP connections can be made directly  between publisher and subscriber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' These direct TCP connections break down when systems are linked over  unreliable mesh networks which necessitates the need for an alternative transport mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  One open source transport layer multimaster fkie (Juan and Cotarelo, 2015) solves the discovery and  advertisement problems using multi-cast packets and specialized nodes on each machine with an architecture  shown in Figure 13b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' However multimaster fkie does nothing to establish prioritization of data flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' With  the standard TCP transport provided by ROS, there is no centralized means of monitoring inter-node  connections to arbitrate data priorities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Prioritization is crucial for monitoring the robot fleet in intermittent  communication situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Mission critical data such as artifacts needs to make it through to the human  supervisor before other auxiliary data such as odometry and maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  One alternative to multimaster fkie, Pound1 (Tardioli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=', 2019), is specifically designed for use in unreliable  mesh networks and implements many of the desired requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' However, Pound relies on hardcoded  topic names and fixed addressing information, which limit the flexibility of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  Alternatively,  1https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='com/dantard/unizar-pound-ros-pkg  Master  Node A  Node B  Node C  (a)  Master α  Node A  Node B  Node C  Master β  Node D  Node E  Node F  (b)  Master α  Node A  Node B  Node C  udp mesh  Master β  Node D  Node E  Node F  udp mesh  (c)  Figure 13: Several different types of ROS architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Red lines indicate data transfer, black lines indicate directory  management, and blue lines are data paths that cross network segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' A basic, single-master ROS network node  graph is shown in (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' A fkie multimaster multi-master ROS network node graph is shown in (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' In contrast to both  (a) and (b), (c) shows how udp mesh creates a single virtual channel between nodes, shown in green, to implement  data prioritization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  nimbro network (Schwarz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=', 2016) implements a similar set of functions with regards to transport over  wireless networks, but omits prioritization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Crucially, nimbro network still utilizes TCP for reliable inter-robot  communication, preventing adaptation of core TCP behavior (particularly retransmits) to unreliable mesh  networks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' UDP links are only used for non-guaranteed data delivery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='2  UDP Mesh  The main innovation in our system is our transport layer, udp mesh which allows for runtime reconfiguration,  implements prioritization, and re-implements reliable communication over UDP to allow for more refined  control over retransmits and fragmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Fundamentally, the udp mesh layer uses only unicast and  broadcast UDP datagrams to implement higher-level services without requiring multicast support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  In  principle, multicasting would offer a performance benefit by reducing broadcast traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' However, in a wireless  mesh environment, these potential gains are offset by multicast group membership management overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='1  Discovery and Address Resolution  Discovery is the process of identifying nodes that are available for communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' We implement discovery  through the use of a periodic heartbeat broadcast that advertises the node’s availability and provides name  resolution information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' In concept, this service is similar to the mcast dns service in Linux, where peers  advertise their naming information to be able to address nodes by hostname instead of layer 2 MAC or layer  3 IP address.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Nodes identified through discovery are added to the list of available nodes for communication as  well as status reporting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' This discovery heartbeat is also used as a lost-communications detector to prevent  higher-level messages from queueing for unreachable nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='2  ROS Message Encapsulation  In the ROS ecosystem, messages are translated from a message definition language specification into internal  representations appropriate to the implementing language2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' This same language specification is used to  serialize and deserialize messages;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' that is, to transform a ROS message into a buffer of bytes suitable for  transmission over an arbitrary channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' udp mesh implements a generic message passing system such that  the message to be transmitted is never deserialized, saving a significant amount of processing time in the case  of complex, large message types such as images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Instead, a generic subscriber is used to acquire the serialized  bytes for direct use to be transmitted to other nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' On the receiver side, the transmitted byte stream is  deserialized to instantiate the message in a format that other ROS ecosystem nodes can readily consume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  These two functions abstract the transport of arbitrary messages over the udp mesh layer and remove any  2http://wiki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='ros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='org/msg  requirement to define a list of acceptable message types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='3  Point to Point Transport  In the udp mesh system, point-to-point transport is implemented via UDP datagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' This envelope contains  provisions for sequence tracking, fragmentation, and message reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' When preparing a message for  transmission, the byte buffer provided by the ROS encapsulation service is split into chunks that fit inside  the underlying medium’s maximum transmit unit (MTU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' We use the standard 802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='11 framing with an MTU  size of 1500 bytes, out of which 100 bytes are reserved for overhead, leaving 1400 bytes for payload.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  In the implementation of our system, a configurable number of message fragments are permitted to be ‘in  flight’ at any given time, similar to TCP congestion window control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' In order for the next fragment to be  transmitted, the receiver must send an acknowledgment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' During unit testing to determine an appropriate  value for the number of in-flight fragments permitted, an initial increase yields improved throughput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' However,  after a certain point, throughput decreases as multiple packets are queued for transmission on the medium and  start to destructively interfere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' As a compromise determined via empirical testing, three packets are permitted  to be in-flight between any two nodes at a time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' With this configuration, our transport-layer throughput is  approximately 20 Mbit/s of payload data, measured using raw images as representative high-density traffic  over a wired gigabit Ethernet link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  Retransmits are automatically queued until either an acknowledgment is received or the host is marked offline  due to non-reception of any heartbeat or acknowledgment messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Once a host is marked offline, any  attempts to send messages are discarded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Hosts may become online once again after receipt of a discovery  message.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' On the receiver side, the message is kept in a temporary state while the fragments arrive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Should  message fragments stop arriving, the partial message is purged after a timeout and the host is once again  marked offline which indicates to higher levels that reliable transport is unavailable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' In this case, the higher  level is BOBCAT, which is discussed in Section 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='4  Quality of Service  Quality of Service (QoS) is the notion that some traffic should be prioritized over other traffic for use of a  limited communications channel, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='g, artifact reports need to arrive before mapping data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Fundamentally,  TCPROS (the default transport used in ROS v1) is not capable of implementing a QoS scheme where a  limited channel is shared between different topics (Figure 13c), as every node subscribing to a topic uses an  individual TCP point-to-point link with no information about other links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' This need to prioritize traffic was  the driving rationale behind the development of the udp mesh layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' As part of the configuration of the layer,  each topic to be transported includes a priority number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Internally, this priority number is used as a sorting  key to order message fragments for transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='5  Point-to-Multipoint Transport  Although udp mesh is based around point-to-point message transfer, mission requirements sometimes necessi-  tate system-wide messaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' For example, broadcast methods are used within the udp mesh layer to manage  name resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' To facilitate these type of messages originated at higher levels, a broadcast mechanism is  provided by the transport layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' For messages that fit within a single MTU, a single, unacknowledged UDP  broadcast is used to distribute the message.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' For larger messages, individual links to each node are used to  send the broadcast as a series of unicast fragments using the same accounting and acknowledgments as the  point-to-point mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='3  Final Solution  The final communication solution used meshmerize as the logical layer with udp mesh as the transport layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  Both robots and beacons acted as nodes in the mesh with robots carrying 1W radios and beacons carrying  2W radios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Beacon drops are controlled by the methodology described in Section 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Table 2 shows the  evolution of our final networking solution from the Tunnel event through the Final Event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Our meshing  solution, including the meshmerize layer 2 software stack, was implemented on ath9k-compatible 802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='11  hardware, while udp mesh was implemented on high-level compute units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Because of this split and radio  hardware commonality, all of our radios ran essentially the same firmware image built off of an OpenWRT3  base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Our beacons only participated in the mesh at layer 2, and as such did not contribute to any broadcast  traffic associated with udp mesh services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' By providing a reliable ROS-compatible mesh networking layer,  higher-level autonomy and human interface via BOBCAT could be provisioned without knowledge of the  underlying infrastructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  Event  Physical  Data Link  Transport  Application  Tunnel  ath9k  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  fkie multimaster  marble multi agent  Urban  ath9k  meshmerize  fkie multimaster  marble multi agent  Final  ath9k  meshmerize  udp mesh  BOBCAT  Table 2:  Evolution of Team MARBLE’s communication system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' As a note, ath9k and meshmerize are commercially  available, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' and fkie multimaster are open-source software, and upd mesh, marble multi agent, and  BOBCAT are custom packages developed for the SubT Challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  9  Mission Management  While the combination of Team MARBLE’s large scale positioning system, mapping, and planning solutions  provide a solid foundation for autonomy, higher level cognition and reasoning is required to take full advantage  of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' For Team MARBLE, this higher level reasoning consists of a flexible mission management  solution which keeps the robots on task and allows for higher level instructions from a human supervisor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  The core of the mission management solution is Behaviors, Objectives and Binary States for Coordinated  Autonomous Tasks (BOBCAT) (Riley and Frew, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' BOBCAT controls the decision-making process for  each individual agent while a separate process known as Multi-Agent Data Collaboration for Autonomous  Teams (MADCAT) controls the data sharing and waypoint deconfliction between robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' In this section we  highlight the design decisions, and algorithm details behind BOBCAT and MADCAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='1  BOBCAT  BOBCAT simplifies the robot and environment states using Monitors such as communication status.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The  Monitors are combined with weighted goals which Objective such as finding artifacts or extending communi-  cations can be fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' BOBCAT then selects the best Behavior such as exploring or deploying a beacon to  fulfill and the most important Objectives to execute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' A full list of implemented Monitors, Objectives, and  Behaviors can be seen in Section 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='7 of the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  Formally, a BOBCAT is defined by the tuple {x, y, w, M, O, B, πB} where  x ∈ X is the system state with state space X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  y ∈ Y are the sensor measurements with measurement space Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  3http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='openwrt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='org  w ∈ W = R|O|  +  is a vector of input weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' These weights are used by the respective Objective  functions and represent the relative importance of the Objective to the overall mission  M is the set of Monitor functions of the form Mi : X × Y → {0, 1} ∀Mi ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Monitor functions Mi  return a binary value based on the robot state and measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  O is the set of Objective functions of the form Oj : W × {0, 1}|M| → {0, Wj} ∀Oj ∈ O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Objective  functions Oj use the input weight Wj and a logical combination of Monitor outputs to return either  the input weight or a 0, which indicates the current preference of the objective to be fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  B is the set of Behavior functions of the form Bk : {0, 1}|M| × {0, Wj}|O| → R≥0 × Fk ∀Bk ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  Behavior functions Bk sum the outputs of the Objectives associated to that Behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Monitor  outputs may be used to selectively inhibit specific Objective weights during evaluation steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The  Behavior function returns a real value that indicates the current utility score of the actions associated  with that Behavior, and a pointer to an execution function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' A Behavior may have a null execution  function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  πB is the policy for selecting the Execution Behavior BE based on each of the Behavior utility scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  BOBCAT can be represented graphically as in Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' States and measurements from both the robot  itself and external agents in a multi-agent scenario feed the various Monitors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' This represents what the robot  “knows”, and provides a binary output to the rest of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The Monitor output lines in Figure 14 and  other figures represent the cases where the Monitor is associated with the respective Objective or Behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  Figure 14: Graphical overview of BOBCAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Numbers represent binary outputs, output weights, and behavior scores,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' A full list of monitors, objectives and behaviors is provided in the Appendix in Tables 8, 9, 10 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='2  MADCAT  The MADCAT framework depicted in Figure 15 provides the multi-agent data sharing capabilities required  for the mission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The framework includes transmission of relevant coordination data and maps, as well as  map merging functionality and decision making for each agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' MADCAT uses BOBCAT to accomplish the  high-level mission management for individual agents with additional higher-level direction provided by the  human supervisor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  Monitor 1  0/1  Objective 1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='0  Monitor 2  0/ 1  Behavior 1  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='0  Objective 2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='0-  States and  Measurements  (Internal and  Monitor 3  0/1  Behavior 2  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='5  External)  Objective 3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='5-  Monitor 4  0/1  BehaviorP  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='0  Objective O  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='0  Monitor N  0 / 1-  Binary Output  OutputWeights  BehaviorScoresFigure 15: Overview of the Multi-Agent Data Collaboration for Autonomous Teams (MADCAT) framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='1  Messages  MADCAT sends most messages by broadcast with no acknowledgement required, and therefore does not  require the sender to needlessly wait.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' This allows any agent who receives the message to act accordingly  without a requirement to respond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' This is helpful in the event the sender leaves communications range shortly  after the broadcast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' An exception to this policy is made for high bandwidth data such as maps and images,  because the receiver can not act on incomplete data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Bandwidth is not strictly managed, but instead uses a  ‘best-available’ strategy consistent with the prioritizations assigned to differing message classes, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' telemetry,  supervisor commands, maps, and FPV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  An agent’s pertinent local messages are concatenated into a single message in order to limit the number of  messages broadcast over the communications channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Messages are re-built and broadcast every second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  Only the most recent message is needed for time-varying data such as odometry or the current goalpoint and  any older messages are discareded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Other data such as artifact reports and relay locations are appended to a  growing list, so any message a remote agent received has all of this type of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Larger data that could  grow to become impractical to transmit repeatedly, such as maps and images, use a point-to-point handshake  transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Messages are deconflicted using sequence numbers, to allow agents to share the messages of  other agents but ensure only the latest data is used by the receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  Each agent’s broadcast message contains not only its own local data, but that of any neighbor agents as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  This allows downstream agents who can communicate with agent A but not agent B to still receive relatively  current information from agent B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='2  Artifact Report Management  Agents keep track of both their own detected artifacts using the procedure described in Section 5 as well any  artifacts they have received from other agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The framework aggregates all of the artifact reports and the  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='Remote Agents  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='MapDiffs  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='Odometry  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='MissionElements  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='SupervisorInputs  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='Goal  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='Mapping  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='Remote Message Aggregation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='Message Deconfliction  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='Map Diffs  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='Monitors  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='Objectives  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='Behaviors  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='Odometry  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='MissionElements  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='Local Message  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='Output Aggregation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='Aggregation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='SupervisorInputs  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='Goal Selection  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='GoalArray  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='Local ControlArtifact monitor determines if the agent needs to return to communications to report the new information  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='to the base station.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The base station further parses these messages for display, selection, and transmission  to the scoring system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' More details of this display can be found in Section 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Images are sent using a  point-to-point request system over a low priority channel to reduce bandwidth requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  The BOBCAT Artifact monitor which is triggered by 3 unreported artifacts or 5 minutes of exploration with  a pending artifact, ultimately determines whether the robot should return to communications for unreported  artifacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The raw artifact reports are always used in this determination, but transmission of images is  configurable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' By default, and as configured during the Final Event Prize Run, artifact images not received by  the base station are considered unreported artifacts, and will force the robot to return to communications  until they are fully transmitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='3  Beacon Deployment  The framework is responsible for identifying locations to deploy communications relays to extend the  communications reach into the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' It uses a combination of communications status, distance and  turn detection to identify potential locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The human supervisor is also able to command drops based on  a robot’s location on the map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='4  Goal Selection  Some behaviors, particularly Explore, require a goal selection step once that behavior has been chosen to  execute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The goals either come from the global planner described in Section 7 or from human supervisor  input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' If two agents goals are found to be conflicting, BOBCAT requests a new goal from the planner which  provides a path to the next closet goal point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='5  Human Supervisor Interface  The human supervisor interacts with BOBCAT using a custom GUI shown in Figure 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' This interface  allows the human supervisor to set a goal point for the robot using an Interactive Marker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' MADCAT then  passes this goal to the robot through the communications network if a connection is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The human  supervisor can also remotely control robots using an Xbox controller when communication systems allow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  Figure 16: Example of the human supervisor interface showing the end of Team MARBLE’s final run  First-person view (FPV) allows the human supervisor to see the environment from the robot’s perspective in  semi-real-time, instead of just through the map representation and infrequent artifact images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' In addition to  increased situational awareness it allows the human supervisor to identify visual artifacts that may be missed  EilePanelsHelp  DARPA  Arifacts Beacons Reset  HO1  Fused Artifacts  HO2  ETO  Reset  31 fps  个stop  H03  Raw Robot Artifacts  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='14m  H03  Reset Left-click: Rotate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Middle-Click: Move X/Y, Right-click: Move Z, shift: More c 31 fps  DO2  Backpaicl  Eile Panels Help  Backpack  Reset Left-click: Rotate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Middle-Click: Move X/Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Right-click: Move Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' shift: More c 31 fps  Submissions  Type  Position  Notes  Response  Drill  21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='83, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='63,-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='73  +1 points  Backpack  27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='28  +1points  New Repot  Transform Transportcustom Arifact  End Mission  TIme  DS Time:1632420024.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='41ROS Elapsed:4940.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='81by the on-board artifact detection system, or identify them more rapidly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Finally, FPV helps the human  supervisor during manual teleoperation in the event it is needed to direct the robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  Flexibility of our udp mesh communication architecture made it possible to rapidly adapt to mission specific  constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' During the Final Event, we observed extra bandwidth in the communication system and decided  to add FPV to our Spot robots to enhance their exploration potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Compressed images from the Spot  forward facing cameras were transmitted as 1 Hz low-priority messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' On the base station computer, these  images were displayed live in RViz, and saved locally so the human supervisor can review that at any time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  The additional scoring potential of FPV is highlighted in Section 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  Several features of the human supervisor GUI were designed to help reduce operator workload.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' First, another  artifact fusion process runs on the Base Station computer, to aid the operator in tracking artifacts reported  by multiple vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' If reports of the same type are within 3m of prior reports, they are fused to the mean  position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Redundant artifact reports already been seen by another robot appear in a light gray color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' In  contrast, new reports flash with large white text to bring attention to the human supervisor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' When submitting  artifact reports, the human supervisor can select from individual or fused reports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' If an artifact is successfully  scored, the submission is locked out to prevent re-submitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' If it does not result in a score, the operator  can utilize additional map, trajectory, and FPV information to improve the estimated position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' New reports  can be submitted by shifting fused artifacts icons in the map or specifying a manual position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  10  Final Event Results  The SubT Final Event Prize Run on September 23, 2021 provided Team MARBLE an excellent opportunity  to evaluate the performance of our complete supervised autonomy solution, and we share our results in this  section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' First, an overview is provided in Section 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='1, which includes an outline of the mission objectives,  a description of the previously unknown course in Section, as well as a high-level summary of the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  Localization and mapping results are presented in Section 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Further analysis of the planner and resulting  exploration effort is described in Section 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Artifact detection results are thoroughly analyzed in Section  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The communication environment was friendlier than expected, and in Section 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='5, we discuss how we  capitalized on that opportunity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Mission management results are detailed in Section 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='6, including the five  instances where the human supervisor manual intervened, as well as the seven artifacts that were scored via  FPV imagery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Together, this section elucidates how our systems worked together to score 18 artifacts, while  also discussing the areas that limited even higher performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Data from Team MARBLE’s deployment  during the Final Event Prize Run is publicly available and discussed in Section 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='1  Overview  The mission objectives were to accurately report as many of the 40 artifacts in the course as possible during  the mission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' There are three hard constraints: the mission is 60 minutes long, there are a total of 40  attempts to report artifacts, and only one human, the human supervisor, is permitted to supervise the mission,  manipulate robotic agents, and submit artifact reports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  The final course was custom constructed as illustrated in Figure 17, which breaks the course out into distinct  tunnel, urban, and cave environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The course contains numerous hazards and challenges: rough terrain,  railroad tracks, slippery surfaces, ramps, stairs, large drop-offs, rocky cliffs, narrow hallways, low-to-the-  ground corridors, wide-open caverns, fog, standing water, dynamic obstacles, trap doors, and a degraded  communications environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' These challenges are discussed further in Section 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  Here, we provide a brief high-level summary of the artifacts scored and the extent of the environment explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  Team MARBLE scored 18 of the 40 artifacts and explored roughly half of the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' For reference,  the top-scoring team scored 23 artifacts, and the performance for all teams is listed in Section 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='9 of the  Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The location and class of all 40 artifacts can be visualized in the context of the course map shown  in Figure 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' This map also indicates which regions of the course that Team MARBLE explored as well  as the 18 scored artifacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The artifacts that Team MARBLE scored are also listed in Table 3, ordered  chronologically from mission start to mission end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Each of the 18 artifacts in Table 3 correspond by ID to the  scored artifacts in Figure 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Further analysis of artifact detection results are detailed in Section 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='Staging Area  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='Explored Cave  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='Unexplored Cave  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='Explored Tunnel  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='Unexplored Tunnel  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='Explored Urban  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
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+page_content='Figure 17: Course map of the 60-minute Final Event Prize Run designed by DARPA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' highlighting the 18 of 40 artifacts  scored by Team MARBLE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' along with the areas of the tunnel,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
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+page_content='  Score  Time [mm:ss]  Type  ID  Error [m]  1  01:08  Drill  L51  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
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+page_content='84  4  12:03  Survivor  L26  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
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+page_content='43  9  36:53  Fire Extinguisher  L38  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='82  10  37:08  Cube  L36  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='94  11  37:58  Backpack  L40  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='40  12  38:47  Rope  L67  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='87  13  47:53  Cube  L11  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='83  14  50:33  Cell Phone  L22  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='06  15  50:45  Cell Phone  L47  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='00  16  51:48  Cell Phone  L59  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='15  17  52:45  Gas  L24  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='55  18  56:33  Helmet  L58  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='74  Table 3:  List of all artifacts scored by Team MARBLE during the 60-minute Final Event Prize Run, along with  the corresponding mission time when artifacts were reported and scored, artifact type, unique DARPA-assigned ID  numbers, and Euclidean distance error between the reported and ground truth location of the artifact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='2  Localization & Mapping  A secondary objective of the 60-minute Final Event Prize Run was to rapidly map the environment and  transmit the real-time map back to DARPA every ten seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The map takes the form of a point cloud, or  a collection of three-dimensional points that represent occupied space in the environment, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' floors, walls,  ceilings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Figure 18 provides a comparison of the DARPA-generated ground truth map in black against the  final map submitted by Team MARBLE, split into inliers in green and outliers in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  Inliers  Outliers  Ground Truth  25  0  50  meters  Figure 18: Final point cloud map submitted by Team MARBLE to DARPA staff during the Final Event Prize Run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  DARPA has generously collected, processed, and shared this map data with participating teams (Schang  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Inliers are defined as points within 1m of ground truth map points, and outliers are defined  as points outside 1m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Map coverage is a metric representing the ratio of the environment explored, and is  defined as  map coverage = ground truth points within 1m of an inlier point  total ground truth points  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  (9)  Map error or deviation is a metric representing the ratio of the submitted map that is inaccurate relative to  the ground truth map, and is defined as  map deviation =  outlier points  total submitted points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  (10)  Figure 19 shows that map coverage steadily increases throughout the mission, with some periods of rapid  exploration, and by the end of the mission, nearly 50% of the environment has been mapped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Map error on  the other hand, increases modestly throughout the mission due to localization drift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' However, it increases  significantly due to a localization failure on D01, which is discussed further in Section 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' This failure  generated erroneous sections of map which are mistakenly appear as a long winding corridor in Figure 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  File  Panels  Help  Reset  21 fp0  10  20  30  40  50  60  Mission Time [minutes]  0  5  10  15  20  25  30  35  40  45  50  Map Statistic [%]  0  2  4  6  8  10  12  14  16  18  Cumulative Score  Map Coverage [%]  Map Error [%]  Cumulative Score  Figure 19: Map coverage, error, and cumulative score throughout Team MARBLE’s deployment during the DARPA  SubT Final Event Prize Run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  (a)  (b)  (c)  (d)  Figure 20: The final maps from (a) D01, (b) H01, (c) D02, and (d) H02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='3  Planning  During the entire 60-minute mission and across a diversity of environments, none of the robots were teleoperated  due to a planner failure or to improve the volumetric gain, which is seen as a successful demonstration of the  planning reliability and flexibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The human supervisor only intervened to complement autonomy with  human-level cognition and intelligence, as discussed further in Section 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  As an overview, Figure 21 presents three scenarios from the Final Event Prize Run, which demonstrate  negative obstacle avoidance, multi-agent coordination based on teammate position histories, and teleoperation  initiation by the human supervisor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Below, Section 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='1 highlights the exploration performance, Section  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='2 shares examples of agents avoiding treacherous terrain, and Section 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='3 demonstrates the planner  adapting to dynamic changes in the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Section 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='4 discusses system limitations that prevented  the agents from exploring the entire course.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  Figure 21: The figure shows three snapshots from the SubT Final Event Prize Round run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The blue lines in the  figure represents the locally sampled RRT∗ tree and the green lines represent the global graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' (a) The robot can be  seen to avoid sampling over the negative obstacles i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=', the edge of the subway platform, owing to the settling-based  collision-checks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' (b) The agent planned away from the position history of a teammate robot that was launched before  it, demonstrating effective multi-agent coordination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' (c) This snapshot shows an instance where teleoperation was  initialized on one of the robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The robot can be seen to plan a path that had sufficient volumetric gain leading the  robot toward a frontier in the urban area that had not been seen by any other robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' However, the human supervisor  decided to teleoperate the robot through the initial section of the tunnel area and then let it autonomously explore  the tunnel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='1  Exploration  The volume of unseen area explored by each agent across the mission is illustrated in Figure 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Unlike the  statistics of the global map submitted to DARPA in Figure 18, Figure 22 presents agent-specific exploration  information stored onboard each robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' D02 was the largest contributor to overall exploration, partly because  it was launched first and had the most time to explore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' D01 played a complementary role by exploring most  of the tunnel environment, which remained unexplored by D02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Because the subset of the environment safely  traversable by the wheel robots was relatively more constrained, it naturally led to less exploration from the  Huskies across the mission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  Besides showing the volume explored by each of the robots, Figure 22 also shows time periods during which  the default planner exploration behavior was paused for higher-level mission management and teleoperation  commands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' These interjections by the autonomous mission management system were primarily triggered  when agents approached each other, and resulted in lower-priority agents pausing and higher-priority agents  resuming their task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' During these encounters, agents appear as fast-approaching dynamic objects, and this  simple procedure was employed rather than incorporating reactive obstacle avoidance into the planner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='2  Treacherous Terrain  The path planning solution successfully kept each agent safe from collision throughout the entire mission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The  Spot robots, which explored more challenging features in the environment, fully avoided negative obstacles,  such as shear drop-offs and rocky slopes, and traversed up and down stairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  (a)  (b)  (c)  Teammate  Teammate Position  Position History  Histories  Subway  Station  PlannedPath  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='00000  Robot  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='00000  Position  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='000000  DO  Planned Path  Negative  Obstacle  Robot Position  Robot Position  Platform0  10  20  30  40  50  60  Time Elapsed (min)  0  500  1000  1500  2000  2500  Explored Volume (  m 3)  D01  D02  H01  H02  Go-Home  Planner-Off  Figure 22: Volumetric gain explored by each robot across the 60-minute Final Event Prize Run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Planner-Off, denoted  by blue lines, represents instances when the planner was paused to allow the autonomous mission-management system  to take over when robots are in close proximity, as well as the several interventions when the human supervisor  manually teleoperated robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Go-Home, denoted by black lines, represents instances when the planner began returning  home to reconnect to the network and report new artifact reports and map data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  The Spot robots successfully traversed up and down the small set of stairs leading up to the subway platform,  as shown in Figure 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' However, when stairs where first encountered from above, agents did not plan down  them due to the limited ±16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='5◦ vertical field of view of onboard Ouster OS-1-64 lidar sensors, as shown in  Figure 24a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' In addition, the Spots can only safely walk down stairs backwards, and therefore additional logic  would be be required to autonomously traverse those stairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  Both Spot robots thoroughly explored the subway platform, approaching the edge, but never stepping and  falling over, as demonstrated by D02 in Figure 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Additionally, the same Spot robot, D02, explored the  entire cavern autonomously without entering treacherous terrain, as shown in Figure 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  (a)  (b)  Figure 23: Instance of (a) D01 planning up stairs with the green edges of the graph, pink planned path, and associated  semantic map with blue voxels representing stairs, along with (b) FPV imagery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='3  Dynamic Environment  The planner has the ability to adapt to dynamic environments, such as closing or opening of doors, falling  rubble, as well as other situations also lead to dynamic changes in the map, including other nearby mobile  agents, and localization and mapping error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  During the Final Event Prize Run, D01 traveled through a side branch of the urban environment, triggering  a trap door.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Figure 26 shows the planner adapting to the dynamic environment by re-assigning previously  +Entry(a)  (b)  Figure 24: Instance of (a) D02 thoroughly exploring the subway platform without planning over the edge, with green  edges of the graph, pink planned path, and associated map, along with (b) FPV imagery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  (a)  (b)  Figure 25: Instance of (a) D02 autonomously exploring the entire cavern, traversing the safer surfaces and avoiding  treacherous areas, such as the one shown in (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  traversable edges as untraversable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  In addition, there were several instances of temporary localization and mapping error, which caused erroneous  new map data to change previously traversable edges of the planning graph to untraversable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' In each case,  the planner adapted to the new scenario, and continuously operated throughout the localization drift and  loop closure correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' An example of this is included in Section 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='10 of the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  When agents pass each other, they appear as fast-moving dynamic obstacles, and cannot re-plan around one  another fast enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Therefore, an agent-based prioritization scheme prevents both collision and deadlock, by  enforcing one agent to wait while the other passes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Examples can be found in Section 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='10 of the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='4  Limitations  However, several limitations did prevent agents from exploring roughly half of the course.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' In total, there  were three types of bottlenecks: constrained corridors, slippery surfaces, and downward sets of stairs, each  exposing unique limitations within the autonomy system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' None of these are limitations of the planner itself,  but rather limitations of mapping, mobility, and perception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  The planner did not plan through all constrained spaces because the selected planning parameters for agent  width and height did not allow the graph to propagate through especially short and narrow spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' These  parameters were intentionally chosen to be conservative to prevent the agent from moving along an unsafe  trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Utilizing a higher-resolution local map and planner could result in a more agile robot that could  safely traverse those spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Additionally, autonomously transitioning into a crouching gait could improve the  Spots ability to traverse spaces with low ceilings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Examples are included in Section 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='11 of the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  Subr(a)  (b)  (c)  (d)  Figure 26: Early in the mission, D01 walked by the corridor with the trap door, as shown by (b) the left camera (1:47).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  The agent later to returns to the corridor, (a) walks under the trap door (10:34), and soon after sees it has closed, as  shown by (d) the right camera (11:01).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' After seeing the trap door close, (c) the updated map and graph show (red)  edges as untraversable (11:11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Had the agent moved closer to the trap door, and fully been within mapping range, all  edges would have updated as untraversable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  The second limitation is slippery surfaces, and led to D01 slipping and falling in the cave section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Some of  these rocky surfaces were intentionally designed to be slippery, and plenty of humans walking through the  course after the event also slipped and fell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' After D01 fell, it also experienced a localization failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Methods  to recovery the system from an event such as this one would involve implementing a fall detection algorithm,  as well as autonomous self-righting and localization reset logic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='4  Artifact Detection  In this section, we present performance results of the artifact detection and reporting system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' During the  60-minute Final Event Prize Run, Team MARBLE scored a total of 18 artifacts out of the 40 artifacts in  the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Figure 27 presents a flow diagram that summarizes how our team scored 18 artifacts and  the limitations that resulted in the remaining 22 from being scored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Of all 40 artifacts, our agents explored  enough of the environment that they were in the vicinity of 25 artifacts, leaving 15 unexplored due to mobility  challenges discussed in Section 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Team MARBLE reported 19 of the 25 artifacts that were explored,  and successfully scored 18 of those 19 reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' A map of the area explored and scored artifacts is shown  in Figure 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Details of these 25 explored artifacts, of which 18 were scored, one was missed, and six were  unreported, are shared in Section 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='13 of the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  Of the 18 artifacts that Team MARBLE successfully scored, 11 were scored by autonomous robot reports, two  were scored by the human supervisor modifying the position of autonomous reports, and five were scored by  the human supervisor via robot FPV imagery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' One artifact was reported but did not score due to localization  error in excess of 5m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' There were six artifacts that agents saw, but did not report due to errors in the  autonomous artifact detection system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The human supervisor received information regarding three of these  six artifacts but missed them due to high workload demands during the mission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The other three artifacts  were located in areas that prevented agents from communicating back to the human supervisor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='1  Visual Detection  The focus of this section is to quantify the performance of the visual artifact detection system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Of the 18  artifacts that Team MARBLE scored, 11 of them were visual artifacts, as shown by Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' First, we focus  on the six artifacts (L51, L53, L26, L34, L40, L67) that were successfully reported by the agents’ autonomous  artifact detection systems in the course.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' All six artifacts were accurately localized to within 5m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' In fact, the  largest error for a visual artifact was 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='87m (L67).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' This eliminated the need for the human supervisor to  spend time trying to correctly localize artifacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  The autonomous artifact detection system filters raw frame-to-frame detections onboard the agent, with the  Figure 27: Flow diagram illustrating how Team MARBLE scored 18 of the 40 artifacts in the course, and the  limitations preventing the remaining 22 artifacts from being scored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  aim to reduce the number of false positive and redundant artifact reports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Because the human supervisor has  limited bandwidth, unnecessary distractions detract from the completing other mission-related tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' In the  process of scoring these six visual artifacts, the human supervisor had to process 21 artifact reports from the  automated artifact detection systems onboard agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Of these, 11 were true positives, six were approved by  the human supervisor and successfully reported, and five were ignored because they were redundant reports  that were previously scored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The other 10 reports were also ignored by the human supervisor because they  were false positives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  In total, there were only four false reports that the human supervisor submitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' One was cause by human  error, the other three were caused by erroneous CO2 reports, as detailed in Section 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='16 of the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  Agents in the course failed to autonomously report the other five artifacts (L55, L32, L31, L38, L58), but did  transmit FPV imagery back to the human supervisor, who manually reported and scored them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The fact  that the autonomous artifact detection system did not detect five of the 11 visual artifacts it saw, indicates  that certain reliability limitations exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Team MARBLE acknowledged this limitation and relied on the the  human supervisor and FPV system to fill in that void, which is further in Section 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='2  Non-Visual Detection  Team MARBLE scored seven non-visual artifacts, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' cell phone, cube, and gas, but as shown in Table  5, required submitting more reports due to difficulty around accurately localizing the source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The main  limitation is that the detection scheme relies on threshold-based logic for RF and CO2 levels, and when  triggered, simply reports the current location of the agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The thresholds were intentionally set low to  increase the probability of detection when agents pass by the vicinity of these non-visual artifacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Of the  seven non-visual artifacts scored, five of them (L08, L36, L22, L47, L24) were scored via the autonomous  robot reports, with an average error of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='27m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The remaining two artifacts (L11, L59) were scored by the  human supervisor manually adjusting the reported artifact locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='5  Communications  The performance of the communication system was evaluated in the final run using both qualitative and  quantitative measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Subjectively, the human operator was able to employ live FPV video from the robots,  a capability that directly contributed to team’s third place finish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The robots were in communication with  Robot Score:11  ReportedArtifacts:19  ScoredArtifacts:18  ExploredArtifacts:25  Hybrid Score:2  AllArtifacts:40  Human Score:5  NotScoredArtifacts:1  Localization Error:1  Unreported Artifacts:6  Robot Detection Error:6  HumanMissed:3  NoCommunications:3  UnexploredArtifacts:15  Limited Mobility: 15Figure 28: Locations of ground truth artifacts in white,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' reports that scored in green,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' and reports that did not score in  red,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' overlaid on ground truth map of the course.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  Artifact Type  Scored  Not Scored  Unreported  Unexplored  Total  Survivor  2  0  0  1  3  Cell Phone  3  0  0  1  4  Backpack  2  0  0  3  5  Drill  2  0  0  2  4  Fire Extinguisher  2  0  1  1  4  Gas  2  0  1  0  3  Vent  0  0  3  1  4  Helmet  1  0  1  3  5  Rope  2  0  0  3  5  Cube  2  1  0  0  3  Total  18  1  6  15  40  Table 4:  Artifact statistics for Team MARBLE during the Final Event Prize Run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' A total of 18 artifacts were scored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  Only one artifact was reported but not scored, which was due to localization error greater than 5m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Agents were in  the vicinity of six artifacts, but they went unreported due to autonomous artifact detection failure and in some cases,  also missed by the human supervisor due to excess workload.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The remaining 15 unexplored artifacts were never seen  by agents because they were located in parts of the course that were never reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  the base station over the majority of the explored regions of the course, as shown by the blue-green hues in  Figure 29a, allowing the human supervisor to monitor and intervene as needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Overall, 125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='2 MB of data  was transferred through the communications network, including all map segments, telemetry, artifact reports,  and other data products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Of that, FPV video comprised 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='6 MB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The latency of the mesh networking  solution was evaluated using inter-message arrival times of a heartbeat message sent from a long-ranging  robot to the base station.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' This mission management message, transmitted regularly from the robot, was  Attempt Type  Scored  Missed  False  Total  Survivor  2  0  0  2  Cell Phone  3  5  0  8  Backpack  2  0  1  3  Drill  2  0  0  2  Fire Extinguisher  2  0  1  3  Gas  2  1  3  6  Vent  0  0  0  0  Helmet  1  0  0  1  Rope  2  0  0  2  Cube  2  5  0  7  Total  18  11  5  34  Table 5: Artifact report statistics for Team MARBLE during the Final Event Prize Run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' A total of 34 reports, or  attempts, were made throughout the mission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' A total of 18 attempts resulted in scores, 11 attempts were misses and  did not result in a score due to localization error greater than 5m, and five attempts were false attempts in that they  were false positives and no artifact of that class was in the vicinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  part of our protocol scheme and is analyzed as a message of convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' A distribution of the inter-message  arrival times of these heartbeat messages from D01 to the base station is shown in Figure 29b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  Since these messages originate from the robot at 1 Hz, an ideal system would observe all inter-message arrival  times to be one second in duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' As our mission management system on the robot does not run with hard  realtime constraints, the 1 Hz publish rate is an estimate that includes noise due to process load, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' On the  base station, there was an approximately normal (N(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='00, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='04)) distribution of arrival times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Since messages  may experience delays, the immediately following message may exhibit an inter-message time of less than one  second, leading to the symmetry apparent in Figure 29b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Our key observation of this plot is that the bulk  of messages arrive within five percent of their expected times across a distance of hundreds of meters and  multiple mesh hops, validating the performance of the entire mesh networking solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  (a)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='85  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='95  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='05  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='15  Time (s)  (b)  Figure 29: Performance results of the communication systems, including (a) the map of the Team MARBLE’s  deployment during the Final Event Prize Run, overlaid with locations of robot connection (C) to the network in  blue-green, and locations of robot disconnection (D) from the network in red-magenta, as well as (b) the distribution  of inter-message arrival times for D01 with a nominal publishing rate of 1 Hz, overlaid with N(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='00, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='04).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The  highlighted red region represents the first σ value which contains 68% of the message times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  C  D  D01  D02  H01  H0210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='6  Mission Management  Overall, the mission management system was able to keep the robots on task with minimal human supervisor  intervention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' However, when needed, the interventions were crucial towards both the exploration capabilities  of the system and the final event performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Figure 30 presents a detailed timeline of the four agents in  the field, as well as the human supervisor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The four robot launches and five robot interventions were the only  times when the human supervisor used teleoperation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' There were only two other types of instructions agents  received from the human supervisor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' One was commanding H02 to drop two communication beacons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The  other was commanding D01 to return home three times, each occurring while the human supervisor was also  teloperating D01 through the fog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  Figure 30: Mission management timeline for all robots and human supervisor during the 60-minute Final Event Prize  Run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' When not manually teleoperating a robot, the human supervisor was monitoring the mission, which includes  watching live FPV streams from D01 and D02, reviewing incoming artifact reports from agents, reporting artifacts to  DARPA, and in the last five minutes of the mission, checking archived FPV images from the Spots for previously  missed artifacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='1  Robot Launches  The human supervisor was under immense pressure to optimally balance many competing tasks during the  60-minute Final Event Prize Run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' With such limited time, the primary objective at the beginning of the  mission is to launch robots into the environment as fast as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Through extensive practice and full-scale  comprehensive field deployments, discussed further in Section 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='2, Team MARBLE launched all four robots,  with a mean launch time of 41 seconds, as shown in Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  Two agents experienced failures late in the mission, reducing overall fleet utilization rate from 92% to 73%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  H02 experienced a hardware failure (36:48), and post-event inspection revealed better vibration isolation  of the computing system would reduce the likelihood of such a failure in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' D01 experienced a  mobility failure (39:46), in which the Spot slipped on a slick rock and fell over.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The agent then experienced  a localization instability due to the large induced velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' To recover from such an incident in the future,  Team MARBLE could implement an autonomous self-righting maneuver and localization reset logic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  Despite the fact that some artifacts were not detected and reported by the autonomous board artifact  detection system, the human supervisor filled in the void.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The human supervisor reported and scored five  artifacts that were seen via robot FPV imagery, but not autonomously detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Of these,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' the human  supervisor saw four (L55,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' L32,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' L31,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' L38) from live FPV streams,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' while one (L58) was found while reviewing  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='aunch  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='narrow  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='D02  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='stairs  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='cave  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='Launch  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='D01  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='fog  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='mobility failure  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='Launch  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='H02  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='entrance  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='hardware failure  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='Launch  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='H01  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='entrance  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='HS  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='15  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='20  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='25  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='30  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='35  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='40  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='45  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='50  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='55  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='60  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='Mission Time  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='[minutes]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='autonomous  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='manual teleoperation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='not in operation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='mission monitoring  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='operationLaunch  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='Agent  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='Duration  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='Teleoperation Window  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='Course Entry  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='[s]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='[mm:ss - mm:ss]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='[mm:ss]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='RL1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='D02  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='75  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='00:37 - 00:38  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='00:02  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='RL2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='D01  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='55  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='01:29 - 02:24  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='01:44  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='RL3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='H02  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='35  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='07:15 - 07:50  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='07:25  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='RL4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='H01  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='50  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='10:57 - 11:32  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='11:08  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='Mean  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='41  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='Table 6: List of the four robot launch (RL) sequences executed by the human supervisor during the 60-minute Final  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='Event Prize Run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Extensive process streamlining and repeated practice deployments resulted in quick and repeatable  launch sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The course entry column represents the mission time at which the agent crossed into the course.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  archived FPV images near the end of the mission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' More details of these artifact reports are presented in  Section 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='17 of the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='2  Robot Interventions  In total, the human supervisor intervened in the autonomous fleet five times during the 60-minute mission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  Table 7 presents the duration of each intervention, with a mean length of 217 seconds, as well as the reason  for intervening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Each of these interventions was in the form of manual teleoperation, in which FPV streams  were available at the base station, and velocity commands from the human supervisor was transmitted to the  remote agent in the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Of the five manual teleoperation interventions, only two had significant impact on  the mission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The first (RI1) was navigating D02 through the narrow cave corridor early in the mission (3:02  7:02), which led the agent to the small cavern and two artifact scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The second (RI2) was navigating  D01 through fog in the tunnel section midway through the mission (21:54 - 31:52), leading to six artifact  scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Figure 31 provides a side-by-side comparison of the full-resolution FPV imagery processed onboard  by the autonomous artifact detection system and the compressed FPV imagery transmitted to the human  supervisor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  Two interventions (RI3, RI5) commanding agents back into the course, was out of a shear abundance of  caution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The planning algorithm is configured so that the staging area is treated as explored, so agents  should not attempt to explore it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' One intervention failed (RI4), in which the human supervisor attempted to  teleoperate D02 down the stairs by the subway platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' This attempt failed because communication to and  from the robot was intermittent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  (a)  (b)  Figure 31: Imagery during robot intervention 2 (RI2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' when human supervisor was pushing D01 through the foggy  area in the tunnel environment, eventually leading to six additional points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Shown is a comparison of (a) full-resolution  FPV imagery onboard the agent and (b) compressed low-resolution FPV imagery transmitted to the human supervisor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  The main takeaways from these results is that our agents are highly autonomous, leaving the human supervisor  to focus on mission monitoring and targeting strategic, high-value intervention opportunities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The mission  management system enabled convenient transition between autonomy and manual operation, while the  communication system enabled visibility and control over the agents in the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  Intervention  Agent  Duration  Window  Goal  Success  Points  [s]  [mm:ss]  RI1  D02  240  03:02 - 07:02  Enter narrow cave corridor  +  6  RI2  D01  598  21:54 - 31:52  Enter foggy tunnel area  +  2  RI3  H02  24  35:04 - 35:28  Avoid course exit  +  0  RI4  D02  200  39:24 - 42:44  Walk down stairs  –  0  RI5  H01  21  49:40 - 50:01  Avoid course exit  +  0  Mean  217  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='6  Table 7: List of the five robot interventions (RI) executed by the human supervisor during the 60-minute Final Event  Prize Run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Our concept of operations relies on autonomous multi-agent exploration, and does not necessitate manual  waypoints or teleoperation from the human supervisor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Therefore, agents were completely autonomous, except for  the human supervisor input during these five instances of teleoperation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The interventions goals varied, but were  all specific scenarios where human intervention would augment autonomous agent capabilities in a mission-relevant  manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='7  Open-Source Data  During the final run at the final event, our team collected a significant amount of data related to autonomous  subterranean exploration in the form of ROS “rosbags”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Our datasets are split up by agent, and each set  contains a rosbag of the system inputs, mostly consisting of raw sensor data, and another rosbag of the  outputs used for visualization and performance monitoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The complete collection of data recorded at the  final event can be found at https://arpg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='io/marble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  11  Lessons Learned  Underground exploration of previously unknown environments, especially in a time and resource-constrained  search-and-rescue context, requires a highly adaptable human-robot team.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The lessons learned presented in  the following sections enhance our proposed system’s flexibility across mobility, communications, human-robot  teaming, and multi-agent coordination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='1  Platform Mobility  Systems with heterogeneous platforms allow for specialization by each platform for both specific environments,  and roles which benefit the entire mission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Specifically, in Team MARBLE’s case, the addition of Spot  platforms into an exploration role enabled rapid multi-story expeditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  This capability was further  augmented with the utility of higher-payload, wheeled Huskies, carrying communication beacons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' When  deployed, these beacons allowed the Spot platforms to report artifacts without the need to traverse “home,”  leading to much more efficient exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' No single, platform is as performant as the combination of  platforms operating in different roles based on each one of their strengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='2  Testing and Validation  Significant testing and validation was conducted for every element of the Team MARBLE system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' These  tests taken over a variety of environments, shown in Figure 32, reduced mission-to-mission variability, and  increased system-wide adaptability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Through these tests, Team MARBLE locked in well tested solutions with  minimal unexplained errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' This reduced changes on the system as we approached the final competition,  given that new solutions had to be verified against similarly stringent tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' To our knowledge our team made  the fewest hardware and software based adjustments from the preliminary runs to the final run to achieve our  score, instead relying on our testing having eliminated all significant errors outside of a so-called ”Poisson  distributed fatal error.” These types of errors were mission ending to any individual robot but difficult to  predict or adapt to in a meaningful sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' These can be seen in our final competition run where D01 fell over  on rough steep terrain, and where H02 suffered a hardware error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  A secondary goal of these full-scale deployments was to reduce human-based variability in performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  They helped both the human supervisor and pit crew prepare for the stressors of operating and responding  quickly to complex interactions between robots, the environment, and potential failure modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' At no point  was a single test considered sufficient for validating a solution as sufficient;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' instead all solutions released  demonstrated repeatability across the same and varied environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' For reference, all full scale deployments  are tabulated in Section 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='18 of the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  (a)  (b)  (c)  Figure 32: Team MARBLE conducted comprehensive field deployments at various sites including (a) the Edgar  Experimental Mine operated by Colorado School of Mines, located in Idaho Springs, CO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' which tested terrain, distance  travelled, and communications;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' (b) the Engineering Center complex also located on University of Colorado Boulder  Main Campus which tested multi-story navigation and repeated features from urban environments;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' and (c) the Folsom  Parking Garage located on University of Colorado Boulder Main Campus in Boulder, CO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' which tested planning in  open spaces and vertical localization across multiple stories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='3  System Adaptability  The challenges posed by operating a system in an unknown environment necessitate a high level of system  adaptability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Predicting every capability that a system will need for a given mission, such as search and  rescue, is impractical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Instead, having a highly flexible system capable of adapting to unknown situations  is crucial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Having already addressed the mobility considerations in Section 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='1, we also found software  adaptability key to our success.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' For instance, the flexible communication network enabled the ability to  pass on FPV to the human supervisor with a simple configuration change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' While our system was designed  with autonomy in mind and the capability was not previously planned for, the adaptation proved invaluable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  Minimal human supervision directly impacted the final score and exploration capabilities of the system as a  whole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  The notion of adaptability extends to other software architecture as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The artifact detection system was  adjusted during the final competition to include SSID information for Bluetooth artifacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' These artifacts  could then be more accurately merged between agents and by the human supervisor, despite inaccurate  positioning from wireless signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  In some cases, adaptability is explicitly accounted for in our design, most notably in the mission management  system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' BOBCAT has many parameters which allowed the human supervisor to adjust exploration activity,  including how long a robot can explore before reporting detected artifacts, whether a robot should find  multiple artifacts before reporting, and how the system should adjust to time constraints in the mission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  STARTHERE11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='4  Autonomy and Human Robot Interaction  Team MARBLE emphasized autonomous system design as our primary goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Since only one person, the  human supervisor, was permitted to interact with agents while they were deployed, having robots controlling  their own paths and decision making was key to reducing the cognitive load required to manage mission  objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Despite the focus on autonomy, the human supervisor is still necessary to maximize system  performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' As Team MARBLE approached the final competition, we found targeted areas where direct  human supervisor control improved results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' This influenced decisions regarding the number of deployed  platforms, how artifacts were passed from agents to the human supervisor, and how we recovered from  anomalous robot behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  To maximize the human supervisor’s ability to track the data streams present across the fleet, only four  robots were deployed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' While a larger fleet might have enabled a rapid exploration of the environment, it  could reduce the human supervisor’s ability to meaningfully address problems arising from any specific robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  The artifact detection system was designed to filter against false positives before passing information to the  human supervisor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Artifacts had to consistently detected in multiple frames, and be a sufficient distance from  existing artifact estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Where possible, artifacts were returned with additional information including  SSID for Bluetooth artifacts and images for visual artifacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' This data allowed the human supervisor to sort  remaining false positives quickly without being distracting from the core mission objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  Specifically, the human supervisor spent most of their operational load solving problems the robots were  incapable of correcting through their autonomy stack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' For example, during the Final Event Prize Run, the  human supervisor used the first person vision to navigate through dense fog, which the robot was incapable  of planning through autonomously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' After this intervention, the human supervisor allowed the robot to return  to autonomous operations, where it explored several new areas, and reported a total of six new artifacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  The human supervisor was able to opportunistically intervene like this, only because the other agents were  operating autonomously without supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='5  Retrospective  It’s important, after a three-year effort of this size and scope, to examine some of the bigger picture questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  What would we do differently next time?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' What did we wish we knew at the start, that we know now?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The  answer to the first two questions is that we would have spent more time at the onset to scope out short-term  and long-term development goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' We excelled at the Tunnel Event, placing fourth amongst a large group  of competitors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' This was mostly due to meeting well-scoped short-term goals within the one year cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  However, during the six-month development period for the Urban Event, we focused on long-term goals that  we ultimately only partially validated, leading to a disappointing performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' In hindsight, we should have  focused on making our already capable platforms more capable, rather than spreading our resources thin  across many thrusts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' However, what our team excelled at after this experience, was pivoting and adopting two  new strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' First, we leaned into student-led project management, which led to greater team coordination,  a key component of rapid and effective system development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Secondly, we focused on long-term development  goals, and given a year and a half, had the time to adequately develop, test, and validate each subsystem,  each fully autonomous robot, as well as our entire fleet in numerous search and rescue missions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Together,  these two powerful changes allowed a small, lean team, produce a highly functional autonomy solution that  could perform under pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  12  Conclusions  In this paper, we showcase our flexible autonomy solution for exploring unknown subterranean environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  The highly performant autonomy solution directly lead to a third place finish at the DARPA SubT Final  Event which was focused on search and rescue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Moreover, the specific innovations presented in graph planning,  flexible communications, and mission management are directly applicable to other multi-robot teaming  applications under limited human supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  Specifically, the combination of legged and wheeled robots allowed for heterogeneous teaming enabling both  rapid exploration and a robust communication network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The deployed mesh network which is described  in Section 8 enables flexible configuration and prioritization of data sent both to other robots, and a  human supervisor for review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' A powerful graph planning framework, as described in Section 7, paired with  semantically encoded Octomaps enabled safe, rapid exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The final system was able to explore a  variety of underground environments, including gold mines and subway stations, with minimal human input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  Human input was reserved for specific situations where higher-level reasoning had the potential to improve  the mission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The flexible mission management system described in Section 9 enables safe transitions between  human input and the underlying autonomy system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' One of the most important lessons learned from the  developed system is that focusing on autonomy is core for human robot teaming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Robots need to be able to  make decisions on their own, enabling the limited human resource to only act in critical situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  Acknowledgments  This work was supported through the DARPA Subterranean Challenge, cooperative agreement number  HR0011-18-2-0043.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' A special thanks to Andrew Beathard, Nikolaas Bender, Cesar Galan, Nicole Gunderson,  Davis Landry, Greg Lund, Cole Radetich, Ben Rautio, Zoe Turin for assisting with the design, testing, and  deployments of our platforms and algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Thank you to the Colorado School of Mines and the Edgar  Experimental Mine for allowing us to conduct mock deployments in the mine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Special thanks also to Simon  Wunderlich and his team at Meshmerize GmBH for their mesh networking support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Finally, thank you to all  of the DARPA staff who have planned and executed absolutely incredible Subterranean Challenge system  track circuit events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
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+page_content=' Pound : A multi-master ROS node for reducing delay  and jitter in wireless multi-robot networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Robotics and Autonomous Systems, 111:73–87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' QC 20181210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  Thrun, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=', Burgard, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=', and Fox, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Probabilistic Robotics (Intelligent Robotics and Autonomous  Agents).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The MIT Press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  Tranzatto, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=', Mascarich, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=', Bernreiter, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=', Godinho, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=', Camurri, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=', Khattak, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=', Dang, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=', Reijgwart,  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=', Loeje, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=', Wisth, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=', Zimmermann, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=', Nguyen, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=', Fehr, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=', Solanka, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=', Buchanan, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=', Bjelonic,  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=', Khedekar, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=', Valceschini, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
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+page_content=', Dharmadhikari, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=', Homberger, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=', De Petris, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=',  Wellhausen, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
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+page_content=', Hirsch, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
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+page_content=', Nieto, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=', Smith, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=', Hutter, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=', Siegwart, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=', Mueller, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=',  Fallon, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
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+page_content=' Cerberus: Autonomous legged and aerial robotic exploration in the  tunnel and urban circuits of the darpa subterranean challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
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+page_content=' Cerberus in the darpa subterranean  challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Science Robotics, 7(66):eabp9742.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
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+page_content=' AprilTag 2: Efficient and robust fiducial detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
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+page_content=' IEEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
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+page_content=' Navigation  planning for legged robots in challenging terrain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' In 2016 IEEE/RSJ International Conference on  Intelligent Robots and Systems (IROS), pages 1184–1189.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' IEEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
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+page_content='  Robust stairway-detection and localization method for mobile robots using a graph-based model and  competing initializations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The International Journal of Robotics Research, 37(12):1463–1483.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  Williams, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
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+page_content=' Online 3d frontier-based ugv and uav exploration using direct point cloud visibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' In 2020  IEEE International Conference on Multisensor Fusion and Integration for Intelligent Systems (MFI),  pages 263–270.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  Wisth, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
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+page_content=' In Proceedings 1997 IEEE  International Symposium on Computational Intelligence in Robotics and Automation CIRA’97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
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+page_content=' (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' A new terrain mapping method for mobile robots obstacle negotiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' In  Unmanned ground vehicle technology V, volume 5083, pages 52–62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' International Society for Optics and  Photonics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  Zhan, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=', Garg, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=', Saroj Weerasekera, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=', Li, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=', Agarwal, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=', and Reid, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Unsupervised learning  of monocular depth estimation and visual odometry with deep feature reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' In The IEEE  Conference on Computer Vision and Pattern Recognition (CVPR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  Zhou, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' and Tuzel, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Voxelnet: End-to-end learning for point cloud based 3d object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' In  Proceedings of the IEEE conference on computer vision and pattern recognition, pages 4490–4499.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  Zlot, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=', Stentz, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=', Dias, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=', and Thayer, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Multi-robot exploration controlled by a market  economy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' In Proceedings of the IEEE International Conference on Robotics and Automation, volume 3,  pages 3016–3023, Washington, DC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  Zou, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=', Shi, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=', Guo, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=', and Ye, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Object detection in 20 years: A survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' arXiv preprint  arXiv:1905.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='05055.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  13  Appendix  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='1  Communication Beacon Design  Each beacon consists of two 3D printed nylon-carbon fiber infused internal brackets serving as structural  support components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Figure 33a shows the top of each beacon that contains a charging port, power button  and LED power status indicator between the antennas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The two stabilizers seen towards the front provide  longitudinal stability for the beacon once deployed on the ground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' These complement the steel counterweights  on the rear of the beacon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The deployment mechanism shown in Figure 33b uses solenoids to hold the beacons  in place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' When the solenoid is released, the beacon falls at a consistent rate using a constant force spring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  (a)  (b)  Figure 33: On the left (a) a perspective view of the beacon design as well as (b) a side view of the beacon attached to  the deployment mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='2  Platform Compute Design  When designing our system, we sought to balance the ease of development with a single, monolithic compute  unit versus potential integration challenges of a distributed system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  Early on, we decided that, given  uncertain compute requirements, we should attempt to pack as much compute as possible into the Husky  platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' This decision had a wide array of collateral consequences, including power system requirements,  cooling requirements, and mechanical considerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Further, the decision to not utilize an industrial-style  motherboard, but rather a consumer gaming motherboard, created integration difficulties that might have  otherwise been avoided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' For example, the power requirements of our Ryzen Threadripper-based system were  roughly 425W at peak consumption of both CPU and GPUs, at the limit of potential commercially available  DC/DC ATX power supplies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Using discrete GPUs mounted in PCI Express slots presented a mechanical  challenge, particularly for shock and vibration mounting, which was discovered as a failure mode late in the  design process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' In contrast, the distributed architecture developed for our Spot platforms utilized significantly  less power and space while delivering similar performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Future system designs are more likely to follow a  distributed approach, rather than a monolithic approach, to ease mechanical and electrical integration efforts  at the expense of only minimal added software effort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  At a deeper level of inter-robot module communications, we underestimated the challenges of differing  ground potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' After shorting out several serial links between components and having unreliable USB  communication, we realized that several ground loops were responsible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' By adding serial optocouplers and  Solenoid Catch  Antennas  Spring  Catch  PowerButton  Counterweight  StabilizerSolenoid Catch  ConstantForce  Spring  Solenoid  C-Channel  Stabilizer  Counterweightconverting to Ethernet-based platform control, these ground loops were eliminated, resulting in highly reliable  platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Future development would rely exclusively on differential signalling such as controller-area network  or Ethernet for inter-module communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='3  Sensor Synchronization Design  To effectively share sensor data between robots, sensor and system timing has to be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Our lidar  solution could utilize IEEE 1588v2 timing (also known as the Precision Time Protocol v2 or PTP), but  our wireless mesh networking solution could not support IEEE-1588v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Therefore, we implement Network  Time Protocol (NTP) between robots and PTP within the same robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' On startup, each robot attempts to  synchronize with the Base Station using NTP over the mesh network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' This synchronization step is critical for  multi-robot operations and coordination to provide a consistent time basis across all nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' In testing, the  relative time drift (a few milliseconds) over the course of a run (1 hour) was not significant enough to cause  problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' If the Base Station is unreachable (say for single-robot testing), the robot falls back to its own  battery-backed realtime clock as a time source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' In either case, after attempted synchronization, no further  attempts are made to match times with any other robot or the Base Station.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' In testing, we observed that  clock slews resulting from attempted time synchronizations as robots entered and left communications range  had a negative impact on localization performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  In the Threadripper monolithic architecture used on the Husky platforms, PTP on the secondary Ethernet  interface functioned perfectly, allowing the Threadripper to become the grandmaster and the lidar to follow  along, as shown in Figure 34a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' However, in the distributed Xavier+NUC architecture (shown in Figure 34b),  designing an effective PTP interface encountered significant challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  Threadripper  Base Station  Lidar  PTP Grandmaster  NTP  PTP  (a)  NUC  Base Station  Xavier AGX  Lidar  PTP Grandmaster  NTP  PTP  (b)  Figure 34: Block diagrams showing Precision Time Protocol (PTP) distribution between system components on the  Threadripper (a) and Xavier+NUC (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  Fundamentally, PTP requires hardware support in order to function by performing sensitive timing operations  as close to transport medium as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The network hardware options on the Spot included a Realtek  r8125, an Nvidia platform SoC module, and a quad-port Intel i210 card.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Realtek r8125 support for PTP was  not functional, as verified by the phc ctl utility;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' this relatively new chipset relies on an out-of-tree kernel  driver for Linux at the time of our development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The NVidia platform module appeared to support PTP  when interrogated by phc ctl, but on further investigation through network traffic inspection, the platform  module was not inserting the correct information into outbound Ethernet traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Our final solution is based  on using a spare port on the Intel i210 card, which had robust, verifiable PTP support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' As the NUC lacks a  port with viable PTP support, we fall back on NTP as a synchronization method, relying on the Xavier as  the robot’s grandmaster time source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  As an aid to the community, we offer the following verification steps to assist in debugging PTP issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' First,  verify that there is hardware support via ethtool -T <iface> to verify kernel-level hardware PTP support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  Second, use phc ctl <iface> cmp to verify that the Ethernet hardware clock is synchronized to the Linux  system clock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Finally, ensure that the hardware timestamps encoded in the PTP network traffic match  the local system by capturing network traffic from a PTP-enabled grandmaster Ethernet interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' These  steps can verify that the PTP software stack is broadcasting the system time via the Ethernet hardware to  downstream consumers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='4  Large-Scale Localization Validation  Validation testing of LIO-SAM onboard Spot and Husky platforms was imperative for ensuring sufficient  accuracy, speed, and stability for long-duration and large-scale missions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Some examples include an outdoor  test at CU South Campus, as shown in Figure 35, and as well as test that begins at the CU Engineering  Center building and treks across campus to the bottom of a three-story underground parking garage, as  shown in Figure 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  (a)  Imagery: @2022 Maxar Technologies, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Geological Survey, USDA/FPAC/GEO, Map data @2022  0  100  200  meters  (b)  Figure 35: On the left (a) photo taken at CU South Campus of a Husky robot during a large-scale, long-duration  localization test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' On the right (b), is an LIO-SAM point cloud map, denoted by small black dots, and LIO-SAM robot  trajectory denoted by larger dots colored by elevation, overlaid with Google Maps satellite imagery of the area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The  Husky was manually controlled, beginning in the CU South parking lot, continuing along a dirt path and up a hill,  looping back, and ending back at the parking lot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='5  Common Reference Frame Alignment Optimization  To further reduce the yaw error of the common reference frame alignment, the lateral spacing of the robot  prisms was increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  In our initial testing, we found that the roughly 120mm plates attached to the robots along with the prisms  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='5mm centering error lead to consistent variance in the resulting yaw of approximately 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='7◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' This aligned  with a calculated value of arcsin(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='5/120) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='72◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' In order to reduce the impact of this error, a mechanical  bar holding two of the prisms at a distance of 655mm, is added to each robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The resulting angle error  after adding the bar was arcsin(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='5/655) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='13◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' This was consistent with external testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' For a rough  comparison see Figure 37a, where a test was conducted using a mock robot plate and the prism separating  bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The difference from the average of each test shows a higher precision for the tests conducted with the  prism bar in place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' An example setup for these test is shown in Figure 37b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  Imagery @2022 CNES / Airbus, Maxar Technologies, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Geological Survey, USDA/FPAC/GEO, Map data @2022  Figure 36: Overlay of LIO-SAM point cloud map, denoted by small black dots, and LIO-SAM robot trajectory  denoted by larger dots colored by elevation, with Google Maps during a large-scale, long-duration localization test at  CU Main Campus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' This test was conducted on a Spot robot, that was manually controlled, beginning in the CU  Engineering Center building on the bottom left corner of the map, continuing across campus, ending at the bottom of  the three-story underground Folsom Parking Garage on the top right corner of the map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  Plate  Bar  Prism Base  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='05  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='1  Yaw (deg)  (a)  (b)  Figure 37: A comparison of transforms generated by LTS using the standard robot sensor plate, and after attaching  prisms to a bar placed on the plate instead (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The difference from the average, and precision of the bar is higher  than without the bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Setup for a prism test with a mock gate highlighted in red (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The mock gate was designed to  have the same dimension as a robot sensor plate  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='6  Artifact Detection Training Procedure  A systematic procedure targeted at low-light conditions is used to train the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' At each location, data was  collected using three different brightness levels to minimize the impact of lighting conditions on the model’s  performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Specifically, images were taken from past circuit events as well as separate field exercises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  Remote data collection sessions took place inside the dark and rocky Edgar Experimental Mine in Idaho  Springs, Colorado.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Local data collection took place on University of Colorado Boulder campus, primarily  within the outdoor courtyard of our Engineering Center building, and during evening hours when there was  0  50  100  meters0  50  100  L  meters0  50  100  meters0  50  100  L  metersno natural illumination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The data was collected with three onboard illumination levels: 0%, 50%, and 100%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  The cameras, FLIR Blackfly PGE-05S2C-CS GigEVision cameras, were mounted in cardinal directions on  the robots as shown in Figure 5b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  Photos in which the artifacts suffered excessive motion blur and occlusions, determined by the ability of the  human reviewer to detect the artifact, were removed from the data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' After the AD pipeline was trained on  this initial data collection effort, we found that it did not generalize well to new environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Therefore, we  augmented the dataset with additional imagery collected from a greater diversity of backgrounds, including a  nearby university loading dock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The datasets used for training are summarized in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The dataset  was later augmented with data from areas where false positives were frequently identified in order to reduce  the identification of these false positives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='7  BOBCAT Components  BOBCAT calculates Objective weights and Behavior scores to select an Execution Behavior whenever one  or more Monitor outputs or Objective input weights change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Behavior execution functions may be either  blocking or non-blocking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' They should be non-blocking to the maximum extent possible, to increase reactivity  and allow Behaviors to change at any time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Behaviors that need to block may be required for certain actions  that must be completed before the robot can do something different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' If a Behavior is blocking, BOBCAT  will delay evaluation until the actions have completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Tables 8, 9, and 10 provide an exhaustive list of the  Monitors, Objective, and Behaviors, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  Monitor  Criteria  ExploreToGoal  Received command to explore to a specific goalpoint, either submitted by the  human supervisor or generated by a node other than the global planner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  iExplore  iGoToGoal  iStop  iDeployBeacon  iGoHome  The associated input command has been sent by the human supervisor from the GUI  or joystick to execute a specific behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  NearbyRobot  This and any other robot’s paths are within 2m of each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  Beacon  A communications beacon is available and other criteria has been met to deploy it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  ReverseDrop  A communications beacon is available and communications have been lost with the  base station for 10 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  Comms  Any message has been received from the base station in the last 3 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  Artifact  There are at least 3 unreported artifacts, or it has been at least 5 minutes since the  first unreported artifact was detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  Table 8: MARBLE Monitors, with a description of robot state requirements for an output of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='8  Artifact Detection Training Data  Table 11 presents a summary of the training data Team MARBLE collected and annotated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Our main data  collection effort was conducted in the Engineering Center Courtyard at University of Colorado Boulder,  during late evening hours when there was no natural illumination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' This location was chosen because we  believed it be more representative of a subterranean environment, with the large amount of concrete and low  illumination levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Another smaller dataset was collected at the Edgar Mine in order to introduce data from  tunnel-level environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Real-world imagery from the Tunnel Event at the NIOSH Mine and the Urban  Event at the Satsop Nuclear Power Plant was incorporated as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' To round out training, a final data set at  Objective  Evaluation Function  Description  FindArtifacts  iwFindArtifacts  Find, identify and localize artifacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Always active.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  Input  iwInput ∗ OR(Input Monitors)  Allow supervisor to override autonomy when  necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  BeSafe  iwBeSafe ∗ NearbyRobot  Safety of the robot, particularly collision with  other robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  ExtendComms  iwExtendComms ∗  (Beacon || ReverseDrop)  Extend communications as far into the  environment as possible to reduce any delay in  reports and minimize robot travel back and forth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  MaintainComms  iwMaintainComms ∗ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='Comms  Communicate with the base station, either by  staying in or returning to communications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  ReportArtifacts  iwReportArtifacts ∗ Artifact  Report artifact types and locations to base station.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  Table 9: MARBLE Objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Evaluation functions calculate the weight for each objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Components prefixed by  “iw” represent the input weight of the objective, while monitor names represent the binary output of the monitor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  Behavior  Evaluation Function  Actions  Explore  owFindArtifacts ∗ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='ExploreToGoal +  owInput ∗ iExplore  Request global planner to plan to  unexplored areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  GoToGoal  owFindArtifacts ∗ ExploreToGoal +  owInput ∗ iGoToGoal  Request global planner to plan a path to a  specific goalpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  Stop  owBeSafe + owInput ∗ iStop  Controller stops using plan generated by  global planner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Causes robot to stop  autonomous movement, but will still  accept manual movement by human  supervisor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  DeployBeacon  owExtendComms ∗ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='ReverseDrop +  owInput ∗ iDeployBeacon  Initiate beacon deployment maneuver,  which positions robot, stops, and deploys a  beacon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  GoHome  owReportArtifacts +  owMaintainComms+owExtendComms∗  ReverseDrop + owInput ∗ iGoHome  Request global planner to plan a path to  the starting point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  Table 10: MARBLE Behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Evaluation functions calculate the score for each behavior for use by the policy πB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  Components prefixed by “ow” represent the output weight of the objective, while monitor names represent the binary  output of the monitor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  the loading dock at the Engineering Center was added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  Dataset Name  FPS  Images  Labels  Courtyard  10  54523  4837  8299  2682  3190  6219  3928  3654  32809  Edgar Mine  10  3603  254  159  148  142  122  187  84  1096  NIOSH Mine  1  13437  118  45  106  173  0  0  0  442  Satsop Nuclear  15  33953  0  630  0  0  302  0  0  932  Loading Dock  10  6923  184  234  143  143  76  329  116  1225  Total  112439  5393  9367  3079  3648  6719  4444  3854  36504  Table 11: List of data sets utilized to train the artifact detection system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Also listed is the frames per second (FPS) of  the imagery, the total number of images, the number of labels for each artifact, and the total number of artifact labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  At the Final Event, our artifact detection system did not correctly detect any vents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' It also falsely detected  many white walls as vents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' One reason for this poor performance is likely because the vent we trained on was  different than the vent at the Final Event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' To support the vent, white sides were added, making it appear as  a box, as shown in Figure 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  Figure 38: Team MARBLE trained on a vent with white-walled sides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='9  Competition Results  Table 12 lists the eight teams that competed in the DARPA Subterranean Challenge Final Event, and the  number of artifacts each team found during the 60-minute Prize Run, out of a maximum possible score of 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  Score  Team  Funding  23  CERBERUS  DARPA  23  CSIRO Data61  DARPA  18  MARBLE  DARPA  17  Explorer  DARPA  13  CoSTAR  DARPA  7  CTU-CRAS-NORLAB  DARPA  2  Coordinated Robotics  Self  2  Robotika  Self  Table 12:  List of final scores of all teams that participated in the 60-minute Final Event Prize Run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='10  Planning in a Dynamic Environment  The planning algorithm presented in this paper has the capability of adapting to dynamic environments,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' closing or opening of doors, falling rubble, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Other situations also lead to dynamic changes in the  map, including other nearly mobile agents, as well as localization and mapping error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The paper presents  the planner response to a trap door, and here in the Appendix, additionally present examples of planner  responses to robot-robot encounters as well as localization and mapping error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  Robots often came withing close proximity of other robots during the mission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Some examples of this  are shown in Figure 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Rather than resolve these interactions through the planner, which operates on a  slower timescale that fast-approaching robots, the autonomous mission management system specifies the  higher-priority agent continue while the lower-priority agent wait.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The planner marks previously traversable  edges as untraversable, and later updates them as traversable again once the other agents leaves the vicinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  (a)  (b)  (c)  (d)  Figure 39: Robot-robot interactions  Figure 40 presents an instance when the planner on D01 adapted to localization and mapping drift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The  erroneous new map data caused many of the previously traversable edges of the graph to change to  untraversable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The planner adapted to the new scenario, helping the agent return to the main corridor,  at which point a loop closure occurred, correcting the agent’s pose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The planner operated continuously  throughout this localization drift and loop closure correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='11  Planning in Constrained Spaces  The planner parameters were configured such that the agents would operate safely and not attempt to traverse  highly confined spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' One limitation to this approach, is that agents could not autonomously plan and  traverse some areas, such as the utility corridor with low ceilings and narrow cave section, as shown in Figures  41 and 42 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The human supervisor was able to teleoperate the Spot through the cave section, and  it autonomously traversed it when later exiting the cavern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='12  Artifact Reports  All true positive 43 and false positive artifact detections can be seen in Figure 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='13  Twenty-Five Explored Artifacts  This section focuses on all 25 artifacts that Team MARBLE agents were in the vicinity of during the  Final Event Prize Run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' To summarize, 18 of these artifacts were scored, one was not scored, and six were  unreported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' These three categories of artifacts are discussed in Section 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='1, Section 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='14, and Section  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='15, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  Each artifact has at least one attempt associated with it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The eighteen scored artifacts all end with a scored  attempt (SA), and some will have multiple missed attempts (MA) before reaching a scored attempt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The  one missed artifact was not scored, so it only has missed attempts associated with it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The six unreported  artifacts have no attempts associated with them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  (a)  (b)  (c)  (d)  (e)  (f)  Figure 40: Instance of D01 experiencing localization drift and erroneous mapping, causing planner to mark traversable  (green) edges in the graph as no longer traversable (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The progression of events is first (a) no localization error  (13:03), (b) initial localization and mapping error (17:12), (c, d) continued localization and mapping error (13:53,  17:16), (e) loop closure correcting the robot pose (17:34), teleporting the agent away from its planned path (pink),  and (f) continuing on the mission (18:23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='1  Eighteen Scored Artifacts  L51 Drill (SA1): This drill was the first artifact scored, within 1 minute and 8 seconds of the mission start.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  D02 reported and scored the drill as it passed through the first junction of the course, splitting it into tunnel,  urban, and cave corridors (SA1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The HS also saw the drill via D02 live FPV view and would have attempted  to score it had D02 not automatically reported it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  L53 Backpack (SA2): D02 quickly continued into the cave corridor, reporting and scoring the backpack  (SA2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' If D02 did not autonomously score the backpack, the HS may have scored it via D02 live FPV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Had  that failed too, H02 and H01 accurately reported the backpack later in the mission and would have scored it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  (a)  (b)  Figure 41: Mobility limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  (a)  (b)  Figure 42: Mobility limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  L55 Rope (SA3): Because D02 had difficulty traversing the narrow cave corridor, the HS manually  teleoperated the agent through this section of the course, and in the process, saw the rope via D02 live FPV  and reported it (SA3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' D02 did not automatically detect the rope, likely due to poor lighting conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  L26 Survivor (SA4): While D02 immediately explored the cave section, D01 began exploring the urban  section and autonomously reported the survivor and scored it (SA4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Later in the mission, H01 accurately  reported L26 and would have scored had D01 missed it at the beginning of the mission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  L32 Survivor (SA5): The HS saw the survivor through D02 live FPV stream, submitted a manual report  and scored the artifact (SA5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Later in the mission, accurate autonomous reports from D01 and H01 would  have scored the artifact, had the HS had not already scored it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  L08 Gas (SA6): D01 autonomously reported and scored the gas artifact as it traversed the urban environment  (SA6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  L31 Fire Extinguisher (SA7): During the middle of the mission, the HS teleoperated D01 through a  foggy section of the tunnel environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' In this process, the HS saw the fire extinguisher through D01 live  FPV stream, and scored the artifact via manual report (SA7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' After this manual intervention, D01 went onto  help score five more artifacts, L34, L38, L36, L40, and L67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The L31 fire extinguisher was also seen at the  end of the mission when the HS was reviewing the D02 archived FPV images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  L34 Drill (SA8): After the HS teleoperated D01 through the fog, D01 autonomously reported and scored  the drill (SA8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The HS also saw the drill through D01 live FPV stream, and would have reported the artifact  manually had D01 not already scored it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  (a)  (b)  (c)  (d)  (e)  (f)  (g)  (h)  (i)  (j)  (k)  Figure 43: All true positive reports of visual artifacts from autonomous artifact detection systems onboard remote  agents D02 (a, b), D01 (c, d, e, f, g), H02 (h), and H01 (i, j, k), in order from mission start to mission end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  L38 Fire Extinguisher (SA9): After teleoperating D01 through the fog, the HS saw the fire extinguisher  though D01 live FPV stream and manually scored the artifact (SA9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  L36 Cube (SA10): After the HS teleoperated D01 through the fog, D01 autonomously reported and scored  the cube (SA10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  L40 Backpack (SA11): After the HS teleoperated D01 through the fog, D01 autonomously reported and  scored the backpack (SA11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  L67 Rope (SA12): After the HS teleoperated D01 through the fog, D01 autonomously reported and scored  the rope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The HS also saw the artifact though D01 live FPV stream, and would have manually reported the  artifact had D01 not already scored it (SA12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  L11 Cube (MA5, MA6, SA13): D01 and D02 both autonomously reported the cube, but were both  missed attempts (MA5, MA6), with corresponding errors of 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='73m and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='42m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The HS then used the  location of the missed attempts to manually submit an adjusted location, scoring the cube (SA13) with an  error of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='83m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  L22 Cell Phone (MA2, MA4, SA14): D01 autonomously reported the cell phone early in the mission,  but was a missed attempt (MA2) with an error of 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='47m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' H02 also autonomously reported the cell phone,  but the report was a missed attempt (MA4) with an error of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='77m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Near the end of the mission, H01  accurately localized the cell phone and scored the artifact (SA14), with an error of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='06m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  drill:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='59backpack:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='90baakpack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='55backpack:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='71survivor:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='81drill:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='94  国rope:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='93backpack:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='99survivor:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='76backpack:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='93survivor:1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='00(a)  (b)  (c)  (d)  (e)  (f)  (g)  (h)  (i)  (j)  Figure 44: All false positive reports of visual artifacts from autonomous artifact detection systems onboard remote  agents D02 (a, b, c, d), D01 (e, f, g, h), H02 (i), and H01 (j), in order from mission start to mission end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  L47 Cell Phone (SA15): D02 autonomously reported the cell phone along the subway platform and scored  it (SA15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  L59 Cell Phone (MA1, MA3, MA10, SA16): Located in the left branch of the cave section, the cell  phone was autonomously reported by D02, but was a missed attempt (MA1) with an error of 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='69m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The HS  then manually submitted the cell phone with an adjusted location, but this was also a missed attempt (MA3)  with an error of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='74m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Later in the mission, a third missed attempt occurred (MA10), as D01 autonomously  reported the cell phone with an error of 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='97m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Immediately after, the HS used the reported locations from  D02 and D01 to submit another manual report with an adjusted location, and scored the artifact (SA16),  with an error of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='15m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  L24 Gas (MA11, SA17): The gas was autonomously reported by H01, but with an error of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='38m, resulted  in a missed attempt (MA11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Soon after, H02 traversed the same space and autonomously reported the gas  independently of H01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' This report better estimated the position of the artifact, with an error of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='55m, and  scored (SA17) the gas artifact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  L58 Helmet (SA18): The HS reviewed the D02 archived FPV images near the end of the mission, and saw  the helmet in the cavern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The report was accurate to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='74m and scored Team MARBLES 18th and final  point of the mission (SA18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' This point was made possible by the HS teleoperation through the narrow cave  corridor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  backpack:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='69helmet:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='91backpack:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='84vent:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='53vevent:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='73vent:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='83vent:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='82backpack:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='81backpack:013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='14  One Missed Artifact  L64 Cube (MA7, MA8, MA9): This cube artifact was located atop a steep slope found along the main  corridor of the cave section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' D01 autonomously reported the cube but was a missed attempt with an error  of 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='24m (MA7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Based on that experience, the HS manually submitted two reports at adjacent locations,  both of which were missed attempts (MA8, MA9), with errors of 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='23m and 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='15m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' This artifact was never  scored during the mission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Figure 45b shows the failed score attempts circled in red, and the actual position  circled in green.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  (a)  (b)  Figure 45: Helmet FPV image transmitted to human supervisor, but missed due to workload (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' L64 Cube location,  circled in green, and incorrect submissions, circled in red (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='15  Six Unreported Artifacts  L02 Vent L02 vent was not detected by the on-board artifact detection system, but was available in the  human supervisor’s FPV feed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' At approximately 20 minutes into the mission, D01 (Spot) was returning home  due to a localization error noticed by the human supervisor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' While it was returning, the Supervisor turned  attention to other robots, and did not scan the FPV feed for some time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The robot stopped for 33 seconds  and transmitted a series of images similar to Figure 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Then the robot moved forward and was stuck on the  corner underneath the vent, due to the localization error, and this is when the Supervisor returned attention  to the robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The FPV images were stored for later review by the Supervisor, but due to workload, these  images were never reviewed during the mission, and thus the artifact remained unscored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  L05 Vent Approximately 7 minutes prior to the end of the mission, D02 reported a vent and transmitted  the image in Figure 47 to the base station.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Unfortunately, due to workload and poor notification design  in the GUI, the human supervisor never noticed the report, and thus it was not submitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Although the  detection system identified the bucket as a vent, the Supervisor could easily see the actual vent above it and  the reported position was 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='24m from the actual position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  L62 Helmet L62 helmet located in the cave section near the tunnel intersection was seen via FPV, as in  Figure 45a but not detected by the robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' This was transmitted by D01 and also available for review by the  human supervisor, but due to workload the images were not reviewed during the mission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  L13 Gas Approximately 10 minutes prior to the end of the mission, D02 passed approximately within 1m of  this gas artifact, but provided no reports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Further analysis shows the only gas detection D02 had was a false  report in an area with no CO2 nearby.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The only other robot to go near this gas was D01, but it only went  near a doorway leading to the area where the gas was located, and had already reported and scored another  gas artifact near that location, so if it did detect L13 it would have assumed it was the same as the previous  artifact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  Figure 46: Vent seen by D01 while stopped and trans-  mitted to human supervisor, but missed due to human  supervisor workload.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  Figure 47: Vent detected and reported by D02 late in  mission but missed by the human supervisor due to work-  load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  L21 Vent This vent was in the subway platform area of the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Both Spot robots viewed the  vent with various cameras, but never detected it, likely due to the white wall background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Additionally,  communications to the base station were limited in this area, and none of the FPV images relayed to the  human supervisor had the vent in view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Interestingly, D01 did provide a false report of a vent in this area, and  transmitted an image seen in Figure 48, which the human supervisor discarded as a false report.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' According to  the truth data, the location reported was 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='93m from the actual vent, and so would have scored if submitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  However, visual analysis indicates the vent position in the ground truth file appears 2m off, which would not  have scored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  L42 Fire Extinguisher This fire extinguisher was seen only with the right-facing camera on D01 for only a  few frames, as seen in Figure 49, which was not enough to trigger a detection and report.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The robot was  outside of communications, so even if the front camera had seen it, it would not have been available for the  human supervisor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  Figure 48: False vent reported close to actual vent by  D01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  Figure 49: Missed fire extinguisher only seen for a few  frames by the right camera of D01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  vent:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='53VeHH13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='16  Two False Artifacts  This section focuses on two false artifacts that Team MARBLE reported, but were actually the result of false  detections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Each false artifact has at least one false attempt (FA) associated with it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' These false attempts  are listed in Table 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  Gas (FA1, FA2, FA5): Gas was autonomously reported by H02 early in the mission (FA1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The HS  modified the location and manually reported again (FA2), but did not score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Later in the mission, D02  reported gas again in a very similar location as D01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' This increased the confidence in the HS that gas was in  the area, so the HS modified the location of D02 report, but this too did not score (FA4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' It remains unknown  why both agents detected elevated levels of CO2, but it is likely that a source in that vicinity existed, even if  it was not an gas artifact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  Fire Extinguisher (FA3): Due to an unknown cause, the HS inadvertantly submitted the same report  twice, which was a fire extinguisher at xyz-coordinates of (20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='60, 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='59, -2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='64).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Because the fire extinguisher  was scored by the last report, this report did not score, but simply wasted a report.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  Backpack (FA4): The image was not clear, but the HS attempted to report it, and it did not score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='17  Tabulated Reports  All reports that Team MARBLE submitted to DARPA are listed in Table 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='18  Field Deployments  Table 14 lists the dates and locations of all full-scale deployments, within the context of the events at the  DARPA SubT Challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  Report  ID  Type  Error  Score  Time  Scorer  Assister  [m]  [mm:ss]  SA1  L51  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='57  1  01:08  D02  SA2  L53  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='23  2  01:23  D02  SA3  L55  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='84  3  06:23  HS†  D02 (live FPV stream)  SA4  L26  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='62  4  12:03  D01  MA1  L59  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='69  14:09  D02  MA2  L22  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='47  14:13  D01  MA3  L59  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='74  14:57  HS  D02 (MA1)  SA5  L32  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='40  5  16:35  HS  D02 (live FPV stream)  SA6  L08  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='80  6  17:23  D01  FA1  —  —  17:33  H02  FA2  —  —  17:59  HS  H02 (FA1)  MA4  L22  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='77  18:38  H02  MA5  L11  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='73  19:49  D01  SA7  L31  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='31  7  28:08  HS†  D01 (live FPV stream)  FA3  —  —  33:38  HS  SA8  L34  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='43  8  35:51  D01†  SA9  L38  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='82  9  36:53  HS†  D01 (live FPV stream)  SA10  L36  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='94  10  37:08  D01†  SA11  L40  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='40  11  37:58  D01†  SA12  L67  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='87  12  38:47  D01†  FA4  —  —  45:18  D02  FA5  —  —  46:44  HS  H02 (FA1), HS (FA2) & D02  MA6  L11  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='42  47:31  D02  SA13  L11  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='83  13  47:53  HS  D01 (MA5), D02 (MA6)  MA7  L64  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='24  48:12  D01  MA8  L64  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='23  48:42  HS  D01 (MA7)  MA9  L64  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='15  49:09  HS  D01 (MA7) & HS (MA8)  SA14  L22  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='06  14  50:33  H01  D01 (MA2), H02 (MA4), D02  SA15  L47  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='00  15  50:45  D02  MA10  L59  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='07  50:57  D01  SA16  L59  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='15  16  51:48  HS  D02 (MA1), HS (MA3), D01 (MA10)  MA11  L24  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='38  52:20  H01  SA17  L24  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='55  17  52:45  H02  SA18  L58  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='74  18  56:33  HS†  D02 (archived FPV images)  Table 13: List of all artifact reports submitted by Team MARBLE during the 60-minute Final Event Prize Run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  Bolded entries represent reports that resulted in a score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' There are three types of reports: a Scored Attempt (SA), a  Missed Attempt (MA) due to error exceeding 5m, and a False Attempt (FA) due to a false positive detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Listed  next are artifact ID, artifact type, error, cumulative score, and time since mission start.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The scorer is the agent that  submitted the report and scored (or attempted to score), and the assister is the agent(s) that provided information  that aided the scorer in scoring (or attempting to score).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The reporting of these artifacts was completely autonomous,  save artifacts scored by the HS as well as those with a (†), denoting artifacts that were seen as a result of the HS  temporarily teleoperating the agent into new areas of the course.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content='  co  2CO  2Date  Deployment  Environment  Onsite  Location  Apr 7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' 2019  STIX Event  Edgar Experimental Mine  Idaho Springs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' CO  Jul 29,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' 2019  Pre-Tunnel 1  Edgar Experimental Mine  Idaho Springs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' CO  Aug 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' 2019  Pre-Tunnel 2  Edgar Experimental Mine  Idaho Springs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' CO  Aug 6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' 2019  Pre-Tunnel 3  Edgar Experimental Mine  Idaho Springs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' CO  Aug 9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' 2019  Pre-Tunnel 4  Edgar Experimental Mine  Idaho Springs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' CO  Aug 12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' 2019  Pre-Tunnel 5  Edgar Experimental Mine  Idaho Springs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' CO  Aug 17,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' 2019  Tunnel Event  NIOSH Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' & Safety Research Mines  Pittsburgh, PA  Feb 4, 2020  Pre-Urban 1  Geotech Warehouse  Denver, CO  Feb 8, 2020  Pre-Urban 2  Geotech Warehouse  Denver, CO  Feb 12, 2020  Pre-Urban 3  Geotech Warehouse  Denver, CO  Feb 21, 2020  Urban Event  Satsop Nuclear Power Plant  Elma, WA  Aug 4, 2020  Pre-Cave 1  Edgar Experimental Mine  Idaho Springs, CO  Sep 17, 2020  Pre-Cave 2  Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Center (L1)  X  Boulder, CO  Sep 19, 2020  Pre-Cave 3  Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Center (L1)  X  Boulder, CO  Sep 21, 2020  Cave Event*  Edgar Experimental Mine  Idaho Springs, CO  Jul 13, 2021  Pre-Final 1  Folsom Parking Garage  X  Boulder, CO  Jul 14, 2021  Pre-Final 2  Folsom Parking Garage  X  Boulder, CO  Jul 15, 2021  Pre-Final 3  Edgar Experimental Mine  Idaho Springs, CO  Aug 13, 2021  Pre-Final 4  Folsom Parking Garage  X  Boulder, CO  Aug 17, 2021  Pre-Final 5  Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Center (LL & Courtyard)  X  Boulder, CO  Aug 19, 2021  Pre-Final 6  Sust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=', Energy, and Env.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Community  X  Boulder, CO  Aug 24, 2021  Pre-Final 7  Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Center (LL & Courtyard)  X  Boulder, CO  Aug 26, 2021  Pre-Final 8  Edgar Experimental Mine  Idaho Springs, CO  Sep 1, 2021  Pre-Final 9  Edgar Experimental Mine  Idaho Springs, CO  Sep 8, 2021  Pre-Final 10  Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Center (LL & Courtyard)  X  Boulder, CO  Sep 10, 2021  Pre-Final 11  Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Center (LL & Courtyard)  Boulder, CO  Sep 12, 2021  Pre-Final 12  Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Center (L2) + Rustandy  X  Boulder, CO  Sep 14, 2021  Pre-Final 13  Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Center (L2) + Rustandy  X  Boulder, CO  Sep 21, 2021  Final Event  Louisville, Megacavern  Louisville, KY  Table 14:  List of all full-scale field deployments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' For the Final Event, Team MARBLE gave greater resources to  system performance validation and human-robot teaming practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' This realized itself as more frequent and diverse  field deployments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' For the circuit events, just two to three weeks were spent on field deployments, whereas for the  Final Event, the team devoted two months.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Instead of practicing in just one or two environments, the team was  asked to perform in five unique environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' Selecting locations that were ”onsite” University of Colorado Boulder  campus enabled the team to be more nimble and operationally efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
+page_content=' The (*) denotes that the Cave Event was a  self-managed mock event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQf3fnG/content/2301.00771v1.pdf'}
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+arXiv:2301.03955v1  [math.PR]  10 Jan 2023
+HEGSELMANN–KRAUSE MODEL WITH ENVIRONMENTAL NOISE
+LI CHEN, PAUL NIKOLAEV, AND DAVID J. PR ¨OMEL
+Abstract. We study a continuous-time version of the Hegelsmann–Krause model describing
+the opinion dynamics of interacting agents subject to random perturbations. Mathematical
+speaking, the opinion of agents is modelled by an interacting particle system with a non-
+Lipschitz continuous interaction force, perturbed by idiosyncratic and environmental noises.
+Sending the number of agent to infinity, we derive a McKean–Vlasov stochastic differential
+equations as the limiting dynamics, by establishing propagation of chaos for regularized
+versions of the noisy opinion dynamics. To that end, we prove the existence of a unique strong
+solution to the McKean–Vlasov stochastic differential equation as well as well-posedness of
+the associated non-local, non-linear stochastic Fokker–Planck equation.
+Key words: interacting particle system, McKean–Vlasov equation, mean-field limit, propa-
+gation of chaos, stochastic Fokker–Planck equation, stochastic partial differential equation.
+MSC 2010 Classification: Primary: 60H15, 60H10, 60K35; Secondary: 91D30.
+1. Introduction
+The theory of opinion dynamics has been around since the 1950s, but over the last few
+decades, the modeling of opinion dynamics has become a rapidly growing area of research.
+With the rapid development of the internet and social networks, we have observed signifi-
+cant changes in how opinion dynamics evolve and by what sources they are effected. For
+example, previous generations were heavily influenced by their geographically nearest social
+group, but nowadays social networks are the primary platforms, for expressing and sharing
+opinions, enabling more people than ever to do so from anywhere in the world. Consequently,
+geographical distance is no longer a significant factor in shaping public opinion. Instead, each
+citizen has a personal filter bubble [Spo17], which affects and in the same time shifts with
+the opinion. This phenomena is described by so called bounded confidence opinion dynamics.
+For an overview of opinion dynamics we refer to the surveys [Lor07, Hos20].
+In the present paper we study the Hegelsmann–Krause model (HK model) [HK02], which
+belongs to the class of bounded confidence opinion dynamics. More precisely, we focus on a
+version of the HK model where the opinions XN := (Xi, i = 1, . . . , N) of N agents are subject
+to idiosyncratic noises as well as an environmental noise, i.e., we consider for i = 1, . . . , N the
+particle system
+(1.1)
+dXi
+t = −
+1
+N − 1
+N
+�
+j=1
+j̸=i
+kHK(Xi
+t − Xj
+t ) dt + σ(t, Xi
+t) dBi
+t + ν dWt,
+Xi
+0 = ζi,
+for t ≥ 0, where Xi
+t is the i-th agent’s opinion at time t, kHK(x) :=
+1[0,R](|x|)x is the
+(non-Lipschitz) interaction force between the agents, σ: [0, T] × R �→ R ane ν > 0 some
+smooth diffusion coefficients, B = ((Bi
+t, t ≥ 0), i ∈ N) is a sequence of one-dimensional
+Date: January 11, 2023.
+1
+
+2
+CHEN, NIKOLAEV, AND PR ¨OMEL
+independent Brownian motions, W = (Wt, t ≥ 0) is a Brownian motion independent of B, and
+(ζi, i ∈ N) is the i.i.d. sequence of initial values independent of all Brownian motions. The local
+interaction kernel kHK represents the insight of bounded confidence opinion dynamics that
+opinions are only influenced in a bounded domain. In the HK model (1.1), the idiosyncratic
+noises B = ((Bi
+t, t ≥ 0), i ∈ N) describe the individual random effects on each agent’s opinion
+and the environmental noise (Wt, t ≥ 0) captures external effects on the agents’ opinions. For
+a more detailed discussion on different types of noises in HK models we refer to [CSDH19]
+and the references therein.
+Our goal is to establish propagation of chaos of the Hegelsmann–Krause model with environ-
+mental noise. More precisely, we show that regularized versions of the particle systems (1.1)
+converge (in a suitable sense) to the McKean–Vlasov stochastic differential equation (SDE)
+�
+dYt = −(kHK ∗ ρt)(Yt) dt + σ(t, Yt) dB1
+t + ν dWt,
+Y0 = X1
+0,
+ρt is the conditional density of Yt given FW
+t ,
+(1.2)
+which comes with the associated stochastic non-linear, non-local Fokker–Planck equation
+(1.3)
+dρt =
+d2
+dx2
+�σ2
+t + ν2
+t
+2
+ρt
+�
+dt + d
+dx((kHK ∗ ρt)ρt) dt − ν d
+dxρt dWt,
+t ≥ 0.
+Let us remark that equation (1.3) is a non-local, non-linear stochastic partial differential
+equation (SPDE), where the stochastic term is a consequence of the environmental noise
+W = (Wt, t ≥ 0). Indeed, as we shall see, if the number of agents tends to infinity the effect
+of the idiosyncratic noises averages out, but the environmental noise does not. Moreover, in
+contrast to many recent works like [CF16, CG19, HvS21, CDFM20, BCD21] on interacting
+particle systems with environmental noise, we deal with density-dependent McKean–Vlasov
+SDEs and the associated Fokker–Planck equations for the conditional densities.
+Our first contribution is to prove the well-posedness of the non-local, non-linear stochastic
+Fokker–Planck equation (1.3). The main challenge in proving existence and uniqueness results
+for (1.3) is the non-linear term (kHK ∗ρt)ρt since this prevents us from applying known results
+in the existing literature on SPDEs, such as those found in textbooks [Kry99, WR15], which
+consider the well-studied case of linear SPDEs. In the case of non-linear SPDEs, one needs
+to take advantage of the specific structure of the considered SPDE to employ a fixed point
+argument, cf. e.g. the recent work [HQ21]. In this line of research, we establish local existence
+and uniqueness of a weak solution to the non-local, non-linear Fokker–Planck equation (1.3).
+Additionally, we show global well-posedness of the Fokker–Planck equation (1.3) assuming a
+sufficiently large diffusion coefficient or a sufficiently small L2-norm of the initial value.
+Our second contribution is to prove the existence of a unique strong solution to the
+McKean–Vlasov stochastic differential equation (1.2), which is essential for showing propaga-
+tion of chaos towards to limiting Fokker–Planck equation (1.3). To obtain the well-posedness
+of the McKean–Vlasov SDE (1.2), a central insight is to introduce suitable stopping times
+to ensure sufficient temporary regularity such that a backward stochastic partial differential
+equation (BSPDE) associated to (1.2) possesses a classical solution, cf. e.g. [DTZ13]. As a
+result, a duality argument in combination with the Itˆo–Wentzell formula allows us to deduce
+the existence of a unique strong solution to the McKean–Vlasov equation (1.3). Let us point
+out that the aforementioned existence and uniqueness results for the stochastic Fokker–Planck
+equation (1.3) and the McKean–Vlasov equation (1.2) also hold for general interaction forces
+k ∈ L1(R) ∩ L2(R).
+
+HEGSELMANN–KRAUSE MODEL WITH ENVIRONMENTAL NOISE
+3
+Our third contribution is to establish propagation of chaos for regularized versions of the
+particle systems (1.1) with environmental noise, which verifies that (1.3) is, indeed, the macro-
+scopic (density based) model corresponding to the microscopic (agent based) opinion dynamics
+described by the Hegelsmann–Krause model with environmental noise. For uniformly Lips-
+chitz interaction forces, propagation of chaos with environmental noise has been showed by
+Coghi and Flandoli [CF16] by utilizing sharp estimates in Kolmogorov’s continuity theorem
+and properties of measure-valued solutions of the associated stochastic Fokker–Planck equa-
+tion.
+Moreover, without environmental noise there is a vast literature on propagation of
+chaos with non-Lipschitz kernels, see [Szn91, GQ15, LY16, JW18]. However, most of these
+works are based on the relative entropy and cannot be simply generalized to a setting with
+environmental noise. For particle systems with environmental noise and non-Lipschitz inter-
+action force k (as in our case), to the best of our knowledge there exists no general theory
+on propagation of chaos. In order to derive propagation of chaos for the HK model (1.2), we
+essentially rely on the well-posedness of the McKean–Vlasov SDE (1.2) and follow [Szn91] as
+well as [LP17, CDHJ21] to prove that (ρt, t ≥ 0) characterizes the measure of the mean-field
+limit.
+Organization of the paper: In Section 2 we introduce the Hegelsmann–Krause model with
+environmental noise and provide necessary definitions and some background information. In
+Section 3 the well-posedness of the stochastic Fokker–Planck equation (1.3) is established
+and in Section 4 of the associated McKean–Vlasov equation (1.2). The mean-field limit and
+propagation of chaos of the HR model with environmental noise are investigated in Section 5.
+2. Hegselmann–Krause model
+In the following section we introduce the Hegselmann–Krause model and briefly review
+some of the related literature on the well studied HK model without environmental noise.
+The Hegselmann–Krause model with environmental noise is set up in Subsection 2.2 with
+its corresponding stochastic Fokker–Plank equation as well as its mean-field equation. We
+conclude this section with some basic definitions and functions spaces necessary to study the
+involved equations.
+2.1. Some background information. In this subsection we present some background on
+the development of the HK model as well as on the terminology of propagation of chaos. We
+start with the original discrete-time Hegelsmann–Krause model [HK02], which is given by
+(2.1)
+xi(t + 1) =
+1
+|Ni(t)|
+�
+j∈Ni(t)
+xj(t),
+t ≥ 0,
+i = 1, . . . , n,
+where xi(t) is the opinion of agent i at time t, Ni(t) := {1 ≤ j ≤ n : |xi(t)−xj(t)| ≤ ri} denotes
+the neighbor set of agent i at time t and |Ni(t)| is the cardinality of the set. The convergence
+and consensus properties of the discrete-time HK model were excessively studied in the past
+years, see for instance [HK02, Lor06, BBCN13, KZPS12, NT12]. The main characteristic
+feature of bounded confidence opinion models, like the HK model (2.1), is that the agents
+interact only locally, which is modelled by the compactly supported interaction force in the
+discrete-time HK model (2.1), i.e. j ∈ Ni(t), and, thus, opinions outside an agent’s moral
+beliefs get ignored through this local interaction kernel. This phenomena is, e.g., observed in
+case of liberal and conservative view points and their respective social media bubbles in the
+USA [ENG+19, GKM17, Spo17]. The discrete-time HK model (2.1) is a fairly simple model
+
+4
+CHEN, NIKOLAEV, AND PR ¨OMEL
+to describe opinion dynamics and by now there are numerous generalization and variants of
+the original HK model, for instance, the HK model with media literacy [XCW+20] or the HK
+model with an opinion leader [WCB15]. For further extensions we refer to [DR10, RD09].
+An important class of extensions of the original HK model captures external random ef-
+fect in opinion dynamics, see e.g. [PTHG13, CSDH19], leading naturally to a system of N
+stochastic processes representing the opinion evolution. In this case, following e.g. [GPY17],
+the opinion dynamics ˆXN := ( ˆXi, i = 1, . . . , N) of N agents are modelled by a system of
+stochastic differential equations
+(2.2)
+d ˆXi
+t = −
+1
+N − 1
+N
+�
+j=1
+j̸=i
+kHK( ˆXi
+t − ˆXj
+t ) dt + σ(t, ˆXi
+t) dBi
+t,
+i = 1, . . . , N,
+ˆXN
+0 ∼
+N⊗
+i=1ρ0,
+for t ≥ 0, where ˆXi
+t is the i-th agent’s opinion at time t, kHK is the interaction force between
+the agents, σ is a smooth diffusion coefficient, ((Bi
+t, t ≥ 0), i ∈ N) is a sequence of one-
+dimensional independent Brownian motion and, as previously, ρ0 is the initial distribution (of
+ζ1). We note that the interaction force kHK has compact support turning the continuous-time
+HK model (2.2) into a bounded confidence model. The continuous-time HK model (2.2) has
+been a topic of active research in the past years, e.g., the convergence to a consensus is studied
+in [GPY17] and the phase transition were investigated in [WLEC17].
+The particle system (2.2) induces in the mean-field limit, i.e.
+as N → ∞, a sequence
+(( ˆY i
+t , t ≥ 0), i ∈ N) of i.i.d. stochastic processes satisfying the non-linear, non-local McKean–
+Vlasov equations
+�
+d ˆY i
+t = −(kHK ∗ ˆρt)( ˆY i
+t ) dt + σ(t, ˆY i
+t ) dBi
+t,
+ˆY i
+0 = ˆXi
+0,
+t ≥ 0,
+ˆρt is the density of ˆY i
+t .
+(2.3)
+For our specific interaction kernel kHK =
+1[0,R](|x|)x, each ˆY i
+t corresponds to a well-posed,
+non-linear Fokker–Planck equation for the agent density profile ˆρ(t, x) given by
+(2.4)
+∂tˆρ(x, t) =
+d2
+dx2
+�σ2
+2 ˆρt(x)
+�
++ d
+dx((kHK ∗ ˆρt)(x)ˆρt(x)),
+t ≥ 0,
+see [CJLW17] for more details.
+The given probability measure ˆρ(t, x) dx is classically acquired as the deterministic limit as
+N → ∞ of the random measures ΠN with values in the space P(C([0, T], R)) of probability
+measures on C([0, T], R), defined as
+(2.5)
+ω �→ ΠN(ω, A) := 1
+N
+N
+�
+i=1
+δ ˆ
+Xi· (ω)(A),
+A ∈ B(C([0, T], R)),
+where δf is the Dirac measure for f ∈ C([0, T], R). This convergence phenomena is known
+as propagation of chaos, see e.g. [Szn91, Kac56]. More precisely, let Z = (Zt, t ≥ 0) be a
+continuous R-valued stochastic process defined on some probability space such that for each
+t ≥ 0, Zt has the law µt. Moreover, let µN
+t be the law of a system ZN
+t := (Z1
+t , . . . , ZN
+t ) of N
+stochastic processes taking values in R. We say that ZN
+t
+is µt-chaotic, if µN
+t
+is a symmetric
+measure and for each fix k ∈ N with k ≥ 2,
+(2.6)
+µk,N
+t
+converges weakly to µt as N → ∞,
+
+HEGSELMANN–KRAUSE MODEL WITH ENVIRONMENTAL NOISE
+5
+where µk,N
+t
+(A1 × · · · × Ak) := µN
+t (A1 × · · · × Ak × R × · · · × R) for A1, . . . , Ak ∈ B(R) is the
+k-th marginal distribution of µN
+t , and we say that propagation of chaos holds if for any time
+point t ≥ 0, ZN
+t
+is µt-chaotic. It is a classical result that condition (2.6) holds for all k ≥ 2
+if and only if condition (2.6) holds for k = 2, which is again equivalent to the convergence of
+the empirical measures (2.5) associated to Z to the deterministic measure µ on C([0, T], R),
+see [Szn91, Proposition 2.2].
+It is well-known that if kHK would satisfy suitable Lispchitz and growth assumptions, the
+particle system (ˆXN
+t , t ≥ 0) satisfies propagation of chaos, see [Szn91, KX99]. However, kHK
+is not Lipschitz continuous and hence, we can no longer apply the classical theory to obtain
+the convergence in law of the empirical measure (2.5). Consequently, it requires novel and
+non-standard methods. An idea, which has been developed for different interacting particle
+systems without environmental noise in recent years, is to introduce a smooth approxima-
+tion kN
+HK, which depend on the number of agents N and converges in some sense to kHK. This
+leads to a regularized version of the system (2.2), which reads as
+d ˆXi,N
+t
+= −
+1
+N − 1
+N
+�
+j=1
+j̸=i
+kN
+HK( ˆXi,N
+t
+− ˆXj,N
+t
+) dt + σ(t, ˆXi,N
+t
+) dBi
+t,
+i = 1, . . . , N,
+ˆXN
+0 ∼
+N⊗
+i=1ˆρ0,
+for t ≥ 0, where ˆXN
+t := ( ˆX1,N
+t
+, . . . , ˆXN,N
+t
+). This allows to introduce an intermediate particle
+system by
+�
+d ˆY i,N
+t
+= −(kN
+HK ∗ ˆρN
+t )( ˆY i,N
+t
+) dt + σ(t, ˆY i,N
+t
+) dBi
+t,
+ˆY i,N
+0
+= ˆXi,N
+0
+,
+i = 1, . . . , N,
+ˆρt is the density of Y i,N
+t
+.
+We note that, similar to (2.3), the aforementioned equation is in general a non-linear, non-
+local McKean–Vlasov SDE, which induces a non-linear Fokker–Planack equation similar to
+the one presented in (2.4). The regularized systems allow to estimates, for all i, terms of
+the form E(| ˆXi,N
+t
+− ˆY i,N
+t
+|) and E(| ˆY i,N
+t
+− ˆY i
+t |) separately via PDE methods by studying the
+associated non-linear Fokker–Planck equations. As a result, one can obtain propagation of
+chaos for the system (2.2) with kN
+HK instead of kHK. Following the above described method,
+various version of propagation of chaos has been shown for a variety of models with general
+kernels k in [LP17, Ana17, CDHJ21] for particle systems without environmental noise, which
+have non-Lipschitz, unbounded and even singular interaction force kernels.
+2.2. Hegselmann–Krause model with environmental noise. In this subsection we in-
+troduce the Hegselmann–Krause model with environmental noise, its corresponding mean-
+field stochastic differential equation with its associated stochastic Fokker–Planack equation.
+Let (Ω, F, (Ft)t≥0, P) be a complete filtered probability space, B = (Bi
+t, t ≥ 0, i ∈ N) be a
+sequence of one-dimensional Brownian motions and W = (Wt, t ≥ 0) be a one-dimensional
+Brownian motion. All Brownian motions (Bi
+t, t ≥ 0, i ∈ N) and (Wt, t ≥ 0) are supposed
+to be independent and measurable with respect to the filtration (Ft, t ≥ 0). Let the initial
+data (ζi, i ∈ N) be i.i.d. random variables with density ρ0 and independent of the Brownian
+motions (Bi
+t, t ≥ 0, i ∈ N), and (Wt, t ≥ 0). Moreover, we denote by FW = (FW
+t , t ≥ 0)
+the augmented filtration generated by W (see [KS91, Section 2.7] for the definition) and by
+PW the predictable σ-algebra with respect to FW . Analogous notation will be used for the
+filtration generated by the Brownian motion B.
+
+6
+CHEN, NIKOLAEV, AND PR ¨OMEL
+The Hegselmann–Krause model with environmental noise is given by the interacting particle
+system XN
+t = (X1
+t , . . . , XN
+t ) following the dynamics
+(2.7) dXi
+t = −
+1
+N − 1
+N
+�
+j=1
+j̸=i
+kHK(Xi
+t −Xj
+t ) dt+σ(t, Xi
+t) dBi
+t +ν dWt,
+i = 1, . . . , N,
+Xi
+0 = ζi,
+for t ∈ [0, T], where σ: [0, T] × R �→ R is the diffusion coefficient, ν > 0 a constant and the
+interaction force kHK(x) =
+1[−R,R](x)x for x ∈ R. We point out, that the kernel kHK always
+stands for the kernel in the Hegselmann–Krause model. On the other hand k will denote a
+general kernel (see Section 3 and Section 4).
+For establishing propagation of chaos we introduce the approximation sequence (kτ
+HK, τ >
+0) of the interaction force kHK such that the following properties hold for each τ > 0:
+• kτ
+HK ∈ C∞
+c (R),
+• supp(kτ
+HK) ⊆ [−R − 2τ, R + 2τ], supp( d
+dxψτ) ⊂ [−R − 2τ, −R + 2τ] ∪ [R − 2τ, R + 2τ],
+• | d
+dxkτ
+HK| ≤ C
+τ for some constant C > 0,
+• ∥kHK∥L∞(R) ≤ 1
+τ ∥kHK∥L2(R).
+We denote the regularized interacting particle system XN,τ
+t
+= (X1,τ
+t
+, . . . , XN,τ
+t
+) by
+(2.8)
+dXi,τ
+t
+= −
+1
+N − 1
+N
+�
+j=1
+j̸=i
+kτ
+HK(Xi,τ
+t
+− Xj,τ
+t
+) dt + σ(t, Xi,τ
+t ) dBi
+t + ν dWt, Xi,τ
+0
+= ζi,
+for t ∈ [0, T] and for i = 1, . . . , N. Although the interaction force kernel kHK is non-Lipschitz
+continuous, the N-particle systems (2.7) and (2.8) possess unique strong solutions, see e.g.
+[MPT19, Theorem 1.1], as kτ
+HK and kHK are bounded and measurable in one dimension.
+Corresponding to the particle systems (2.7) and (2.8), for i ∈ N, the system of mean-field
+SDEs is given by
+�
+dY i
+t = −(kHK ∗ ρt)(Y i
+t ) dt + σ(t, Y i
+t ) dBi
+t + ν dWt,
+Y i
+0 = Xi
+0,
+ρt is the conditional density of Y i
+t given FW
+t ,
+(2.9)
+and the system of regularized mean-field SDEs is defined by
+�
+dY i,τ
+t
+= −(kτ
+HK ∗ ρτ
+t )(Y i,τ
+t
+) dt + σ(t, Y i,τ
+t
+) dBi
+t + ν dWt,
+Y i,τ
+0
+= Xi,τ
+0 ,
+ρτ
+t is the conditional density of Y i,τ
+t
+given FW
+t ,
+(2.10)
+for t ∈ [0, T], where ρt denotes the conditional density of Y i
+t given FW
+t , that is, for every
+bounded continuous function ϕ, ρt satisfies
+E(ϕ(Y i
+t ) | FW
+t ) =
+�
+R
+ϕ(x)ρt(x) dx,
+P-a.e.
+The same holds for the regularized conditional density ρτ
+t of Y i,τ
+t
+given FW
+t . Let us remark
+that ρτ
+t , ρt have no superscript i since there are independent of i ∈ N. Indeed, the (regularized)
+mean-filed particles are conditionally independent given FW and identically distributed, thus,
+the conditional density is the same for each i ∈ N.
+
+HEGSELMANN–KRAUSE MODEL WITH ENVIRONMENTAL NOISE
+7
+Associated to the mean-field SDEs (2.9) and (2.10), the stochastic Fokker–Planck equation
+reads as
+(2.11)
+dρt =
+d2
+dx2
+�σ2
+t + ν2
+2
+ρt
+�
+dt + d
+dx((kHK ∗ ρt)ρt) dt − ν d
+dxρt dWt,
+and the regularized stochastic Fokker–Planck equation as
+(2.12)
+dρτ
+t =
+d2
+dx2
+�σ2
+t + ν2
+2
+ρτ
+t
+�
+dt + d
+dx((kτ
+HK ∗ ρτ
+t )ρτ
+t ) dt − ν d
+dxρτ
+t dWt,
+t ∈ [0, T].
+Let us remark that we purposely use the same unknown functions ρ, ρτ for the solutions
+of the stochastic Fokker–Planak equation (2.11) and (2.12) and for the conditional density of
+the mean-field SDEs (2.9) and (2.10) since, as we will see in Theorem 4.3, they both coincide
+under enough regularity assumptions on the initial condition ρ0. Nevertheless, the meaning
+of ρ, ρτ will always be clear from context. We make the following assumptions on the diffusion
+coefficient σ.
+Assumption 2.1. Let T > 0 and σ: [0, T] × R → R the diffusion coefficient, which satisfies:
+(i) There exists a constant λ > 0 such that
+σ2(t, x) ≥ λ
+for all x ∈ R and t ∈ [0, T].
+(ii) There exists a constant Λ > 0 such that for all t ∈ [0, T] we have
+σ(t, ·) ∈ C3(R)
+and
+sup
+t∈[0,T]
+3
+�
+i=1
+����
+di
+dxi σ(t, ·)
+����
+L∞(R)
+≤ Λ.
+The well-posedness of the stochastic Fokker–Planak equation (2.11) and (2.12) is presented
+in Section 3 and the well-posedness of the mean-field SDEs (2.9) and (2.10) in Section 4.
+For this purpose, we first need to fix some basic definitions and function space in the next
+subsection.
+2.3. Function spaces and basic definitions. In this subsection we collect some basic
+definitions and introduce the required function spaces. For 1 ≤ p ≤ ∞ we denote by Lp(Rd)
+with norm ∥·∥Lp(Rd) the vector space of measurable functions whose p-th power is Lebesgue
+integrable (with the standard modification for p = ∞), by C∞
+c (Rd) the set of all infinitely
+differentiable functions with compact support on Rd and by S(Rd) the set of all Schwartz
+functions, see [Yos80, Chapter 6] for more details.
+We note that C∞
+c (Rd) and S(Rd) are
+endowed with their standard topologies. Let
+A := {α = (α1, . . . , αd) : α1, . . . , αd ∈ N0}
+be the set of all multi-indices and |α| := α1 + . . . + αd. The derivative will be denoted by
+∂α :=
+∂|α|
+∂xα1
+1 ∂xα2
+2 · · · ∂xαd
+d
+.
+In one dimension (d = 1), we also write
+dn
+dxn f for the n-th derivative with respect to x ∈ R
+of a smooth function f defined on R. We drop the superscripts α, n in the case α = n = 1.
+Moreover, as an inductive limit, we have C∞
+c (R) =
+∞
+�
+M=1
+C∞
+c (B(0, M)), where B(0, M) is a
+
+8
+CHEN, NIKOLAEV, AND PR ¨OMEL
+ball with radius M > 0 in Rd and (C∞
+c (B(0, M)), pα,M) is the complete metrizable space of
+smooth functions with compact support in B(0, M) and semi-norm
+pα,M(f) := sup
+|x|≤M
+|(∂αf)(x)|
+for f ∈ C∞
+c (B(0, M)) and α ∈ A. We note that this characterization and Baire category
+theorem immediately imply that C∞
+c (R) is not metrizable. Similar, for each α, β ∈ A we
+define the semi-norms
+pα,β(f) := sup
+x∈Rd |xα(∂βf)(x)|.
+Equipped with these seminorms, S(Rd) is a Fr´echet space [Abe12, Appendix A.5]. Further-
+more, we introduce the space of Schwartz distributions S′(Rd). We denote dual parings by
+⟨·, ·⟩. For instance, for f ∈ S′, u ∈ S we have ⟨u, f⟩ = u[f] and for a probability measure µ
+we have ⟨f, µ⟩ =
+�
+f dµ. The correct interpretation will be clear from the context but should
+not be confused with scalar product ⟨·, ·⟩L2(R) in L2.
+The Fourier transform F[u] and the inverse Fourier transform F−1[u] for u ∈ S′(Rd) and
+f ∈ S(Rd) are defined by
+⟨F[u], f⟩ := ⟨u, F[f]⟩,
+where F[f] and F−1[f] is given by
+F[f](ξ) :=
+1
+(2π)d/2
+�
+e−iη·xf(x) dx
+and
+F−1[f](ξ) :=
+1
+(2π)d/2
+�
+eiη·xf(x) dx.
+The Bessel potential for each s ∈ R is denoted by Js := (1−∆)s/2u := F−1[(1+|ξ|2)s/2F[u]]
+for u ∈ S′(Rd). We define the Bessel potential space Hs
+p for p ∈ (1, ∞) and s ∈ R by
+Hs
+p := {u ∈ S′(Rd) : (1 − ∆)s/2u ∈ Lp(Rd)}
+with the norm
+∥u∥Hsp :=
+���(1 − ∆)s/2u
+���
+Lp(Rd) ,
+u ∈ Hs
+p.
+For 1 < p < ∞, m ∈ N we can characterize the above Bessel potential spaces Hm
+p as Sobolev
+spaces
+W m,p(Rd) :=
+�
+f ∈ Lp(Rd) : ∥f∥W m,p(Rd) :=
+�
+α∈A, |α|≤m
+∥∂αf∥Lp(Rd) < ∞
+�
+,
+where ∂αf is to be understood as weak derivatives [AF03]. We refer to [Tri83, Theorem 2.5.6]
+for the proof of the above characterization. As a result, we use Sobolev spaces, which in our
+context are easier to handle, instead of Bessel potential spaces, whenever possible.
+Finally, we introduce general Lp-spaces, which will serve as the solution space for the
+SPDEs (2.11) and (2.12). For a Banach space (Z, ∥·∥Z), some filtration (Ft)t≥0, 1 ≤ p ≤ ∞
+and 0 ≤ s < t ≤ T we denote by Sp
+F([s, t]; Z) the set of Z-valued (Ft)-adapted continuous
+processes (Xu, u ∈ [s, t]) such that
+∥X∥Sp
+F([s,t];Z) :=
+
+
+
+
+
+
+
+�
+E
+�
+sup
+u∈[s,t]
+∥Xu∥p
+Z
+�� 1
+p
+,
+p ∈ [1, ∞)
+sup
+ω∈Ω
+sup
+u∈[s,t]
+∥Xu∥Z ,
+p = ∞
+
+HEGSELMANN–KRAUSE MODEL WITH ENVIRONMENTAL NOISE
+9
+is finite. Similar, Lp
+F([s, t]; Z) denotes the set of Z-valued predictable processes (Xu, u ∈ [s, t])
+such that
+∥X∥Lp
+F([s,t];Z) :=
+
+
+
+
+
+
+
+�
+E
+� t�
+s
+∥Xu∥p
+Z du
+�� 1
+p
+,
+p ∈ [1, ∞)
+sup
+(ω,u)∈Ω×[s,t]
+∥Xu∥Z ,
+p = ∞
+is finite.
+In most case Z will be the Bessel potential space Hn
+p , as it is mainly used by
+Krylov [Kry10] in treating SPDEs.
+For a more detail introduction to the above function
+spaces we refer to [Kry99, Section 3].
+3. Well-posedness of the stochastic Fokker–Planck equations
+This section is dedicated to establish the global existence and uniqueness of weak solutions
+of the stochastic Fokker–Planck equations (2.11) and (2.12) under suitable conditions on the
+initial condition and coefficients. Instead of treating the special case kHK, we will take a more
+general approach and prove existence and uniqueness for general interaction force k: R → R
+under some integrability conditions.
+Before we start our analysis, we introduce the concept of weak solutions.
+Definition 3.1. For a general interaction force k ∈ L2(R), a non-negative stochastic process
+(ρt, t ≥ 0) is called a (weak) solution of the SPDE
+(3.1)
+dρt =
+d2
+dx2
+�σ2
+t + ν2
+2
+ρt
+�
+dt + d
+dx((k ∗ ρt)ρt) dt − ν d
+dxρt dWt,
+t ∈ [0, T],
+if
+(ρt, t ∈ [0, T]) ∈ L2
+FW ([0, T]; W 1,2(R)) ∩ S∞
+FW ([0, T]; L1(R) ∩ L2(R))
+and, for any ϕ ∈ C∞
+c (R), ρ satisfies almost surely the equation, for all t ∈ [0, T],
+⟨ρt, ϕ⟩L2(R) = ⟨ρ0, ϕ⟩L2(R) +
+t
+�
+0
+�σ2
+s + ν2
+2
+ρs, d2
+dx2 ϕ
+�
+L2(R)
+ds −
+t
+�
+0
+�
+(k ∗ ρs)ρs, d
+dxϕ
+�
+L2(R)
+ds
++
+t
+�
+0
+ν
+�
+ρs, d
+dxϕ
+�
+L2(R)
+dWs
+(3.2)
+Remark 3.2. A solution to the stochastic partial differential equation (2.11) and (2.12) is
+defined analogously by replacing k with kHK or kτ
+HK, respectively.
+Remark 3.3. There are multiple solution concepts for SPDEs, see for example [WR15] for
+strong solutions in general separable Hilbert spaces or [DPZ14] for mild solutions with respect
+to a infinitesimal generator. In the present work, we use the concept presented in [Kry99].
+This has the advantage that we can use Itˆo’s formula for Lp-norms [Kry10] as well as the
+linear SPDE theory in [Kry99].
+
+10
+CHEN, NIKOLAEV, AND PR ¨OMEL
+Remark 3.4. Under the assumption that for all t ∈ [0, T], σt ∈ C2(R) we can rewrite formally
+equation (3.2) such that the leading coefficient is in non-divergence form, i.e.
+⟨ρt, ϕ⟩L2(R)
+= ⟨ρ0, ϕ⟩L2(R) + 1
+2
+t
+�
+0
+�
+(σ2
+s + ν2) d2
+dx2 ρs, ϕ
+�
+L2(R)
++ 2
+� d
+dx(σ2
+s + ν2) d
+dxρs, ϕ
+�
+L2(R)
++
+� d2
+dx2 (σ2
+s + ν2)ρs, ϕ
+�
+L2(R)
+ds −
+t
+�
+0
+�
+(k ∗ ρs)ρs, d
+dxϕ
+�
+L2(R)
+ds
+−
+t
+�
+0
+ν
+� d
+dxρs, ϕ
+�
+L2(R)
+dWs.
+Hence, (ρt, t ≥ 0) solves the following SPDE
+dρt = σ2
+t + ν2
+2
+d2
+dx2 ρt dt + d
+dx(σ2
+t + ν2) d
+dxρt dt
++ 1
+2
+d2
+dx2 (σ2
+t + ν2)ρt dt + d
+dx((k ∗ ρt)ρt) dt − ν d
+dxρt dWt,
+t ∈ [0, T].
+In the next theorem we establish uniqueness and local existence of weak solutions to the
+non-local stochastic Fokker–Planck equation (3.1). Furthermore, we are going to see in Corol-
+lary 3.8 that the existence will not depend on the L2-norm of the initial condition ρ0, allowing
+us to extend the local solution obtained in Theorem 3.5 to a global solution on an arbitrary
+interval [0, T].
+Theorem 3.5. Let Assumption 2.1 hold. Further, assume 0 ≤ ρ0 ∈ L1(R) ∩ L2(R) with
+∥ρ0∥L1(R) = 1 and k ∈ L2(R). Then, there exists a T ∗ > 0 and a unique non-negative solution
+of the SPDE (3.1) in the space
+B := L2
+FW ([0, T ∗]; W 1,2(R)) ∩ S∞
+FW ([0, T ∗]; L1(R) ∩ L2(R)).
+Moreover, the solution ρ has the property of mass conservation
+∥ρt∥L1(R) = 1,
+P-a.s.,
+for all t ∈ [0, T].
+Proof. Let us define the metric space
+F T,M :=
+�
+X ∈ S∞
+FW ([0, T]; L2(R)) : ∥X∥S∞
+FW ([0,T];L2(R)) ≤ M
+�
+for some constant M > ∥ρ0∥L2(R), for instance M = 2 ∥ρ0∥L2(R). The metric on F T,M is
+induced by the norm on S∞
+FW ([0, T]; L2(R)). The solution map T : F T,M → F T,M is defined
+as follows. For each ζ ∈ F T,M we define T (ζ) as the solution of the following linear SPDE
+dρt = σ2
+t + ν2
+2
+d2
+dx2 ρt dt + d
+dx(σ2
+t + ν2) d
+dxρt dt
++ 1
+2
+d2
+dx2 (σ2
+t + ν2)ρt dt + d
+dx((k ∗ ζt)ρt) dt − ν d
+dxρt dWt,
+t ∈ [0, T].
+(3.3)
+
+HEGSELMANN–KRAUSE MODEL WITH ENVIRONMENTAL NOISE
+11
+The L2-bound on k and H¨older’s inequality imply
+(3.4)
+|k ∗ ζt| ≤ ∥k∥L2(R) ∥ζt∥L2(R) ≤ ∥k∥L2(R) M,
+which allows us to check the conditions of the Lp-theory of SPDEs [Kry99, Theorem 5.1 and
+Theorem 7.1] for the case n = −1 therein. For instance, if we define for q ∈ W 1,2(R) the
+function
+f(q, t, x) =
+d
+dx((k ∗ ζt)qt),
+then obviously f(0, ·, ·) ∈ L2
+FW ([0, T]; H−1,2(R)) and, since x/(1 + |x|2)1/2 is bounded (see
+[Tri78, Theorem 2.3.8] for the lifting property), we have
+∥f(q, t, ·)∥H−1,2(R)) ≤ C ∥(k ∗ ζt)qt∥L2(R) ≤ ∥k∥L2(R) M ∥qt∥L2(R) .
+By [Kry99, Remark 5.5] this is sufficient to verify [Kry99, Assumption 5.6]. The other as-
+sumptions are proven similarly.
+Hence, we can deduce that the linear SPDE (3.3) admits a unique solution
+ρζ ∈ L2
+FW ([0, T]; W 1,2(R)) ∩ S2
+FW ([0, T]; L2(R)).
+In the next step we want to demonstrate the non-negativity of the solution ρζ with the
+regularity of the solution ρζ. Let us denote by km the mollification of k and let
+ρζ,m ∈ L2
+FW ([0, T]; W 1,2(R)) ∩ S2
+FW ([0, T]; L2(R))
+be the solution of the SPDE (3.3) with (km ∗ ζ)ρζ instead of (k ∗ ζ)ρζ. Then we can write the
+SPDE in the form
+dρζ,m
+t
+= am(t, x) d2
+dx2ρζ,m
+t
+dt + bm(t, x) d
+dxρζ,m
+t
+dt + cm(t, x)ρζ,m
+t
+dt − ν d
+dxρζ,m
+t
+dWt,
+for t ∈ [0, T], with
+am(t, x) := σ2
+t + ν2
+2
+, bm(t, x) :=
+d
+dx(σ2
+t +ν2)+km∗ζt, cm(t, x) := 1
+2
+d2
+dx2 (σ2
+t +ν2)+ d
+dxkm∗ζt.
+Now, by Assumption 2.1 the coefficients am, bm, cm and the coefficient in the stochastic part
+is bounded. Hence, by the maximum principle [Kry99, Theorem 5.12] the solution ρζ,m is
+non-negative. On the other hand, we have
+����
+d
+dx
+�
+(km ∗ ζt)ρζ
+t − (k ∗ ζt)ρζ
+t
+�����
+2
+L2
+FW ([0,T];H−1,2(R))
+≤ C
+���((km − k) ∗ ζt)ρζ
+t
+���
+2
+L2
+FW ([0,T];L2(R))
+≤ E
+�
+T
+�
+0
+∥((km − k) ∗ ζt)∥2
+L∞(R)
+���ρζ
+t
+���
+2
+L2(R)
+�
+≤ E
+�
+T
+�
+0
+∥((km − k)∥2
+L2(R) ∥ζt∥2
+L2(R)
+���ρζ
+t
+���
+2
+L2(R)
+�
+≤ ∥((km − k)∥2
+L2(R) M2 ���ρζ
+t
+���
+2
+L2
+FW ([0,T];L2(R))
+m→∞
+−−−−→ 0.
+
+12
+CHEN, NIKOLAEV, AND PR ¨OMEL
+Consequently, by [Kry99, Theorem 5.7] we have
+lim
+m→∞
+���ρζ,m − ρζ���
+L2
+FW ([0,T];W 1,2(R)) = 0
+and therefore ρζ
+t (·) ≥ 0 for all t ∈ [0, T] almost surely (by intersecting all sets of measure one,
+where ρζ,m is non-negative).
+The non-negativity of the solution ρζ and the divergence structure of the equation provides
+us with the normalization condition/mass conservation, that is
+���ρζ
+t
+���
+L1(R) = ∥ρ0∥L1(R) = 1,
+P-a.s.,
+for t ∈ [0, T]. This follows immediately by plugging in a cut-off sequence (ξn, n ∈ N) for our
+test function ϕ and taking the limit n → ∞ (see [Bre11, p. 212] for properties of the cut-off
+sequence). Therefore, the map T (ζ) = ρζ will be well-defined if we can obtain a bound on
+the S∞
+FW ([0, T]; L2(R))-norm. For readability we will from now on drop the superscript ζ in
+the following. Applying Itˆo’s formula [Kry10], we obtain
+∥ρt∥2
+L2(R) − ∥ρ0∥2
+L2(R)
+= 2ν
+t
+�
+0
+�
+ρs, d
+dxρs
+�
+L2(R)
+dWs +
+t
+�
+0
+�
+ρs, d
+dx(σ2
+s + ν2) d
+dxρs + ρs
+d2
+dx2 (σ2
+s + ν2)
+�
+L2(R)
+ds
+−
+t
+�
+0
+�
+(σ2
+s + ν2) d
+dxρs, d
+dxρs
+�
+L2(R)
+ds − 2
+t
+�
+0
+�
+(k ∗ ζs)ρs, d
+dxρs
+�
+L2(R)
+ds
++ ν2
+t
+�
+0
+����
+d
+dxρs
+����
+2
+L2(R)
+ds
+=
+t
+�
+0
+�
+ρs, d
+dx(σ2
+s + ν2) d
+dxρs + ρs
+d2
+dx2 (σ2
+s + ν2)
+�
+L2(R)
+ds − 2
+t
+�
+0
+�
+(k ∗ ζs)ρs, d
+dxρs
+�
+L2(R)
+ds
+−
+t
+�
+0
+�
+σ2
+s
+d
+dxρs, d
+dxρs
+�
+L2(R)
+ds
+≤
+t
+�
+0
+�
+ρs, d
+dx(σ2
+s + ν2) d
+dxρs + ρs
+d2
+dx2 (σ2
+s + ν2)
+�
+L2(R)
+ds − 2
+t
+�
+0
+�
+(k ∗ ζs)ρs, d
+dxρs
+�
+L2(R)
+ds
+− λ
+t
+�
+0
+����
+d
+dxρs
+����
+2
+L2(R)
+ds,
+for 0 ≤ t ≤ T, where we used the fact that ρs d
+dxρs = 1
+2
+d
+dx(ρ2
+s) to get rid of the stochastic
+integral. At this step, it is crucial that we only have additive common noise. Otherwise the
+stochastic integral will not vanish and the above estimate will not achieve the L∞-bound in ω.
+
+HEGSELMANN–KRAUSE MODEL WITH ENVIRONMENTAL NOISE
+13
+For the first term we can use Assumption 2.1 and Young’s inequality to find
+����
+t
+�
+0
+�
+ρs, d
+dx(σ2
+s + ν2) d
+dxρs + ρs
+d2
+dx2 (σ2
+s + ν2)
+�
+L2(R)
+ds
+����
+(3.5)
+≤ Λ
+t
+�
+0
+����
+�
+ρs, d
+dxρs + ρs
+�
+L2(R)
+���� ds
+≤ λ
+4
+t
+�
+0
+����
+d
+dxρs
+����
+2
+L2(R)
+ds +
+�Λ2
+λ + Λ
+�
+t
+�
+0
+∥ρs∥2
+L2(R) ds.
+On the other hand, using (3.4) and Young’s inequality, we obtain
+����
+�
+(k ∗ ζs)ρs, d
+dxρs
+�
+L2(R)
+���� ≤ ∥k ∗ ζs∥L∞(R)
+�
+|ρs|,
+����
+d
+dxρs
+����
+�
+L2(R)
+≤ ∥k∥L2(R) M ∥ρs∥L2(R)
+����
+d
+dxρs
+����
+L2(R)
+≤
+∥k∥2
+L2(R) M2
+λ
+∥ρs∥2
+L2(R) + λ
+4
+����
+d
+dxρs
+����
+2
+L2(R)
+.
+After absorbing the terms, we find
+∥ρt∥2
+L2(R) − ∥ρ0∥2
+L2(R) ≤
+�∥k∥2
+L2(R) M2 + Λ2
+λ
++ Λ
+�
+t
+�
+0
+∥ρs∥2
+L2(R) ds.
+For the rest of the proof we define the constant
+C(λ, Λ, k, M) :=
+∥k∥2
+L2(R) M2 + Λ2
+λ
++ Λ
+and conclude
+(3.6)
+∥ρt∥2
+L2(R) ≤ ∥ρ0∥2
+L2(R) exp
+�
+C(λ, Λ, k, M)T
+�
+,
+by Gronwall’s inequality.
+Choosing ˆT ∗ < ln(M/ ∥ρ0∥2
+L2(R))C(λ, Λ, k, M)−1, we have ρ ∈
+F ˆT ∗,M and the map
+T : F
+ˆT ∗,M → F
+ˆT ∗,M,
+ζ → ρζ,
+is well-defined up to time ˆT ∗.
+The next step is to show that T is a contraction in a small time span (T ≤ ˆT ∗) and,
+therefore, has a fixed point. For ζ, ˜ζ ∈ F T,M let ρ := T (ζ), ˜ρ := T (˜ζ) be the associated
+solutions of the linear SPDE (3.3). Then, we have
+d(ρt − ˜ρt) = σ2
+t + ν2
+2
+d2
+dx2 (ρt − ˜ρt) dt + (σ2
+t + ν2) d
+dx(ρt − ˜ρt) dt + 1
+2
+d2
+dx2 (σ2
+t + ν2)(ρt − ˜ρt) dt
++ d
+dx((k ∗ ζt)ρt) dt − d
+dx((k ∗ ˜ζt)˜ρt) dt − ν d
+dx(ρt − ˜ρt) dWt,
+t ∈ [0, T].
+
+14
+CHEN, NIKOLAEV, AND PR ¨OMEL
+Applying Itˆo’s formula [Kry10] and multiple Young’s inequality again (see (3.5)), we obtain
+∥ρt − ˜ρt∥2
+L2(R)
+= −
+t
+�
+0
+�
+(σ2
+s + ν2) d
+dxρs − d
+dx ˜ρs, d
+dxρs − d
+dx ˜ρs
+�
+L2(R)
+ds
+− 2
+t
+�
+0
+�
+(k ∗ ζs)ρs − (k ∗ ˜ζs)˜ρs, d
+dxρs − d
+dx ˜ρs
+�
+L2(R)
+ds + ν2
+t
+�
+0
+����
+d
+dx(ρs − ˜ρs)
+����
+2
+L2(R)
+ds
++
+t
+�
+0
+�
+ρs − ˜ρs, d
+dx(σ2
+s + ν2)
+� d
+dxρs − d
+dx ˜ρs
+�
++ (ρs − ˜ρs) d2
+dx2 (σ2
+s + ν2)
+�
+L2(R)
+ds
+≤ −λ
+t
+�
+0
+����
+d
+dxρs − d
+dx ˜ρs
+����
+2
+L2(R)
+ds
+− 2
+t
+�
+0
+�
+(k ∗ (ζs − ˜ζs))ρs + (k ∗ ˜ζs)(ρs − ˜ρs), d
+dxρs − d
+dx ˜ρs
+�
+L2(R)
+ds
++ λ
+4
+t
+�
+0
+����
+d
+dxρs − d
+dx ˜ρs
+����
+2
+L2(R)
+ds +
+�Λ2
+λ + Λ
+�
+t
+�
+0
+∥ρs − ˜ρs∥2
+L2(R) ds
+≤ −3λ
+4
+t
+�
+0
+����
+d
+dxρs − d
+dx ˜ρs
+����
+2
+L2(R)
+ds +
+�Λ2
+λ + Λ
+�
+t
+�
+0
+∥ρs − ˜ρs∥2
+L2(R) ds
++
+t
+�
+0
+∥k∥L2(R)
+���ζs − ˜ζs
+���
+L2(R) ∥ρs∥L2(R)
+����
+d
+dxρs − d
+dx ˜ρs
+����
+L2(R)
+ds
++ ∥k∥L2(R)
+t
+�
+0
+∥ρs − ˜ρs∥L2(R)
+���˜ζs
+���
+L2(R)
+����
+d
+dxρs − d
+dx ˜ρs
+����
+L2(R)
+ds
+≤ −3λ
+4
+t
+�
+0
+����
+d
+dxρs − d
+dx ˜ρs
+����
+2
+L2(R)
+ds +
+�Λ2
+λ + Λ
+�
+t
+�
+0
+∥ρs − ˜ρs∥2
+L2(R) ds
++
+t
+�
+0
+∥k∥2
+L2(R) M2
+λ
+���ζs − ˜ζs
+���
+2
+L2(R) + λ
+4
+����
+d
+dxρs − d
+dx ˜ρs
+����
+2
+L2(R)
+ds
++
+t
+�
+0
+∥k∥2
+L2(R) M2
+λ
+∥ρs − ˜ρs∥2
+L2(R) + λ
+4
+����
+d
+dxρs − d
+dx ˜ρs
+����
+2
+L2(R)
+ds
+
+HEGSELMANN–KRAUSE MODEL WITH ENVIRONMENTAL NOISE
+15
+≤
+t
+�
+0
+∥k∥2
+L2(R) M2
+λ
+���ζs − ˜ζs
+���
+2
+L2(R) ds +
+t
+�
+0
+C(λ, Λ, k, M) ∥ρs − ˜ρs∥2
+L2(R) ds
+≤ T
+∥k∥2
+L2(R) M2
+λ
+���ζ − ˜ζ
+���
+2
+S∞
+FW ([0,T];L2(R)) + C(λ, Λ, k, M)
+t
+�
+0
+∥ρs − ˜ρs∥2
+L2(R) ds.
+Gronwall’s inequality provide us with the estimate
+∥ρ − ˜ρ∥S∞
+FW ([0,T];L2(R)) ≤
+�
+T ∥k∥2
+L2(R) M2
+λ
+exp
+�
+T
+2 C(λ, Λ, k, M)
+�
+∥ζ − ˜ζ∥S∞
+FW ([0,T];L2(R)).
+Now, choosing T ∗ such that
+�
+T ∗ ∥k∥2
+L2(R) M2
+λ
+λ exp
+�
+T ∗
+2 C(λ, Λ, k, M)
+�
+< 1
+and T ∗ ≤ ˆT ∗, we see that the map T : F T ∗,M → F T ∗,M is a contraction and consequently we
+obtain a fixed point ρ, which is a local weak solution up to the time T ∗ of the SPDE (3.1).
+□
+We notice that in the proof of Theorem 3.5, we only use H¨older inequality, Young’s convo-
+lution and product inequality. Hence, the statement of Theorem 3.5 holds also for arbitrary
+dimensions. We state this observation in the following corollary.
+Corollary 3.6. Assume 0 ≤ ρ0 ∈ L1(Rd)∩L2(Rd) with ∥ρ0∥L1(Rd) = 1 and consider a general
+interaction force k ∈ L2(Rd). Then, there exists a T ∗ > 0 and a unique non-negative solution
+of the SPDE (3.1) in the space
+B := L2
+FW ([0, T ∗]; W 1,2(Rd)) ∩ S∞
+FW ([0, T ∗]; L1(Rd) ∩ L2(Rd)).
+The solution should be understood in the sense of Definition 3.1, where Definition 3.1 is
+modified for arbitrary dimension d in the obvious way, see also [Kry99, Definition 3.5].
+Remark 3.7. Following the steps of the proof of Theorem 3.5, we see that we can obtain not
+only a local solution but a (global) solution for any T > 0 by requiring a very small L2-norm
+on the initial condition ρ0. In particular, we can choose a constant M > 0 such that
+�
+T ∥k∥2
+L2(R) M2
+λ
+λ exp
+�
+T
+2
+�
+∥k∥2
+L2(R) M2 + Λ2
+λ
++ Λ
+��
+< 1
+and then the condition
+∥ρ0∥L2(R) ≤ M exp
+�
+− T
+�
+∥k∥2
+L2(R) M2 + Λ2
+λ
++ Λ
+��
+guarantees a unique non-negative solution of the SPDE (3.1) on the interval [0, T].
+Next, we establish another global existence and uniqueness result. We emphasize that in the
+following result we do not need any further assumptions on ρ0 besides being in L1(R)∩L2(R).
+Instead, we impose a lower bound on the diffusion coefficient σ. Hence, we require a sufficiently
+high randomness in stochastic Fokker–Planck equation.
+We also assert the fact that the
+continuation of the solution (ρt, t ≥ 0) is a direct consequence of the L2-theory of SPDEs.
+
+16
+CHEN, NIKOLAEV, AND PR ¨OMEL
+Corollary 3.8. Let Assumption 2.1 hold. Further, assume 0 ≤ ρ0 ∈ L1(R) ∩ L2(R) with
+∥ρ0∥L1(R) = 1 and k ∈ L1(R) ∩ L2(R). Furthermore, assume that the diffusion coefficient σ
+has a derivative
+d
+dxσ with compact support [−L, L] and satisfies
+(3.7)
+2L2 sup
+0≤t≤T
+����
+d
+dx(σ2
+t )
+����
+L∞(R)
++
+�
+C4
+GNS ∥k∥2
+L1(R) + 4L4 sup
+0≤t≤T
+����
+d
+dx(σ2
+t )
+����
+2
+L∞(R)
+�1/2
+≤ λ,
+where CGNS is the constant given by the Gagliardo–Nirenberg–Sobolev interpolation in one
+dimension [Leo17, Theorem 12.83], i.e. CGNS =
+� 4π2
+9
+�−1/4. Then, for each T > 0 there exist
+unique global non-negative solutions of the stochastic Fokker–Planck equations (3.1) in the
+space B.
+Proof. Let ρ be the solution given by Theorem 3.5. Following the proof of Theorem 3.5, we
+apply Itˆo’s formula [Kry10] and obtain, for 0 ≤ t ≤ T ∗,
+∥ρt∥2
+L2(R) − ∥ρ0∥2
+L2(R)
+= −
+t
+�
+0
+����σs
+d
+dxρs
+����
+2
+L2(R)
+ds +
+t
+�
+0
+�
+ρs, d
+dx(σ2
+s + ν2) d
+dxρs + ρs
+d2
+dx2 (σ2
+s + ν2)
+�
+ds
+− 2
+t
+�
+0
+�
+(k ∗ ρs)ρs, d
+dxρs
+�
+L2(R)
+ds
+= −
+t
+�
+0
+����σs
+d
+dxρs
+����
+2
+L2(R)
+ds −
+t
+�
+0
+�
+ρs
+d
+dx(σ2
+s), d
+dxρs
+�
+L2(R)
+ds
+− 2
+t
+�
+0
+�
+(k ∗ ρs)ρs, d
+dxρs
+�
+L2(R)
+ds
+≤ −λ
+2
+t
+�
+0
+����
+d
+dxρs
+����
+2
+L2(R)
+ds + sup
+0≤t≤T
+����
+d
+dx(σ2
+t )
+����
+L∞(R)
+t
+�
+0
+∥ρs∥L2([−L,L])
+����
+d
+dxρs
+����
+L2(R)
+ds
++ 1
+λ
+t
+�
+0
+∥(k ∗ ρs)ρs∥2
+L2(R) ds
+≤
+�
+− λ
+2 + 2L2 sup
+0≤t≤T
+����
+d
+dx(σ2
+t )
+����
+L∞(R)
+�
+t
+�
+0
+����
+d
+dxρs
+����
+2
+L2(R)
+ds
++ 1
+λ
+t
+�
+0
+∥k ∗ ρs∥2
+L4(R) ∥ρs∥2
+L4(R) ds
+≤
+�
+− λ
+2 + 2L2 sup
+0≤t≤T
+����
+d
+dx(σ2
+t )
+����
+L∞(R)
+�
+t
+�
+0
+����
+d
+dxρs
+����
+2
+L2(R)
+ds
+
+HEGSELMANN–KRAUSE MODEL WITH ENVIRONMENTAL NOISE
+17
++ 1
+λ
+t
+�
+0
+∥k∥2
+L1(R) ∥ρs∥2
+L4(R) ∥ρs∥2
+L4(R) ds
+=
+�
+− λ
+2 + 2L2 sup
+0≤t≤T
+����
+d
+dx(σ2
+t )
+����
+L∞(R)
+�
+t
+�
+0
+����
+d
+dxρs
+����
+2
+L2(R)
+ds +
+∥k∥2
+L1(R)
+λ
+t
+�
+0
+∥ρs∥4
+L4(R) ds.
+We remark that we used integration by parts in the first step, Young’s and H¨older’s inequality
+in the third step, H¨older’s and Poincar´e inequaity [Leo17, Theorem 13.19] in the forth step and
+Young’s inequality for convolutions in the fifth step. Let us recall the Gagliardo–Nirenberg–
+Sobolev interpolation [Leo17, Theorem 12.83], which states that for u ∈ L1(R) ∩ W 1,2(R) we
+have
+∥u∥L4(R) ≤ CGNS ∥u∥1/2
+L1(R)
+����
+d
+dxu
+����
+1/2
+L2(R)
+.
+Consequently, applying this inequality on the last term in our estimate and having mass
+conservation in mind we find
+∥ρt∥2
+L2(R) − ∥ρ0∥2
+L2(R)
+≤
+�
+− λ
+2 + 2L2 sup
+0≤t≤T
+����
+d
+dx(σ2
+t )
+����
+L∞(R)
++
+C4
+GNS ∥k∥2
+L1(R)
+λ
+�
+t
+�
+0
+����
+d
+dxρs
+����
+2
+L2(R)
+ds.
+Hence, if
+2L2 sup
+0≤t≤T
+����
+d
+dx(σ2
+t )
+����
+L∞(R)
++
+�
+C4
+GNS ∥k∥2
+L1(R) + 4L4 sup
+0≤t≤T
+����
+d
+dx(σ2
+t )
+����
+2
+L∞(R)
+�1/2
+≤ λ,
+we discover
+(3.8)
+∥ρ∥S∞
+FW ([0,T ∗];L2(R)) ≤ ∥ρ0∥2
+L2(R) .
+Since ρ ∈ L2
+FW ([0, T ∗]; W 1,2(R)), we may apply [Kry99, Theorem 7.1], which tells us that
+ρ ∈ C([0, T ∗], L2(R)), P-a.s., and
+E(∥ρT ∗∥2
+L2(R)) < ∞.
+As a result, we can take ρT ∗ as the new initial value and apply [Kry99, Theorem 5.1] in
+combination with our above arguments in proof of Theorem 3.5 to obtain a solution on
+[T ∗, 2T ∗], since the estimate (3.8) and the condition (3.7) are independent of T ∗. Hence, after
+finitely many steps we have a global solution ρ on [0, T]. The uniqueness and ρ ∈ B follows
+by repeating the inequalities derived in the contraction argument in Theorem 3.5 or using the
+uniqueness of the SPDE presented in [Kry99, Theorem 5.1 and Corollary 5.11].
+□
+Remark 3.9. In particular, for a constant diffusion σ > 0 the condition (3.7) reads simply
+as
+C2
+GNS ∥k∥L1(R) ≤ σ,
+which can be interpreted such that for a given integrable kernel k the system needs a certain
+amount of idiosyncratic noise to stay alive for arbitrary T > 0. In other word, the diffusion
+needs to be dominant.
+Next, we are going to improve the regularity of the solution ρ by a bootstrap argument.
+
+18
+CHEN, NIKOLAEV, AND PR ¨OMEL
+Lemma 3.10. Let ρ0 ∈ L1(R) ∩ W 2,2(R) with ∥ρ0∥L1(R) = 1. Moreover, let Assumption 2.1
+hold and k ∈ L2(R). Assume we have a solution
+ρ ∈ L2
+FW ([0, T]; W 1,2(R)) ∩ S∞
+FW ([0, T]; L1(R) ∩ L2(R))
+of the SPDE (3.1) on [0, T]. Then ρ has the following regularity
+ρ ∈ L2
+FW ([0, T]; W 3,2(R)) ∩ S2
+FW ([0, T], W 2,2(R)) ∩ S∞
+FW ([0, T]; L1(R) ∩ L2(R)).
+Proof. Let us explore the following bootstrap argument.
+By assumptions we know ρ ∈
+L2
+FW ([0, T]; W 1,2(R)) ∩ S∞
+FW ([0, T]; L1(R) ∩ L2(R)) and solves the SPDE
+(3.9)
+dρt =
+d2
+dx2
+�σ2
+t + ν2
+2
+ρt
+�
+dt + d
+dx((k ∗ ρt)ρt) dt − ν d
+dxρt dWt.
+Furthermore,
+d
+dx(kτ ∗ ρt) = kτ ∗
+d
+dxρt for the smooth approximation kτ of k and consequently
+the dominated convergence theorem implies
+d
+dx(k∗ρt) = k∗ d
+dxρt in the sense of distributions.
+As a result
+d
+dx((k ∗ ρt)ρt) =
+�
+k ∗ d
+dxρt
+�
+ρt + (k ∗ ρt) d
+dxρt
+is well-defined as a function in L1(R). Moreover, we find
+����
+d
+dx((k ∗ ρt)ρt)
+����
+L2(R)
+≤
+����
+�
+k ∗ d
+dxρt
+�
+ρt
+����
+L2(R)
++
+����(k ∗ ρt) d
+dxρt
+����
+L2(R)
+≤
+����k ∗ d
+dxρt
+����
+L∞(R)
+∥ρt∥L2(R) + ∥k ∗ ρt∥L∞(R)
+����
+d
+dxρt
+����
+L2(R)
+≤ ∥k∥L2(R) ∥ρt∥W 1,2(R) ∥ρt∥L2(R) + ∥k∥L2(R) ∥ρt∥L2(R) ∥ρt∥W 1,2(R)
+≤ 2 ∥k∥L2(R) ∥ρt∥W 1,2(R) ∥ρt∥L2(R) ,
+which implies
+����
+d
+dx((k ∗ ρ)ρ)
+����
+L2
+FW ([0,T];L2(R))
+≤ 2 ∥k∥L2(R) ∥ρ∥S∞
+FW ([0,T];L2(R)) ∥ρ∥L2
+FW ([0,T];W 1,2(R)) .
+From the uniqueness of the SPDE (3.9), ρ0 ∈ W 1,2(R) and [Kry99, Theorem 5.1 and Theo-
+rem 7.1] we obtain
+ρ ∈ L2
+FW ([0, T]; W 2,2(R)) ∩ S2
+FW ([0, T]; W 1,2(R)).
+With the same strategy and the discovered regularity of ρ one obtains
+d2
+dx2((k ∗ ρt)ρt) =
+�
+k ∗ d2
+dx2ρt
+�
+ρt + 2
+�
+k ∗ d
+dxρt
+� d
+dxρt + (k ∗ ρt) d2
+dx2 ρt
+and consequently
+����
+d2
+dx2 ((k ∗ ρt)ρt)
+����
+L2(R)
+≤ 2 ∥k∥L2(R) ∥ρt∥W 2,2(R) ∥ρt∥L2(R) + 2 ∥k∥L2(R)
+����
+d
+dxρt
+����
+2
+L2(R)
+≤ 4 ∥k∥L2(R) ∥ρt∥W 2,2(R) ∥ρt∥L2(R) ,
+
+HEGSELMANN–KRAUSE MODEL WITH ENVIRONMENTAL NOISE
+19
+where we used Gagliardo–Nirenberg–Sobolev interpolation inequality [Leo17, Theorem 7.41]
+in the last step. Therefore, we have
+����
+d
+dx((k ∗ ρ)ρ)
+����
+L2
+FW ([0,T];W 1,2(R))
+≤ 4 ∥k∥L2(R) ∥ρ∥S∞
+FW ([0,T];L2(R)) ∥ρ∥L2
+FW ([0,T];W 2,2(R)) .
+Again, from the uniqueness of the SPDE (3.9), ρ0 ∈ W 2,2(R) and [Kry99, Theorem 5.1,
+Corollary 5.11, Theorem 7.1] we obtain
+ρ ∈ L2
+FW ([0, T]; W 3,2(R)) ∩ S2
+FW ([0, T], W 2,2(R)).
+□
+As a consequence of Theorem 3.5, Corollary 3.8 and the fact that kHK, kτ
+HK ∈ L1(R)∩L2(R)
+for all τ > 0, we obtain the following corollary.
+Corollary 3.11. Let Assumption 2.1 hold. Further, assume 0 ≤ ρ0 ∈ L1(R) ∩ L2(R) with
+∥ρ0∥L1(R) = 1. Then, there exists a T ∗ > 0 and a unique non-negative solution ρ, ρτ of the
+SPDE (2.11) and (2.12) in the space
+B = L2
+FW ([0, T ∗]; W 1,2(R)) ∩ S∞
+FW ([0, T ∗]; L1(R) ∩ L2(R))
+Furthermore, if
+d
+dxσt(x) has compact support and inequality (3.7) holds, then we can extend
+ρ, ρτ to a global solution.
+4. Well-posedness of the mean-field SDEs
+In this section we establish the existence of unique strong solutions of the mean-field sto-
+chastic differential equations (2.9) and (2.10). Analogously to the classical theory of ordinary
+SDEs, it turns out that the mean-field SDEs (2.9) and (2.10) are linked to the stochastic
+Fokker–Planck equations (2.11) and (2.12) in the same way as ordinary SDEs are linked to
+deterministic Fokker–Planck equations (also called Kolmogorov forward equations).
+Similar to Section 3, we prove the existence of strong solutions for general interaction
+force k. In the following we consider the mean-field SDE
+�
+dYt = −(k ∗ ρt)(Yt) dt + σ(t, Yt) dBt + ν dWt,
+Y0 = X0,
+ρt is the conditional density of Yt given FW
+t ,
+(4.1)
+for a general interaction force k ∈ L1(R) ∩ L2(R). We notice that (4.1) is just one of the
+identically distributed SDE’s of the system (2.9), if we set k = kHK. In order to guarantee
+the well-posedness of the SPDE (3.1) we make the assumption:
+Assumption 4.1. Let 0 ≤ ρ0 ∈ L1(R) ∩ W 2,2(R) with ∥ρ0∥L1(R) = 1. For T > 0 there exists
+a unique solution ρ in L2
+FW ([0, T]; W 1,2(R)) of the SPDE (3.1) on the interval [0, T] with
+∥ρ∥L2
+Fw([0,T];W 1,2(R)) + ∥ρ∥S∞
+Fw([0,T];L1(R)∩L2(R)) ≤ C
+for some finite constant C > 0.
+Remark 4.2. The existence of a unique solution to the SPDE (3.1) in the above assumption
+is satisfied, for instance, if the conditions stated in Remark 3.7, Theorem 3.5 or Corollary 3.8
+are satisfied.
+
+20
+CHEN, NIKOLAEV, AND PR ¨OMEL
+Theorem 4.3. Let Assumption 2.1 as well as Assumption 4.1 hold and k ∈ L1(R) ∩ L2(R).
+Then, the mean-field SDE (4.1) has a unique strong solution (Yt, t ∈ [0, T]) and ρt is the
+conditional density of Yt given FW
+t
+for every t ∈ [0, T].
+The idea to prove Theorem 4.3 is to freeze the measure ρt in the SDE (4.1) and use a duality
+argument by introducing a dual backward stochastic partial differential equation (BSPDE)
+in Lemma 4.4 in order to prove that ρt is the conditional density of Y i
+t for given FW
+t .
+Proof. Let ρ be the unique solution of the SPDE (3.1) as in Assumption 4.1. We recall that
+by the regularity result presented in Lemma 3.10 we have
+ρ ∈ L2
+FW ([0, T]; W 3,2(R)) ∩ S2
+FW ([0, T], W 2,2(R)).
+Step 1. Fix ρ in the mean-field SDE (4.1) and notice that we are dealing with a standard
+SDE with random coefficients. Hence, we can apply classical results if the drift coefficient
+k ∗ ρ is Lipschitz continuous. The regularity of the solution, Sobolev embedding theorem and
+Morrey’s inequality yields
+sup
+0≤t≤T
+sup
+x,y∈R
+x̸=y
+|(k ∗ ρt)(ω, x) − (k ∗ ρt)(ω, y)|
+|x − y|
+≤ sup
+0≤t≤T
+∥k ∗ ρt(ω)∥W 2,2(R)
+≤ ∥k∥L1(R) sup
+0≤t≤T
+∥ρt(ω)∥W 2,2(R)
+and
+sup
+0≤t≤T
+|(k ∗ ρt)(ω, 0)| ≤ sup
+0≤t≤T
+sup
+x∈R
+|(k ∗ ρt)(ω, x)|
+≤ ∥k∥L1(R) sup
+0≤t≤T
+∥ρt(ω)∥W 1,2(R) .
+Furthermore, the maps ω �→ sup
+0≤t≤T
+∥ρt(ω)∥W 2,2(R) and ω �→ sup
+0≤t≤T
+∥ρt(ω)∥W 1,2(R) are measur-
+able. Therefore, standard results for the existence of SDEs with Lipschitz continuous drift,
+see e.g. [KRZ99, Theorem 1.1] or [KHLN97, Theorem 2.2], imply that the following SDE has
+a unique strong solution
+(4.2)
+�
+dY t = −(k ∗ ρt)(Y t) dt + σ(t, Y t) dB1
+t + ν dWt,
+Y 0 ∼ ρ0 .
+Step 2. We are going to use a dual argument (see Lemma 4.4 below) to show that ρt is
+the conditional density of Y 1
+t with respect to FW
+t . Hence, let T1 > 0 and (ut, t ∈ [0, T1])
+be the solution of the BSPDE (4.3) below with terminal condition G ∈ L∞(Ω, FT1, C∞
+c (R)).
+Utilizing the dual equation from Lemma 4.4, the dual analysis [Zho92, Corollary 3.1] and the
+fact that u0 is FW
+0 -measurable, we find
+⟨ρ0, u0⟩ = E(⟨G, ρT1⟩).
+On the other hand we can use the explicit representation of u0 given by Lemma 4.4 to obtain
+⟨ρ0, u0⟩ =
+�
+R
+u0(y)ρ0(y) dy = E(u0(Y 0)) = E(E(G(Y T1)| σ(FW
+0 , σ(Y 0)))) = E(G(Y T1)).
+Now, let G = φξ with φ ∈ C∞
+c (R) and ξ ∈ L∞(Ω, FT1). Consequently, we obtain
+E(ξ⟨φ, ρT1⟩) = E(ξφ(Y T1)) = E(ξE(φ(Y T1) | FW
+T1 )),
+
+HEGSELMANN–KRAUSE MODEL WITH ENVIRONMENTAL NOISE
+21
+which proves
+⟨φ, ρT1⟩ = E(φ(Y T1) | FW
+T1 ),
+P-a.e.,
+and, therefore, ρT1 is the conditional density of Y
+1
+T1 given FT1. Since T1 is arbitrary, we have
+proven the existence of a strong solution Y 1 of the mean-field SDE (4.1).
+On the other hand, if (4.1) has a strong solution with conditional density
+ρ ∈ L2
+FW ([0, T ∗]; W 1,2(R)) ∩ S∞
+FW ([0, T ∗]; L1(R) ∩ L2(R)),
+then the conditional density process of Y 1 is the solution to the SPDE (3.1). Indeed, if we first
+apply Itˆo’s formula with a function ϕ ∈ C∞
+c (R), then take the conditional expectation with
+respect to the filtration FW and subsequently applying stochastic Fubini theorem [HvS21,
+Lemma A.5], we conclude that density process of Y 1
+t satisfies (3.2). By the uniqueness of
+the SPDE (3.1) we obtain that ρt, which is the solution constructed in Theorem 3.5, is the
+conditional density of Y 1
+t given FW
+t
+for all t ∈ [0, T].
+□
+In the following lemma, we close the gap in the above proof by demonstrating the existence
+of a solution of the BSPDE (4.3) and the explicit representation of u0.
+Lemma 4.4 (Dual BSPDE). Let Assumption 2.1 and Assumption 4.1 hold along with k ∈
+L1(R)∩L2(R). Then, for every T1 ∈ (0, T] and G ∈ L∞(Ω, FT1, C∞
+c (R)) the following BSPDE
+dut = −
+�σ2
+t + ν2
+2
+d2
+dx2 ut − (k ∗ ρt) d
+dxut + ν d
+dxvt
+�
+dt + vt dWt,
+t ∈ [0, T],
+uT1 = G,
+(4.3)
+admits a unique solution
+(u, v) ∈ (L2
+FW ([0, T]; W 2,2(R)) ∩ S2
+FW ([0, T]; W 1,2(R))) × L2
+FW ([0, T]; W 1,2(R)),
+i.e. for any ϕ ∈ C∞
+c (R) the equality
+⟨ut, ϕ⟩L2(R) = ⟨G, ϕ⟩L2(R) +
+T1
+�
+t
+�σ2
+s + ν2
+2
+d2
+dx2us − (k ∗ ρs) d
+dxus + ν d
+dxvs, ϕ
+�
+L2(R)
+ds
+−
+T1
+�
+t
+⟨vs, ϕ⟩L2(R) dWs
+holds for all t ∈ [0, T] with probability one. Moreover, we have
+(4.4)
+u0(Y 0) = E(G(Y T1) | σ(σ(Y 0), FW
+0 )),
+where (Y t, t ∈ [0, T]) is the solution of the linearised SDE (4.2) in the proof of Theorem 4.3.
+Proof. We note that by Theorem 4.3 we have ρ ∈ L2
+F([0, T]; W 3,2(R)) ∩ S2
+F([0, T]; W 2,2(R)).
+Our approach is to verify the assumptions of the L2-theory (see for example [DQT12, Theo-
+rem 5.5]) for BSPDEs. Let u1, u2 ∈ W 2,2(R), then
+����(k ∗ ρt) d
+dxu1 − (k ∗ ρt) d
+dxu2
+����
+L2(R)
+≤ ∥k ∗ ρt∥L∞(R)
+����
+d
+dx(u1 − u2)
+����
+L2(R)
+≤ ∥k∥L2(R) ∥ρt∥L2(R)
+����
+d
+dx(u1 − u2)
+����
+L2(R)
+≤ ∥k∥L2(R) ∥ρt∥L2(R) ∥u1 − u2∥W 1,2(R) .
+
+22
+CHEN, NIKOLAEV, AND PR ¨OMEL
+Now, by Theorem 3.5, ∥ρt∥L2(R) is uniformly bounded in (ω, t) ∈ Ω × [0, T] and the interpo-
+lation theorem [AF03, Theorem 5.2] implies for all ε > 0,
+����(k ∗ ρt) d
+dxu1 − (k ∗ ρt) d
+dxu2
+����
+L2(R)
+≤ ε ∥u1 − u2∥W 2,2(R) + C(k, ∥ρ∥S∞
+FW ([0,T];L2(R)))κ(ε) ∥u1 − u2∥L2(R)
+for some non-negative decreasing function κ. Hence, Assumption 5.4 in [DQT12, Theorem 5.5]
+is satisfied. The other assumptions are also easily verified. As a result we obtain a solution
+(u, v) ∈ (L2
+FW ([0, T]; W 2,2(R)) ∩ S2
+FW ([0, T]; L2(R))) × L2
+FW ([0, T]; W 1,2(R))
+of the BSPDE (4.3). Here, the fact that u ∈ S2
+FW ([0, T]; L2(R)) is a direct consequence of
+[DM10, Theorem 2.2].
+It remains to show that the equality (4.4) holds. By the bound
+E
+�
+sup
+t≤T1
+∥ρt∥2
+W 1,2(R)
+�
+< ∞
+given by ρ ∈ S2
+FW ([0, T]; W 1,2(R)), we observe that there exists a set Ω′ with P(Ω′) = 1 and
+for all ω ∈ Ω′ we have
+(4.5)
+sup
+t≤T1
+∥ρt(ω, ·)∥W 1,2(R) < ∞.
+Also, the map (ω, t) → ∥ρt(ω, ·)∥W 1,2(R) is predictable with respect to FW by the L2-SPDE
+theory. Consequently, we can define for each m ∈ N the stopping time
+τm(ω) = inf{t ∈ [0, T1] : ∥ρt(ω, ·)∥W 1,2(R) ≥ m}
+and τm ↑ T1 by (4.5). Furthermore, let us define
+F(t, x) := (k ∗ ρt)(x) d
+dxut(x)
+and
+Fm(t, x) := F(t, x)1(0,τm](t),
+and note that Fm ∈ L2
+FW ([0, T]; L2(R)) still satisfies all assumptions of the L2-BSPDE theory
+([DQT12, Theorem 5.5]) and therefore there exists a solution
+(um, vm) ∈ (L2
+FW ([0, T]; W 2,2(R)) ∩ S2
+FW ([0, T]; L2(R))) × L2
+FW ([0, T]; W 1,2(R))
+of the following BSPDE
+dum
+t = −
+�σ2
+t + ν2
+2
+d2
+dx2 um
+t − Fm(t) + ν d
+dxvm
+t
+�
+dt + vm
+t dWt,
+um
+T1 = G,
+for each m ∈ N.
+
+HEGSELMANN–KRAUSE MODEL WITH ENVIRONMENTAL NOISE
+23
+In the next step we want to obtain a L2
+FW ([0, T]; W 1,2(R))-bound for Fm. The L2-estimate
+follows directly from the above computations. For the weak derivative we compute
+����
+d
+dx
+�
+(k ∗ ρt) d
+dxut
+�
+1(0,τm]
+����
+L2(R)
+≤
+����
+1(0,τm]
+�
+k ∗ d
+dxρt
+� d
+dxut
+����
+L2(R)
++
+����(k ∗ ρt) d2
+dx2 ut
+����
+L2(R)
+≤
+1(0,τm]
+����
+�
+k ∗ d
+dxρt
+� d
+dxut
+����
+L2(R)
++ ∥k∥L2(R) ∥ρt∥S∞
+F ([0,T];L2(R))) ∥ut∥W 2,2(R) .
+Since ρ ∈ S∞
+FW ([0, T]; L2(R))), the last term behaves nicely. However, the first term would
+be problematic because without the stopping time we do not have a similar L∞-estimate for
+the derivative, i.e. ρ ∈ S∞
+FW ([0, T]; W 1,2(R))). Hence, in order to overcome this problem we
+introduced the stopping time τm and, therefore, we discover
+����
+d
+dx
+�
+(k ∗ ρt) d
+dxut
+�
+1(0,τm]
+����
+L2
+FW ([0,T1];L2(R))
+≤ ∥k∥2
+L2(R) E
+� T1
+�
+0
+1(0,τm](t)
+����
+d
+dxρt
+����
+2
+L2(R)
+����
+d
+dxut
+����
+2
+L2(R)
+dt
+�
++ ∥k∥L2(R) ∥ρt∥S∞
+F ([0,T];L2(R))) ∥ut∥L2
+FW ([0,T];W 2,2(R))
+≤ ∥k∥L2(R) m2 ∥u∥L2
+FW ([0,T];W 1,2(R))
++ ∥k∥L2(R) ∥ρt∥S∞
+F ([0,T];L2(R))) ∥ut∥L2
+FW ([0,T];W 2,2(R)) .
+As a result, we obtain
+∥Fm∥L2
+FW ([0,T];W 1,2(R)) < ∞
+for each m ∈ N. Applying [DQT12, Theorem 5.5] again, we find
+(um, vm) ∈ (L2
+FW ([0, T]; W 3,2(R)) ∩ S2
+FW ([0, T]; W 1,2(R))) × L2
+FW ([0, T]; W 2,2(R)).
+The above regularity (p(m − 2) > 1 with p = 2, m = 3) allows us to apply [DTZ13, Corol-
+lary 2.2], which tells us that there exists a set of full measure Ω′′
+m maybe different from Ω′
+such that
+um(t, x) = G(x) +
+T1
+�
+t
+σ2
+t + ν2
+2
+d2
+dx2 um(s, x) −
+1(0,τm](s)(k ∗ ρ)(s, x) d
+dxu(s, x)
++ ν d
+dxvm(s, x) ds −
+T1
+�
+t
+vm(s, x) dWs
+holds for all (t, x) ∈ [0, T1] × R on Ω′′
+m. We use the subscript m to indicate that even though
+the set Ω′′
+m is independent of (t, x) it still may depend on m ∈ N.
+Besides, to be precise, [Du20, Corollary 2.2] actually requires Fm ∈ L2
+FW ([0, T]; W 3,2(R)),
+which is more regularity than we have.
+However, one can modify the proof of [DTZ13,
+Corollary 2.2] to obtain the same result with Fm ∈ L2
+FW ([0, T]; W 1,2(R)). The crucial part
+is that a mollification of Fm with the standard mollifier converges in the supremum norm
+
+24
+CHEN, NIKOLAEV, AND PR ¨OMEL
+to Fm, which follows from Morrey’s inequality even in our case Fm ∈ L2
+FW ([0, T]; W 1,2(R)).
+For a similar result in the SPDE setting we refer to [Roz90, Lemma 4.1].
+Next, we want to apply an Itˆo–Wentzell type formula [YT13, Theorem 3.1] (with V =
+u, X = Y therein). Hence, we need to verify the required assumption. First, we can view um
+as a jointly continuous Itˆo process in (t, x) by [DTZ13, Corollary 2.2] on the set Ω′′
+m. We also
+recall that um ∈ L2
+FW ([0, T]; W 2,2(R)), vm ∈ L2
+FW ([0, T]; W 1,2(R)), Fm ∈ L2
+FW ([0, T]; L2(R))
+and Y is a strong solution and therefore a continuous semimartingale.
+Moreover, we note that ρ,
+d
+dxu are PW × B(R)-measurable and τm is PW -measurable.
+Hence, the same holds true for Fm. Also as previously mentioned (k ∗ρt) is bounded in x ∈ R
+for almost all (ω, t) ∈ Ω × [0, T1] and as we have seen in the proof of Theorem 4.3 (Step 1)
+is Lipschitz continuous for almost all (ω, t) ∈ Ω × [0, T1]. Thus, all assumptions of [YT13,
+Theorem 3.1] are fulfilled and we obtain
+um
+T1(Y T1)
+= um
+0 (Y 0) +
+T1
+�
+0
+�σ2
+t + ν2
+2
+d2
+dx2 um
+t − (k ∗ ρt) d
+dxut + ν d
+dxvm
+t − σ2
+t + ν2
+2
+d2
+dx2 um
+t
++
+1(0,τm](t)(k ∗ ρt) d
+dxut − ν d
+dxvm
+t
+�
+(Y t) dt
++
+T1
+�
+0
+�
+vm
+t + ν d
+dxum
+t
+�
+(Y t) dWt +
+T1
+�
+0
+σt(Y t) d
+dxum
+t (Y t) dBt
+= um
+0 (Y 0) +
+T1
+�
+0
+F(t, Y t)(1(0,τm](t) − 1) dt +
+T1
+�
+0
+�
+vm
+t + ν d
+dxum
+t
+�
+(Y t) dWt
++
+T1
+�
+0
+σt(Y t) d
+dxum
+t (Y t) dBt.
+(4.6)
+With this formula at hand, let us introduce the filtration Gt = σ(σ(Y t), FW
+t ), t ∈ [0, T].
+Our aim is to take the conditional expectation with respect to G0 on both sides of the above
+equation in order to cancel both the stochastic integrals. We observe that Gt ⊂ Ft and the
+solution (Y t, t ∈ [0, T]) is predictable with respect to the filtration G. Moreover, B1 and W
+are still per definition Brownian motions under the filtration (Ft, t ∈ [0, T]). Hence, both
+stochastic integrals are martingales with respect to the filtration (Ft, t ∈ [0, T]), if we can
+prove an L2-bound on the integrands.
+By Sobolev’s embedding or Morrey’s inequality and the bound on σt we have
+E
+� T1
+�
+0
+����σt(Y t) d
+dxum
+t (Y t)
+����
+2
+dt
+�
+≤ ΛE
+� T1
+�
+0
+����
+d
+dxum
+t
+����
+2
+L∞(R)
+dt
+�
+≤ ΛE
+� T1
+�
+0
+∥um
+t ∥2
+W 2,2(R) dt
+�
+< ∞,
+
+HEGSELMANN–KRAUSE MODEL WITH ENVIRONMENTAL NOISE
+25
+which verifies that the second stochastic integral of (4.6) is a martingale with respect to the
+filtration F. Hence, we discover
+E
+� T1
+�
+0
+σt(Y t) d
+dxum
+t (Y t) dBt
+���� G0
+�
+= E
+�
+E
+� T1
+�
+0
+d
+dxum
+t (Y t) dBt |F0
+� ���� G0
+�
+= 0.
+Furthermore, we have the estimate
+E
+� T1
+�
+0
+����vm
+t (Y t) + ν d
+dxum
+t (Y t)
+����
+2
+dt
+�
+≤ 2E
+� T1
+�
+0
+∥vm
+t ∥2
+L∞(R) + ν2
+����
+d
+dxum
+t
+����
+2
+L∞(R)
+dt
+�
+≤ CE
+� T1
+�
+0
+∥vm
+t ∥2
+W 1,2(R) + ∥um
+t ∥2
+W 2,2(R) dt
+�
+< ∞,
+where we used Morrey’s inequality in the second step. Hence, the first stochastic integral
+appearing in (4.6) is also a martingale with respect to the filtration G starting at zero. Taking
+the conditional expectation with respect to G0 in (4.6) and having in mind that Y 0 = Y0, we
+obtain
+E(um
+T1(Y T1) | G0) = um
+0 (Y0) + E
+� T1
+�
+0
+F(t, Y t)(1(0,τm](t) − 1) dt
+���� G0
+�
+.
+It remains to show that
+lim
+m→∞(E(um
+T1(Y T1) | G0) − um
+0 (Y0)) = E(uT1(Y T1) | G0) − u0(Y0),
+P-a.e.,
+(4.7)
+and
+(4.8)
+lim
+m→∞ E
+� T1
+�
+0
+F(t, Y t)(1(0,τm](t) − 1) dt
+���� G0
+�
+= 0,
+P-a.e.
+We first show (4.8) but we prove the L1-convergence, which then implies (4.8) along a subse-
+quence. We compute
+E
+�����E
+� T1
+�
+0
+F(t, Y t)(1(0,τm](t) − 1) dt
+���� G0
+�����
+�
+≤ E
+�
+E
+�����
+T1
+�
+0
+F(t, Y t)(1(0,τm](t) − 1) dt
+����
+���� G0
+��
+≤ E
+� T1
+�
+0
+|F(t, Y t)(1(0,τm](t) − 1)| dt
+�
+≤ E
+� T1
+�
+0
+∥F(t, ·)∥L∞(R) |1(0,τm](t) − 1| dt
+�
+
+26
+CHEN, NIKOLAEV, AND PR ¨OMEL
+≤ E
+� T1
+�
+0
+∥F(t, ·)∥W 1,2(R) |1(0,τm](t) − 1| dt
+�
+≤ CE
+� T1
+�
+0
+� ����(k ∗ ρt) d
+dxut
+����
+L2(R)
++
+����(k ∗ ρt) d2
+dx2ut
+����
+L2(R)
++
+����(k ∗ d
+dxρt) d
+dxut
+����
+L2(R)
+�
+|1(0,τm](t) − 1| dt
+�
+≤ C(T)E
+� T1
+�
+0
+�
+2 ∥ρt∥L2(R) +
+����
+d
+dxρt
+����
+L2(R)
+�
+∥ut∥W 2,2(R) |1(0,τm](t) − 1| dt
+�
+≤ C(T)E
+� T1
+�
+0
+(∥ρt∥2
+W 1,2(R) + ∥ut∥2
+W 2,2(R))|1(0,τm](t) − 1| dt
+�
+,
+where we used Morrey’s inequality in the fourth step and H¨older’s inequality as well as (3.4)
+in the sixth step. Finally, ρ ∈ L2
+FW ([0, T]; W 1,2(R)), u ∈ L2
+FW ([0, T]; W 2,2(R)), dominated
+convergence theorem and τm ↑ T1 tell us that the last term vanishes for m → ∞.
+Taking the above subsequence, which we do not rename, we demonstrate (4.7) along a
+further subsequence by proving L2-convergence of (4.7). Let us define �um = u − um and
+�vm = v − vm, which solve the following BSPDE
+d�um
+t = −
+�σ2
+t + ν2
+2
+d2
+dx2 �um
+t − �Fm + ν d
+dx�vm
+t
+�
+dt + �vm
+t dWt
+with terminal condition �G = 0, free term
+�Fm(t.x) = (k ∗ ρ)(t, x) d
+dxu(t, x)(1 −
+1(0,τm](t)) = F(t, x)(1 −
+1(0,τm](t))
+and �Fm ∈ L2
+FW ([0, T]; L2(R)). Hence, by [DQT12, Theorem 5.5] the solution of the BSPDE
+is unique and by [DM10, Proposition 3.2 and Proposition 4.1, Step 1] satisfies the estimate
+(4.9)
+E
+�
+sup
+t≤T1
+∥�um
+t ∥2
+W 1,2(R)
+�
+≤ C(T)
+��� �Fm
+���
+L2
+FW ([0,T];L2(R)) .
+Consequently, using Jensen inequality, the G0-measurability of u0(Y0), Morrey’s inequality
+and (4.9) we find
+E(|E(um
+T1(Y T1) | G0) − um
+0 (Y0)) − (E(uT1(Y T1) | G0) − u0(Y0))|2)
+≤ 2E(|�um
+T1(Y T1)|2 + |�um
+0 (Y0)|2)
+≤ 2E(
+���um
+T1
+��2
+L∞(R) + ∥�um
+0 ∥2
+L∞(R))
+≤ 2E(
+���um
+T1
+��2
+W 1,2(R) + ∥�um
+0 ∥2
+W 1,2(R))
+≤ 4E
+�
+sup
+t≤T1
+∥�um
+t ∥2
+W 1,2(R)
+�
+
+HEGSELMANN–KRAUSE MODEL WITH ENVIRONMENTAL NOISE
+27
+≤ C(T)E
+� T1
+�
+0
+���( �Fm)t
+���
+2
+L2(R) dt
+�
+≤ C(T)E
+� T1
+�
+0
+|1 −
+1(0,τm](t)|2 ∥F(t, x)∥2
+L2(R) dt
+�
+.
+But F ∈ L2
+FW ([0, T]; L2(R)) and, therefore, an application of the dominated convergence
+theorem proves (4.7) along a subsequence. As a result, the last inequality together with (4.8)
+implies (4.4) for all ω ∈ �Ω := �
+m∈N
+Ω′′
+m ∩ Ω′. Hence, the lemma is proven.
+□
+As an application of Theorem (4.3), we obtain a solution for the non-regularized and the
+regularized mean-field SDEs (2.9) and (2.10), respectively.
+Corollary 4.5. Let 0 ≤ ρ0 ∈ L1(R) ∩ W 2,2(R) with ∥ρ0∥L1(R) = 1. Suppose that for T > 0
+there exists unique solutions ρ and ρτ in L2
+FW ([0, T]; W 1,2(R)) of the SPDEs (2.11) and (2.12),
+respectively, on the interval [0, T] with
+∥ρ∥L2
+Fw([0,T];W 1,2(R)) + ∥ρ∥S∞
+Fw([0,T];L1(R)∩L2(R)) ≤ C
+and
+∥ρτ∥L2
+Fw([0,T];W 1,2(R)) + ∥ρτ∥S∞
+Fw([0,T];L1(R)∩L2(R)) ≤ C,
+for some finite constant C > 0. Moreover, let the diffusion coefficient σ: [0, T] × R → R
+satisfy Assumption 2.1. Then, for τ > 0 there exists unique solution (Y i, t ∈ [0, T]) and
+(Y i,τ, t ∈ [0, T]) for the mean-field SDEs (2.9) and (2.10), respectively. Moreover, ρt is the
+conditional density of Y i
+t given FW
+t
+and ρτ
+t is the conditional density of Y i,τ
+t
+given FW
+t , for
+every t ∈ [0, T] and for all i ∈ N.
+5. Mean-field limits of the interacting particle systems
+In this section we establish propagation of chaos for the regularized Hegelsmann–Krause
+models (in particular we recall kτ
+HK(x) is the approximation of kHK(x) =
+1[−R,R](x)x) with en-
+vironmental noise (2.8) towards the (non-regularized) mean-field stochastic differential equa-
+tions (2.9) resp. the (non-regularized) stochastic Fokker–Planck equation (2.11), presenting
+the density based model of the opinion dynamics, see Theorem 5.5. To prove propagation of
+chaos, we first derive estimates of the difference of the regularized interacting particle sys-
+tem (2.8) and the regularized mean-field SDE (2.10) (see Proposition 5.2) as well as of the
+difference of the solutions to the regularized stochastic Fokker–Planack equation (2.12) and
+of the non-regularized stochastic Fokker–Planack equation (2.11) (see Proposition 5.3). As
+preparation, we need the following auxiliary lemma.
+Lemma 5.1. Let (Ω, F, P) be a probability space, G ⊆ F a sub-σ-algebra and X, Y con-
+ditionally independent random variables with values in R given G. Moreover, let X have a
+conditional density f : Ω×R → R such that f is G ⊗B(R)/B(R) measurable and in L1(Ω×R).
+Then, for every bounded measurable functions h: R × R → R, we have
+(5.1)
+E(h(X, Y ) | σ(G, σ(Y )))(ω) =
+�
+R
+h(z, Y (ω))f(ω, z) dz,
+ω ∈ Ω′,
+on a set Ω′ ⊂ Ω of full probability.
+
+28
+CHEN, NIKOLAEV, AND PR ¨OMEL
+Proof. First, we notice that by Fubini’s theorem the right-hand side of (5.1) is σ(G, σ(Y ))-
+measurable.
+By the standard Lebesgue integral approximation technique we may assume
+h =
+1B×B′(x, y) for some measurable sets B, B′ ∈ B(R) in order to prove (5.1). Hence, we
+need to show
+E(1A
+1B×B′(X, Y )) = E
+�
+1A
+�
+R
+1B×B′(z, Y (ω))f(ω, z) dz
+�
+for all A ∈ σ(G, σ(Y )). Now, we reduce the problem again to A = C ∩ C′′ with C ∈ G and
+C′′ = Y −1(B′′) for some B′′ ∈ B(R). Consequently, using the conditional independence we
+find
+E(1C∩C′′
+1B×B′(X, Y )) = E(1CE(1C′′
+1B×B′(X, Y ) | G))
+= E(1CE(1B′′∩B′(Y )1B(X) | G))
+= E(1CE(1B′′∩B′(Y ) | G)E(1B(X) | G))
+= E(1C
+1B′′∩B′(Y )E(1B(X) | G))
+= E
+�
+1C∩C′′(ω)1B′(Y (ω))
+�
+R
+1B(z)f(ω, z) dz
+�
+= E
+�
+1C∩C′′(ω)
+�
+R
+1B×B′(z, Y (ω))f(ω, z) dz
+�
+and the lemma is proven.
+□
+The next proposition provides an estimate of the difference of the regularized particle
+system and the regularized mean-field SDE.
+Proposition 5.2. Suppose Assumption 2.1 and Assumption 4.1 hold. For each N ∈ N, let
+((Y i,τ
+t
+, t ∈ [0, T]), i = 1, . . . , N) be the solutions to the regularized mean-field SDEs (2.10), as
+provided by Corollary 4.5, and let ((Xi,τ
+t , t ∈ [0, T]), i = 1, . . . , N) be the solution to regularized
+interaction particle system (2.8). Then, for any τ > 0 and N ∈ N we have
+sup
+t∈[0,T]
+sup
+i=1,...,N
+E(|Xi,τ
+t
+− Y i,τ
+t
+|2) ≤
+2x ∥kHK∥2
+L2(R) T
+(N − 1)τ
+exp
+�(C + Λ)T
+τ
+�
+,
+where C is some finite generic constant.
+Proof. Applying Itˆo’s formula, we find
+|Xi,τ
+t
+− Y i,τ
+t
+|2
+= 2
+t
+�
+0
+(Xi,τ
+s
+− Y i,τ
+s
+)
+1
+N − 1
+N
+�
+j=1
+j̸=i
+−kτ
+HK(Xi,τ
+s
+− Xj,τ
+s ) + (kτ
+HK ∗ ρτ
+s)(Y i,τ
+s
+) ds
++ 2
+t
+�
+0
+(Xi,τ
+s
+− Y i,τ
+s
+)(σ(s, Xi,τ
+s ) − σ(s, Y i,τ
+s
+)) dBi
+s +
+t
+�
+0
+(σ(s, Xi,τ
+s ) − σ(s, Y i,τ
+s
+))2 ds.
+(5.2)
+
+HEGSELMANN–KRAUSE MODEL WITH ENVIRONMENTAL NOISE
+29
+Splitting the sum we have
+1
+N − 1
+N
+�
+j=1
+j̸=i
+−kτ
+HK(Xi,τ
+s
+− Xj,τ
+s ) + (kτ
+HK ∗ ρτ
+s)(Y i,τ
+s
+)
+=
+1
+N − 1
+N
+�
+j=1
+j̸=i
+(kτ
+HK ∗ ρτ
+s)(Y i,τ
+s
+) − kτ
+HK(Y i,τ
+s
+− Y j,τ
+s
+)
++
+1
+N − 1
+N
+�
+j=1
+j̸=i
+kτ
+HK(Y i,τ
+s
+− Y j,τ
+s
+) − kτ
+HK(Xi,τ
+s
+− Xj,τ
+s )
+= Is
+1 + Is
+2.
+For Is
+2, we use the property of our approximation sequence to discover
+|Is
+2| ≤
+1
+N − 1
+N
+�
+j=1
+j̸=i
+|kτ
+HK(Y i,τ
+s
+− Y j,τ
+s
+) − kτ
+HK(Xi,τ
+s
+− Xj,τ
+s )|
+≤
+C
+(N − 1)τ
+N
+�
+j=1
+|Xj,τ
+s
+− Y j,τ
+s
+| + |Xi,τ
+s
+− Y i,τ
+s
+|
+and consequently
+E
+�����2
+t
+�
+0
+(Xi,τ
+s
+− Y i,τ
+s
+)Is
+2 ds
+����
+�
+≤
+C
+(N − 1)τ
+t
+�
+0
+N
+�
+j=1
+E(|Xj,τ
+s
+− Y j,τ
+s
+|2 + |Xi,τ
+s
+− Y i,τ
+s
+|2) ds
+≤ C
+τ
+t
+�
+0
+sup
+i=1,...,N
+E(|Xi,τ
+s
+− Y i,τ
+s
+|2) ds,
+where we used Young’s inequality. Next, let us rewrite Is
+1 such that
+Is
+1 =
+1
+N − 1
+N
+�
+j=1
+j̸=i
+(kτ
+HK ∗ ρτ
+s)(Y i,τ
+s
+) − kτ
+HK(Y i,τ
+s
+− Y j,τ
+s
+) =
+1
+N − 1
+N
+�
+j=1
+j̸=i
+Zs
+i,j
+with
+Zs
+i,j = (kτ
+HK ∗ ρτ
+s)(Y i,τ
+s
+) − kτ
+HK(Y i,τ
+s
+− Y j,τ
+s
+)
+for i ̸= j. Furthermore,
+E(|Is
+1|2) =
+1
+(N − 1)2 E
+�
+E
+� N
+�
+j=1
+j̸=i
+Zs
+i,j
+N
+�
+k=1
+j̸=i
+Zs
+i,k
+���� σ(FW
+s , σ(Y i,τ
+s
+))
+��
+=
+1
+(N − 1)2
+N
+�
+j=1
+j̸=i
+N
+�
+k=1
+j̸=i
+E(E(Zs
+i,jZs
+i,k | σ(FW
+s , σ(Y i,τ
+s
+))).
+
+30
+CHEN, NIKOLAEV, AND PR ¨OMEL
+It easy to verify that (Y i,τ
+s
+, i = 1, . . . , N) are conditionally independent given FW
+s
+and by
+Theorem 4.3 have conditional density ρs. Hence, we apply Lemma 5.1 to obtain
+E(kτ
+HK(Y i,τ
+s
+− Y j,τ
+s
+) | σ(FW
+s , σ(Y i
+s , τ))) = (kτ
+HK ∗ ρτ
+s)(Y i,τ
+s
+)
+and therefore E(Zs
+i,j | σ(FW
+s , σ(Y i,τ
+s
+))) = 0 since (kτ
+HK∗ρτ
+s)(Y i,τ
+s
+) is σ(FW
+s , σ(Y i,τ
+s
+))-measurable.
+Consequently, for the cross terms j ̸= k one can verify that
+E(Zs
+i,jZs
+i,k | σ(FW
+s , σ(Y ,τ
+s ))) = E(Zs
+i,j | σ(FW
+s , σ(Y ,τ
+s )))E(Zs
+i,k | σ(FW
+s , σ(Y ,τ
+s ))) = 0
+by the previous findings. Hence, we have
+E(|Is
+1|2) =
+1
+(N − 1)2
+N
+�
+j=1
+j̸=i
+E(|Zs
+i,j|2)
+and using the boundedness of kτ, the structure of our approximation and mass conservation,
+we obtain
+E(|Zs
+i,j|2) = E(|(kτ
+HK ∗ ρτ
+s)(Y i,τ
+s
+) − kτ
+HK(Y i,τ
+s
+− Y j,τ
+s
+)|2) ≤ 2 ∥kτ
+HK∥2
+L∞(R) ≤ 2
+τ ∥kHK∥2
+L2(R) .
+Combining all the above facts, we get
+E(|Is
+1|2) ≤
+2 ∥kHK∥2
+L2(R)
+(N − 1)τ
+and
+E
+�
+2
+t
+�
+0
+(Xi,τ
+t
+− Y i,τ
+t
+)Is
+1 ds
+�
+≤ E
+�
+t
+�
+0
+|Xi,τ
+t
+− Y i,τ
+t
+|2 ds +
+t
+�
+0
+|Is
+1|2 ds
+�
+≤
+t
+�
+0
+E(|Xi,τ
+t
+− Y i,τ
+t
+|2) ds +
+2 ∥kHK∥2
+L2(R) T
+(N − 1)τ
+.
+Moreover, using the Lipschitz continuity of our coefficients σ we obtain
+E
+�
+t
+�
+0
+(σ(s, Xi,τ
+s ) − σ(s, Y i,τ
+s
+))2 ds
+�
+≤ Λ
+t
+�
+0
+E
+�
+|Xi,τ
+s
+− Y i,τ
+s
+|2�
+≤ Λ
+t
+�
+0
+sup
+i=1,...,N
+E
+�
+|Xi,τ
+s
+− Y i,τ
+s
+|2�
+.
+Now, combining this with the estimate of Is
+2, as wells as the fact that the stochastic integral
+in equation (5.2) are martingales (Assumption 2.1), we obtain
+sup
+i=1,...,N
+E(|Xi,τ
+t
+− Y i,τ
+t
+|2) ≤ C
+τ
+t
+�
+0
+sup
+i=1,...,N
+E(|Xi,τ
+t
+− Y i,τ
+t
+|2) ds
++ Λ
+t
+�
+0
+sup
+i=1,...,N
+E(|Xi,τ
+t
+− Y i,τ
+t
+|2) ds +
+2 ∥kHK∥2
+L2(R) T
+(N − 1)τ
+≤ C + Λ
+τ
+t
+�
+0
+sup
+i=1,...,N
+E(|Xi,τ
+t
+− Y i,τ
+t
+|2) ds +
+2 ∥kHK∥2
+L2(R) T
+(N − 1)τ
+.
+
+HEGSELMANN–KRAUSE MODEL WITH ENVIRONMENTAL NOISE
+31
+Applying Gronwall’s inequality yields
+sup
+t∈[0,T]
+sup
+i=1,...,N
+E(|Xi,τ
+t
+− Y i,τ
+t
+|2) ≤
+2 ∥kHK∥2
+L2(R) T
+(N − 1)τ
+exp
+�(C + Λ)T
+τ
+�
+,
+which proves the proposition.
+□
+As next step we need an estimate of the difference of the solutions to the regularized
+mean-field SDEs and the non-regularized mean-field SDE. Recall that, by the stochastic
+Fokker–Planack equations, it is sufficient to consider the associated solutions ρτ and ρ of the
+SPDEs (2.12) and (2.11). For more details regarding this observation we refer to the proof of
+Theorem 4.3.
+Proposition 5.3. Suppose Assumption 2.1 and Assumption 4.1 hold. Let ρτ and ρ be the so-
+lutions to the regularized stochastic Fokker–Planack equation (2.12) and to the non-regularized
+stochastic Fokker–Planack equation (2.11), respectively, as provided in Corollary 3.11. Then,
+∥ρτ − ρ∥S∞
+FW ([0,T];L2(R))
+≤ C(λ, Λ, T, ∥kHK∥L2(R) , ∥ρ∥S∞
+FW ([0,T];L2(R)) ∥ρτ∥S∞
+FW ([0,T];L2(R))) ∥kτ
+HK − kHK∥L2(R) .
+Proof. To estimate the difference ρt − ρτ
+t , we notice that
+ρτ
+t − ρt =
+d2
+dx2
+�
+σ2
+t + ν2
+2
+(ρτ
+t − ρt)
+�
+dt + d
+dx((kτ
+HK ∗ ρτ
+t )ρτ
+t ) dt
+− d
+dx((kHK ∗ ρt)ρt) dt − ν d
+dx(ρτ
+t − ρt) dWt.
+Performing similar computations as in the proof of Theorem 3.5 by using Young’s inequality,
+we get
+∥ρτ
+t − ρt∥2
+L2(R)
+≤ −λ
+t
+�
+0
+����
+d
+dxρτ
+s − d
+dxρs
+����
+2
+L2(R)
+ds −
+t
+�
+0
+�
+(ρs − ρτ
+s) d
+dx(σ2
+s), d
+dxρs − Dρτ
+s
+�
+L2(R)
+ds
+− 2
+t
+�
+0
+�
+(kτ
+HK ∗ ρτ
+s)ρτ
+s − (kHK ∗ ρs)ρs, d
+dxρτ
+s − d
+dxρs
+�
+L2(R)
+ds
+≤ −3λ
+4
+t
+�
+0
+����
+d
+dxρτ
+s − d
+dxρs
+����
+2
+L2(R)
+ds + Λ2
+λ
+t
+�
+0
+∥ρs − ρτ
+s∥2
+L2(R) ds
+− 2
+t
+�
+0
+�
+(kτ
+HK ∗ ρτ
+s)ρτ
+s − (kHK ∗ ρs)ρs, d
+dxρτ
+s − Dρs
+�
+L2(R)
+ds.
+Rewriting the last term gives
+(kτ
+HK ∗ ρτ
+s)ρτ
+s − (kHK ∗ ρs)ρs
+= ((kτ
+HK − kHK) ∗ ρτ
+s)ρτ
+s + (kHK ∗ ρτ
+s)ρτ
+s − (kHK ∗ ρs)ρs
+= ((kτ
+HK − kHK) ∗ ρτ
+s)ρτ
+s + (kHK ∗ (ρτ
+s − ρs))ρτ
+s + ((kHK ∗ ρs)(ρτ
+s − ρs).
+
+32
+CHEN, NIKOLAEV, AND PR ¨OMEL
+Hence, for the last two terms we can use Young’s inequality, Young’s inequality for convolu-
+tion, mass conservation and (3.4) to obtain
+�
+(kHK ∗ ρs)(ρτ
+s − ρs) + (kHK ∗ (ρτ
+s − ρs))ρτ
+s, d
+dxρτ
+s − d
+dxρs
+�
+L2(R)
+≤ ∥kHK∥L2(R) ∥ρs∥L2(R) ∥ρτ
+s − ρs∥L2(R)
+����
+d
+dxρτ
+s − d
+dxρs
+����
+L2(R)
++ ∥kHK∥L2(R) ∥ρτ
+s − ρs∥L2(R) ∥ρτ
+s∥L2(R)
+����
+d
+dxρτ
+s − d
+dxρs
+����
+L2(R)
+≤ λ
+4
+����
+d
+dxρτ
+s − d
+dxρs
+����
+2
+L2(R)
++ 1
+λ(∥kHK∥L2(R) ∥ρs∥L2(R) ∥ρτ
+s − ρs∥L2(R) + ∥kHK∥L2(R) ∥ρτ
+s − ρs∥L2(R) ∥ρτ
+s∥L2(R))2
+≤ λ
+2
+����
+d
+dxρτ
+s − d
+dxρs
+����
+2
+L2(R)
++ 2
+λ ∥kHK∥2
+L2(R) ∥ρτ
+s − ρs∥2
+L2(R) (∥ρs∥2
+L2(R) + ∥ρτ
+s∥2
+L2(R)).
+Moreover,
+�
+((kτ
+HK − kHK) ∗ ρτ
+s)ρτ
+s, d
+dxρτ
+s − d
+dxρs
+�
+L2(R)
+≤ ∥(kτ
+HK − kHK) ∗ ρτ
+s∥L∞(R) ∥ρτ
+s∥L2(R)
+����
+d
+dxρτ
+s − d
+dxρs
+����
+L2(R)
+≤ ∥kτ
+HK − kHK∥L2(R) ∥ρτ
+s∥L2(R) ∥ρτ
+s∥L2(R)
+����
+d
+dxρτ
+s − d
+dxρs
+����
+L2(R)
+≤ λ
+4
+����
+d
+dxρτ
+s − d
+dxρs
+����
+2
+L2(R)
++ 1
+λ ∥kτ
+HK − kHK∥2
+L2(R) ∥ρτ
+s∥4
+L2(R) ,
+where we used Young’s inequality for convolutions in the second inequality and Young’s
+inequality for the last step. Consequently, combining the last two estimates with our previous
+L2-norm inequality and absorbing the L2-norm of the derivatives we obtain
+∥ρτ
+t − ρt∥2
+L2(R)
+≤
+2 ∥kHK∥2
+L2(R) + 1
+λ
+t
+�
+0
+(∥ρs∥2
+L2(R) + ∥ρτ
+s∥2
+L2(R) + Λ2) ∥ρτ
+s − ρs∥2
+L2(R) ds
++ 1
+λ
+t
+�
+0
+∥kτ
+HK − kHK∥2
+L2(R) ∥ρτ
+s∥4
+L2(R) ds
+≤
+2 ∥kHK∥2
+L2(R) + 1
+λ
+t
+�
+0
+(∥ρs∥2
+L2(R) ∥ρτ
+s∥2
+L2(R) + Λ2) ∥ρτ
+s − ρs∥2
+L2(R) ds
++ T
+λ ∥kτ
+HK − kHK∥2
+L2(R) sup
+t∈[0,T]
+∥ρτ
+t ∥4
+L2(R) .
+
+HEGSELMANN–KRAUSE MODEL WITH ENVIRONMENTAL NOISE
+33
+Applying Gronwall’s inequality and using the uniform bound (3.6), we obtain
+sup
+t∈[0,T]
+∥ρτ
+t − ρt∥2
+L2(R)
+≤ T
+λ ∥kτ
+HK − kHK∥2
+L2(R) sup
+t∈[0,T]
+∥ρτ
+t ∥4
+L2(R)
+× exp
+�2 ∥kHK∥2
+L2(R) + 1
+λ
+T
+�
+0
+(∥ρs∥2
+L2(R) + ∥ρτ
+s∥2
+L2(R) + Λ) ds
+�
+≤ T
+λ ∥kτ
+HK − kHK∥2
+L2(R) sup
+t∈[0,T]
+∥ρτ
+t ∥4
+L2(R)
+× exp
+�2T(∥kHK∥2
+L2(R) + 1)
+λ
+(∥ρ∥2
+S∞
+FW ([0,T];L2(R)) + ∥ρτ∥2
+S∞
+FW ([0,T];L2(R)) + Λ)
+�
+.
+After taking the supremum over ω ∈ Ω, we arrive at
+∥ρτ − ρ∥S∞
+FW ([0,T];L2(R))
+≤ C(λ, Λ, T, ∥kHK∥L2(R) , ∥ρ∥S∞
+FW ([0,T];L2(R)) ∥ρτ∥S∞
+FW ([0,T];L2(R))) ∥kτ − k∥L2(R) .
+□
+Remark 5.4. Due to Proposition 5.3, we know that the solutions ρτ of the regularized sto-
+chastic Fokker–Planack equations converges to the solution ρ of the non-regularized stochastic
+Fokker–Planack equation as the interaction force kernels converge in the L2-norm for τ → 0.
+Finally, we are in a position to state and prove the main theorem of this section.
+Theorem 5.5 (Propagation of chaos). Suppose Assumption 2.1 and Assumption 4.1 hold.
+Let ρ be the solution of the stochastic Fokker–Planack equation (2.11) and let us denote by
+ΠN
+t (ω) = 1
+N
+N
+�
+i=1
+δXi,τ
+t
+(ω)
+the empirical measure of our regularized interaction system ((Xi,τ
+t , τ > 0), i = 1, . . . , N) given
+by (2.8). Then, we have, for all t ∈ [0, T],
+E(|⟨ΠN
+t , ϕ⟩ − ⟨ρt, ϕ⟩|2)
+≤ C
+�
+λ, Λ, T, ∥kHK∥L2(R) , ∥ϕ∥C1(R) , ∥ϕ∥L2(R) , ∥ρ∥S∞
+FW ([0,T];L2(R)) , ∥ρτ∥S∞
+FW ([0,T];L2(R))
+�
+×
+� 1
+N +
+1
+(N − 1)τ exp
+�(C + Λ)T
+τ
+�
++ ∥kτ − k∥2
+L2(R)
+�
+for any ϕ ∈ C∞
+c (R) and a finite constant C.
+
+34
+CHEN, NIKOLAEV, AND PR ¨OMEL
+Proof. We compute
+E(|⟨ΠN
+t , ϕ⟩ − ⟨ρτ
+t , ϕ⟩|2)
+= E
+�� 1
+N
+N
+�
+i=1
+ϕ(Xi,τ
+t ) −
+�
+R
+ρτ
+t (x)ϕ(x) dx
+�2�
+= 2E
+�� 1
+N
+N
+�
+i=1
+ϕ(Xi,τ
+t ) − 1
+N
+N
+�
+i=1
+ϕ(Y i,τ
+t
+)
+�2�
++ 2E
+�� 1
+N
+N
+�
+i=1
+ϕ(Y i,τ
+t
+) −
+�
+R
+ρτ
+t (x)ϕ(x) dx
+�2�
+≤
+2
+N 2
+� N
+�
+i=1
+E(|ϕ(Xi,τ
+t ) − ϕ(Y i,τ
+t
+)|2)1/2
+�2
++ 2E
+� 1
+N
+N
+�
+i=1
+ϕ(Y i,τ
+t
+) −
+�
+R
+ρτ
+t (x)ϕ(x) dx
+�2�
+≤ 2
+sup
+i=1,...,N
+E(|ϕ(Xi,τ
+t ) − ϕ(Y i,τ
+t
+)|2) + 2E
+�� 1
+N
+N
+�
+i=1
+ϕ(Y i,τ
+t
+) −
+�
+R
+ρτ
+t (x)ϕ(x) dx
+�2�
+,
+(5.3)
+where we used Minkwoski’s inequality in the third step. Now, by Proposition 5.2 and
+|ϕ(Xi,τ
+t ) − ϕ(Y i,τ
+t
+)|2 ≤
+����
+d
+dxϕ
+����
+2
+L∞ |Xi,τ
+t
+− Y i,τ
+t
+|2,
+we can estimate the first term by
+�� d
+dxϕ
+��2
+L∞
+2∥kHK∥2
+L2(R)T
+(N−1)τ
+exp
+�
+(C+Λ)T
+τ
+�
+. For the second term
+we write out the square to obtain
+2
+N 2
+N
+�
+i,j=1
+E
+��
+ϕ(Y i,τ
+t
+) −
+�
+R
+ρτ
+t (y)ϕ(y) dy
+��
+ϕ(Y j,τ
+t
+) −
+�
+R
+ρτ
+t (y)ϕ(y) dy
+��
+=
+2
+N 2
+N
+�
+i,j=1
+E
+�
+ϕ(Y i,τ
+t
+)ϕ(Y j,τ
+t
+) − ϕ(Y i,τ
+t
+)
+�
+R
+ρτ
+t (y)ϕ(y) dy
+− ϕ(Y j,τ
+t
+)
+�
+R
+ρτ
+t (y)ϕ(y) dy +
+� �
+R
+ρτ
+t (y)ϕ(y) dy
+�2�
+.
+Now, using the fact that ρτ
+t is the conditional distribution of Y i,τ with respect to FW
+t , we
+find
+E
+�
+ϕ(Y i,τ
+t
+)
+�
+R
+ρτ
+t (y)ϕ(y) dy
+�
+= E
+�
+E
+�
+ϕ(Y i,τ
+t
+)
+�
+R
+ρτ
+t (y)ϕ(y) dy
+����FW
+t
+��
+= E
+� �
+R
+ρτ
+t (y)ϕ(y) dy E(ϕ(Y i,τ
+t
+)|FW
+t )
+�
+= E
+�� �
+R
+ρτ
+t (y)ϕ(y) dy
+�2�
+.
+
+HEGSELMANN–KRAUSE MODEL WITH ENVIRONMENTAL NOISE
+35
+Since (Y i,τ
+t
+, i = 1, . . . , N) have identical distribution given FW
+t , the same equality holds for j
+instead of i. Additionally, using the fact that Y i,τ
+t
+, Y j,τ
+t
+are conditionally independent for
+i ̸= j, we obtain
+E
+�
+ϕ(Y i,τ
+t
+)ϕ(Y j,τ
+t
+)
+�
+= E
+�
+E(ϕ(Y i,τ
+t
+)ϕ(Y j,τ
+t
+)|FW
+t )
+�
+= E
+�
+E(ϕ(Y i,τ
+t
+)|FW
+t )E(ϕ(Y j,τ
+t
+)|FW
+t )
+�
+= E
+�� �
+R
+ρτ
+t (y)ϕ(y) dy
+�2�
+for the cross terms. Hence, the cross terms vanish and we can estimate the second term
+in (5.3) by
+2
+N 2
+N
+�
+i=1
+E
+��
+ϕ(Y i,τ
+t
+) −
+�
+R
+ρτ
+t (y)ϕ(y) dy
+�2�
+≤
+C(∥φ∥L∞(R))
+N
+for some finite constant C(∥ϕ∥L∞(R)), which depends only on ϕ. Putting everything together,
+we find
+E(|⟨ΠN
+t , ϕ⟩ − ⟨ρτ
+t , ϕ⟩|2) ≤
+����
+d
+dxϕ
+����
+2
+L∞
+2 ∥kHK∥2
+L2(R) T
+(N − 1)τ
+exp
+�(C + Λ)T
+τ
+�
++
+C(∥ϕ∥L∞(R))
+N
+≤ C(∥kHK∥L2(R) , T, ∥ϕ∥C1(R))
+� 1
+N +
+1
+(N − 1)τ exp
+�CT
+τ
+��
+.
+Next, using H¨older’s inequality and Proposition 5.3, we discover
+E(|⟨ρτ
+t , ϕ⟩ − ⟨ρt, ϕ⟩|2) ≤ E(∥ϕ∥2
+L2(R) ∥ρτ
+t − ρt∥2
+L2(R)) ≤ ∥ϕ∥2
+L2(R) ∥ρτ − ρ∥2
+S∞
+F ([0,T];L2(R))
+≤ ∥ϕ∥2
+L2(R) C(λ, Λ, T, ∥ρ∥S∞
+FW ([0,T];L2(R)) ∥ρτ∥S∞
+FW ([0,T];L2(R))) ∥kτ − k∥2
+L2(R) .
+Therefore, combining this estimate with the previous one we obtain
+E(|⟨ΠN
+t , ϕ⟩ − ⟨ρt, ϕ⟩|2)
+≤ 2E(|⟨ΠN
+t , ϕ⟩ − ⟨ρτ
+t , ϕ⟩|2) + 2E(|⟨ρτ
+t , ϕ⟩ − ⟨ρt, ϕ⟩|2)
+≤ C
+�
+λ, Λ, T, ∥kHK∥L2(R) , ∥ϕ∥C1(R) , ∥ϕ∥L2(R) , ∥ρ∥S∞
+FW ([0,T];L2(R)) , ∥ρτ∥S∞
+FW ([0,T];L2(R))
+�
+×
+� 1
+N +
+1
+(N − 1)τ exp
+�(C + Λ)T
+τ
+�
++ ∥kτ − k∥2
+L2(R)
+�
+.
+□
+Remark 5.6. While we focus in the present section on the interaction force kHK(x) =
+1[0,R](|x|)x, as used in the HK model, all results of Section 5 extend verbatim to general
+interaction forces k ∈ L1(R) ∩ L2(R) such that there exists a sequence (kτ)τ∈N ⊂ C∞
+c (R)
+satisfying
+• ∥kτ − k∥L2(R) → 0 as τ → ∞,
+• supp(kτ) ⊂ K for τ ∈ N and supp( d
+dxkτ) ⊂ K for some compact set K ⊂ R,
+• 0 ≤ kτ ≤ C, | d
+dxkτ| ≤ C
+τ for some constant C > 0.
+Remark 5.7. In the spacial case of a constant diffusion coefficient σ > 0, one can also derive
+propagation of chaos without introducing a regularized interacting particle system. Indeed, by
+considering the shifted interacting particle system ( ˜Xi, i = 1, . . . , N) with ˜Xi := Xi−σBi, one
+
+36
+CHEN, NIKOLAEV, AND PR ¨OMEL
+could exploit the results of propagation of chaos provided in the recent work [HRZ22] applied
+to this shifted system and transfer the so obtained propagation of chose back to the original
+interacting particle system.
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+
diff --git a/0NE2T4oBgHgl3EQfiQcC/content/tmp_files/load_file.txt b/0NE2T4oBgHgl3EQfiQcC/content/tmp_files/load_file.txt
new file mode 100644
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf,len=1523
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='03955v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='PR]  10 Jan 2023  HEGSELMANN–KRAUSE MODEL WITH ENVIRONMENTAL NOISE  LI CHEN, PAUL NIKOLAEV, AND DAVID J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' PR ¨OMEL  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' We study a continuous-time version of the Hegelsmann–Krause model describing  the opinion dynamics of interacting agents subject to random perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Mathematical  speaking, the opinion of agents is modelled by an interacting particle system with a non-  Lipschitz continuous interaction force, perturbed by idiosyncratic and environmental noises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  Sending the number of agent to infinity, we derive a McKean–Vlasov stochastic differential  equations as the limiting dynamics, by establishing propagation of chaos for regularized  versions of the noisy opinion dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' To that end, we prove the existence of a unique strong  solution to the McKean–Vlasov stochastic differential equation as well as well-posedness of  the associated non-local, non-linear stochastic Fokker–Planck equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  Key words: interacting particle system, McKean–Vlasov equation, mean-field limit, propa-  gation of chaos, stochastic Fokker–Planck equation, stochastic partial differential equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  MSC 2010 Classification: Primary: 60H15, 60H10, 60K35;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Secondary: 91D30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Introduction  The theory of opinion dynamics has been around since the 1950s, but over the last few  decades, the modeling of opinion dynamics has become a rapidly growing area of research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  With the rapid development of the internet and social networks, we have observed signifi-  cant changes in how opinion dynamics evolve and by what sources they are effected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' For  example, previous generations were heavily influenced by their geographically nearest social  group, but nowadays social networks are the primary platforms, for expressing and sharing  opinions, enabling more people than ever to do so from anywhere in the world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Consequently,  geographical distance is no longer a significant factor in shaping public opinion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Instead, each  citizen has a personal filter bubble [Spo17], which affects and in the same time shifts with  the opinion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' This phenomena is described by so called bounded confidence opinion dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  For an overview of opinion dynamics we refer to the surveys [Lor07, Hos20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  In the present paper we study the Hegelsmann–Krause model (HK model) [HK02], which  belongs to the class of bounded confidence opinion dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' More precisely, we focus on a  version of the HK model where the opinions XN := (Xi, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' , N) of N agents are subject  to idiosyncratic noises as well as an environmental noise, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=', we consider for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' , N the  particle system  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='1)  dXi  t = −  1  N − 1  N  �  j=1  j̸=i  kHK(Xi  t − Xj  t ) dt + σ(t, Xi  t) dBi  t + ν dWt,  Xi  0 = ζi,  for t ≥ 0, where Xi  t is the i-th agent’s opinion at time t, kHK(x) :=  1[0,R](|x|)x is the  (non-Lipschitz) interaction force between the agents, σ: [0, T] × R �→ R ane ν > 0 some  smooth diffusion coefficients, B = ((Bi  t, t ≥ 0), i ∈ N) is a sequence of one-dimensional  Date: January 11, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  1  2  CHEN, NIKOLAEV, AND PR ¨OMEL  independent Brownian motions, W = (Wt, t ≥ 0) is a Brownian motion independent of B, and  (ζi, i ∈ N) is the i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' sequence of initial values independent of all Brownian motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' The local  interaction kernel kHK represents the insight of bounded confidence opinion dynamics that  opinions are only influenced in a bounded domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' In the HK model (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='1), the idiosyncratic  noises B = ((Bi  t, t ≥ 0), i ∈ N) describe the individual random effects on each agent’s opinion  and the environmental noise (Wt, t ≥ 0) captures external effects on the agents’ opinions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' For  a more detailed discussion on different types of noises in HK models we refer to [CSDH19]  and the references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  Our goal is to establish propagation of chaos of the Hegelsmann–Krause model with environ-  mental noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' More precisely, we show that regularized versions of the particle systems (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='1)  converge (in a suitable sense) to the McKean–Vlasov stochastic differential equation (SDE)  �  dYt = −(kHK ∗ ρt)(Yt) dt + σ(t, Yt) dB1  t + ν dWt,  Y0 = X1  0,  ρt is the conditional density of Yt given FW  t ,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='2)  which comes with the associated stochastic non-linear, non-local Fokker–Planck equation  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='3)  dρt =  d2  dx2  �σ2  t + ν2  t  2  ρt  �  dt + d  dx((kHK ∗ ρt)ρt) dt − ν d  dxρt dWt,  t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  Let us remark that equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='3) is a non-local, non-linear stochastic partial differential  equation (SPDE), where the stochastic term is a consequence of the environmental noise  W = (Wt, t ≥ 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Indeed, as we shall see, if the number of agents tends to infinity the effect  of the idiosyncratic noises averages out, but the environmental noise does not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Moreover, in  contrast to many recent works like [CF16, CG19, HvS21, CDFM20, BCD21] on interacting  particle systems with environmental noise, we deal with density-dependent McKean–Vlasov  SDEs and the associated Fokker–Planck equations for the conditional densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  Our first contribution is to prove the well-posedness of the non-local, non-linear stochastic  Fokker–Planck equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' The main challenge in proving existence and uniqueness results  for (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='3) is the non-linear term (kHK ∗ρt)ρt since this prevents us from applying known results  in the existing literature on SPDEs, such as those found in textbooks [Kry99, WR15], which  consider the well-studied case of linear SPDEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' In the case of non-linear SPDEs, one needs  to take advantage of the specific structure of the considered SPDE to employ a fixed point  argument, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' the recent work [HQ21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' In this line of research, we establish local existence  and uniqueness of a weak solution to the non-local, non-linear Fokker–Planck equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  Additionally, we show global well-posedness of the Fokker–Planck equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='3) assuming a  sufficiently large diffusion coefficient or a sufficiently small L2-norm of the initial value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  Our second contribution is to prove the existence of a unique strong solution to the  McKean–Vlasov stochastic differential equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='2), which is essential for showing propaga-  tion of chaos towards to limiting Fokker–Planck equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' To obtain the well-posedness  of the McKean–Vlasov SDE (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='2), a central insight is to introduce suitable stopping times  to ensure sufficient temporary regularity such that a backward stochastic partial differential  equation (BSPDE) associated to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='2) possesses a classical solution, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' [DTZ13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' As a  result, a duality argument in combination with the Itˆo–Wentzell formula allows us to deduce  the existence of a unique strong solution to the McKean–Vlasov equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Let us point  out that the aforementioned existence and uniqueness results for the stochastic Fokker–Planck  equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='3) and the McKean–Vlasov equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='2) also hold for general interaction forces  k ∈ L1(R) ∩ L2(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  HEGSELMANN–KRAUSE MODEL WITH ENVIRONMENTAL NOISE  3  Our third contribution is to establish propagation of chaos for regularized versions of the  particle systems (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='1) with environmental noise, which verifies that (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='3) is, indeed, the macro-  scopic (density based) model corresponding to the microscopic (agent based) opinion dynamics  described by the Hegelsmann–Krause model with environmental noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' For uniformly Lips-  chitz interaction forces, propagation of chaos with environmental noise has been showed by  Coghi and Flandoli [CF16] by utilizing sharp estimates in Kolmogorov’s continuity theorem  and properties of measure-valued solutions of the associated stochastic Fokker–Planck equa-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  Moreover, without environmental noise there is a vast literature on propagation of  chaos with non-Lipschitz kernels, see [Szn91, GQ15, LY16, JW18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' However, most of these  works are based on the relative entropy and cannot be simply generalized to a setting with  environmental noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' For particle systems with environmental noise and non-Lipschitz inter-  action force k (as in our case), to the best of our knowledge there exists no general theory  on propagation of chaos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' In order to derive propagation of chaos for the HK model (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='2), we  essentially rely on the well-posedness of the McKean–Vlasov SDE (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='2) and follow [Szn91] as  well as [LP17, CDHJ21] to prove that (ρt, t ≥ 0) characterizes the measure of the mean-field  limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  Organization of the paper: In Section 2 we introduce the Hegelsmann–Krause model with  environmental noise and provide necessary definitions and some background information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' In  Section 3 the well-posedness of the stochastic Fokker–Planck equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='3) is established  and in Section 4 of the associated McKean–Vlasov equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' The mean-field limit and  propagation of chaos of the HR model with environmental noise are investigated in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Hegselmann–Krause model  In the following section we introduce the Hegselmann–Krause model and briefly review  some of the related literature on the well studied HK model without environmental noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  The Hegselmann–Krause model with environmental noise is set up in Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='2 with  its corresponding stochastic Fokker–Plank equation as well as its mean-field equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' We  conclude this section with some basic definitions and functions spaces necessary to study the  involved equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Some background information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' In this subsection we present some background on  the development of the HK model as well as on the terminology of propagation of chaos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' We  start with the original discrete-time Hegelsmann–Krause model [HK02], which is given by  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='1)  xi(t + 1) =  1  |Ni(t)|  �  j∈Ni(t)  xj(t),  t ≥ 0,  i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' , n,  where xi(t) is the opinion of agent i at time t, Ni(t) := {1 ≤ j ≤ n : |xi(t)−xj(t)| ≤ ri} denotes  the neighbor set of agent i at time t and |Ni(t)| is the cardinality of the set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' The convergence  and consensus properties of the discrete-time HK model were excessively studied in the past  years, see for instance [HK02, Lor06, BBCN13, KZPS12, NT12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' The main characteristic  feature of bounded confidence opinion models, like the HK model (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='1), is that the agents  interact only locally, which is modelled by the compactly supported interaction force in the  discrete-time HK model (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='1), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' j ∈ Ni(t), and, thus, opinions outside an agent’s moral  beliefs get ignored through this local interaction kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' This phenomena is, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=', observed in  case of liberal and conservative view points and their respective social media bubbles in the  USA [ENG+19, GKM17, Spo17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' The discrete-time HK model (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='1) is a fairly simple model  4  CHEN, NIKOLAEV, AND PR ¨OMEL  to describe opinion dynamics and by now there are numerous generalization and variants of  the original HK model, for instance, the HK model with media literacy [XCW+20] or the HK  model with an opinion leader [WCB15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' For further extensions we refer to [DR10, RD09].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  An important class of extensions of the original HK model captures external random ef-  fect in opinion dynamics, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' [PTHG13, CSDH19], leading naturally to a system of N  stochastic processes representing the opinion evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' In this case, following e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' [GPY17],  the opinion dynamics ˆXN := ( ˆXi, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' , N) of N agents are modelled by a system of  stochastic differential equations  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='2)  d ˆXi  t = −  1  N − 1  N  �  j=1  j̸=i  kHK( ˆXi  t − ˆXj  t ) dt + σ(t, ˆXi  t) dBi  t,  i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' , N,  ˆXN  0 ∼  N⊗  i=1ρ0,  for t ≥ 0, where ˆXi  t is the i-th agent’s opinion at time t, kHK is the interaction force between  the agents, σ is a smooth diffusion coefficient, ((Bi  t, t ≥ 0), i ∈ N) is a sequence of one-  dimensional independent Brownian motion and, as previously, ρ0 is the initial distribution (of  ζ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' We note that the interaction force kHK has compact support turning the continuous-time  HK model (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='2) into a bounded confidence model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' The continuous-time HK model (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='2) has  been a topic of active research in the past years, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=', the convergence to a consensus is studied  in [GPY17] and the phase transition were investigated in [WLEC17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  The particle system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='2) induces in the mean-field limit, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  as N → ∞, a sequence  (( ˆY i  t , t ≥ 0), i ∈ N) of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' stochastic processes satisfying the non-linear, non-local McKean–  Vlasov equations  �  d ˆY i  t = −(kHK ∗ ˆρt)( ˆY i  t ) dt + σ(t, ˆY i  t ) dBi  t,  ˆY i  0 = ˆXi  0,  t ≥ 0,  ˆρt is the density of ˆY i  t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='3)  For our specific interaction kernel kHK =  1[0,R](|x|)x, each ˆY i  t corresponds to a well-posed,  non-linear Fokker–Planck equation for the agent density profile ˆρ(t, x) given by  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='4)  ∂tˆρ(x, t) =  d2  dx2  �σ2  2 ˆρt(x)  �  + d  dx((kHK ∗ ˆρt)(x)ˆρt(x)),  t ≥ 0,  see [CJLW17] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  The given probability measure ˆρ(t, x) dx is classically acquired as the deterministic limit as  N → ∞ of the random measures ΠN with values in the space P(C([0, T], R)) of probability  measures on C([0, T], R), defined as  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='5)  ω �→ ΠN(ω, A) := 1  N  N  �  i=1  δ ˆ  Xi· (ω)(A),  A ∈ B(C([0, T], R)),  where δf is the Dirac measure for f ∈ C([0, T], R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' This convergence phenomena is known  as propagation of chaos, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' [Szn91, Kac56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' More precisely, let Z = (Zt, t ≥ 0) be a  continuous R-valued stochastic process defined on some probability space such that for each  t ≥ 0, Zt has the law µt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Moreover, let µN  t be the law of a system ZN  t := (Z1  t , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' , ZN  t ) of N  stochastic processes taking values in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' We say that ZN  t  is µt-chaotic, if µN  t  is a symmetric  measure and for each fix k ∈ N with k ≥ 2,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='6)  µk,N  t  converges weakly to µt as N → ∞,  HEGSELMANN–KRAUSE MODEL WITH ENVIRONMENTAL NOISE  5  where µk,N  t  (A1 × · · · × Ak) := µN  t (A1 × · · · × Ak × R × · · · × R) for A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' , Ak ∈ B(R) is the  k-th marginal distribution of µN  t , and we say that propagation of chaos holds if for any time  point t ≥ 0, ZN  t  is µt-chaotic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' It is a classical result that condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='6) holds for all k ≥ 2  if and only if condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='6) holds for k = 2, which is again equivalent to the convergence of  the empirical measures (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='5) associated to Z to the deterministic measure µ on C([0, T], R),  see [Szn91, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  It is well-known that if kHK would satisfy suitable Lispchitz and growth assumptions, the  particle system (ˆXN  t , t ≥ 0) satisfies propagation of chaos, see [Szn91, KX99].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' However, kHK  is not Lipschitz continuous and hence, we can no longer apply the classical theory to obtain  the convergence in law of the empirical measure (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Consequently, it requires novel and  non-standard methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' An idea, which has been developed for different interacting particle  systems without environmental noise in recent years, is to introduce a smooth approxima-  tion kN  HK, which depend on the number of agents N and converges in some sense to kHK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' This  leads to a regularized version of the system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='2), which reads as  d ˆXi,N  t  = −  1  N − 1  N  �  j=1  j̸=i  kN  HK( ˆXi,N  t  − ˆXj,N  t  ) dt + σ(t, ˆXi,N  t  ) dBi  t,  i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' , N,  ˆXN  0 ∼  N⊗  i=1ˆρ0,  for t ≥ 0, where ˆXN  t := ( ˆX1,N  t  , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' , ˆXN,N  t  ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' This allows to introduce an intermediate particle  system by  �  d ˆY i,N  t  = −(kN  HK ∗ ˆρN  t )( ˆY i,N  t  ) dt + σ(t, ˆY i,N  t  ) dBi  t,  ˆY i,N  0  = ˆXi,N  0  ,  i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' , N,  ˆρt is the density of Y i,N  t  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  We note that, similar to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='3), the aforementioned equation is in general a non-linear, non-  local McKean–Vlasov SDE, which induces a non-linear Fokker–Planack equation similar to  the one presented in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' The regularized systems allow to estimates, for all i, terms of  the form E(| ˆXi,N  t  − ˆY i,N  t  |) and E(| ˆY i,N  t  − ˆY i  t |) separately via PDE methods by studying the  associated non-linear Fokker–Planck equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' As a result, one can obtain propagation of  chaos for the system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='2) with kN  HK instead of kHK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Following the above described method,  various version of propagation of chaos has been shown for a variety of models with general  kernels k in [LP17, Ana17, CDHJ21] for particle systems without environmental noise, which  have non-Lipschitz, unbounded and even singular interaction force kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Hegselmann–Krause model with environmental noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' In this subsection we in-  troduce the Hegselmann–Krause model with environmental noise, its corresponding mean-  field stochastic differential equation with its associated stochastic Fokker–Planack equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  Let (Ω, F, (Ft)t≥0, P) be a complete filtered probability space, B = (Bi  t, t ≥ 0, i ∈ N) be a  sequence of one-dimensional Brownian motions and W = (Wt, t ≥ 0) be a one-dimensional  Brownian motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' All Brownian motions (Bi  t, t ≥ 0, i ∈ N) and (Wt, t ≥ 0) are supposed  to be independent and measurable with respect to the filtration (Ft, t ≥ 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Let the initial  data (ζi, i ∈ N) be i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' random variables with density ρ0 and independent of the Brownian  motions (Bi  t, t ≥ 0, i ∈ N), and (Wt, t ≥ 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Moreover, we denote by FW = (FW  t , t ≥ 0)  the augmented filtration generated by W (see [KS91, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='7] for the definition) and by  PW the predictable σ-algebra with respect to FW .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Analogous notation will be used for the  filtration generated by the Brownian motion B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  6  CHEN, NIKOLAEV, AND PR ¨OMEL  The Hegselmann–Krause model with environmental noise is given by the interacting particle  system XN  t = (X1  t , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' , XN  t ) following the dynamics  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='7) dXi  t = −  1  N − 1  N  �  j=1  j̸=i  kHK(Xi  t −Xj  t ) dt+σ(t, Xi  t) dBi  t +ν dWt,  i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' , N,  Xi  0 = ζi,  for t ∈ [0, T], where σ: [0, T] × R �→ R is the diffusion coefficient, ν > 0 a constant and the  interaction force kHK(x) =  1[−R,R](x)x for x ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' We point out, that the kernel kHK always  stands for the kernel in the Hegselmann–Krause model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' On the other hand k will denote a  general kernel (see Section 3 and Section 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  For establishing propagation of chaos we introduce the approximation sequence (kτ  HK, τ >  0) of the interaction force kHK such that the following properties hold for each τ > 0:  kτ  HK ∈ C∞  c (R),  supp(kτ  HK) ⊆ [−R − 2τ, R + 2τ], supp( d  dxψτ) ⊂ [−R − 2τ, −R + 2τ] ∪ [R − 2τ, R + 2τ],  | d  dxkτ  HK| ≤ C  τ for some constant C > 0,  ∥kHK∥L∞(R) ≤ 1  τ ∥kHK∥L2(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  We denote the regularized interacting particle system XN,τ  t  = (X1,τ  t  , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' , XN,τ  t  ) by  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='8)  dXi,τ  t  = −  1  N − 1  N  �  j=1  j̸=i  kτ  HK(Xi,τ  t  − Xj,τ  t  ) dt + σ(t, Xi,τ  t ) dBi  t + ν dWt, Xi,τ  0  = ζi,  for t ∈ [0, T] and for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' , N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Although the interaction force kernel kHK is non-Lipschitz  continuous, the N-particle systems (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='7) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='8) possess unique strong solutions, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  [MPT19, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='1], as kτ  HK and kHK are bounded and measurable in one dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  Corresponding to the particle systems (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='7) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='8), for i ∈ N, the system of mean-field  SDEs is given by  �  dY i  t = −(kHK ∗ ρt)(Y i  t ) dt + σ(t, Y i  t ) dBi  t + ν dWt,  Y i  0 = Xi  0,  ρt is the conditional density of Y i  t given FW  t ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='9)  and the system of regularized mean-field SDEs is defined by  �  dY i,τ  t  = −(kτ  HK ∗ ρτ  t )(Y i,τ  t  ) dt + σ(t, Y i,τ  t  ) dBi  t + ν dWt,  Y i,τ  0  = Xi,τ  0 ,  ρτ  t is the conditional density of Y i,τ  t  given FW  t ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='10)  for t ∈ [0, T], where ρt denotes the conditional density of Y i  t given FW  t , that is, for every  bounded continuous function ϕ, ρt satisfies  E(ϕ(Y i  t ) | FW  t ) =  �  R  ϕ(x)ρt(x) dx,  P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  The same holds for the regularized conditional density ρτ  t of Y i,τ  t  given FW  t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Let us remark  that ρτ  t , ρt have no superscript i since there are independent of i ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Indeed, the (regularized)  mean-filed particles are conditionally independent given FW and identically distributed, thus,  the conditional density is the same for each i ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  HEGSELMANN–KRAUSE MODEL WITH ENVIRONMENTAL NOISE  7  Associated to the mean-field SDEs (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='9) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='10), the stochastic Fokker–Planck equation  reads as  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='11)  dρt =  d2  dx2  �σ2  t + ν2  2  ρt  �  dt + d  dx((kHK ∗ ρt)ρt) dt − ν d  dxρt dWt,  and the regularized stochastic Fokker–Planck equation as  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='12)  dρτ  t =  d2  dx2  �σ2  t + ν2  2  ρτ  t  �  dt + d  dx((kτ  HK ∗ ρτ  t )ρτ  t ) dt − ν d  dxρτ  t dWt,  t ∈ [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  Let us remark that we purposely use the same unknown functions ρ, ρτ for the solutions  of the stochastic Fokker–Planak equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='11) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='12) and for the conditional density of  the mean-field SDEs (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='9) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='10) since, as we will see in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='3, they both coincide  under enough regularity assumptions on the initial condition ρ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Nevertheless, the meaning  of ρ, ρτ will always be clear from context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' We make the following assumptions on the diffusion  coefficient σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Let T > 0 and σ: [0, T] × R → R the diffusion coefficient, which satisfies:  (i) There exists a constant λ > 0 such that  σ2(t, x) ≥ λ  for all x ∈ R and t ∈ [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  (ii) There exists a constant Λ > 0 such that for all t ∈ [0, T] we have  σ(t, ·) ∈ C3(R)  and  sup  t∈[0,T]  3  �  i=1  ����  di  dxi σ(t, ·)  ����  L∞(R)  ≤ Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  The well-posedness of the stochastic Fokker–Planak equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='11) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='12) is presented  in Section 3 and the well-posedness of the mean-field SDEs (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='9) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='10) in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  For this purpose, we first need to fix some basic definitions and function space in the next  subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Function spaces and basic definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' In this subsection we collect some basic  definitions and introduce the required function spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' For 1 ≤ p ≤ ∞ we denote by Lp(Rd)  with norm ∥·∥Lp(Rd) the vector space of measurable functions whose p-th power is Lebesgue  integrable (with the standard modification for p = ∞), by C∞  c (Rd) the set of all infinitely  differentiable functions with compact support on Rd and by S(Rd) the set of all Schwartz  functions, see [Yos80, Chapter 6] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  We note that C∞  c (Rd) and S(Rd) are  endowed with their standard topologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Let  A := {α = (α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' , αd) : α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' , αd ∈ N0}  be the set of all multi-indices and |α| := α1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' + αd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' The derivative will be denoted by  ∂α :=  ∂|α|  ∂xα1  1 ∂xα2  2 · · · ∂xαd  d  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  In one dimension (d = 1), we also write  dn  dxn f for the n-th derivative with respect to x ∈ R  of a smooth function f defined on R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' We drop the superscripts α, n in the case α = n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  Moreover, as an inductive limit, we have C∞  c (R) =  ∞  �  M=1  C∞  c (B(0, M)), where B(0, M) is a  8  CHEN, NIKOLAEV, AND PR ¨OMEL  ball with radius M > 0 in Rd and (C∞  c (B(0, M)), pα,M) is the complete metrizable space of  smooth functions with compact support in B(0, M) and semi-norm  pα,M(f) := sup  |x|≤M  |(∂αf)(x)|  for f ∈ C∞  c (B(0, M)) and α ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' We note that this characterization and Baire category  theorem immediately imply that C∞  c (R) is not metrizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Similar, for each α, β ∈ A we  define the semi-norms  pα,β(f) := sup  x∈Rd |xα(∂βf)(x)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  Equipped with these seminorms, S(Rd) is a Fr´echet space [Abe12, Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Further-  more, we introduce the space of Schwartz distributions S′(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' We denote dual parings by  ⟨·, ·⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' For instance, for f ∈ S′, u ∈ S we have ⟨u, f⟩ = u[f] and for a probability measure µ  we have ⟨f, µ⟩ =  �  f dµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' The correct interpretation will be clear from the context but should  not be confused with scalar product ⟨·, ·⟩L2(R) in L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  The Fourier transform F[u] and the inverse Fourier transform F−1[u] for u ∈ S′(Rd) and  f ∈ S(Rd) are defined by  ⟨F[u], f⟩ := ⟨u, F[f]⟩,  where F[f] and F−1[f] is given by  F[f](ξ) :=  1  (2π)d/2  �  e−iη·xf(x) dx  and  F−1[f](ξ) :=  1  (2π)d/2  �  eiη·xf(x) dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  The Bessel potential for each s ∈ R is denoted by Js := (1−∆)s/2u := F−1[(1+|ξ|2)s/2F[u]]  for u ∈ S′(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' We define the Bessel potential space Hs  p for p ∈ (1, ∞) and s ∈ R by  Hs  p := {u ∈ S′(Rd) : (1 − ∆)s/2u ∈ Lp(Rd)}  with the norm  ∥u∥Hsp :=  ���(1 − ∆)s/2u  ���  Lp(Rd) ,  u ∈ Hs  p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  For 1 < p < ∞, m ∈ N we can characterize the above Bessel potential spaces Hm  p as Sobolev  spaces  W m,p(Rd) :=  �  f ∈ Lp(Rd) : ∥f∥W m,p(Rd) :=  �  α∈A, |α|≤m  ∥∂αf∥Lp(Rd) < ∞  �  ,  where ∂αf is to be understood as weak derivatives [AF03].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' We refer to [Tri83, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='6]  for the proof of the above characterization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' As a result, we use Sobolev spaces, which in our  context are easier to handle, instead of Bessel potential spaces, whenever possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  Finally, we introduce general Lp-spaces, which will serve as the solution space for the  SPDEs (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='11) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' For a Banach space (Z, ∥·∥Z), some filtration (Ft)t≥0, 1 ≤ p ≤ ∞  and 0 ≤ s < t ≤ T we denote by Sp  F([s, t];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Z) the set of Z-valued (Ft)-adapted continuous  processes (Xu, u ∈ [s, t]) such that  ∥X∥Sp  F([s,t];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='Z) :=  \uf8f1  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f3  �  E  �  sup  u∈[s,t]  ∥Xu∥p  Z  �� 1  p  ,  p ∈ [1, ∞)  sup  ω∈Ω  sup  u∈[s,t]  ∥Xu∥Z ,  p = ∞  HEGSELMANN–KRAUSE MODEL WITH ENVIRONMENTAL NOISE  9  is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Similar, Lp  F([s, t];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Z) denotes the set of Z-valued predictable processes (Xu, u ∈ [s, t])  such that  ∥X∥Lp  F([s,t];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='Z) :=  \uf8f1  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f3  �  E  � t�  s  ∥Xu∥p  Z du  �� 1  p  ,  p ∈ [1, ∞)  sup  (ω,u)∈Ω×[s,t]  ∥Xu∥Z ,  p = ∞  is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  In most case Z will be the Bessel potential space Hn  p , as it is mainly used by  Krylov [Kry10] in treating SPDEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  For a more detail introduction to the above function  spaces we refer to [Kry99, Section 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Well-posedness of the stochastic Fokker–Planck equations  This section is dedicated to establish the global existence and uniqueness of weak solutions  of the stochastic Fokker–Planck equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='11) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='12) under suitable conditions on the  initial condition and coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Instead of treating the special case kHK, we will take a more  general approach and prove existence and uniqueness for general interaction force k: R → R  under some integrability conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  Before we start our analysis, we introduce the concept of weak solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' For a general interaction force k ∈ L2(R), a non-negative stochastic process  (ρt, t ≥ 0) is called a (weak) solution of the SPDE  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='1)  dρt =  d2  dx2  �σ2  t + ν2  2  ρt  �  dt + d  dx((k ∗ ρt)ρt) dt − ν d  dxρt dWt,  t ∈ [0, T],  if  (ρt, t ∈ [0, T]) ∈ L2  FW ([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' W 1,2(R)) ∩ S∞  FW ([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' L1(R) ∩ L2(R))  and, for any ϕ ∈ C∞  c (R), ρ satisfies almost surely the equation, for all t ∈ [0, T],  ⟨ρt, ϕ⟩L2(R) = ⟨ρ0, ϕ⟩L2(R) +  t  �  0  �σ2  s + ν2  2  ρs, d2  dx2 ϕ  �  L2(R)  ds −  t  �  0  �  (k ∗ ρs)ρs, d  dxϕ  �  L2(R)  ds  +  t  �  0  ν  �  ρs, d  dxϕ  �  L2(R)  dWs  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='2)  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' A solution to the stochastic partial differential equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='11) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='12) is  defined analogously by replacing k with kHK or kτ  HK, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' There are multiple solution concepts for SPDEs, see for example [WR15] for  strong solutions in general separable Hilbert spaces or [DPZ14] for mild solutions with respect  to a infinitesimal generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' In the present work, we use the concept presented in [Kry99].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  This has the advantage that we can use Itˆo’s formula for Lp-norms [Kry10] as well as the  linear SPDE theory in [Kry99].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  10  CHEN, NIKOLAEV, AND PR ¨OMEL  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Under the assumption that for all t ∈ [0, T], σt ∈ C2(R) we can rewrite formally  equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='2) such that the leading coefficient is in non-divergence form, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  ⟨ρt, ϕ⟩L2(R)  = ⟨ρ0, ϕ⟩L2(R) + 1  2  t  �  0  �  (σ2  s + ν2) d2  dx2 ρs, ϕ  �  L2(R)  + 2  � d  dx(σ2  s + ν2) d  dxρs, ϕ  �  L2(R)  +  � d2  dx2 (σ2  s + ν2)ρs, ϕ  �  L2(R)  ds −  t  �  0  �  (k ∗ ρs)ρs, d  dxϕ  �  L2(R)  ds  −  t  �  0  ν  � d  dxρs, ϕ  �  L2(R)  dWs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  Hence, (ρt, t ≥ 0) solves the following SPDE  dρt = σ2  t + ν2  2  d2  dx2 ρt dt + d  dx(σ2  t + ν2) d  dxρt dt  + 1  2  d2  dx2 (σ2  t + ν2)ρt dt + d  dx((k ∗ ρt)ρt) dt − ν d  dxρt dWt,  t ∈ [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  In the next theorem we establish uniqueness and local existence of weak solutions to the  non-local stochastic Fokker–Planck equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Furthermore, we are going to see in Corol-  lary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='8 that the existence will not depend on the L2-norm of the initial condition ρ0, allowing  us to extend the local solution obtained in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='5 to a global solution on an arbitrary  interval [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Let Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='1 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Further, assume 0 ≤ ρ0 ∈ L1(R) ∩ L2(R) with  ∥ρ0∥L1(R) = 1 and k ∈ L2(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Then, there exists a T ∗ > 0 and a unique non-negative solution  of the SPDE (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='1) in the space  B := L2  FW ([0, T ∗];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' W 1,2(R)) ∩ S∞  FW ([0, T ∗];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' L1(R) ∩ L2(R)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  Moreover, the solution ρ has the property of mass conservation  ∥ρt∥L1(R) = 1,  P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=',  for all t ∈ [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Let us define the metric space  F T,M :=  �  X ∈ S∞  FW ([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' L2(R)) : ∥X∥S∞  FW ([0,T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='L2(R)) ≤ M  �  for some constant M > ∥ρ0∥L2(R), for instance M = 2 ∥ρ0∥L2(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' The metric on F T,M is  induced by the norm on S∞  FW ([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' L2(R)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' The solution map T : F T,M → F T,M is defined  as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' For each ζ ∈ F T,M we define T (ζ) as the solution of the following linear SPDE  dρt = σ2  t + ν2  2  d2  dx2 ρt dt + d  dx(σ2  t + ν2) d  dxρt dt  + 1  2  d2  dx2 (σ2  t + ν2)ρt dt + d  dx((k ∗ ζt)ρt) dt − ν d  dxρt dWt,  t ∈ [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='3)  HEGSELMANN–KRAUSE MODEL WITH ENVIRONMENTAL NOISE  11  The L2-bound on k and H¨older’s inequality imply  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='4)  |k ∗ ζt| ≤ ∥k∥L2(R) ∥ζt∥L2(R) ≤ ∥k∥L2(R) M,  which allows us to check the conditions of the Lp-theory of SPDEs [Kry99, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='1 and  Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='1] for the case n = −1 therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' For instance, if we define for q ∈ W 1,2(R) the  function  f(q, t, x) =  d  dx((k ∗ ζt)qt),  then obviously f(0, ·, ·) ∈ L2  FW ([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' H−1,2(R)) and, since x/(1 + |x|2)1/2 is bounded (see  [Tri78, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='8] for the lifting property), we have  ∥f(q, t, ·)∥H−1,2(R)) ≤ C ∥(k ∗ ζt)qt∥L2(R) ≤ ∥k∥L2(R) M ∥qt∥L2(R) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  By [Kry99, Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='5] this is sufficient to verify [Kry99, Assumption 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' The other as-  sumptions are proven similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  Hence, we can deduce that the linear SPDE (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='3) admits a unique solution  ρζ ∈ L2  FW ([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' W 1,2(R)) ∩ S2  FW ([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' L2(R)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  In the next step we want to demonstrate the non-negativity of the solution ρζ with the  regularity of the solution ρζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Let us denote by km the mollification of k and let  ρζ,m ∈ L2  FW ([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' W 1,2(R)) ∩ S2  FW ([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' L2(R))  be the solution of the SPDE (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='3) with (km ∗ ζ)ρζ instead of (k ∗ ζ)ρζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Then we can write the  SPDE in the form  dρζ,m  t  = am(t, x) d2  dx2ρζ,m  t  dt + bm(t, x) d  dxρζ,m  t  dt + cm(t, x)ρζ,m  t  dt − ν d  dxρζ,m  t  dWt,  for t ∈ [0, T], with  am(t, x) := σ2  t + ν2  2  , bm(t, x) :=  d  dx(σ2  t +ν2)+km∗ζt, cm(t, x) := 1  2  d2  dx2 (σ2  t +ν2)+ d  dxkm∗ζt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  Now, by Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='1 the coefficients am, bm, cm and the coefficient in the stochastic part  is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Hence, by the maximum principle [Kry99, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='12] the solution ρζ,m is  non-negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' On the other hand, we have  ����  d  dx  �  (km ∗ ζt)ρζ  t − (k ∗ ζt)ρζ  t  �����  2  L2  FW ([0,T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='H−1,2(R))  ≤ C  ���((km − k) ∗ ζt)ρζ  t  ���  2  L2  FW ([0,T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='L2(R))  ≤ E  �  T  �  0  ∥((km − k) ∗ ζt)∥2  L∞(R)  ���ρζ  t  ���  2  L2(R)  �  ≤ E  �  T  �  0  ∥((km − k)∥2  L2(R) ∥ζt∥2  L2(R)  ���ρζ  t  ���  2  L2(R)  �  ≤ ∥((km − k)∥2  L2(R) M2 ���ρζ  t  ���  2  L2  FW ([0,T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='L2(R))  m→∞  −−−−→ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  12  CHEN, NIKOLAEV, AND PR ¨OMEL  Consequently, by [Kry99, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='7] we have  lim  m→∞  ���ρζ,m − ρζ���  L2  FW ([0,T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='W 1,2(R)) = 0  and therefore ρζ  t (·) ≥ 0 for all t ∈ [0, T] almost surely (by intersecting all sets of measure one,  where ρζ,m is non-negative).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  The non-negativity of the solution ρζ and the divergence structure of the equation provides  us with the normalization condition/mass conservation, that is  ���ρζ  t  ���  L1(R) = ∥ρ0∥L1(R) = 1,  P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=',  for t ∈ [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' This follows immediately by plugging in a cut-off sequence (ξn, n ∈ N) for our  test function ϕ and taking the limit n → ∞ (see [Bre11, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' 212] for properties of the cut-off  sequence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Therefore, the map T (ζ) = ρζ will be well-defined if we can obtain a bound on  the S∞  FW ([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' L2(R))-norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' For readability we will from now on drop the superscript ζ in  the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Applying Itˆo’s formula [Kry10],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' we obtain  ∥ρt∥2  L2(R) − ∥ρ0∥2  L2(R)  = 2ν  t  �  0  �  ρs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' d  dxρs  �  L2(R)  dWs +  t  �  0  �  ρs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' d  dx(σ2  s + ν2) d  dxρs + ρs  d2  dx2 (σ2  s + ν2)  �  L2(R)  ds  −  t  �  0  �  (σ2  s + ν2) d  dxρs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' d  dxρs  �  L2(R)  ds − 2  t  �  0  �  (k ∗ ζs)ρs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' d  dxρs  �  L2(R)  ds  + ν2  t  �  0  ����  d  dxρs  ����  2  L2(R)  ds  =  t  �  0  �  ρs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' d  dx(σ2  s + ν2) d  dxρs + ρs  d2  dx2 (σ2  s + ν2)  �  L2(R)  ds − 2  t  �  0  �  (k ∗ ζs)ρs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' d  dxρs  �  L2(R)  ds  −  t  �  0  �  σ2  s  d  dxρs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' d  dxρs  �  L2(R)  ds  ≤  t  �  0  �  ρs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' d  dx(σ2  s + ν2) d  dxρs + ρs  d2  dx2 (σ2  s + ν2)  �  L2(R)  ds − 2  t  �  0  �  (k ∗ ζs)ρs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' d  dxρs  �  L2(R)  ds  − λ  t  �  0  ����  d  dxρs  ����  2  L2(R)  ds,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  for 0 ≤ t ≤ T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' where we used the fact that ρs d  dxρs = 1  2  d  dx(ρ2  s) to get rid of the stochastic  integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' At this step, it is crucial that we only have additive common noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Otherwise the  stochastic integral will not vanish and the above estimate will not achieve the L∞-bound in ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  HEGSELMANN–KRAUSE MODEL WITH ENVIRONMENTAL NOISE  13  For the first term we can use Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='1 and Young’s inequality to find  ����  t  �  0  �  ρs, d  dx(σ2  s + ν2) d  dxρs + ρs  d2  dx2 (σ2  s + ν2)  �  L2(R)  ds  ����  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='5)  ≤ Λ  t  �  0  ����  �  ρs, d  dxρs + ρs  �  L2(R)  ���� ds  ≤ λ  4  t  �  0  ����  d  dxρs  ����  2  L2(R)  ds +  �Λ2  λ + Λ  �  t  �  0  ∥ρs∥2  L2(R) ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  On the other hand, using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='4) and Young’s inequality, we obtain  ����  �  (k ∗ ζs)ρs, d  dxρs  �  L2(R)  ���� ≤ ∥k ∗ ζs∥L∞(R)  �  |ρs|,  ����  d  dxρs  ����  �  L2(R)  ≤ ∥k∥L2(R) M ∥ρs∥L2(R)  ����  d  dxρs  ����  L2(R)  ≤  ∥k∥2  L2(R) M2  λ  ∥ρs∥2  L2(R) + λ  4  ����  d  dxρs  ����  2  L2(R)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  After absorbing the terms, we find  ∥ρt∥2  L2(R) − ∥ρ0∥2  L2(R) ≤  �∥k∥2  L2(R) M2 + Λ2  λ  + Λ  �  t  �  0  ∥ρs∥2  L2(R) ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  For the rest of the proof we define the constant  C(λ, Λ, k, M) :=  ∥k∥2  L2(R) M2 + Λ2  λ  + Λ  and conclude  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='6)  ∥ρt∥2  L2(R) ≤ ∥ρ0∥2  L2(R) exp  �  C(λ, Λ, k, M)T  �  ,  by Gronwall’s inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  Choosing ˆT ∗ < ln(M/ ∥ρ0∥2  L2(R))C(λ, Λ, k, M)−1, we have ρ ∈  F ˆT ∗,M and the map  T : F  ˆT ∗,M → F  ˆT ∗,M,  ζ → ρζ,  is well-defined up to time ˆT ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  The next step is to show that T is a contraction in a small time span (T ≤ ˆT ∗) and,  therefore, has a fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' For ζ, ˜ζ ∈ F T,M let ρ := T (ζ), ˜ρ := T (˜ζ) be the associated  solutions of the linear SPDE (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Then, we have  d(ρt − ˜ρt) = σ2  t + ν2  2  d2  dx2 (ρt − ˜ρt) dt + (σ2  t + ν2) d  dx(ρt − ˜ρt) dt + 1  2  d2  dx2 (σ2  t + ν2)(ρt − ˜ρt) dt  + d  dx((k ∗ ζt)ρt) dt − d  dx((k ∗ ˜ζt)˜ρt) dt − ν d  dx(ρt − ˜ρt) dWt,  t ∈ [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  14  CHEN, NIKOLAEV, AND PR ¨OMEL  Applying Itˆo’s formula [Kry10] and multiple Young’s inequality again (see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='5)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' we obtain  ∥ρt − ˜ρt∥2  L2(R)  = −  t  �  0  �  (σ2  s + ν2) d  dxρs − d  dx ˜ρs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' d  dxρs − d  dx ˜ρs  �  L2(R)  ds  − 2  t  �  0  �  (k ∗ ζs)ρs − (k ∗ ˜ζs)˜ρs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' d  dxρs − d  dx ˜ρs  �  L2(R)  ds + ν2  t  �  0  ����  d  dx(ρs − ˜ρs)  ����  2  L2(R)  ds  +  t  �  0  �  ρs − ˜ρs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' d  dx(σ2  s + ν2)  � d  dxρs − d  dx ˜ρs  �  + (ρs − ˜ρs) d2  dx2 (σ2  s + ν2)  �  L2(R)  ds  ≤ −λ  t  �  0  ����  d  dxρs − d  dx ˜ρs  ����  2  L2(R)  ds  − 2  t  �  0  �  (k ∗ (ζs − ˜ζs))ρs + (k ∗ ˜ζs)(ρs − ˜ρs),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' d  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='dxρs − d  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='dx ˜ρs  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='L2(R)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='ds  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='+ λ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='t  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='����  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='d  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='dxρs − d  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='dx ˜ρs  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='����  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='L2(R)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='ds +  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='�Λ2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='λ + Λ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='t  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='∥ρs − ˜ρs∥2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='L2(R) ds  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='≤ −3λ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='t  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='����  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='d  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='dxρs − d  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='dx ˜ρs  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='����  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='L2(R)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='ds +  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='�Λ2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='λ + Λ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='t  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='∥ρs − ˜ρs∥2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='L2(R) ds  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='t  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='∥k∥L2(R)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='���ζs − ˜ζs  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='���  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='L2(R) ∥ρs∥L2(R)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='����  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='d  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='dxρs − d  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='dx ˜ρs  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
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+page_content='���ζs − ˜ζs  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='���  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
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+page_content='L2(R) + λ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
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+page_content='+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
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+page_content='λ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='∥ρs − ˜ρs∥2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='L2(R) + λ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
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+page_content='����  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
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+page_content='dxρs − d  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='dx ˜ρs  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='����  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='L2(R)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='ds  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='HEGSELMANN–KRAUSE MODEL WITH ENVIRONMENTAL NOISE  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='15  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='≤  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='t  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='∥k∥2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='L2(R) M2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='λ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='���ζs − ˜ζs  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='���  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='L2(R) ds +  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='t  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='C(λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' M) ∥ρs − ˜ρs∥2  L2(R) ds  ≤ T  ∥k∥2  L2(R) M2  λ  ���ζ − ˜ζ  ���  2  S∞  FW ([0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='L2(R)) + C(λ, Λ, k, M)  t  �  0  ∥ρs − ˜ρs∥2  L2(R) ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  Gronwall’s inequality provide us with the estimate  ∥ρ − ˜ρ∥S∞  FW ([0,T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='L2(R)) ≤  �  T ∥k∥2  L2(R) M2  λ  exp  �  T  2 C(λ, Λ, k, M)  �  ∥ζ − ˜ζ∥S∞  FW ([0,T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='L2(R)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  Now, choosing T ∗ such that  �  T ∗ ∥k∥2  L2(R) M2  λ  λ exp  �  T ∗  2 C(λ, Λ, k, M)  �  < 1  and T ∗ ≤ ˆT ∗, we see that the map T : F T ∗,M → F T ∗,M is a contraction and consequently we  obtain a fixed point ρ, which is a local weak solution up to the time T ∗ of the SPDE (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  □  We notice that in the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='5, we only use H¨older inequality, Young’s convo-  lution and product inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Hence, the statement of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='5 holds also for arbitrary  dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' We state this observation in the following corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Assume 0 ≤ ρ0 ∈ L1(Rd)∩L2(Rd) with ∥ρ0∥L1(Rd) = 1 and consider a general  interaction force k ∈ L2(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Then, there exists a T ∗ > 0 and a unique non-negative solution  of the SPDE (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='1) in the space  B := L2  FW ([0, T ∗];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' W 1,2(Rd)) ∩ S∞  FW ([0, T ∗];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' L1(Rd) ∩ L2(Rd)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  The solution should be understood in the sense of Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='1, where Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='1 is  modified for arbitrary dimension d in the obvious way, see also [Kry99, Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Following the steps of the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='5, we see that we can obtain not  only a local solution but a (global) solution for any T > 0 by requiring a very small L2-norm  on the initial condition ρ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' In particular, we can choose a constant M > 0 such that  �  T ∥k∥2  L2(R) M2  λ  λ exp  �  T  2  �  ∥k∥2  L2(R) M2 + Λ2  λ  + Λ  ��  < 1  and then the condition  ∥ρ0∥L2(R) ≤ M exp  �  − T  �  ∥k∥2  L2(R) M2 + Λ2  λ  + Λ  ��  guarantees a unique non-negative solution of the SPDE (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='1) on the interval [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  Next, we establish another global existence and uniqueness result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' We emphasize that in the  following result we do not need any further assumptions on ρ0 besides being in L1(R)∩L2(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  Instead, we impose a lower bound on the diffusion coefficient σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Hence, we require a sufficiently  high randomness in stochastic Fokker–Planck equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  We also assert the fact that the  continuation of the solution (ρt, t ≥ 0) is a direct consequence of the L2-theory of SPDEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  16  CHEN, NIKOLAEV, AND PR ¨OMEL  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Let Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='1 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Further, assume 0 ≤ ρ0 ∈ L1(R) ∩ L2(R) with  ∥ρ0∥L1(R) = 1 and k ∈ L1(R) ∩ L2(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Furthermore, assume that the diffusion coefficient σ  has a derivative  d  dxσ with compact support [−L, L] and satisfies  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='7)  2L2 sup  0≤t≤T  ����  d  dx(σ2  t )  ����  L∞(R)  +  �  C4  GNS ∥k∥2  L1(R) + 4L4 sup  0≤t≤T  ����  d  dx(σ2  t )  ����  2  L∞(R)  �1/2  ≤ λ,  where CGNS is the constant given by the Gagliardo–Nirenberg–Sobolev interpolation in one  dimension [Leo17, Theorem 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='83], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' CGNS =  � 4π2  9  �−1/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Then, for each T > 0 there exist  unique global non-negative solutions of the stochastic Fokker–Planck equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='1) in the  space B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Let ρ be the solution given by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Following the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' we  apply Itˆo’s formula [Kry10] and obtain,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' for 0 ≤ t ≤ T ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  ∥ρt∥2  L2(R) − ∥ρ0∥2  L2(R)  = −  t  �  0  ����σs  d  dxρs  ����  2  L2(R)  ds +  t  �  0  �  ρs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' d  dx(σ2  s + ν2) d  dxρs + ρs  d2  dx2 (σ2  s + ν2)  �  ds  − 2  t  �  0  �  (k ∗ ρs)ρs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' d  dxρs  �  L2(R)  ds  = −  t  �  0  ����σs  d  dxρs  ����  2  L2(R)  ds −  t  �  0  �  ρs  d  dx(σ2  s),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' d  dxρs  �  L2(R)  ds  − 2  t  �  0  �  (k ∗ ρs)ρs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' d  dxρs  �  L2(R)  ds  ≤ −λ  2  t  �  0  ����  d  dxρs  ����  2  L2(R)  ds + sup  0≤t≤T  ����  d  dx(σ2  t )  ����  L∞(R)  t  �  0  ∥ρs∥L2([−L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='L])  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='����  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='d  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='dxρs  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='����  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='L2(R)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='ds  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='+ 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='λ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='t  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='∥(k ∗ ρs)ρs∥2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='L2(R) ds  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='≤  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='− λ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='2 + 2L2 sup  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='0≤t≤T  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='����  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='d  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='dx(σ2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='t )  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='����  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='L∞(R)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='t  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='����  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='d  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='dxρs  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='����  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='L2(R)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='ds  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='+ 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='λ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='t  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='∥k ∗ ρs∥2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='L4(R) ∥ρs∥2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='L4(R) ds  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='≤  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='− λ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='2 + 2L2 sup  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='0≤t≤T  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='����  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='d  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='dx(σ2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='t )  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='����  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='L∞(R)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='t  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='����  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='d  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='dxρs  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='����  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='L2(R)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='ds  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='HEGSELMANN–KRAUSE MODEL WITH ENVIRONMENTAL NOISE  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='17  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='+ 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='λ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='t  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='∥k∥2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='L1(R) ∥ρs∥2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='L4(R) ∥ρs∥2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='L4(R) ds  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='=  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='− λ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='2 + 2L2 sup  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='0≤t≤T  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='����  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='d  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='dx(σ2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='t )  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='����  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='L∞(R)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='t  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='����  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='d  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='dxρs  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='����  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='L2(R)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='ds +  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='∥k∥2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='L1(R)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='λ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='t  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='∥ρs∥4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='L4(R) ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  We remark that we used integration by parts in the first step, Young’s and H¨older’s inequality  in the third step, H¨older’s and Poincar´e inequaity [Leo17, Theorem 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='19] in the forth step and  Young’s inequality for convolutions in the fifth step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Let us recall the Gagliardo–Nirenberg–  Sobolev interpolation [Leo17, Theorem 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='83], which states that for u ∈ L1(R) ∩ W 1,2(R) we  have  ∥u∥L4(R) ≤ CGNS ∥u∥1/2  L1(R)  ����  d  dxu  ����  1/2  L2(R)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  Consequently, applying this inequality on the last term in our estimate and having mass  conservation in mind we find  ∥ρt∥2  L2(R) − ∥ρ0∥2  L2(R)  ≤  �  − λ  2 + 2L2 sup  0≤t≤T  ����  d  dx(σ2  t )  ����  L∞(R)  +  C4  GNS ∥k∥2  L1(R)  λ  �  t  �  0  ����  d  dxρs  ����  2  L2(R)  ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  Hence, if  2L2 sup  0≤t≤T  ����  d  dx(σ2  t )  ����  L∞(R)  +  �  C4  GNS ∥k∥2  L1(R) + 4L4 sup  0≤t≤T  ����  d  dx(σ2  t )  ����  2  L∞(R)  �1/2  ≤ λ,  we discover  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='8)  ∥ρ∥S∞  FW ([0,T ∗];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='L2(R)) ≤ ∥ρ0∥2  L2(R) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  Since ρ ∈ L2  FW ([0, T ∗];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' W 1,2(R)), we may apply [Kry99, Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='1], which tells us that  ρ ∈ C([0, T ∗], L2(R)), P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=', and  E(∥ρT ∗∥2  L2(R)) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  As a result, we can take ρT ∗ as the new initial value and apply [Kry99, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='1] in  combination with our above arguments in proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='5 to obtain a solution on  [T ∗, 2T ∗], since the estimate (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='8) and the condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='7) are independent of T ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Hence, after  finitely many steps we have a global solution ρ on [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' The uniqueness and ρ ∈ B follows  by repeating the inequalities derived in the contraction argument in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='5 or using the  uniqueness of the SPDE presented in [Kry99, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='1 and Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  □  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' In particular, for a constant diffusion σ > 0 the condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='7) reads simply  as  C2  GNS ∥k∥L1(R) ≤ σ,  which can be interpreted such that for a given integrable kernel k the system needs a certain  amount of idiosyncratic noise to stay alive for arbitrary T > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' In other word, the diffusion  needs to be dominant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  Next, we are going to improve the regularity of the solution ρ by a bootstrap argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  18  CHEN, NIKOLAEV, AND PR ¨OMEL  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Let ρ0 ∈ L1(R) ∩ W 2,2(R) with ∥ρ0∥L1(R) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Moreover, let Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='1  hold and k ∈ L2(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Assume we have a solution  ρ ∈ L2  FW ([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' W 1,2(R)) ∩ S∞  FW ([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' L1(R) ∩ L2(R))  of the SPDE (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='1) on [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Then ρ has the following regularity  ρ ∈ L2  FW ([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' W 3,2(R)) ∩ S2  FW ([0, T], W 2,2(R)) ∩ S∞  FW ([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' L1(R) ∩ L2(R)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Let us explore the following bootstrap argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  By assumptions we know ρ ∈  L2  FW ([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' W 1,2(R)) ∩ S∞  FW ([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' L1(R) ∩ L2(R)) and solves the SPDE  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='9)  dρt =  d2  dx2  �σ2  t + ν2  2  ρt  �  dt + d  dx((k ∗ ρt)ρt) dt − ν d  dxρt dWt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  Furthermore,  d  dx(kτ ∗ ρt) = kτ ∗  d  dxρt for the smooth approximation kτ of k and consequently  the dominated convergence theorem implies  d  dx(k∗ρt) = k∗ d  dxρt in the sense of distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  As a result  d  dx((k ∗ ρt)ρt) =  �  k ∗ d  dxρt  �  ρt + (k ∗ ρt) d  dxρt  is well-defined as a function in L1(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Moreover, we find  ����  d  dx((k ∗ ρt)ρt)  ����  L2(R)  ≤  ����  �  k ∗ d  dxρt  �  ρt  ����  L2(R)  +  ����(k ∗ ρt) d  dxρt  ����  L2(R)  ≤  ����k ∗ d  dxρt  ����  L∞(R)  ∥ρt∥L2(R) + ∥k ∗ ρt∥L∞(R)  ����  d  dxρt  ����  L2(R)  ≤ ∥k∥L2(R) ∥ρt∥W 1,2(R) ∥ρt∥L2(R) + ∥k∥L2(R) ∥ρt∥L2(R) ∥ρt∥W 1,2(R)  ≤ 2 ∥k∥L2(R) ∥ρt∥W 1,2(R) ∥ρt∥L2(R) ,  which implies  ����  d  dx((k ∗ ρ)ρ)  ����  L2  FW ([0,T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='L2(R))  ≤ 2 ∥k∥L2(R) ∥ρ∥S∞  FW ([0,T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='L2(R)) ∥ρ∥L2  FW ([0,T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='W 1,2(R)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  From the uniqueness of the SPDE (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='9), ρ0 ∈ W 1,2(R) and [Kry99, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='1 and Theo-  rem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='1] we obtain  ρ ∈ L2  FW ([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' W 2,2(R)) ∩ S2  FW ([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' W 1,2(R)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  With the same strategy and the discovered regularity of ρ one obtains  d2  dx2((k ∗ ρt)ρt) =  �  k ∗ d2  dx2ρt  �  ρt + 2  �  k ∗ d  dxρt  � d  dxρt + (k ∗ ρt) d2  dx2 ρt  and consequently  ����  d2  dx2 ((k ∗ ρt)ρt)  ����  L2(R)  ≤ 2 ∥k∥L2(R) ∥ρt∥W 2,2(R) ∥ρt∥L2(R) + 2 ∥k∥L2(R)  ����  d  dxρt  ����  2  L2(R)  ≤ 4 ∥k∥L2(R) ∥ρt∥W 2,2(R) ∥ρt∥L2(R) ,  HEGSELMANN–KRAUSE MODEL WITH ENVIRONMENTAL NOISE  19  where we used Gagliardo–Nirenberg–Sobolev interpolation inequality [Leo17, Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='41]  in the last step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Therefore, we have  ����  d  dx((k ∗ ρ)ρ)  ����  L2  FW ([0,T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='W 1,2(R))  ≤ 4 ∥k∥L2(R) ∥ρ∥S∞  FW ([0,T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='L2(R)) ∥ρ∥L2  FW ([0,T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='W 2,2(R)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  Again, from the uniqueness of the SPDE (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='9), ρ0 ∈ W 2,2(R) and [Kry99, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='1,  Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='11, Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='1] we obtain  ρ ∈ L2  FW ([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' W 3,2(R)) ∩ S2  FW ([0, T], W 2,2(R)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  □  As a consequence of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='5, Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='8 and the fact that kHK, kτ  HK ∈ L1(R)∩L2(R)  for all τ > 0, we obtain the following corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Let Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='1 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Further, assume 0 ≤ ρ0 ∈ L1(R) ∩ L2(R) with  ∥ρ0∥L1(R) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Then, there exists a T ∗ > 0 and a unique non-negative solution ρ, ρτ of the  SPDE (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='11) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='12) in the space  B = L2  FW ([0, T ∗];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' W 1,2(R)) ∩ S∞  FW ([0, T ∗];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' L1(R) ∩ L2(R))  Furthermore, if  d  dxσt(x) has compact support and inequality (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='7) holds, then we can extend  ρ, ρτ to a global solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Well-posedness of the mean-field SDEs  In this section we establish the existence of unique strong solutions of the mean-field sto-  chastic differential equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='9) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Analogously to the classical theory of ordinary  SDEs, it turns out that the mean-field SDEs (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='9) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='10) are linked to the stochastic  Fokker–Planck equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='11) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='12) in the same way as ordinary SDEs are linked to  deterministic Fokker–Planck equations (also called Kolmogorov forward equations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  Similar to Section 3, we prove the existence of strong solutions for general interaction  force k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' In the following we consider the mean-field SDE  �  dYt = −(k ∗ ρt)(Yt) dt + σ(t, Yt) dBt + ν dWt,  Y0 = X0,  ρt is the conditional density of Yt given FW  t ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='1)  for a general interaction force k ∈ L1(R) ∩ L2(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' We notice that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='1) is just one of the  identically distributed SDE’s of the system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='9), if we set k = kHK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' In order to guarantee  the well-posedness of the SPDE (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='1) we make the assumption:  Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Let 0 ≤ ρ0 ∈ L1(R) ∩ W 2,2(R) with ∥ρ0∥L1(R) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' For T > 0 there exists  a unique solution ρ in L2  FW ([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' W 1,2(R)) of the SPDE (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='1) on the interval [0, T] with  ∥ρ∥L2  Fw([0,T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='W 1,2(R)) + ∥ρ∥S∞  Fw([0,T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='L1(R)∩L2(R)) ≤ C  for some finite constant C > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' The existence of a unique solution to the SPDE (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='1) in the above assumption  is satisfied, for instance, if the conditions stated in Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='7, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='5 or Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='8  are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  20  CHEN, NIKOLAEV, AND PR ¨OMEL  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Let Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='1 as well as Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='1 hold and k ∈ L1(R) ∩ L2(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  Then, the mean-field SDE (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='1) has a unique strong solution (Yt, t ∈ [0, T]) and ρt is the  conditional density of Yt given FW  t  for every t ∈ [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  The idea to prove Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='3 is to freeze the measure ρt in the SDE (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='1) and use a duality  argument by introducing a dual backward stochastic partial differential equation (BSPDE)  in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='4 in order to prove that ρt is the conditional density of Y i  t for given FW  t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Let ρ be the unique solution of the SPDE (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='1) as in Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' We recall that  by the regularity result presented in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='10 we have  ρ ∈ L2  FW ([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' W 3,2(R)) ∩ S2  FW ([0, T], W 2,2(R)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Fix ρ in the mean-field SDE (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='1) and notice that we are dealing with a standard  SDE with random coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Hence, we can apply classical results if the drift coefficient  k ∗ ρ is Lipschitz continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' The regularity of the solution, Sobolev embedding theorem and  Morrey’s inequality yields  sup  0≤t≤T  sup  x,y∈R  x̸=y  |(k ∗ ρt)(ω, x) − (k ∗ ρt)(ω, y)|  |x − y|  ≤ sup  0≤t≤T  ∥k ∗ ρt(ω)∥W 2,2(R)  ≤ ∥k∥L1(R) sup  0≤t≤T  ∥ρt(ω)∥W 2,2(R)  and  sup  0≤t≤T  |(k ∗ ρt)(ω, 0)| ≤ sup  0≤t≤T  sup  x∈R  |(k ∗ ρt)(ω, x)|  ≤ ∥k∥L1(R) sup  0≤t≤T  ∥ρt(ω)∥W 1,2(R) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  Furthermore, the maps ω �→ sup  0≤t≤T  ∥ρt(ω)∥W 2,2(R) and ω �→ sup  0≤t≤T  ∥ρt(ω)∥W 1,2(R) are measur-  able.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Therefore, standard results for the existence of SDEs with Lipschitz continuous drift,  see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' [KRZ99, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='1] or [KHLN97, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='2], imply that the following SDE has  a unique strong solution  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='2)  �  dY t = −(k ∗ ρt)(Y t) dt + σ(t, Y t) dB1  t + ν dWt,  Y 0 ∼ ρ0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' We are going to use a dual argument (see Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='4 below) to show that ρt is  the conditional density of Y 1  t with respect to FW  t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Hence, let T1 > 0 and (ut, t ∈ [0, T1])  be the solution of the BSPDE (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='3) below with terminal condition G ∈ L∞(Ω, FT1, C∞  c (R)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  Utilizing the dual equation from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='4, the dual analysis [Zho92, Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='1] and the  fact that u0 is FW  0 -measurable, we find  ⟨ρ0, u0⟩ = E(⟨G, ρT1⟩).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  On the other hand we can use the explicit representation of u0 given by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='4 to obtain  ⟨ρ0, u0⟩ =  �  R  u0(y)ρ0(y) dy = E(u0(Y 0)) = E(E(G(Y T1)| σ(FW  0 , σ(Y 0)))) = E(G(Y T1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  Now, let G = φξ with φ ∈ C∞  c (R) and ξ ∈ L∞(Ω, FT1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Consequently, we obtain  E(ξ⟨φ, ρT1⟩) = E(ξφ(Y T1)) = E(ξE(φ(Y T1) | FW  T1 )),  HEGSELMANN–KRAUSE MODEL WITH ENVIRONMENTAL NOISE  21  which proves  ⟨φ, ρT1⟩ = E(φ(Y T1) | FW  T1 ),  P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=',  and, therefore, ρT1 is the conditional density of Y  1  T1 given FT1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Since T1 is arbitrary, we have  proven the existence of a strong solution Y 1 of the mean-field SDE (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  On the other hand, if (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='1) has a strong solution with conditional density  ρ ∈ L2  FW ([0, T ∗];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' W 1,2(R)) ∩ S∞  FW ([0, T ∗];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' L1(R) ∩ L2(R)),  then the conditional density process of Y 1 is the solution to the SPDE (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Indeed, if we first  apply Itˆo’s formula with a function ϕ ∈ C∞  c (R), then take the conditional expectation with  respect to the filtration FW and subsequently applying stochastic Fubini theorem [HvS21,  Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='5], we conclude that density process of Y 1  t satisfies (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' By the uniqueness of  the SPDE (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='1) we obtain that ρt, which is the solution constructed in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='5, is the  conditional density of Y 1  t given FW  t  for all t ∈ [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  □  In the following lemma, we close the gap in the above proof by demonstrating the existence  of a solution of the BSPDE (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='3) and the explicit representation of u0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='4 (Dual BSPDE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Let Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='1 and Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='1 hold along with k ∈  L1(R)∩L2(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Then, for every T1 ∈ (0, T] and G ∈ L∞(Ω, FT1, C∞  c (R)) the following BSPDE  dut = −  �σ2  t + ν2  2  d2  dx2 ut − (k ∗ ρt) d  dxut + ν d  dxvt  �  dt + vt dWt,  t ∈ [0, T],  uT1 = G,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='3)  admits a unique solution  (u, v) ∈ (L2  FW ([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' W 2,2(R)) ∩ S2  FW ([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' W 1,2(R))) × L2  FW ([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' W 1,2(R)),  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' for any ϕ ∈ C∞  c (R) the equality  ⟨ut, ϕ⟩L2(R) = ⟨G, ϕ⟩L2(R) +  T1  �  t  �σ2  s + ν2  2  d2  dx2us − (k ∗ ρs) d  dxus + ν d  dxvs, ϕ  �  L2(R)  ds  −  T1  �  t  ⟨vs, ϕ⟩L2(R) dWs  holds for all t ∈ [0, T] with probability one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Moreover, we have  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='4)  u0(Y 0) = E(G(Y T1) | σ(σ(Y 0), FW  0 )),  where (Y t, t ∈ [0, T]) is the solution of the linearised SDE (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='2) in the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' We note that by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='3 we have ρ ∈ L2  F([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' W 3,2(R)) ∩ S2  F([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' W 2,2(R)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  Our approach is to verify the assumptions of the L2-theory (see for example [DQT12, Theo-  rem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='5]) for BSPDEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Let u1, u2 ∈ W 2,2(R), then  ����(k ∗ ρt) d  dxu1 − (k ∗ ρt) d  dxu2  ����  L2(R)  ≤ ∥k ∗ ρt∥L∞(R)  ����  d  dx(u1 − u2)  ����  L2(R)  ≤ ∥k∥L2(R) ∥ρt∥L2(R)  ����  d  dx(u1 − u2)  ����  L2(R)  ≤ ∥k∥L2(R) ∥ρt∥L2(R) ∥u1 − u2∥W 1,2(R) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  22  CHEN, NIKOLAEV, AND PR ¨OMEL  Now, by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='5, ∥ρt∥L2(R) is uniformly bounded in (ω, t) ∈ Ω × [0, T] and the interpo-  lation theorem [AF03, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='2] implies for all ε > 0,  ����(k ∗ ρt) d  dxu1 − (k ∗ ρt) d  dxu2  ����  L2(R)  ≤ ε ∥u1 − u2∥W 2,2(R) + C(k, ∥ρ∥S∞  FW ([0,T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='L2(R)))κ(ε) ∥u1 − u2∥L2(R)  for some non-negative decreasing function κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Hence, Assumption 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='4 in [DQT12, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='5]  is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' The other assumptions are also easily verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' As a result we obtain a solution  (u, v) ∈ (L2  FW ([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' W 2,2(R)) ∩ S2  FW ([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' L2(R))) × L2  FW ([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' W 1,2(R))  of the BSPDE (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Here, the fact that u ∈ S2  FW ([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' L2(R)) is a direct consequence of  [DM10, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  It remains to show that the equality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='4) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' By the bound  E  �  sup  t≤T1  ∥ρt∥2  W 1,2(R)  �  < ∞  given by ρ ∈ S2  FW ([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' W 1,2(R)), we observe that there exists a set Ω′ with P(Ω′) = 1 and  for all ω ∈ Ω′ we have  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='5)  sup  t≤T1  ∥ρt(ω, ·)∥W 1,2(R) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  Also, the map (ω, t) → ∥ρt(ω, ·)∥W 1,2(R) is predictable with respect to FW by the L2-SPDE  theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Consequently, we can define for each m ∈ N the stopping time  τm(ω) = inf{t ∈ [0, T1] : ∥ρt(ω, ·)∥W 1,2(R) ≥ m}  and τm ↑ T1 by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Furthermore, let us define  F(t, x) := (k ∗ ρt)(x) d  dxut(x)  and  Fm(t, x) := F(t, x)1(0,τm](t),  and note that Fm ∈ L2  FW ([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' L2(R)) still satisfies all assumptions of the L2-BSPDE theory  ([DQT12, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='5]) and therefore there exists a solution  (um, vm) ∈ (L2  FW ([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' W 2,2(R)) ∩ S2  FW ([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' L2(R))) × L2  FW ([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' W 1,2(R))  of the following BSPDE  dum  t = −  �σ2  t + ν2  2  d2  dx2 um  t − Fm(t) + ν d  dxvm  t  �  dt + vm  t dWt,  um  T1 = G,  for each m ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  HEGSELMANN–KRAUSE MODEL WITH ENVIRONMENTAL NOISE  23  In the next step we want to obtain a L2  FW ([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' W 1,2(R))-bound for Fm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' The L2-estimate  follows directly from the above computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' For the weak derivative we compute  ����  d  dx  �  (k ∗ ρt) d  dxut  �  1(0,τm]  ����  L2(R)  ≤  ����  1(0,τm]  �  k ∗ d  dxρt  � d  dxut  ����  L2(R)  +  ����(k ∗ ρt) d2  dx2 ut  ����  L2(R)  ≤  1(0,τm]  ����  �  k ∗ d  dxρt  � d  dxut  ����  L2(R)  + ∥k∥L2(R) ∥ρt∥S∞  F ([0,T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='L2(R))) ∥ut∥W 2,2(R) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  Since ρ ∈ S∞  FW ([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' L2(R))), the last term behaves nicely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' However, the first term would  be problematic because without the stopping time we do not have a similar L∞-estimate for  the derivative, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' ρ ∈ S∞  FW ([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' W 1,2(R))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Hence, in order to overcome this problem we  introduced the stopping time τm and, therefore, we discover  ����  d  dx  �  (k ∗ ρt) d  dxut  �  1(0,τm]  ����  L2  FW ([0,T1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='L2(R))  ≤ ∥k∥2  L2(R) E  � T1  �  0  1(0,τm](t)  ����  d  dxρt  ����  2  L2(R)  ����  d  dxut  ����  2  L2(R)  dt  �  + ∥k∥L2(R) ∥ρt∥S∞  F ([0,T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='L2(R))) ∥ut∥L2  FW ([0,T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='W 2,2(R))  ≤ ∥k∥L2(R) m2 ∥u∥L2  FW ([0,T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='W 1,2(R))  + ∥k∥L2(R) ∥ρt∥S∞  F ([0,T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='L2(R))) ∥ut∥L2  FW ([0,T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='W 2,2(R)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  As a result, we obtain  ∥Fm∥L2  FW ([0,T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='W 1,2(R)) < ∞  for each m ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Applying [DQT12, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='5] again, we find  (um, vm) ∈ (L2  FW ([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' W 3,2(R)) ∩ S2  FW ([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' W 1,2(R))) × L2  FW ([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' W 2,2(R)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  The above regularity (p(m − 2) > 1 with p = 2, m = 3) allows us to apply [DTZ13, Corol-  lary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='2], which tells us that there exists a set of full measure Ω′′  m maybe different from Ω′  such that  um(t, x) = G(x) +  T1  �  t  σ2  t + ν2  2  d2  dx2 um(s, x) −  1(0,τm](s)(k ∗ ρ)(s, x) d  dxu(s, x)  + ν d  dxvm(s, x) ds −  T1  �  t  vm(s, x) dWs  holds for all (t, x) ∈ [0, T1] × R on Ω′′  m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' We use the subscript m to indicate that even though  the set Ω′′  m is independent of (t, x) it still may depend on m ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  Besides, to be precise, [Du20, Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='2] actually requires Fm ∈ L2  FW ([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' W 3,2(R)),  which is more regularity than we have.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  However, one can modify the proof of [DTZ13,  Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='2] to obtain the same result with Fm ∈ L2  FW ([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' W 1,2(R)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' The crucial part  is that a mollification of Fm with the standard mollifier converges in the supremum norm  24  CHEN, NIKOLAEV, AND PR ¨OMEL  to Fm, which follows from Morrey’s inequality even in our case Fm ∈ L2  FW ([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' W 1,2(R)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  For a similar result in the SPDE setting we refer to [Roz90, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  Next, we want to apply an Itˆo–Wentzell type formula [YT13, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='1] (with V =  u, X = Y therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Hence, we need to verify the required assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' First, we can view um  as a jointly continuous Itˆo process in (t, x) by [DTZ13, Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='2] on the set Ω′′  m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' We also  recall that um ∈ L2  FW ([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' W 2,2(R)), vm ∈ L2  FW ([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' W 1,2(R)), Fm ∈ L2  FW ([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' L2(R))  and Y is a strong solution and therefore a continuous semimartingale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  Moreover, we note that ρ,  d  dxu are PW × B(R)-measurable and τm is PW -measurable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  Hence, the same holds true for Fm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Also as previously mentioned (k ∗ρt) is bounded in x ∈ R  for almost all (ω, t) ∈ Ω × [0, T1] and as we have seen in the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='3 (Step 1)  is Lipschitz continuous for almost all (ω, t) ∈ Ω × [0, T1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Thus, all assumptions of [YT13,  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='1] are fulfilled and we obtain  um  T1(Y T1)  = um  0 (Y 0) +  T1  �  0  �σ2  t + ν2  2  d2  dx2 um  t − (k ∗ ρt) d  dxut + ν d  dxvm  t − σ2  t + ν2  2  d2  dx2 um  t  +  1(0,τm](t)(k ∗ ρt) d  dxut − ν d  dxvm  t  �  (Y t) dt  +  T1  �  0  �  vm  t + ν d  dxum  t  �  (Y t) dWt +  T1  �  0  σt(Y t) d  dxum  t (Y t) dBt  = um  0 (Y 0) +  T1  �  0  F(t, Y t)(1(0,τm](t) − 1) dt +  T1  �  0  �  vm  t + ν d  dxum  t  �  (Y t) dWt  +  T1  �  0  σt(Y t) d  dxum  t (Y t) dBt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='6)  With this formula at hand, let us introduce the filtration Gt = σ(σ(Y t), FW  t ), t ∈ [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  Our aim is to take the conditional expectation with respect to G0 on both sides of the above  equation in order to cancel both the stochastic integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' We observe that Gt ⊂ Ft and the  solution (Y t, t ∈ [0, T]) is predictable with respect to the filtration G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Moreover, B1 and W  are still per definition Brownian motions under the filtration (Ft, t ∈ [0, T]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Hence, both  stochastic integrals are martingales with respect to the filtration (Ft, t ∈ [0, T]), if we can  prove an L2-bound on the integrands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  By Sobolev’s embedding or Morrey’s inequality and the bound on σt we have  E  � T1  �  0  ����σt(Y t) d  dxum  t (Y t)  ����  2  dt  �  ≤ ΛE  � T1  �  0  ����  d  dxum  t  ����  2  L∞(R)  dt  �  ≤ ΛE  � T1  �  0  ∥um  t ∥2  W 2,2(R) dt  �  < ∞,  HEGSELMANN–KRAUSE MODEL WITH ENVIRONMENTAL NOISE  25  which verifies that the second stochastic integral of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='6) is a martingale with respect to the  filtration F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Hence, we discover  E  � T1  �  0  σt(Y t) d  dxum  t (Y t) dBt  ���� G0  �  = E  �  E  � T1  �  0  d  dxum  t (Y t) dBt |F0  � ���� G0  �  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  Furthermore, we have the estimate  E  � T1  �  0  ����vm  t (Y t) + ν d  dxum  t (Y t)  ����  2  dt  �  ≤ 2E  � T1  �  0  ∥vm  t ∥2  L∞(R) + ν2  ����  d  dxum  t  ����  2  L∞(R)  dt  �  ≤ CE  � T1  �  0  ∥vm  t ∥2  W 1,2(R) + ∥um  t ∥2  W 2,2(R) dt  �  < ∞,  where we used Morrey’s inequality in the second step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Hence, the first stochastic integral  appearing in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='6) is also a martingale with respect to the filtration G starting at zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Taking  the conditional expectation with respect to G0 in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='6) and having in mind that Y 0 = Y0, we  obtain  E(um  T1(Y T1) | G0) = um  0 (Y0) + E  � T1  �  0  F(t, Y t)(1(0,τm](t) − 1) dt  ���� G0  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  It remains to show that  lim  m→∞(E(um  T1(Y T1) | G0) − um  0 (Y0)) = E(uT1(Y T1) | G0) − u0(Y0),  P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=',  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='7)  and  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='8)  lim  m→∞ E  � T1  �  0  F(t, Y t)(1(0,τm](t) − 1) dt  ���� G0  �  = 0,  P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  We first show (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='8) but we prove the L1-convergence, which then implies (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='8) along a subse-  quence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' We compute  E  �����E  � T1  �  0  F(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Y t)(1(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='τm](t) − 1) dt  ���� G0  �����  �  ≤ E  �  E  �����  T1  �  0  F(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Y t)(1(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='τm](t) − 1) dt  ����  ���� G0  ��  ≤ E  � T1  �  0  |F(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Y t)(1(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='τm](t) − 1)| dt  �  ≤ E  � T1  �  0  ∥F(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' ·)∥L∞(R) |1(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='τm](t) − 1| dt  �  26  CHEN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' NIKOLAEV,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' AND PR ¨OMEL  ≤ E  � T1  �  0  ∥F(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' ·)∥W 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='2(R) |1(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='τm](t) − 1| dt  �  ≤ CE  � T1  �  0  � ����(k ∗ ρt) d  dxut  ����  L2(R)  +  ����(k ∗ ρt) d2  dx2ut  ����  L2(R)  +  ����(k ∗ d  dxρt) d  dxut  ����  L2(R)  �  |1(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='τm](t) − 1| dt  �  ≤ C(T)E  � T1  �  0  �  2 ∥ρt∥L2(R) +  ����  d  dxρt  ����  L2(R)  �  ∥ut∥W 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='2(R) |1(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='τm](t) − 1| dt  �  ≤ C(T)E  � T1  �  0  (∥ρt∥2  W 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='2(R) + ∥ut∥2  W 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='2(R))|1(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='τm](t) − 1| dt  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  where we used Morrey’s inequality in the fourth step and H¨older’s inequality as well as (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='4)  in the sixth step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Finally, ρ ∈ L2  FW ([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' W 1,2(R)), u ∈ L2  FW ([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' W 2,2(R)), dominated  convergence theorem and τm ↑ T1 tell us that the last term vanishes for m → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  Taking the above subsequence, which we do not rename, we demonstrate (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='7) along a  further subsequence by proving L2-convergence of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Let us define �um = u − um and  �vm = v − vm, which solve the following BSPDE  d�um  t = −  �σ2  t + ν2  2  d2  dx2 �um  t − �Fm + ν d  dx�vm  t  �  dt + �vm  t dWt  with terminal condition �G = 0, free term  �Fm(t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='x) = (k ∗ ρ)(t, x) d  dxu(t, x)(1 −  1(0,τm](t)) = F(t, x)(1 −  1(0,τm](t))  and �Fm ∈ L2  FW ([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' L2(R)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Hence, by [DQT12, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='5] the solution of the BSPDE  is unique and by [DM10, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='2 and Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='1, Step 1] satisfies the estimate  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='9)  E  �  sup  t≤T1  ∥�um  t ∥2  W 1,2(R)  �  ≤ C(T)  ��� �Fm  ���  L2  FW ([0,T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='L2(R)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  Consequently, using Jensen inequality, the G0-measurability of u0(Y0), Morrey’s inequality  and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='9) we find  E(|E(um  T1(Y T1) | G0) − um  0 (Y0)) − (E(uT1(Y T1) | G0) − u0(Y0))|2)  ≤ 2E(|�um  T1(Y T1)|2 + |�um  0 (Y0)|2)  ≤ 2E(  ���um  T1  ��2  L∞(R) + ∥�um  0 ∥2  L∞(R))  ≤ 2E(  ���um  T1  ��2  W 1,2(R) + ∥�um  0 ∥2  W 1,2(R))  ≤ 4E  �  sup  t≤T1  ∥�um  t ∥2  W 1,2(R)  �  HEGSELMANN–KRAUSE MODEL WITH ENVIRONMENTAL NOISE  27  ≤ C(T)E  � T1  �  0  ���( �Fm)t  ���  2  L2(R) dt  �  ≤ C(T)E  � T1  �  0  |1 −  1(0,τm](t)|2 ∥F(t, x)∥2  L2(R) dt  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  But F ∈ L2  FW ([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' L2(R)) and, therefore, an application of the dominated convergence  theorem proves (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='7) along a subsequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' As a result, the last inequality together with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='8)  implies (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='4) for all ω ∈ �Ω := �  m∈N  Ω′′  m ∩ Ω′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Hence, the lemma is proven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  □  As an application of Theorem (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='3), we obtain a solution for the non-regularized and the  regularized mean-field SDEs (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='9) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='10), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Let 0 ≤ ρ0 ∈ L1(R) ∩ W 2,2(R) with ∥ρ0∥L1(R) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Suppose that for T > 0  there exists unique solutions ρ and ρτ in L2  FW ([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' W 1,2(R)) of the SPDEs (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='11) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='12),  respectively, on the interval [0, T] with  ∥ρ∥L2  Fw([0,T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='W 1,2(R)) + ∥ρ∥S∞  Fw([0,T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='L1(R)∩L2(R)) ≤ C  and  ∥ρτ∥L2  Fw([0,T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='W 1,2(R)) + ∥ρτ∥S∞  Fw([0,T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='L1(R)∩L2(R)) ≤ C,  for some finite constant C > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Moreover, let the diffusion coefficient σ: [0, T] × R → R  satisfy Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Then, for τ > 0 there exists unique solution (Y i, t ∈ [0, T]) and  (Y i,τ, t ∈ [0, T]) for the mean-field SDEs (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='9) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='10), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Moreover, ρt is the  conditional density of Y i  t given FW  t  and ρτ  t is the conditional density of Y i,τ  t  given FW  t , for  every t ∈ [0, T] and for all i ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Mean-field limits of the interacting particle systems  In this section we establish propagation of chaos for the regularized Hegelsmann–Krause  models (in particular we recall kτ  HK(x) is the approximation of kHK(x) =  1[−R,R](x)x) with en-  vironmental noise (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='8) towards the (non-regularized) mean-field stochastic differential equa-  tions (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='9) resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' the (non-regularized) stochastic Fokker–Planck equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='11), presenting  the density based model of the opinion dynamics, see Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' To prove propagation of  chaos, we first derive estimates of the difference of the regularized interacting particle sys-  tem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='8) and the regularized mean-field SDE (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='10) (see Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='2) as well as of the  difference of the solutions to the regularized stochastic Fokker–Planack equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='12) and  of the non-regularized stochastic Fokker–Planack equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='11) (see Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' As  preparation, we need the following auxiliary lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Let (Ω, F, P) be a probability space, G ⊆ F a sub-σ-algebra and X, Y con-  ditionally independent random variables with values in R given G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Moreover, let X have a  conditional density f : Ω×R → R such that f is G ⊗B(R)/B(R) measurable and in L1(Ω×R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  Then, for every bounded measurable functions h: R × R → R, we have  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='1)  E(h(X, Y ) | σ(G, σ(Y )))(ω) =  �  R  h(z, Y (ω))f(ω, z) dz,  ω ∈ Ω′,  on a set Ω′ ⊂ Ω of full probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  28  CHEN, NIKOLAEV, AND PR ¨OMEL  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' First, we notice that by Fubini’s theorem the right-hand side of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='1) is σ(G, σ(Y ))-  measurable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  By the standard Lebesgue integral approximation technique we may assume  h =  1B×B′(x, y) for some measurable sets B, B′ ∈ B(R) in order to prove (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Hence, we  need to show  E(1A  1B×B′(X, Y )) = E  �  1A  �  R  1B×B′(z, Y (ω))f(ω, z) dz  �  for all A ∈ σ(G, σ(Y )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Now, we reduce the problem again to A = C ∩ C′′ with C ∈ G and  C′′ = Y −1(B′′) for some B′′ ∈ B(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Consequently, using the conditional independence we  find  E(1C∩C′′  1B×B′(X, Y )) = E(1CE(1C′′  1B×B′(X, Y ) | G))  = E(1CE(1B′′∩B′(Y )1B(X) | G))  = E(1CE(1B′′∩B′(Y ) | G)E(1B(X) | G))  = E(1C  1B′′∩B′(Y )E(1B(X) | G))  = E  �  1C∩C′′(ω)1B′(Y (ω))  �  R  1B(z)f(ω, z) dz  �  = E  �  1C∩C′′(ω)  �  R  1B×B′(z, Y (ω))f(ω, z) dz  �  and the lemma is proven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  □  The next proposition provides an estimate of the difference of the regularized particle  system and the regularized mean-field SDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Suppose Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='1 and Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='1 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' For each N ∈ N, let  ((Y i,τ  t  , t ∈ [0, T]), i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' , N) be the solutions to the regularized mean-field SDEs (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='10), as  provided by Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='5, and let ((Xi,τ  t , t ∈ [0, T]), i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' , N) be the solution to regularized  interaction particle system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Then, for any τ > 0 and N ∈ N we have  sup  t∈[0,T]  sup  i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=',N  E(|Xi,τ  t  − Y i,τ  t  |2) ≤  2x ∥kHK∥2  L2(R) T  (N − 1)τ  exp  �(C + Λ)T  τ  �  ,  where C is some finite generic constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Applying Itˆo’s formula, we find  |Xi,τ  t  − Y i,τ  t  |2  = 2  t  �  0  (Xi,τ  s  − Y i,τ  s  )  1  N − 1  N  �  j=1  j̸=i  −kτ  HK(Xi,τ  s  − Xj,τ  s ) + (kτ  HK ∗ ρτ  s)(Y i,τ  s  ) ds  + 2  t  �  0  (Xi,τ  s  − Y i,τ  s  )(σ(s, Xi,τ  s ) − σ(s, Y i,τ  s  )) dBi  s +  t  �  0  (σ(s, Xi,τ  s ) − σ(s, Y i,τ  s  ))2 ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='2)  HEGSELMANN–KRAUSE MODEL WITH ENVIRONMENTAL NOISE  29  Splitting the sum we have  1  N − 1  N  �  j=1  j̸=i  −kτ  HK(Xi,τ  s  − Xj,τ  s ) + (kτ  HK ∗ ρτ  s)(Y i,τ  s  )  =  1  N − 1  N  �  j=1  j̸=i  (kτ  HK ∗ ρτ  s)(Y i,τ  s  ) − kτ  HK(Y i,τ  s  − Y j,τ  s  )  +  1  N − 1  N  �  j=1  j̸=i  kτ  HK(Y i,τ  s  − Y j,τ  s  ) − kτ  HK(Xi,τ  s  − Xj,τ  s )  = Is  1 + Is  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  For Is  2, we use the property of our approximation sequence to discover  |Is  2| ≤  1  N − 1  N  �  j=1  j̸=i  |kτ  HK(Y i,τ  s  − Y j,τ  s  ) − kτ  HK(Xi,τ  s  − Xj,τ  s )|  ≤  C  (N − 1)τ  N  �  j=1  |Xj,τ  s  − Y j,τ  s  | + |Xi,τ  s  − Y i,τ  s  |  and consequently  E  �����2  t  �  0  (Xi,τ  s  − Y i,τ  s  )Is  2 ds  ����  �  ≤  C  (N − 1)τ  t  �  0  N  �  j=1  E(|Xj,τ  s  − Y j,τ  s  |2 + |Xi,τ  s  − Y i,τ  s  |2) ds  ≤ C  τ  t  �  0  sup  i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=',N  E(|Xi,τ  s  − Y i,τ  s  |2) ds,  where we used Young’s inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Next, let us rewrite Is  1 such that  Is  1 =  1  N − 1  N  �  j=1  j̸=i  (kτ  HK ∗ ρτ  s)(Y i,τ  s  ) − kτ  HK(Y i,τ  s  − Y j,τ  s  ) =  1  N − 1  N  �  j=1  j̸=i  Zs  i,j  with  Zs  i,j = (kτ  HK ∗ ρτ  s)(Y i,τ  s  ) − kτ  HK(Y i,τ  s  − Y j,τ  s  )  for i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Furthermore,  E(|Is  1|2) =  1  (N − 1)2 E  �  E  � N  �  j=1  j̸=i  Zs  i,j  N  �  k=1  j̸=i  Zs  i,k  ���� σ(FW  s , σ(Y i,τ  s  ))  ��  =  1  (N − 1)2  N  �  j=1  j̸=i  N  �  k=1  j̸=i  E(E(Zs  i,jZs  i,k | σ(FW  s , σ(Y i,τ  s  ))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  30  CHEN, NIKOLAEV, AND PR ¨OMEL  It easy to verify that (Y i,τ  s  , i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' , N) are conditionally independent given FW  s  and by  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='3 have conditional density ρs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Hence, we apply Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='1 to obtain  E(kτ  HK(Y i,τ  s  − Y j,τ  s  ) | σ(FW  s , σ(Y i  s , τ))) = (kτ  HK ∗ ρτ  s)(Y i,τ  s  )  and therefore E(Zs  i,j | σ(FW  s , σ(Y i,τ  s  ))) = 0 since (kτ  HK∗ρτ  s)(Y i,τ  s  ) is σ(FW  s , σ(Y i,τ  s  ))-measurable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  Consequently, for the cross terms j ̸= k one can verify that  E(Zs  i,jZs  i,k | σ(FW  s , σ(Y ,τ  s ))) = E(Zs  i,j | σ(FW  s , σ(Y ,τ  s )))E(Zs  i,k | σ(FW  s , σ(Y ,τ  s ))) = 0  by the previous findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Hence, we have  E(|Is  1|2) =  1  (N − 1)2  N  �  j=1  j̸=i  E(|Zs  i,j|2)  and using the boundedness of kτ, the structure of our approximation and mass conservation,  we obtain  E(|Zs  i,j|2) = E(|(kτ  HK ∗ ρτ  s)(Y i,τ  s  ) − kτ  HK(Y i,τ  s  − Y j,τ  s  )|2) ≤ 2 ∥kτ  HK∥2  L∞(R) ≤ 2  τ ∥kHK∥2  L2(R) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  Combining all the above facts, we get  E(|Is  1|2) ≤  2 ∥kHK∥2  L2(R)  (N − 1)τ  and  E  �  2  t  �  0  (Xi,τ  t  − Y i,τ  t  )Is  1 ds  �  ≤ E  �  t  �  0  |Xi,τ  t  − Y i,τ  t  |2 ds +  t  �  0  |Is  1|2 ds  �  ≤  t  �  0  E(|Xi,τ  t  − Y i,τ  t  |2) ds +  2 ∥kHK∥2  L2(R) T  (N − 1)τ  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  Moreover, using the Lipschitz continuity of our coefficients σ we obtain  E  �  t  �  0  (σ(s, Xi,τ  s ) − σ(s, Y i,τ  s  ))2 ds  �  ≤ Λ  t  �  0  E  �  |Xi,τ  s  − Y i,τ  s  |2�  ≤ Λ  t  �  0  sup  i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=',N  E  �  |Xi,τ  s  − Y i,τ  s  |2�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  Now, combining this with the estimate of Is  2, as wells as the fact that the stochastic integral  in equation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='2) are martingales (Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='1), we obtain  sup  i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=',N  E(|Xi,τ  t  − Y i,τ  t  |2) ≤ C  τ  t  �  0  sup  i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=',N  E(|Xi,τ  t  − Y i,τ  t  |2) ds  + Λ  t  �  0  sup  i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=',N  E(|Xi,τ  t  − Y i,τ  t  |2) ds +  2 ∥kHK∥2  L2(R) T  (N − 1)τ  ≤ C + Λ  τ  t  �  0  sup  i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=',N  E(|Xi,τ  t  − Y i,τ  t  |2) ds +  2 ∥kHK∥2  L2(R) T  (N − 1)τ  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  HEGSELMANN–KRAUSE MODEL WITH ENVIRONMENTAL NOISE  31  Applying Gronwall’s inequality yields  sup  t∈[0,T]  sup  i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=',N  E(|Xi,τ  t  − Y i,τ  t  |2) ≤  2 ∥kHK∥2  L2(R) T  (N − 1)τ  exp  �(C + Λ)T  τ  �  ,  which proves the proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  □  As next step we need an estimate of the difference of the solutions to the regularized  mean-field SDEs and the non-regularized mean-field SDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Recall that, by the stochastic  Fokker–Planack equations, it is sufficient to consider the associated solutions ρτ and ρ of the  SPDEs (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='12) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' For more details regarding this observation we refer to the proof of  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Suppose Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='1 and Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='1 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Let ρτ and ρ be the so-  lutions to the regularized stochastic Fokker–Planack equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='12) and to the non-regularized  stochastic Fokker–Planack equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='11), respectively, as provided in Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Then,  ∥ρτ − ρ∥S∞  FW ([0,T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='L2(R))  ≤ C(λ, Λ, T, ∥kHK∥L2(R) , ∥ρ∥S∞  FW ([0,T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='L2(R)) ∥ρτ∥S∞  FW ([0,T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='L2(R))) ∥kτ  HK − kHK∥L2(R) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' To estimate the difference ρt − ρτ  t , we notice that  ρτ  t − ρt =  d2  dx2  �  σ2  t + ν2  2  (ρτ  t − ρt)  �  dt + d  dx((kτ  HK ∗ ρτ  t )ρτ  t ) dt  − d  dx((kHK ∗ ρt)ρt) dt − ν d  dx(ρτ  t − ρt) dWt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  Performing similar computations as in the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='5 by using Young’s inequality,  we get  ∥ρτ  t − ρt∥2  L2(R)  ≤ −λ  t  �  0  ����  d  dxρτ  s − d  dxρs  ����  2  L2(R)  ds −  t  �  0  �  (ρs − ρτ  s) d  dx(σ2  s), d  dxρs − Dρτ  s  �  L2(R)  ds  − 2  t  �  0  �  (kτ  HK ∗ ρτ  s)ρτ  s − (kHK ∗ ρs)ρs, d  dxρτ  s − d  dxρs  �  L2(R)  ds  ≤ −3λ  4  t  �  0  ����  d  dxρτ  s − d  dxρs  ����  2  L2(R)  ds + Λ2  λ  t  �  0  ∥ρs − ρτ  s∥2  L2(R) ds  − 2  t  �  0  �  (kτ  HK ∗ ρτ  s)ρτ  s − (kHK ∗ ρs)ρs, d  dxρτ  s − Dρs  �  L2(R)  ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  Rewriting the last term gives  (kτ  HK ∗ ρτ  s)ρτ  s − (kHK ∗ ρs)ρs  = ((kτ  HK − kHK) ∗ ρτ  s)ρτ  s + (kHK ∗ ρτ  s)ρτ  s − (kHK ∗ ρs)ρs  = ((kτ  HK − kHK) ∗ ρτ  s)ρτ  s + (kHK ∗ (ρτ  s − ρs))ρτ  s + ((kHK ∗ ρs)(ρτ  s − ρs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  32  CHEN, NIKOLAEV, AND PR ¨OMEL  Hence, for the last two terms we can use Young’s inequality, Young’s inequality for convolu-  tion, mass conservation and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='4) to obtain  �  (kHK ∗ ρs)(ρτ  s − ρs) + (kHK ∗ (ρτ  s − ρs))ρτ  s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' d  dxρτ  s − d  dxρs  �  L2(R)  ≤ ∥kHK∥L2(R) ∥ρs∥L2(R) ∥ρτ  s − ρs∥L2(R)  ����  d  dxρτ  s − d  dxρs  ����  L2(R)  + ∥kHK∥L2(R) ∥ρτ  s − ρs∥L2(R) ∥ρτ  s∥L2(R)  ����  d  dxρτ  s − d  dxρs  ����  L2(R)  ≤ λ  4  ����  d  dxρτ  s − d  dxρs  ����  2  L2(R)  + 1  λ(∥kHK∥L2(R) ∥ρs∥L2(R) ∥ρτ  s − ρs∥L2(R) + ∥kHK∥L2(R) ∥ρτ  s − ρs∥L2(R) ∥ρτ  s∥L2(R))2  ≤ λ  2  ����  d  dxρτ  s − d  dxρs  ����  2  L2(R)  + 2  λ ∥kHK∥2  L2(R) ∥ρτ  s − ρs∥2  L2(R) (∥ρs∥2  L2(R) + ∥ρτ  s∥2  L2(R)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  Moreover,  �  ((kτ  HK − kHK) ∗ ρτ  s)ρτ  s, d  dxρτ  s − d  dxρs  �  L2(R)  ≤ ∥(kτ  HK − kHK) ∗ ρτ  s∥L∞(R) ∥ρτ  s∥L2(R)  ����  d  dxρτ  s − d  dxρs  ����  L2(R)  ≤ ∥kτ  HK − kHK∥L2(R) ∥ρτ  s∥L2(R) ∥ρτ  s∥L2(R)  ����  d  dxρτ  s − d  dxρs  ����  L2(R)  ≤ λ  4  ����  d  dxρτ  s − d  dxρs  ����  2  L2(R)  + 1  λ ∥kτ  HK − kHK∥2  L2(R) ∥ρτ  s∥4  L2(R) ,  where we used Young’s inequality for convolutions in the second inequality and Young’s  inequality for the last step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Consequently, combining the last two estimates with our previous  L2-norm inequality and absorbing the L2-norm of the derivatives we obtain  ∥ρτ  t − ρt∥2  L2(R)  ≤  2 ∥kHK∥2  L2(R) + 1  λ  t  �  0  (∥ρs∥2  L2(R) + ∥ρτ  s∥2  L2(R) + Λ2) ∥ρτ  s − ρs∥2  L2(R) ds  + 1  λ  t  �  0  ∥kτ  HK − kHK∥2  L2(R) ∥ρτ  s∥4  L2(R) ds  ≤  2 ∥kHK∥2  L2(R) + 1  λ  t  �  0  (∥ρs∥2  L2(R) ∥ρτ  s∥2  L2(R) + Λ2) ∥ρτ  s − ρs∥2  L2(R) ds  + T  λ ∥kτ  HK − kHK∥2  L2(R) sup  t∈[0,T]  ∥ρτ  t ∥4  L2(R) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  HEGSELMANN–KRAUSE MODEL WITH ENVIRONMENTAL NOISE  33  Applying Gronwall’s inequality and using the uniform bound (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='6), we obtain  sup  t∈[0,T]  ∥ρτ  t − ρt∥2  L2(R)  ≤ T  λ ∥kτ  HK − kHK∥2  L2(R) sup  t∈[0,T]  ∥ρτ  t ∥4  L2(R)  × exp  �2 ∥kHK∥2  L2(R) + 1  λ  T  �  0  (∥ρs∥2  L2(R) + ∥ρτ  s∥2  L2(R) + Λ) ds  �  ≤ T  λ ∥kτ  HK − kHK∥2  L2(R) sup  t∈[0,T]  ∥ρτ  t ∥4  L2(R)  × exp  �2T(∥kHK∥2  L2(R) + 1)  λ  (∥ρ∥2  S∞  FW ([0,T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='L2(R)) + ∥ρτ∥2  S∞  FW ([0,T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='L2(R)) + Λ)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  After taking the supremum over ω ∈ Ω, we arrive at  ∥ρτ − ρ∥S∞  FW ([0,T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='L2(R))  ≤ C(λ, Λ, T, ∥kHK∥L2(R) , ∥ρ∥S∞  FW ([0,T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='L2(R)) ∥ρτ∥S∞  FW ([0,T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='L2(R))) ∥kτ − k∥L2(R) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  □  Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Due to Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='3, we know that the solutions ρτ of the regularized sto-  chastic Fokker–Planack equations converges to the solution ρ of the non-regularized stochastic  Fokker–Planack equation as the interaction force kernels converge in the L2-norm for τ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  Finally, we are in a position to state and prove the main theorem of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='5 (Propagation of chaos).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Suppose Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='1 and Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='1 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  Let ρ be the solution of the stochastic Fokker–Planack equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='11) and let us denote by  ΠN  t (ω) = 1  N  N  �  i=1  δXi,τ  t  (ω)  the empirical measure of our regularized interaction system ((Xi,τ  t , τ > 0), i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' , N) given  by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Then, we have, for all t ∈ [0, T],  E(|⟨ΠN  t , ϕ⟩ − ⟨ρt, ϕ⟩|2)  ≤ C  �  λ, Λ, T, ∥kHK∥L2(R) , ∥ϕ∥C1(R) , ∥ϕ∥L2(R) , ∥ρ∥S∞  FW ([0,T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='L2(R)) , ∥ρτ∥S∞  FW ([0,T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='L2(R))  �  ×  � 1  N +  1  (N − 1)τ exp  �(C + Λ)T  τ  �  + ∥kτ − k∥2  L2(R)  �  for any ϕ ∈ C∞  c (R) and a finite constant C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  34  CHEN, NIKOLAEV, AND PR ¨OMEL  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' We compute  E(|⟨ΠN  t , ϕ⟩ − ⟨ρτ  t , ϕ⟩|2)  = E  �� 1  N  N  �  i=1  ϕ(Xi,τ  t ) −  �  R  ρτ  t (x)ϕ(x) dx  �2�  = 2E  �� 1  N  N  �  i=1  ϕ(Xi,τ  t ) − 1  N  N  �  i=1  ϕ(Y i,τ  t  )  �2�  + 2E  �� 1  N  N  �  i=1  ϕ(Y i,τ  t  ) −  �  R  ρτ  t (x)ϕ(x) dx  �2�  ≤  2  N 2  � N  �  i=1  E(|ϕ(Xi,τ  t ) − ϕ(Y i,τ  t  )|2)1/2  �2  + 2E  � 1  N  N  �  i=1  ϕ(Y i,τ  t  ) −  �  R  ρτ  t (x)ϕ(x) dx  �2�  ≤ 2  sup  i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=',N  E(|ϕ(Xi,τ  t ) − ϕ(Y i,τ  t  )|2) + 2E  �� 1  N  N  �  i=1  ϕ(Y i,τ  t  ) −  �  R  ρτ  t (x)ϕ(x) dx  �2�  ,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='3)  where we used Minkwoski’s inequality in the third step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Now, by Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='2 and  |ϕ(Xi,τ  t ) − ϕ(Y i,τ  t  )|2 ≤  ����  d  dxϕ  ����  2  L∞ |Xi,τ  t  − Y i,τ  t  |2,  we can estimate the first term by  �� d  dxϕ  ��2  L∞  2∥kHK∥2  L2(R)T  (N−1)τ  exp  �  (C+Λ)T  τ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' For the second term  we write out the square to obtain  2  N 2  N  �  i,j=1  E  ��  ϕ(Y i,τ  t  ) −  �  R  ρτ  t (y)ϕ(y) dy  ��  ϕ(Y j,τ  t  ) −  �  R  ρτ  t (y)ϕ(y) dy  ��  =  2  N 2  N  �  i,j=1  E  �  ϕ(Y i,τ  t  )ϕ(Y j,τ  t  ) − ϕ(Y i,τ  t  )  �  R  ρτ  t (y)ϕ(y) dy  − ϕ(Y j,τ  t  )  �  R  ρτ  t (y)ϕ(y) dy +  � �  R  ρτ  t (y)ϕ(y) dy  �2�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  Now, using the fact that ρτ  t is the conditional distribution of Y i,τ with respect to FW  t , we  find  E  �  ϕ(Y i,τ  t  )  �  R  ρτ  t (y)ϕ(y) dy  �  = E  �  E  �  ϕ(Y i,τ  t  )  �  R  ρτ  t (y)ϕ(y) dy  ����FW  t  ��  = E  � �  R  ρτ  t (y)ϕ(y) dy E(ϕ(Y i,τ  t  )|FW  t )  �  = E  �� �  R  ρτ  t (y)ϕ(y) dy  �2�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  HEGSELMANN–KRAUSE MODEL WITH ENVIRONMENTAL NOISE  35  Since (Y i,τ  t  , i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' , N) have identical distribution given FW  t , the same equality holds for j  instead of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Additionally, using the fact that Y i,τ  t  , Y j,τ  t  are conditionally independent for  i ̸= j, we obtain  E  �  ϕ(Y i,τ  t  )ϕ(Y j,τ  t  )  �  = E  �  E(ϕ(Y i,τ  t  )ϕ(Y j,τ  t  )|FW  t )  �  = E  �  E(ϕ(Y i,τ  t  )|FW  t )E(ϕ(Y j,τ  t  )|FW  t )  �  = E  �� �  R  ρτ  t (y)ϕ(y) dy  �2�  for the cross terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Hence, the cross terms vanish and we can estimate the second term  in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='3) by  2  N 2  N  �  i=1  E  ��  ϕ(Y i,τ  t  ) −  �  R  ρτ  t (y)ϕ(y) dy  �2�  ≤  C(∥φ∥L∞(R))  N  for some finite constant C(∥ϕ∥L∞(R)), which depends only on ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Putting everything together,  we find  E(|⟨ΠN  t , ϕ⟩ − ⟨ρτ  t , ϕ⟩|2) ≤  ����  d  dxϕ  ����  2  L∞  2 ∥kHK∥2  L2(R) T  (N − 1)τ  exp  �(C + Λ)T  τ  �  +  C(∥ϕ∥L∞(R))  N  ≤ C(∥kHK∥L2(R) , T, ∥ϕ∥C1(R))  � 1  N +  1  (N − 1)τ exp  �CT  τ  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  Next, using H¨older’s inequality and Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='3, we discover  E(|⟨ρτ  t , ϕ⟩ − ⟨ρt, ϕ⟩|2) ≤ E(∥ϕ∥2  L2(R) ∥ρτ  t − ρt∥2  L2(R)) ≤ ∥ϕ∥2  L2(R) ∥ρτ − ρ∥2  S∞  F ([0,T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='L2(R))  ≤ ∥ϕ∥2  L2(R) C(λ, Λ, T, ∥ρ∥S∞  FW ([0,T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='L2(R)) ∥ρτ∥S∞  FW ([0,T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='L2(R))) ∥kτ − k∥2  L2(R) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  Therefore, combining this estimate with the previous one we obtain  E(|⟨ΠN  t , ϕ⟩ − ⟨ρt, ϕ⟩|2)  ≤ 2E(|⟨ΠN  t , ϕ⟩ − ⟨ρτ  t , ϕ⟩|2) + 2E(|⟨ρτ  t , ϕ⟩ − ⟨ρt, ϕ⟩|2)  ≤ C  �  λ, Λ, T, ∥kHK∥L2(R) , ∥ϕ∥C1(R) , ∥ϕ∥L2(R) , ∥ρ∥S∞  FW ([0,T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='L2(R)) , ∥ρτ∥S∞  FW ([0,T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='L2(R))  �  ×  � 1  N +  1  (N − 1)τ exp  �(C + Λ)T  τ  �  + ∥kτ − k∥2  L2(R)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  □  Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' While we focus in the present section on the interaction force kHK(x) =  1[0,R](|x|)x, as used in the HK model, all results of Section 5 extend verbatim to general  interaction forces k ∈ L1(R) ∩ L2(R) such that there exists a sequence (kτ)τ∈N ⊂ C∞  c (R)  satisfying  ∥kτ − k∥L2(R) → 0 as τ → ∞,  supp(kτ) ⊂ K for τ ∈ N and supp( d  dxkτ) ⊂ K for some compact set K ⊂ R,  0 ≤ kτ ≤ C, | d  dxkτ| ≤ C  τ for some constant C > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' In the spacial case of a constant diffusion coefficient σ > 0, one can also derive  propagation of chaos without introducing a regularized interacting particle system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Indeed, by  considering the shifted interacting particle system ( ˜Xi, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' , N) with ˜Xi := Xi−σBi, one  36  CHEN, NIKOLAEV, AND PR ¨OMEL  could exploit the results of propagation of chaos provided in the recent work [HRZ22] applied  to this shifted system and transfer the so obtained propagation of chose back to the original  interacting particle system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  References  [Abe12]  Helmut Abels, Pseudodifferential and singular integral operators, De Gruyter Graduate Lectures,  De Gruyter, Berlin, 2012, An introduction with applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  [AF03]  Robert A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
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+page_content=' Funct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' 103  (1992), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' 2, 275–293.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='  Li Chen, University of Mannheim, Germany  Email address: chen@uni-mannheim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='de  Paul Nikolaev, University of Mannheim, Germany  Email address: pnikolae@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='uni-mannheim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='de  David J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content=' Pr¨omel, University of Mannheim, Germany  Email address: proemel@uni-mannheim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
+page_content='de' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQfiQcC/content/2301.03955v1.pdf'}
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+arXiv:2301.02827v1  [physics.chem-ph]  7 Jan 2023
+Lower bounds on par with upper bounds for few-electron atomic energies
+Miklos Ronto,1, ∗ Peter Jeszenszki,2, † Edit M´atyus,2, ‡ and Eli Pollak1, §
+1Chemical and Biological Physics Department, Weizmann Institute of Science, 76100 Rehovot, Israel
+2ELTE, E¨otv¨os Lor´and University, Institute of Chemistry,
+P´azm´any P´eter s´et´any 1/A, Budapest, H-1117, Hungary
+(Dated: January 10, 2023)
+The development of computational resources has made it possible to determine upper bounds
+for atomic and molecular energies with high precision. Yet, error bounds to the computed energies
+have been available only as estimates. In this paper, the Pollak–Martinazzo lower bound theory,
+in conjunction with correlated Gaussian basis sets, is elaborated and implemented to provide sub-
+parts-per-million convergence of the ground and excited state energies for the He, Li, and Be atoms.
+The quality of the lower bounds is comparable to that of the upper bounds obtained from the Ritz
+method. These results exemplify the power of lower bounds to provide tight estimates of atomic
+energies.
+I.
+INTRODUCTION
+A century ago, the development of quantum theory
+was motivated, among others, by the stability of atoms
+and molecules. Schr¨odinger’s Coulomb Hamiltonian for
+the hydrogen atom has a finite, lowest energy eigenvalue,
+i.e., quantum theory correctly predicted its stability. Re-
+garding poly-electronic and poly-atomic systems, the an-
+alytic solution is unknown, but it has been demonstrated
+by formal tools that the many-particle Coulomb Hamil-
+tonian is bounded from below [1,2]. In this sense, formal
+lower bound theory played an essential role in showing
+that non-relativistic quantum theory was qualitatively
+correct.
+The evaluation of the ground and excited state en-
+ergy levels of the Hamiltonian has been of central im-
+portance during the course of the practical application
+of quantum theory to molecular physics and chemistry.
+The Schr¨odinger equation of atomic and molecular sys-
+tems has been solved by various numerical techniques;
+the most accurate energy values have been obtained by
+variational methods.
+Variational methods are based on the variational prin-
+ciple, formulated for an energy upper bound, and pro-
+vide systematic numerical means to converge from above
+to the (unknown) exact energy (and the corresponding
+wave function) by using computer power.
+In spite of the essential role lower bounds played in the
+formal theory, they were rarely used as practical com-
+putational tools and for good reason. Computed lower
+bounds have been orders of magnitude less accurate than
+the upper bound; thus, the computational effort was con-
+centrated on converging the upper bound. The conver-
+gence rate of the upper bound has been used to estimate
+the exact non-relativistic energy (within some estimated
+∗ miklos.ronto@weizmann.ac.il
+† jeszenszki.peter@ttk.elte.hu
+‡ edit.matyus@ttk.elte.hu
+§ eli.pollak@weizmann.ac.il
+energy interval). Such an extrapolation is approximate
+and may fail. The energy uncertainties derived from basis
+set extrapolation have sometimes turned out to be overly
+optimistic, making conclusions based on estimated error
+bars to the computed energies unreliable.
+The present work aims to turn the formal lower bound
+theory into a practical computational tool that provides
+an energy lower bound converging to the (unknown) ex-
+act energy value from below at a rate comparable to the
+upper bound. Thereby, it becomes possible to compute
+and systematically narrow the energy interval within
+which the exact non-relativistic energy resides. This pro-
+cedure allows us to take the first step towards ensure-
+computing error intervals, instead of estimating them.
+A computed error interval to the computed atomic (or
+molecular) energy is necessary for a good comparison
+with experimental data, when we aim to test and further
+develop the fundamental theory of atomic and molecular
+matter.
+In this work, we present algorithmic develop-
+ments and computations for few-electron atoms. Further
+work is planned to generalize the procedure for molecular
+energies.
+Several lower bound methods have been introduced
+based on the Temple [3] and the Weinstein [4] approaches.
+The Weinstein lower bound was further elaborated and
+generalized by Stevenson and Kato [5–7]. Several the-
+oretical [8–15] and practical improvements [16–18] have
+been developed with respect to the Temple bound. The
+optimal inclusion intervals introduced by Lehmann [19–
+21] were a significant development in relation to the orig-
+inal Temple bound. Further approaches of lower bound
+methods are based on bracketing functions [22–27] and
+on the method of intermediate operators [28,29]. Lower
+bounds are also of importance in the context of phys-
+ical properties of few-electron atoms such as oscillator
+strengths [30–34].
+In the past few years, a novel class of lower bound
+methods [35,36] based on the L´anczos construction of ba-
+sis sets has been proposed. A self-consistent lower bound
+theory (SCLBT) [37,38] was developed and successfully
+applied to quartic [38] and double-well [39] potentials as
+well as lattice models [37,40]; however, these methods are
+
+2
+typically not applicable for Coulomb-interacting systems
+due to the divergence of matrix elements of cubic and
+higher powers of the Coulombic Hamiltonian, unless one
+can devise basis sets, which like the true eigenfunctions,
+prevent such divergence and still the basis is in principle
+complete.
+While a tight Temple lower bound was computed for
+the helium atom [41], the quality of this lower bound is
+several orders of magnitude worse than the correspond-
+ing upper bound. As alternatives to Temple’s approach,
+expanding on Aronszajn’s work [29], Bazely [42,43] and
+later Bazely and Fox [44] obtained lower bounds using in-
+termediate Hamiltonians by introducing a special choice
+for the finite-dimensional space used to represent the
+Hamiltonian operator. This class of methods has been
+further elaborated and applied to Coulombic systems as
+well as other potentials [45–48], most recently by Mar-
+morino [49,50].
+Lower bounds to the energy eigenval-
+ues of the helium atom have been computed using fur-
+ther strategies [51,52] and an energy lower bound to the
+ground state of the lithium atom was computed using
+Lehmann’s method [53]. However, none of these studies
+resulted in lower bounds with comparable accuracy to
+those obtained with the Ritz variational method.
+Most recently, a different approach has been intro-
+duced by Pollak and Martinazzo applicable for Coulom-
+bic potentials, which was successfully used to compute
+lower bounds to the energy levels of hydrogen [54], and
+the two-electron helium and the three-electron lithium
+atoms [55].
+This work presents a further development
+and application of the method based on the use of explic-
+itly correlated Gaussian basis sets. We report algorithmic
+and computational strategies and present numerical re-
+sults for lower bounds to the (ground and excited state)
+energies of the helium, lithium, and beryllium atoms with
+a relative precision comparable to the corresponding up-
+per bound obtained in the same series of computations.
+II.
+LOWER BOUND THEORY
+Schr¨odinger’s
+formulation
+of
+the
+non-relativistic
+Hamiltonian of atoms with a fixed nucleus of charge num-
+ber Z is written in Hartree atomic units as
+H = −1
+2
+N
+�
+i=1
+∆i +
+N
+�
+i=1
+N
+�
+j>i
+1
+rij
+−
+N
+�
+i=1
+Z
+ri
+.
+(1)
+with ri denoting the distance of the ith electron from the
+nucleus, rij denoting the distance of the ith electron from
+the jth one, and ∆i is the kinetic energy operator for the
+ith electron. The nucleus is assumed to be stationary,
+with infinite mass, located at the origin of the spatial
+coordinate system. A central branch of molecular physics
+revolves about the computation of stationary states of H
+by (numerical) solution of the eigenvalue equation
+Hψn = εnψn ,
+n = 1, 2, . . . .
+(2)
+According to the Ritz–Macdonald variational princi-
+ple [56,57], the energy functional
+λn = ⟨ϕn|H|ϕn⟩
+⟨ϕn|ϕn⟩
+≥ εn,
+(3)
+provides an upper bound to the exact energy εn for an ap-
+propriate ϕn trial function. For a linear parametrization
+of the trial function in terms of “primitive” basis func-
+tions, fi, ϕn = �L
+i=1 cnifi, the minimization problem is
+turned into a matrix eigenvalue equation
+Hcn = λ(L)
+n Scn ,
+(4)
+where the H Hamiltonian and S overlap matrices are
+calculated using the basis functions. The matrix elements
+are Hij = ⟨fi|H|fj⟩ and Sij = ⟨fi|fj⟩.
+A central difficulty in computing lower bounds to
+Coulombic systems using L´anczos basis sets or more
+generally the Krylov algorithm [58] is due to the fact
+that with most basis sets, H2 is the highest power of
+the Coulomb Hamiltonian that can be handled; powers
+greater than 2 usually diverge. The expectation value of
+H2, ⟨H2⟩n = ⟨ϕn|H2|ϕn⟩, and the corresponding vari-
+ance, σ2
+n =
+�
+⟨H2⟩n − ⟨H⟩2n, can be computed and used
+in relation with several lower bound theories. However,
+until recently, all lower bound theories returned numeri-
+cal values that were several orders of magnitude less accu-
+rate than the upper bound obtained in a similar computa-
+tional setup; hence, the practical utility of the computed
+lower bounds remained limited.
+The recently formulated Pollak–Martinazzo (PM)
+lower bound theory addresses this problem by construct-
+ing a special matrix used in conjunction with the Cauchy
+interlacing theorem. According to the interlacing theo-
+rem, which can be derived from the Courant–Fisher the-
+orem [59], if the eigenvalues of an n×n Hermitian matrix
+A are given in ascending order as a1 ≤ a2 ≤ . . . ≤ an−1 ≤
+an, and the eigenvalues of its (n − 1) × (n − 1) principal
+submatrix B are b1 ≤ b2 ≤ . . . ≤ bn−2 ≤ bn−1, then
+a1 ≤ b1 ≤ a2 ≤ b2 ≤ . . . ≤ an−1 ≤ bn−1 ≤ an. This theo-
+rem is used to obtain lower bounds to the eigenvalues of
+B as follows. The matrix B with dimension L = n − 1
+is substituted with the diagonal Hamiltonian matrix ob-
+tained by diagonalizing the L × L Hamiltonian matrix,
+Eq. (4), with eigenvalues denoted in an ascending order
+as λ(L)
+j
+.
+Then, a “big” [(L + 1) × (L + 1)]-dimensional matrix
+is defined, motivated by the matrix A in the previous
+paragraph, as [54]
+KL(ε) =
+
+
+
+
+
+
+
+
+
+
+
+
+λ(L)
+1
+0
+. . .
+0
+σ(L)
+1
+0
+λ(L)
+2
+. . .
+0
+σ(L)
+2
+...
+...
+...
+0
+...
+0
+0
+. . . λ(L)
+L
+σ(L)
+L
+σ(L)
+1
+σ(L)
+2
+. . . σ(L)
+L
+ε +
+L
+�
+k=1
+�
+σ(L)
+k
+�2
+λ(L)
+k
+− ε
+
+
+
+
+
+
+
+
+
+
+
+
+,
+(5)
+
+3
+where λ(L)
+n
+labels the nth Ritz eigenvalue and σ(L)
+n
+is the
+associated standard deviation. We refer to the matrix
+KL(ε) as the PM matrix with parameter ε.
+By construction, the parameter ε is an eigenvalue of
+the PM matrix. The remaining eigenvalues of the matrix
+are the L solutions of the polynomial equation
+1 =
+L
+�
+k=1
+σ2
+k
+(λk − ε) (x − λk) .
+(6)
+They are denoted in ascending order as xj(j = 1, . . . , L)
+and have the important property that xj (ε) is a mono-
+tonically increasing function of the parameter ε.
+Suppose that we choose ε to equal the unknown ground
+state energy denoted as ε1. According to Cauchy’s in-
+terlacing theorem, the eigenvalues xk of the KL(ε) ma-
+trix are interlaced by the Ritz eigenvalues λk as follows:
+ε1 ≤ λ(L)
+1
+≤ x1 ≤ λ(L)
+2
+≤ . . . ≤ λ(L)
+L
+≤ xL. Then, if
+we have a lower bound for x1 and compute from Eq. (6)
+the value of ε that would give the same value of x1, then
+due to the monotonicity property this value of ε would
+necessarily be a lower bound to the ground state energy.
+If the basis set used is “good” in the sense that both λ(L)
+1
+and λ(L)
+2
+are not too far from the exact eigenvalues ε1
+and ε2, then, barring special circumstances such as de-
+scribed below for the He atom, one finds that x1 ≥ ε2,
+so that bounding x1 from below by a lower bound to the
+excited state energy gives a lower bound to the ground
+state energy. Since the eigenvalues of the PM matrix are
+not very sensitive to the precise value of x1 used, this
+leads to accurate lower bounds, as shown below.
+This procedure may then be continued. For example,
+if λ(L)
+3
+is also not too far from the exact eigenvalue ε3
+then x2 will be larger than ε3, so that replacing it with a
+lower bound to ε3 and finding the two lowest eigenvalues
+of the PM equation, yields lower bounds to the ground
+and first excited state energies. This procedure may then
+be continued for the next excited state, etc.
+III.
+COMPUTATIONAL SETUP WITH AN
+EXPLICITLY CORRELATED GAUSSIAN BASIS
+Explicitly correlated Gaussian (ECG) functions [60–
+64] are commonly used as a spatial basis for atomic and
+molecular problems; however, unlike orthogonal polyno-
+mials, they do not provide uniform coverage of space by
+simply increasing the polynomial order. ECGs can be
+powerfully used in relation with parametrization by opti-
+mization (minimization) of some appropriate target func-
+tional. (Regarding nodeless harmonic oscillator functions
+used as a basis, see Ref. [63].) The parametrization of
+ECGs with respect to minimization of the energy func-
+tional is a powerful means of obtaining and systemati-
+cally improving energy upper bounds. While ECGs fail
+to satisfy the cusp condition [65], they have general, an-
+alytic N-particle integrals for most physically relevant
+operators, which can also be generalized for molecular
+computations.
+In this work, trial functions corresponding to the S
+ground-state symmetries of the helium, lithium, and
+beryllium atoms are expressed as anti-symmetrized (A)
+products of φ spatial and χ spin functions
+fi(r, σ) = A{φL,ML(r, Ai)χS,MS(σ, ϑi)} .
+(7)
+χS,MS(σ, ϑi) corresponds to the two-, three-, and four-
+electron spin functions coupled to spin states with to-
+tal spin quantum numbers (S, MS) = (1, 0) for helium
+and beryllium and to (2, 0) for lithium. The total spin
+functions for lithium and beryllium correspond to a two-
+dimensional spin space, which is parametrized by one
+free parameter (θi) [63]. We used ECGs as spatial ba-
+sis functions corresponding to (L, ML) = (0, 0) orbital
+momentum quantum numbers (suppressed in the rest of
+the paper),
+φ(r, Ai) = e−rT(Ai⊗I3)r ,
+(8)
+centered at the origin (where the nucleus is fixed) and r ∈
+R3N collects the electronic coordinates. The Ai ∈ RN×N
+positive-definite, symmetric matrix determines the width
+of the Gaussian and the correlation length of the parti-
+cles, and is determined by optimization of some appro-
+priate target function.
+All computations were performed using a computer
+program named QUANTEN (QUANTum mechanical de-
+scription of electrons and atomic nuclei) and developed
+by the Budapest group.
+QUANTEN has a (stochas-
+tic and deterministic) variational engine and an ex-
+tensive ECG library with recent applications includ-
+ing non-adiabatic, pre-Born–Oppenheimer, perturbative,
+and variational relativistic computations [65–74]. It can
+be efficiently run with double precision arithmetic, but
+a quadruple precision mode is also available. The first
+implementation of the H2 integrals and assembling the
+variance computation in QUANTEN, which was recently
+reported for two- and three-electron atoms [55,75], is fur-
+ther developed and extended to four (and, in general, N)
+particle systems in the present work.
+A.
+Strategy for converging the PM lower bound to
+the energy
+Atomic PM lower bounds have been reported for the
+helium and lithium ground states using the computa-
+tional setup described above [55].
+Although, the PM
+bounds were tighter than the Weinstein, Temple, or
+Lehmann bounds obtained with the same basis set [55],
+even the best PM bound was (at least) three orders of
+magnitude less precise (a relative precision of 0.55 ppm
+for helium and 4.0 ppm for lithium was achieved), than
+the corresponding upper bound (with a relative precision
+of 0.000 17 ppm for helium and 0.002 ppm for lithium).
+The natural question arises:
+How can we improve
+the convergence of the PM bounds? The plausible idea
+
+4
+of fine-tuning the ECG basis parametrization based on
+a simple PM energy ensuremaximization condition was
+found to be impractical in Ref. [55]. If one is not careful,
+then a simple minded application of the PM method may
+lead to energy values which are higher rather than lower
+than the true eigenvalue under study.
+To better understand the conditions for which the PM
+method leads to lower bounds, it is necessary to consider
+that all lower bound theories based on the variances of
+the Hamiltonian are only valid under certain conditions.
+These conditions are typically connected to the quality
+of the variances and the ε parameter. When the upper
+bounds are “well behaved,” in the sense that the λj − εj
+distance may be considered as small, one may expand the
+PM equation, Eq. (6), to leading order in the distance to
+find [54]
+xj(ε) − εj+1 ≃ λj+1 − εj+1 − σ2
+j+1
+σ2
+j
+(λj − εj) ≥ 0 .
+(9)
+This relation implies that the left-hand side of the equa-
+tion will be positive if the ratio of the variances of the
+(j + 1)th to the jth state is sufficiently small. This sug-
+gests that if we are interested, for example, in a high-
+quality lower bound to the ground-state energy, and we
+already have a fairly good description of the ground-state
+upper bound, then we should continue improving not the
+ground but the first-excited state’s description, i.e., con-
+tinue with the minimization of the first-excited-state en-
+ergy and associated reduction of its variance. This is the
+core idea for the computational developments presented
+in this paper. Furthermore, Eq. (9) will also be used to
+rationalize some further observations regarding the nu-
+merical results [sensitivity of the computed lower bounds
+to the ε parameter of the PM matrix, Eq. (5), and pos-
+sible failure of obtaining a lower bound].
+The implementation of the core idea, i.e., improvement
+of the description of excited states to have a better lower
+bound for the ground state, was not readily available in
+the existing computational setup. Although ECG basis
+sets generated based on the energy minimization condi-
+tion for a selected state provide a very compact represen-
+tation, they do not guarantee a high-quality description
+of other states (unlike a set of orthogonal polynomials,
+for which increasing the number of functions, i.e., the
+polynomial order, automatically ensures more complete
+coverage of the space, and, hence, improved convergence
+of excited-state energies).
+1.
+Implementation of the multi-state energy minimization
+strategy in an ECG-based procedure
+A usual energy minimization procedure, e.g., for the
+ground state, is initiated by random basis generation and
+selection [63], which is followed by repeated refinement
+cycles of the already existing basis set, for which we use
+the Powell method [76].
+Both steps are based on the
+energy minimization condition (and the variational prin-
+ciple for Hamiltonians bounded from below).
+The same procedure can be repeated for the first- (nth)
+excited state (even long-lived states embedded in the con-
+tinuum [69] in combination with a stabilization-like pro-
+cedure). In this fashion, separate near-optimal basis sets
+for separate states can be straightforwardly generated.
+One could then try and merge the basis sets optimized
+for the ground and for the first-excited states, but this
+procedure would result in a gigantic basis and, more im-
+portantly, near-linear dependency problems in the finite
+precision arithmetic used for the computations.
+Instead, we have implemented a multi-state procedure
+in a single computation as follows. The usual basis gen-
+eration and refinement using the energy minimization
+condition for the ground state is implemented up to a
+certain number of basis functions. This number is de-
+termined based on the convergence of the Ritz ground-
+state energy. This results in the first “block” of our basis
+set. The computation is then continued with the genera-
+tion and refinement of additional basis functions (second
+block of the basis set), for which the energy minimiza-
+tion condition for the second state (first-excited state)
+was implemented. We have regularly refined (using the
+Powell method) the entire basis set, one function after the
+other, by using the energy minimization condition for the
+ground-state energy for functions belonging to the first
+basis block, and the energy minimization condition for
+the second state for functions belonging to the second
+basis block.
+The repeated full-basis refinement cycles allow us to re-
+lax functions in the first block (optimized to the ground
+state) while the ground-state energy is also (partly) de-
+scribed by the second-block functions (optimized to the
+first-excited state). Therefore, small deviations from a
+monotonic decrease of the energy may occur upon en-
+largement of the basis set.
+For sufficiently large basis
+sets and with further, extended optimizations these small
+deviations from monotonicity can be smoothed out.
+By construction, the procedure generates a basis set
+which is (near) optimal for both the ground and the
+first-excited states, and the linear–dependency problem is
+automatically avoided (a new basis function that would
+have a too large overlap with the existing basis set is
+discarded or “weighted down” with a “penalty” correc-
+tion to the value of the energy functional). Furthermore,
+the procedure can be straightforwardly extended to addi-
+tional states, and thus, applicable also beyond the ground
+state.
+2.
+Numerical demonstration of the multi-state optimization
+strategy for lithium and beryllium
+The computational strategy described above has been
+implemented in QUANTEN. It is highlighted for the case
+of lithium in Fig. 1 (see also Table I). The computations
+are more expensive for beryllium, so for this case, we
+
+5
+−10
+−9
+−8
+−7
+−6
+−5
+−4
+−3
+−2
+−1
+ 500
+ 1000
+ 1500
+ 2000
+∆εj
+L
+∆ε1
+∆ε2
+−10
+−9
+−8
+−7
+−6
+−5
+−4
+−3
+−2
+−1
+∆λj
+∆λ1
+∆λ2
+∆λ3
+FIG. 1. Upper and lower bound gaps [Eq. (10)] for the lithium
+atom with respect to the εref,j reference energies taken from
+Ref. [77]. The first, second, and third basis blocks with L ∈
+[1, 980], (980, 1700], and (1700, 2175] basis indexes, in short
+[1:980:1700:2175], were optimized according to minimization
+of the ground (yellow), the first- (white), and the second-
+excited state (gray) energies. (See also Table I.)
+report only the final results (Tables II and III). Unex-
+pectedly, helium turned out to be a very special case, for
+which the strategy does not work (the condition x1 ≥ ε2
+fails), and this can be rationalized on the basis of Eq. (9)
+as explained in the last paragraphs of this section.
+As a measure of the quality of the lower-bound energy
+for a given basis set, we compare the relative upper and
+lower bound gaps defined as
+∆λj = log10 (|λj − εref,j|/εref,j)
+∆εj = log10 (|εj,− − εref,j|/εref,j) ,
+(10)
+where εref,j is a reference value (expected to be very close
+to the exact value and available from the literature for
+the computed examples), λj and εj,− are the computed
+upper and lower bounds for the jth state (j = 1, 2, . . .),
+respectively. (We note that ε−
+j is used in Tables I–IV to
+label the estimated lower bounds used in the PM equa-
+tion.) If the gap ratio,
+ηj = ∆εj/∆λj ,
+(11)
+approaches one, we may say that the lower (and upper)
+bound computation is useful in terms of bracketing the
+exact energy.
+In Fig. 1, showcasing our computation for lithium, the
+performance of the various energy estimates in the yel-
+low region is comparable to the best gap ratio achieved
+in Ref. [55]. Then, we continue with the multi-state opti-
+mization procedure. During the generation of the second
+basis block (white region in the figure), we see a signifi-
+cant improvement for the ground-state lower bound, and
+the first-excited state lower bound also improves (lower
+part of the figure), in parallel with the improvement of
+the first- and second-excited-state upper bounds (upper
+part of the figure).
+As can be seen in the figure, the optimization for one
+state does not necessarily guarantee the monotonic im-
+provement of the ground and other states; however, any
+increase in a state energy can be minimized by applying
+subsequent refinement cycles to the already generated
+basis set. The figure also shows the generation and op-
+timization of a third basis block (gray-shaded area), in
+which the basis functions are optimized using the energy
+minimization condition for the second-excited state.
+The resulting best upper and lower bound values ob-
+tained for the lithium atom corresponding to a total
+basis size of L = 2175 are collected in Table I. While
+previous PM computations carried out for the lithium
+atom ground state [55] (with a single basis block) already
+improved upon the Lehmann bound obtained using a
+Hylleraas basis [53], the present PM lower bounds signif-
+icantly outperform both. The lower bounds are at most
+one order of magnitude worse than the upper bounds.
+This may be improved upon if one has a better estimate
+for the excited state energies as discussed in further de-
+tail below. At this point it suffices to say that the values
+used as estimates for the excited state energies upon im-
+plementing the PM equations are rather conservative.
+As might be expected, the ground-state lower bound
+TABLE I. Results for the lithium atom. Tabulation of lower-
+and upper-bound energies, εj,− and λj, respectively, in units
+of Eh, resulting from multi-state optimization with three
+blocks (with basis sizes [1:980:1700:2175], cf. Fig. 1). The
+relevant variances are also given in units of E2
+h.
+The rela-
+tive deviation from the εref,j reference energy adapted from
+Ref. [77] is shown in parentheses in ppb (parts per billion).
+The PM parameter in Eq. (5) was ε−
+2 = εref,2−5·10−8 Eh and
+ε−
+3 = εref,3 − 10−7 Eh for obtaining the lower bounds for the
+ground and first excited states, respectively (see also Fig. 2
+and corresponding text).
+Ground state
+ε1,−
+−7.478 060 364
+(5.3)
+εref,1
+−7.478 060 324 [77]
+λ1
+−7.478 060 316
+(1.0)
+First excited state
+ε2,−
+−7.354 098 569
+(20.1)
+εref,2
+−7.354 098 421 [77]
+λ2
+−7.354 098 404
+(2.4)
+Second excited state
+εref,3
+−7.318 530 846 [77]
+λ3
+−7.318 530 751
+(12.9)
+Variances
+σ2
+1
+0.158 934 586
+σ2
+2
+0.225 395 309
+σ2
+3
+0.265 772 122
+
+6
+is more accurate than the first excited state lower bound
+and the same ordering of accuracy is true for the upper
+bounds. The plateauing of the ground-state lower bound
+for L ≥ 1500 reflects the plateauing of the ground-state
+upper bound. The ground-state lower bound will improve
+as the PM eigenvalue x1 converges to the first excited
+state energy.
+As seen from Eq. (9), for this to occur
+one needs an improvement of the upper bound. Since
+this does not happen, the lower bound reaches a plateau
+value. The same occurs for the first excited state lower
+bound.
+Similarly good results are obtained with multi-state
+optimization for the ground- and first-excited states of
+the beryllium atom (Tables II and III). The multi-state
+optimization strategy was essential to arrive at good
+lower-bounds also for beryllium. In this case, the qual-
+ity of the ground-state lower bound is comparable to the
+Ritz upper bound while it is somewhat worse, a factor of
+≃ 6, for the excited state, This reflects to some extent
+the lower bound values used for the excited states when
+implementing the PM equation.
+TABLE II. Results for the beryllium atom: ground state.
+Tabulation of lower- and upper-bound energies, εj,− and λj,
+respectively, in units of Eh resulting from multi-state opti-
+mization with two blocks (with basis set sizes [1:2000:4500]).
+The relevant variances are also given in units of E2
+h. The rela-
+tive deviation from the εref,j reference energy given in Ref. [78]
+is shown in parentheses in ppb (parts per billion). The PM
+parameter in Eq. (5) was ε−
+2 = εref,2−10−7 Eh (see also Fig. 3
+and corresponding text).
+Ground state
+ε1,−
+−14.667 356 917
+(28)
+εref,1
+−14.667 356 507 [78]
+λ1
+−14.667 356 191
+(22)
+First excited state
+εref,2
+−14.418 240 364 [78]
+λ2
+−14.418 239 479
+(61)
+Variances
+σ2
+1
+0.506 745 221
+σ2
+2
+0.698 028 906
+B.
+Stability and sensitivity of the results to the ε
+parameter of the PM matrix
+The PM lower-bound computation (similarly to Tem-
+ple’s bound or other lower-bound methods) requires some
+knowledge about the higher-energy state(s). This infor-
+mation (estimate) is encoded in the ε parameter of the
+PM matrix, [Eq. (5)]: for the computation of a lower
+bound to the nth eigenvalue, the ε value in the PM ma-
+trix must be a lower estimate to the (n + 1)th energy
+eigenvalue.
+The computed lower-bound results (in Tables I–IV)
+have been reported with a specific ε value (estimated
+from a known precise reference value) used in the PM
+TABLE III.
+Results for the beryllium atom: first-excited
+state. Tabulation of lower- and upper-bound energies, εj,−
+and λj, respectively, in units of Eh resulting from multi-
+state optimization with three blocks (with basis set sizes
+[1:2000:4000:4500]). The relevant variances are also given in
+units of E2
+h. The relative deviation from the εref,j reference
+energy [78] is shown in parentheses in ppb (parts per billion).
+The PM parameter in Eq. (5) was ε−
+3 = εref,3 − 10−6 Eh (see
+also Fig. 3 and corresponding text).
+First-excited state
+ε2,−
+−14.418 248 205
+(543)
+εref,2
+−14.418 240 364 [78]
+λ2
+−14.418 238 971
+(97)
+Second excited state
+εref,3
+−14.370 087 938 [78]
+λ3
+−14.370 062 140
+(1795)
+Variances
+σ2
+2
+0.698 028 906
+σ2
+3
+5.389 904 253
+calculation. The critical reader might comment that ob-
+taining a tight lower bound which is based on knowledge
+of a different tight lower bound is problematic. Hence, it
+is necessary to address the “stability” of the results with
+respect to the precise choice of this value. The PM results
+obtained in previous computations reported in Refs. [54]
+and [55] have been found to be relatively insensitive to ε.
+In this work, we repeated the PM computations for
+the largest basis set results of lithium and beryllium (Ta-
+bles I and II) using various ε parameters. Figures 2 and 3
+present the lower bound gap defined with respect to the
+Ritz eigenvalue
+δεj = log10 (|εj,− − λj|/λj) ,
+(12)
+which is defined analogously to ∆εj in Eq. (10), but
+free from the knowledge of an “external” reference value,
+εref,j.
+The gap for the (estimated) ε−
+j
+parameter in
+Eq. (5)—which is a lower bound to the respective ex-
+cited state—is defined with respect to the reference value
+exactly the same way as the lower bound gap in Eq. (10):
+∆ε−
+j = log10
+�
+|ε−
+j − εref,j|/εref,j
+�
+.
+(13)
+In Fig. 2, the red and blue lines show the ground-
+and first-excited-state PM lower bound gaps, δεj, re-
+spectively, plotted with respect to the lower-bound gap
+∆ε−
+j+1, whereas the black line shows the (orders of mag-
+nitude worse) gap for the Temple lower bound, defined
+as
+εTemple
+1,−
+= λ1 −
+σ2
+1
+ε−
+2 − λ1
+.
+(14)
+As can be seen in Figs. 2 and 3, the PM lower bounds are
+sensitive to the precision of the lower estimate to the jth
+energy (ε−
+j = εref,j − ∆ε−
+j is the parameter used in the
+PM matrix) while the Temple lower bound is not, due
+
+7
+−10
+−8
+−6
+−4
+−2
+ 0
+−10
+−8
+−6
+−4
+−2
+ 0
+δεj
+∆εj+1
+−
+PM: δε1  vs. ∆ε2
+−
+PM: δε2  vs. ∆ε3
+−
+Temple: δε1  vs. ∆ε2
+−
+FIG. 2. Sensitivity of the PM ground- (red) and first-excited
+(blue) state lower-bound gap [Eq. (12)] for the lithium atom
+at a basis size of L = 2175 (cf. Fig. 1) with respect to the
+precision of the ε PM parameter used in Eq. (5) given in
+Eq. (13). The ground-state Temple gap is also shown (black).
+The ratios of the variances are σ2
+2/σ2
+1 = 1.42 and σ2
+3/σ2
+2 =
+1.18.
+to its poor quality. In the ∆ε−
+j range used to compute
+the data reported in Tables I–III, the functions in Figs. 2
+and 3 are nearly linear, i.e., the precision of the PM lower
+bound is determined by the precision of the excited-state
+estimate used in the PM matrix. This observation can
+be rationalized on the basis of Eq. (9). In contrast to
+the results presented in Refs. [54] and [55], the ratio of
+variances σ2
+j+1/σ2
+j is of the order of unity, due to the
+optimization of the excited states, and the accuracy of the
+(j + 1)th excited state Ritz eigenvalue is much improved,
+leading to the linear dependence.
+How then does one know the correct value of ε to be
+used in the lower bound calculation? The strategy we
+employed was to use a value that is substantially lower
+than the accuracy expected from the convergence proper-
+ties of the relevant Ritz eigenvalue. These are the values
+reported in Tables
+I–III. The high accuracy of the re-
+sulting lower bounds demonstrates that this strategy is
+robust and that the linear dependence is not really a se-
+rious problem.
+1.
+The special case of the helium atom
+We applied the multi-state optimization strategy also
+for the helium atom.
+When applied naively, the PM
+equation gave values for the ground-state energy which,
+−10
+−8
+−6
+−4
+−2
+ 0
+−10
+−8
+−6
+−4
+−2
+ 0
+δεj
+∆εj+1
+−
+PM: δε1  vs. ∆ε2
+−
+Temple: δε1  vs. ∆ε2
+−
+FIG. 3. Sensitivity of the PM ground-state lower-bound gap
+[Eq. (12)] for the beryllium atom at a basis size of L = 4500
+(2000 states for the ground and 2500 for the first-excited state,
+(cf. Table II) with respect to the precision of the ε PM pa-
+rameter used in Eq. (5) given in Eq. (13). The ground-state
+Temple gap is also shown (black). The ratio of the variances
+is σ2
+2/σ2
+1 = 1.37.
+in the limit of a large basis set, were larger than the
+known ground-state energy. What went awry?
+This is related to the use of a correlated Gaussian
+basis set rather than an orthogonal polynomial basis.
+We observed in convergence figures (similar to Fig. 1)
+that the upper bounds (Ritz eigenvalues) to the first-
+and second-excited states, optimized in the second and
+third basis blocks, converged faster than the ground-state
+eigenvalue. As may be then reasoned from Eq. (9), this
+causes the right-hand side of the equation to be negative,
+that is, the eigenvalue x1 is no longer greater than the
+first excited state energy. Using the first excited state
+energy in the PM equation will then naturally no longer
+give a lower bound.
+It can also be understood that this behavior is unique
+to the helium atom, which is a two-electron system. The
+ground state is dominated by a 1s2 configuration, the
+first-excited state is 1s2s, the second-excited state is
+1s3s, etc. The correlation of the electrons, which is de-
+scribed increasingly more accurately during the course of
+the variational computation, is less important for excited
+states, than for the ground state, and hence their Ritz
+eigenvalues for the excited states converge faster than
+for the ground state.
+Does this mean that one cannot get meaningful and
+accurate lower bounds for the He atom using correlated
+Gaussian basis sets? Not necessarily. If one forces the ba-
+
+8
+TABLE IV. Lower and upper bounds for the He atom energy
+levels, in Eh, computed in this work for the ground, first-, and
+second-excited states using L = 510 ECG basis functions.
+The relative deviation from the reference energies [41,79] is
+shown in parentheses in parts per billion (ppb).
+The PM
+matrix parameters used in Eq. (5) were ε−
+i = εref,i − 2 · 10−9
+Eh for both i = 2 and 3. The variances are in units of E2
+h.
+Ground state
+ε1,−
+−2.903 724 379
+(0.7)
+εref,1
+−2.903 724 377 [41]
+λ1
+−2.903 724 376
+(0.3)
+First excited state
+ε2,−
+−2.145 974 048
+(0.9)
+εref,2
+−2.145 974 046 [79]
+λ2
+−2.145 974 045
+(0.5)
+Second excited state
+εref,3
+−2.061 271 990 [79]
+λ3
+−2.061 271 989
+(0.4)
+Variances
+σ2
+1
+0.013 300 747
+σ2
+2
+0.129 998 619
+σ2
+3
+0.121 216 647
+sis set so that the excited-state eigenvalues are not better
+than the ground-state level, one may expect the method
+to work. This is demonstrated in Table IV, where us-
+ing the known excited-state energy values, we can ensure
+that the accuracy of all three levels is similar. However,
+this does not answer the question as to how would it be
+possible, without the knowledge of the numerically ex-
+act values, to ensure that the PM equation leads to a
+lower bound. Fortunately, for larger atoms, the problem
+does not exist, and as we showed, it is straightforward
+to obtain high-quality lower bounds for the Li and Be
+atoms.
+IV.
+SUMMARY AND DISCUSSION
+A multi-state optimization strategy is developed to
+systematically converge the Pollak–Martinazzo energy
+lower bound with an explicitly correlated Gaussian basis
+set. Lower bounds to the ground- and first-excited state
+energies of the lithium and beryllium atoms are com-
+puted. The resulting lower bounds are the most precise
+to date, and their relative precision is comparable to that
+of the energy upper bound in the same basis.
+In view of the performance of the multi-state optimiza-
+tion and the PM lower bound theory, the following con-
+clusions can be drawn:
+• The multi-state optimization of ECG bases pro-
+vides a systematic and robust improvement of the
+low-lying eigenvalues.
+• The optimization of higher lying states does not
+affect the already converged states adversely.
+• The optimization of the energy of the (n+1)th state
+improves the quality of the lower bound to the nth
+state.
+• The PM theory is able to provide lower bounds
+with ppb relative precision for the energy levels of
+few-electron systems.
+The presented computational procedure and numeri-
+cal results are for non-relativistic energies. Relativistic
+and leading-order quantum electrodynamic effects have
+been traditionally accounted for as perturbative correc-
+tions to the non-relativistic energy, e. g., [70]. The iden-
+tification of a many-particle relativistic wave equation
+based on relativistic quantum electrodynamics (QED)
+is more challenging. Most recently, it became possible
+(for two particles) to start out from the Bethe–Salpeter
+QED wave equation, exploit that interactions in atoms
+and molecules are dominantly instantaneous, and arrive
+at an eigenvalue equation for a no-pair Dirac–Coulomb–
+Breit Hamiltonian [80].
+This Hamiltonian appears to
+be bounded from below, and robust variational proce-
+dures could be developed to compute its eigenvalues,
+which have an α fine-structure constant dependence that
+is in agreement with the known α orders of the well-
+established perturbative procedures [71–74].
+This theoretical approach provides variational ensur-
+erelativistic upper bounds (including also some of the
+so-called “non-radiative” QED corrections of the per-
+turbative framework), and (with further development to
+many-particle systems), it will be relevant to ask for en-
+surerelativistic lower bounds in a spirit similar to this
+work.
+ACKNOWLEDGMENTS
+Financial support of the European Research Council
+through a Starting Grant (No. 851421) is gratefully ac-
+knowledged.
+This work has also been graciously sup-
+ported by the Ben May Center for Chemical Theory and
+Computation at the Weizmann Institute of Science.
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+
diff --git a/0tE0T4oBgHgl3EQf_gJj/content/tmp_files/load_file.txt b/0tE0T4oBgHgl3EQf_gJj/content/tmp_files/load_file.txt
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf,len=955
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='02827v1  [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='chem-ph]  7 Jan 2023  Lower bounds on par with upper bounds for few-electron atomic energies  Miklos Ronto,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' ∗ Peter Jeszenszki,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' † Edit M´atyus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' ‡ and Eli Pollak1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' §  1Chemical and Biological Physics Department,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' Weizmann Institute of Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' 76100 Rehovot,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' Israel  2ELTE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' E¨otv¨os Lor´and University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' Institute of Chemistry,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='  P´azm´any P´eter s´et´any 1/A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' Budapest,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' H-1117,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' Hungary  (Dated: January 10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' 2023)  The development of computational resources has made it possible to determine upper bounds  for atomic and molecular energies with high precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' Yet, error bounds to the computed energies  have been available only as estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' In this paper, the Pollak–Martinazzo lower bound theory,  in conjunction with correlated Gaussian basis sets, is elaborated and implemented to provide sub-  parts-per-million convergence of the ground and excited state energies for the He, Li, and Be atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='  The quality of the lower bounds is comparable to that of the upper bounds obtained from the Ritz  method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' These results exemplify the power of lower bounds to provide tight estimates of atomic  energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='  INTRODUCTION  A century ago, the development of quantum theory  was motivated, among others, by the stability of atoms  and molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' Schr¨odinger’s Coulomb Hamiltonian for  the hydrogen atom has a finite, lowest energy eigenvalue,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=', quantum theory correctly predicted its stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' Re-  garding poly-electronic and poly-atomic systems, the an-  alytic solution is unknown, but it has been demonstrated  by formal tools that the many-particle Coulomb Hamil-  tonian is bounded from below [1,2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' In this sense, formal  lower bound theory played an essential role in showing  that non-relativistic quantum theory was qualitatively  correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='  The evaluation of the ground and excited state en-  ergy levels of the Hamiltonian has been of central im-  portance during the course of the practical application  of quantum theory to molecular physics and chemistry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='  The Schr¨odinger equation of atomic and molecular sys-  tems has been solved by various numerical techniques;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='  the most accurate energy values have been obtained by  variational methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='  Variational methods are based on the variational prin-  ciple, formulated for an energy upper bound, and pro-  vide systematic numerical means to converge from above  to the (unknown) exact energy (and the corresponding  wave function) by using computer power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='  In spite of the essential role lower bounds played in the  formal theory, they were rarely used as practical com-  putational tools and for good reason.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' Computed lower  bounds have been orders of magnitude less accurate than  the upper bound;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' thus, the computational effort was con-  centrated on converging the upper bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' The conver-  gence rate of the upper bound has been used to estimate  the exact non-relativistic energy (within some estimated  ∗ miklos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='ronto@weizmann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='il  † jeszenszki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='peter@ttk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='elte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='hu  ‡ edit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='matyus@ttk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='elte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='hu  § eli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='pollak@weizmann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='il  energy interval).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' Such an extrapolation is approximate  and may fail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' The energy uncertainties derived from basis  set extrapolation have sometimes turned out to be overly  optimistic, making conclusions based on estimated error  bars to the computed energies unreliable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='  The present work aims to turn the formal lower bound  theory into a practical computational tool that provides  an energy lower bound converging to the (unknown) ex-  act energy value from below at a rate comparable to the  upper bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' Thereby, it becomes possible to compute  and systematically narrow the energy interval within  which the exact non-relativistic energy resides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' This pro-  cedure allows us to take the first step towards ensure-  computing error intervals, instead of estimating them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='  A computed error interval to the computed atomic (or  molecular) energy is necessary for a good comparison  with experimental data, when we aim to test and further  develop the fundamental theory of atomic and molecular  matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='  In this work, we present algorithmic develop-  ments and computations for few-electron atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' Further  work is planned to generalize the procedure for molecular  energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='  Several lower bound methods have been introduced  based on the Temple [3] and the Weinstein [4] approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='  The Weinstein lower bound was further elaborated and  generalized by Stevenson and Kato [5–7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' Several the-  oretical [8–15] and practical improvements [16–18] have  been developed with respect to the Temple bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' The  optimal inclusion intervals introduced by Lehmann [19–  21] were a significant development in relation to the orig-  inal Temple bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' Further approaches of lower bound  methods are based on bracketing functions [22–27] and  on the method of intermediate operators [28,29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' Lower  bounds are also of importance in the context of phys-  ical properties of few-electron atoms such as oscillator  strengths [30–34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='  In the past few years, a novel class of lower bound  methods [35,36] based on the L´anczos construction of ba-  sis sets has been proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' A self-consistent lower bound  theory (SCLBT) [37,38] was developed and successfully  applied to quartic [38] and double-well [39] potentials as  well as lattice models [37,40];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' however, these methods are  2  typically not applicable for Coulomb-interacting systems  due to the divergence of matrix elements of cubic and  higher powers of the Coulombic Hamiltonian, unless one  can devise basis sets, which like the true eigenfunctions,  prevent such divergence and still the basis is in principle  complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='  While a tight Temple lower bound was computed for  the helium atom [41], the quality of this lower bound is  several orders of magnitude worse than the correspond-  ing upper bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' As alternatives to Temple’s approach,  expanding on Aronszajn’s work [29], Bazely [42,43] and  later Bazely and Fox [44] obtained lower bounds using in-  termediate Hamiltonians by introducing a special choice  for the finite-dimensional space used to represent the  Hamiltonian operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' This class of methods has been  further elaborated and applied to Coulombic systems as  well as other potentials [45–48], most recently by Mar-  morino [49,50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='  Lower bounds to the energy eigenval-  ues of the helium atom have been computed using fur-  ther strategies [51,52] and an energy lower bound to the  ground state of the lithium atom was computed using  Lehmann’s method [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' However, none of these studies  resulted in lower bounds with comparable accuracy to  those obtained with the Ritz variational method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='  Most recently, a different approach has been intro-  duced by Pollak and Martinazzo applicable for Coulom-  bic potentials, which was successfully used to compute  lower bounds to the energy levels of hydrogen [54], and  the two-electron helium and the three-electron lithium  atoms [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='  This work presents a further development  and application of the method based on the use of explic-  itly correlated Gaussian basis sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' We report algorithmic  and computational strategies and present numerical re-  sults for lower bounds to the (ground and excited state)  energies of the helium, lithium, and beryllium atoms with  a relative precision comparable to the corresponding up-  per bound obtained in the same series of computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='  LOWER BOUND THEORY  Schr¨odinger’s  formulation  of  the  non-relativistic  Hamiltonian of atoms with a fixed nucleus of charge num-  ber Z is written in Hartree atomic units as  H = −1  2  N  �  i=1  ∆i +  N  �  i=1  N  �  j>i  1  rij  −  N  �  i=1  Z  ri  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='  (1)  with ri denoting the distance of the ith electron from the  nucleus, rij denoting the distance of the ith electron from  the jth one, and ∆i is the kinetic energy operator for the  ith electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' The nucleus is assumed to be stationary,  with infinite mass, located at the origin of the spatial  coordinate system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' A central branch of molecular physics  revolves about the computation of stationary states of H  by (numerical) solution of the eigenvalue equation  Hψn = εnψn ,  n = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='  (2)  According to the Ritz–Macdonald variational princi-  ple [56,57], the energy functional  λn = ⟨ϕn|H|ϕn⟩  ⟨ϕn|ϕn⟩  ≥ εn,  (3)  provides an upper bound to the exact energy εn for an ap-  propriate ϕn trial function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' For a linear parametrization  of the trial function in terms of “primitive” basis func-  tions, fi, ϕn = �L  i=1 cnifi, the minimization problem is  turned into a matrix eigenvalue equation  Hcn = λ(L)  n Scn ,  (4)  where the H Hamiltonian and S overlap matrices are  calculated using the basis functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' The matrix elements  are Hij = ⟨fi|H|fj⟩ and Sij = ⟨fi|fj⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='  A central difficulty in computing lower bounds to  Coulombic systems using L´anczos basis sets or more  generally the Krylov algorithm [58] is due to the fact  that with most basis sets, H2 is the highest power of  the Coulomb Hamiltonian that can be handled;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' powers  greater than 2 usually diverge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' The expectation value of  H2, ⟨H2⟩n = ⟨ϕn|H2|ϕn⟩, and the corresponding vari-  ance, σ2  n =  �  ⟨H2⟩n − ⟨H⟩2n, can be computed and used  in relation with several lower bound theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' However,  until recently, all lower bound theories returned numeri-  cal values that were several orders of magnitude less accu-  rate than the upper bound obtained in a similar computa-  tional setup;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' hence, the practical utility of the computed  lower bounds remained limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='  The recently formulated Pollak–Martinazzo (PM)  lower bound theory addresses this problem by construct-  ing a special matrix used in conjunction with the Cauchy  interlacing theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' According to the interlacing theo-  rem, which can be derived from the Courant–Fisher the-  orem [59], if the eigenvalues of an n×n Hermitian matrix  A are given in ascending order as a1 ≤ a2 ≤ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' ≤ an−1 ≤  an, and the eigenvalues of its (n − 1) × (n − 1) principal  submatrix B are b1 ≤ b2 ≤ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' ≤ bn−2 ≤ bn−1, then  a1 ≤ b1 ≤ a2 ≤ b2 ≤ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' ≤ an−1 ≤ bn−1 ≤ an.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' This theo-  rem is used to obtain lower bounds to the eigenvalues of  B as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' The matrix B with dimension L = n − 1  is substituted with the diagonal Hamiltonian matrix ob-  tained by diagonalizing the L × L Hamiltonian matrix,  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' (4), with eigenvalues denoted in an ascending order  as λ(L)  j  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='  Then, a “big” [(L + 1) × (L + 1)]-dimensional matrix  is defined, motivated by the matrix A in the previous  paragraph, as [54]  KL(ε) =  \uf8eb  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ed  λ(L)  1  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='  0  σ(L)  1  0  λ(L)  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='  0  σ(L)  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='  0  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' λ(L)  L  σ(L)  L  σ(L)  1  σ(L)  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' σ(L)  L  ε +  L  �  k=1  �  σ(L)  k  �2  λ(L)  k  − ε  \uf8f6  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f8  ,  (5)  3  where λ(L)  n  labels the nth Ritz eigenvalue and σ(L)  n  is the  associated standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' We refer to the matrix  KL(ε) as the PM matrix with parameter ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='  By construction, the parameter ε is an eigenvalue of  the PM matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' The remaining eigenvalues of the matrix  are the L solutions of the polynomial equation  1 =  L  �  k=1  σ2  k  (λk − ε) (x − λk) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='  (6)  They are denoted in ascending order as xj(j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' , L)  and have the important property that xj (ε) is a mono-  tonically increasing function of the parameter ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='  Suppose that we choose ε to equal the unknown ground  state energy denoted as ε1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' According to Cauchy’s in-  terlacing theorem, the eigenvalues xk of the KL(ε) ma-  trix are interlaced by the Ritz eigenvalues λk as follows:  ε1 ≤ λ(L)  1  ≤ x1 ≤ λ(L)  2  ≤ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' ≤ λ(L)  L  ≤ xL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' Then, if  we have a lower bound for x1 and compute from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' (6)  the value of ε that would give the same value of x1, then  due to the monotonicity property this value of ε would  necessarily be a lower bound to the ground state energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='  If the basis set used is “good” in the sense that both λ(L)  1  and λ(L)  2  are not too far from the exact eigenvalues ε1  and ε2, then, barring special circumstances such as de-  scribed below for the He atom, one finds that x1 ≥ ε2,  so that bounding x1 from below by a lower bound to the  excited state energy gives a lower bound to the ground  state energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' Since the eigenvalues of the PM matrix are  not very sensitive to the precise value of x1 used, this  leads to accurate lower bounds, as shown below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='  This procedure may then be continued.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' For example,  if λ(L)  3  is also not too far from the exact eigenvalue ε3  then x2 will be larger than ε3, so that replacing it with a  lower bound to ε3 and finding the two lowest eigenvalues  of the PM equation, yields lower bounds to the ground  and first excited state energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' This procedure may then  be continued for the next excited state, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='  COMPUTATIONAL SETUP WITH AN  EXPLICITLY CORRELATED GAUSSIAN BASIS  Explicitly correlated Gaussian (ECG) functions [60–  64] are commonly used as a spatial basis for atomic and  molecular problems;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' however, unlike orthogonal polyno-  mials, they do not provide uniform coverage of space by  simply increasing the polynomial order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' ECGs can be  powerfully used in relation with parametrization by opti-  mization (minimization) of some appropriate target func-  tional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' (Regarding nodeless harmonic oscillator functions  used as a basis, see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=') The parametrization of  ECGs with respect to minimization of the energy func-  tional is a powerful means of obtaining and systemati-  cally improving energy upper bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' While ECGs fail  to satisfy the cusp condition [65], they have general, an-  alytic N-particle integrals for most physically relevant  operators, which can also be generalized for molecular  computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='  In this work, trial functions corresponding to the S  ground-state symmetries of the helium, lithium, and  beryllium atoms are expressed as anti-symmetrized (A)  products of φ spatial and χ spin functions  fi(r, σ) = A{φL,ML(r, Ai)χS,MS(σ, ϑi)} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='  (7)  χS,MS(σ, ϑi) corresponds to the two-, three-, and four-  electron spin functions coupled to spin states with to-  tal spin quantum numbers (S, MS) = (1, 0) for helium  and beryllium and to (2, 0) for lithium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' The total spin  functions for lithium and beryllium correspond to a two-  dimensional spin space, which is parametrized by one  free parameter (θi) [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' We used ECGs as spatial ba-  sis functions corresponding to (L, ML) = (0, 0) orbital  momentum quantum numbers (suppressed in the rest of  the paper),  φ(r, Ai) = e−rT(Ai⊗I3)r ,  (8)  centered at the origin (where the nucleus is fixed) and r ∈  R3N collects the electronic coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' The Ai ∈ RN×N  positive-definite, symmetric matrix determines the width  of the Gaussian and the correlation length of the parti-  cles, and is determined by optimization of some appro-  priate target function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='  All computations were performed using a computer  program named QUANTEN (QUANTum mechanical de-  scription of electrons and atomic nuclei) and developed  by the Budapest group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='  QUANTEN has a (stochas-  tic and deterministic) variational engine and an ex-  tensive ECG library with recent applications includ-  ing non-adiabatic, pre-Born–Oppenheimer, perturbative,  and variational relativistic computations [65–74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' It can  be efficiently run with double precision arithmetic, but  a quadruple precision mode is also available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' The first  implementation of the H2 integrals and assembling the  variance computation in QUANTEN, which was recently  reported for two- and three-electron atoms [55,75], is fur-  ther developed and extended to four (and, in general, N)  particle systems in the present work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='  Strategy for converging the PM lower bound to  the energy  Atomic PM lower bounds have been reported for the  helium and lithium ground states using the computa-  tional setup described above [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='  Although, the PM  bounds were tighter than the Weinstein, Temple, or  Lehmann bounds obtained with the same basis set [55],  even the best PM bound was (at least) three orders of  magnitude less precise (a relative precision of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='55 ppm  for helium and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='0 ppm for lithium was achieved), than  the corresponding upper bound (with a relative precision  of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='000 17 ppm for helium and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='002 ppm for lithium).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='  The natural question arises:  How can we improve  the convergence of the PM bounds?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' The plausible idea  4  of fine-tuning the ECG basis parametrization based on  a simple PM energy ensuremaximization condition was  found to be impractical in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' If one is not careful,  then a simple minded application of the PM method may  lead to energy values which are higher rather than lower  than the true eigenvalue under study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='  To better understand the conditions for which the PM  method leads to lower bounds, it is necessary to consider  that all lower bound theories based on the variances of  the Hamiltonian are only valid under certain conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='  These conditions are typically connected to the quality  of the variances and the ε parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' When the upper  bounds are “well behaved,” in the sense that the λj − εj  distance may be considered as small, one may expand the  PM equation, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' (6), to leading order in the distance to  find [54]  xj(ε) − εj+1 ≃ λj+1 − εj+1 − σ2  j+1  σ2  j  (λj − εj) ≥ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='  (9)  This relation implies that the left-hand side of the equa-  tion will be positive if the ratio of the variances of the  (j + 1)th to the jth state is sufficiently small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' This sug-  gests that if we are interested, for example, in a high-  quality lower bound to the ground-state energy, and we  already have a fairly good description of the ground-state  upper bound, then we should continue improving not the  ground but the first-excited state’s description, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=', con-  tinue with the minimization of the first-excited-state en-  ergy and associated reduction of its variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' This is the  core idea for the computational developments presented  in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' Furthermore, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' (9) will also be used to  rationalize some further observations regarding the nu-  merical results [sensitivity of the computed lower bounds  to the ε parameter of the PM matrix, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' (5), and pos-  sible failure of obtaining a lower bound].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='  The implementation of the core idea, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=', improvement  of the description of excited states to have a better lower  bound for the ground state, was not readily available in  the existing computational setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' Although ECG basis  sets generated based on the energy minimization condi-  tion for a selected state provide a very compact represen-  tation, they do not guarantee a high-quality description  of other states (unlike a set of orthogonal polynomials,  for which increasing the number of functions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=', the  polynomial order, automatically ensures more complete  coverage of the space, and, hence, improved convergence  of excited-state energies).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='  Implementation of the multi-state energy minimization  strategy in an ECG-based procedure  A usual energy minimization procedure, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=', for the  ground state, is initiated by random basis generation and  selection [63], which is followed by repeated refinement  cycles of the already existing basis set, for which we use  the Powell method [76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='  Both steps are based on the  energy minimization condition (and the variational prin-  ciple for Hamiltonians bounded from below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='  The same procedure can be repeated for the first- (nth)  excited state (even long-lived states embedded in the con-  tinuum [69] in combination with a stabilization-like pro-  cedure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' In this fashion, separate near-optimal basis sets  for separate states can be straightforwardly generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='  One could then try and merge the basis sets optimized  for the ground and for the first-excited states, but this  procedure would result in a gigantic basis and, more im-  portantly, near-linear dependency problems in the finite  precision arithmetic used for the computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='  Instead, we have implemented a multi-state procedure  in a single computation as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' The usual basis gen-  eration and refinement using the energy minimization  condition for the ground state is implemented up to a  certain number of basis functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' This number is de-  termined based on the convergence of the Ritz ground-  state energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' This results in the first “block” of our basis  set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' The computation is then continued with the genera-  tion and refinement of additional basis functions (second  block of the basis set), for which the energy minimiza-  tion condition for the second state (first-excited state)  was implemented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' We have regularly refined (using the  Powell method) the entire basis set, one function after the  other, by using the energy minimization condition for the  ground-state energy for functions belonging to the first  basis block, and the energy minimization condition for  the second state for functions belonging to the second  basis block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='  The repeated full-basis refinement cycles allow us to re-  lax functions in the first block (optimized to the ground  state) while the ground-state energy is also (partly) de-  scribed by the second-block functions (optimized to the  first-excited state).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' Therefore, small deviations from a  monotonic decrease of the energy may occur upon en-  largement of the basis set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='  For sufficiently large basis  sets and with further, extended optimizations these small  deviations from monotonicity can be smoothed out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='  By construction, the procedure generates a basis set  which is (near) optimal for both the ground and the  first-excited states, and the linear–dependency problem is  automatically avoided (a new basis function that would  have a too large overlap with the existing basis set is  discarded or “weighted down” with a “penalty” correc-  tion to the value of the energy functional).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' Furthermore,  the procedure can be straightforwardly extended to addi-  tional states, and thus, applicable also beyond the ground  state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='  Numerical demonstration of the multi-state optimization  strategy for lithium and beryllium  The computational strategy described above has been  implemented in QUANTEN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' It is highlighted for the case  of lithium in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' 1 (see also Table I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' The computations  are more expensive for beryllium, so for this case, we  5  −10  −9  −8  −7  −6  −5  −4  −3  −2  −1  500  1000  1500  2000  ∆εj  L  ∆ε1  ∆ε2  −10  −9  −8  −7  −6  −5  −4  −3  −2  −1  ∆λj  ∆λ1  ∆λ2  ∆λ3  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' Upper and lower bound gaps [Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' (10)] for the lithium  atom with respect to the εref,j reference energies taken from  Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' [77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' The first, second, and third basis blocks with L ∈  [1, 980], (980, 1700], and (1700, 2175] basis indexes, in short  [1:980:1700:2175], were optimized according to minimization  of the ground (yellow), the first- (white), and the second-  excited state (gray) energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' (See also Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=')  report only the final results (Tables II and III).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' Unex-  pectedly, helium turned out to be a very special case, for  which the strategy does not work (the condition x1 ≥ ε2  fails), and this can be rationalized on the basis of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' (9)  as explained in the last paragraphs of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='  As a measure of the quality of the lower-bound energy  for a given basis set, we compare the relative upper and  lower bound gaps defined as  ∆λj = log10 (|λj − εref,j|/εref,j)  ∆εj = log10 (|εj,− − εref,j|/εref,j) ,  (10)  where εref,j is a reference value (expected to be very close  to the exact value and available from the literature for  the computed examples), λj and εj,− are the computed  upper and lower bounds for the jth state (j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' ),  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' (We note that ε−  j is used in Tables I–IV to  label the estimated lower bounds used in the PM equa-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=') If the gap ratio,  ηj = ∆εj/∆λj ,  (11)  approaches one, we may say that the lower (and upper)  bound computation is useful in terms of bracketing the  exact energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' 1, showcasing our computation for lithium, the  performance of the various energy estimates in the yel-  low region is comparable to the best gap ratio achieved  in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' Then, we continue with the multi-state opti-  mization procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' During the generation of the second  basis block (white region in the figure), we see a signifi-  cant improvement for the ground-state lower bound, and  the first-excited state lower bound also improves (lower  part of the figure), in parallel with the improvement of  the first- and second-excited-state upper bounds (upper  part of the figure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='  As can be seen in the figure, the optimization for one  state does not necessarily guarantee the monotonic im-  provement of the ground and other states;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' however, any  increase in a state energy can be minimized by applying  subsequent refinement cycles to the already generated  basis set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' The figure also shows the generation and op-  timization of a third basis block (gray-shaded area), in  which the basis functions are optimized using the energy  minimization condition for the second-excited state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='  The resulting best upper and lower bound values ob-  tained for the lithium atom corresponding to a total  basis size of L = 2175 are collected in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' While  previous PM computations carried out for the lithium  atom ground state [55] (with a single basis block) already  improved upon the Lehmann bound obtained using a  Hylleraas basis [53], the present PM lower bounds signif-  icantly outperform both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' The lower bounds are at most  one order of magnitude worse than the upper bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='  This may be improved upon if one has a better estimate  for the excited state energies as discussed in further de-  tail below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' At this point it suffices to say that the values  used as estimates for the excited state energies upon im-  plementing the PM equations are rather conservative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='  As might be expected, the ground-state lower bound  TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' Results for the lithium atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' Tabulation of lower-  and upper-bound energies, εj,− and λj, respectively, in units  of Eh, resulting from multi-state optimization with three  blocks (with basis sizes [1:980:1700:2175], cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' The  relevant variances are also given in units of E2  h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='  The rela-  tive deviation from the εref,j reference energy adapted from  Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' [77] is shown in parentheses in ppb (parts per billion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='  The PM parameter in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' (5) was ε−  2 = εref,2−5·10−8 Eh and  ε−  3 = εref,3 − 10−7 Eh for obtaining the lower bounds for the  ground and first excited states, respectively (see also Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' 2  and corresponding text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='  Ground state  ε1,−  −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='478 060 364  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='3)  εref,1  −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='478 060 324 [77]  λ1  −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='478 060 316  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='0)  First excited state  ε2,−  −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='354 098 569  (20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='1)  εref,2  −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='354 098 421 [77]  λ2  −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='354 098 404  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='4)  Second excited state  εref,3  −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='318 530 846 [77]  λ3  −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='318 530 751  (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='9)  Variances  σ2  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='158 934 586  σ2  2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='225 395 309  σ2  3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='265 772 122  6  is more accurate than the first excited state lower bound  and the same ordering of accuracy is true for the upper  bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' The plateauing of the ground-state lower bound  for L ≥ 1500 reflects the plateauing of the ground-state  upper bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' The ground-state lower bound will improve  as the PM eigenvalue x1 converges to the first excited  state energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='  As seen from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' (9), for this to occur  one needs an improvement of the upper bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' Since  this does not happen, the lower bound reaches a plateau  value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' The same occurs for the first excited state lower  bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='  Similarly good results are obtained with multi-state  optimization for the ground- and first-excited states of  the beryllium atom (Tables II and III).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' The multi-state  optimization strategy was essential to arrive at good  lower-bounds also for beryllium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' In this case, the qual-  ity of the ground-state lower bound is comparable to the  Ritz upper bound while it is somewhat worse, a factor of  ≃ 6, for the excited state, This reflects to some extent  the lower bound values used for the excited states when  implementing the PM equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='  TABLE II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' Results for the beryllium atom: ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='  Tabulation of lower- and upper-bound energies, εj,− and λj,  respectively, in units of Eh resulting from multi-state opti-  mization with two blocks (with basis set sizes [1:2000:4500]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='  The relevant variances are also given in units of E2  h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' The rela-  tive deviation from the εref,j reference energy given in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' [78]  is shown in parentheses in ppb (parts per billion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' The PM  parameter in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' (5) was ε−  2 = εref,2−10−7 Eh (see also Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' 3  and corresponding text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='  Ground state  ε1,−  −14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='667 356 917  (28)  εref,1  −14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='667 356 507 [78]  λ1  −14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='667 356 191  (22)  First excited state  εref,2  −14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='418 240 364 [78]  λ2  −14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='418 239 479  (61)  Variances  σ2  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='506 745 221  σ2  2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='698 028 906  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='  Stability and sensitivity of the results to the ε  parameter of the PM matrix  The PM lower-bound computation (similarly to Tem-  ple’s bound or other lower-bound methods) requires some  knowledge about the higher-energy state(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' This infor-  mation (estimate) is encoded in the ε parameter of the  PM matrix, [Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' (5)]: for the computation of a lower  bound to the nth eigenvalue, the ε value in the PM ma-  trix must be a lower estimate to the (n + 1)th energy  eigenvalue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='  The computed lower-bound results (in Tables I–IV)  have been reported with a specific ε value (estimated  from a known precise reference value) used in the PM  TABLE III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='  Results for the beryllium atom: first-excited  state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' Tabulation of lower- and upper-bound energies, εj,−  and λj, respectively, in units of Eh resulting from multi-  state optimization with three blocks (with basis set sizes  [1:2000:4000:4500]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' The relevant variances are also given in  units of E2  h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' The relative deviation from the εref,j reference  energy [78] is shown in parentheses in ppb (parts per billion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='  The PM parameter in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' (5) was ε−  3 = εref,3 − 10−6 Eh (see  also Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' 3 and corresponding text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='  First-excited state  ε2,−  −14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='418 248 205  (543)  εref,2  −14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='418 240 364 [78]  λ2  −14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='418 238 971  (97)  Second excited state  εref,3  −14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='370 087 938 [78]  λ3  −14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='370 062 140  (1795)  Variances  σ2  2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='698 028 906  σ2  3  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='389 904 253  calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' The critical reader might comment that ob-  taining a tight lower bound which is based on knowledge  of a different tight lower bound is problematic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' Hence, it  is necessary to address the “stability” of the results with  respect to the precise choice of this value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' The PM results  obtained in previous computations reported in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' [54]  and [55] have been found to be relatively insensitive to ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='  In this work, we repeated the PM computations for  the largest basis set results of lithium and beryllium (Ta-  bles I and II) using various ε parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' Figures 2 and 3  present the lower bound gap defined with respect to the  Ritz eigenvalue  δεj = log10 (|εj,− − λj|/λj) ,  (12)  which is defined analogously to ∆εj in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' (10), but  free from the knowledge of an “external” reference value,  εref,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='  The gap for the (estimated) ε−  j  parameter in  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' (5)—which is a lower bound to the respective ex-  cited state—is defined with respect to the reference value  exactly the same way as the lower bound gap in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' (10):  ∆ε−  j = log10  �  |ε−  j − εref,j|/εref,j  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='  (13)  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' 2, the red and blue lines show the ground-  and first-excited-state PM lower bound gaps, δεj, re-  spectively, plotted with respect to the lower-bound gap  ∆ε−  j+1, whereas the black line shows the (orders of mag-  nitude worse) gap for the Temple lower bound, defined  as  εTemple  1,−  = λ1 −  σ2  1  ε−  2 − λ1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='  (14)  As can be seen in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' 2 and 3, the PM lower bounds are  sensitive to the precision of the lower estimate to the jth  energy (ε−  j = εref,j − ∆ε−  j is the parameter used in the  PM matrix) while the Temple lower bound is not, due  7  −10  −8  −6  −4  −2  0  −10  −8  −6  −4  −2  0  δεj  ∆εj+1  −  PM: δε1  vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' ∆ε2  −  PM: δε2  vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' ∆ε3  −  Temple: δε1  vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' ∆ε2  −  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' Sensitivity of the PM ground- (red) and first-excited  (blue) state lower-bound gap [Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' (12)] for the lithium atom  at a basis size of L = 2175 (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' 1) with respect to the  precision of the ε PM parameter used in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' (5) given in  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' The ground-state Temple gap is also shown (black).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='  The ratios of the variances are σ2  2/σ2  1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='42 and σ2  3/σ2  2 =  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='  to its poor quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' In the ∆ε−  j range used to compute  the data reported in Tables I–III, the functions in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' 2  and 3 are nearly linear, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=', the precision of the PM lower  bound is determined by the precision of the excited-state  estimate used in the PM matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' This observation can  be rationalized on the basis of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' In contrast to  the results presented in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' [54] and [55], the ratio of  variances σ2  j+1/σ2  j is of the order of unity, due to the  optimization of the excited states, and the accuracy of the  (j + 1)th excited state Ritz eigenvalue is much improved,  leading to the linear dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='  How then does one know the correct value of ε to be  used in the lower bound calculation?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' The strategy we  employed was to use a value that is substantially lower  than the accuracy expected from the convergence proper-  ties of the relevant Ritz eigenvalue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' These are the values  reported in Tables  I–III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' The high accuracy of the re-  sulting lower bounds demonstrates that this strategy is  robust and that the linear dependence is not really a se-  rious problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='  The special case of the helium atom  We applied the multi-state optimization strategy also  for the helium atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='  When applied naively, the PM  equation gave values for the ground-state energy which,  −10  −8  −6  −4  −2  0  −10  −8  −6  −4  −2  0  δεj  ∆εj+1  −  PM: δε1  vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' ∆ε2  −  Temple: δε1  vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' ∆ε2  −  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' Sensitivity of the PM ground-state lower-bound gap  [Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' (12)] for the beryllium atom at a basis size of L = 4500  (2000 states for the ground and 2500 for the first-excited state,  (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' Table II) with respect to the precision of the ε PM pa-  rameter used in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' (5) given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' The ground-state  Temple gap is also shown (black).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' The ratio of the variances  is σ2  2/σ2  1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='  in the limit of a large basis set, were larger than the  known ground-state energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' What went awry?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='  This is related to the use of a correlated Gaussian  basis set rather than an orthogonal polynomial basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='  We observed in convergence figures (similar to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' 1)  that the upper bounds (Ritz eigenvalues) to the first-  and second-excited states, optimized in the second and  third basis blocks, converged faster than the ground-state  eigenvalue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' As may be then reasoned from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' (9), this  causes the right-hand side of the equation to be negative,  that is, the eigenvalue x1 is no longer greater than the  first excited state energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' Using the first excited state  energy in the PM equation will then naturally no longer  give a lower bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='  It can also be understood that this behavior is unique  to the helium atom, which is a two-electron system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' The  ground state is dominated by a 1s2 configuration, the  first-excited state is 1s2s, the second-excited state is  1s3s, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' The correlation of the electrons, which is de-  scribed increasingly more accurately during the course of  the variational computation, is less important for excited  states, than for the ground state, and hence their Ritz  eigenvalues for the excited states converge faster than  for the ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='  Does this mean that one cannot get meaningful and  accurate lower bounds for the He atom using correlated  Gaussian basis sets?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' Not necessarily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' If one forces the ba-  8  TABLE IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' Lower and upper bounds for the He atom energy  levels, in Eh, computed in this work for the ground, first-, and  second-excited states using L = 510 ECG basis functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='  The relative deviation from the reference energies [41,79] is  shown in parentheses in parts per billion (ppb).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='  The PM  matrix parameters used in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' (5) were ε−  i = εref,i − 2 · 10−9  Eh for both i = 2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' The variances are in units of E2  h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='  Ground state  ε1,−  −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='903 724 379  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='7)  εref,1  −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='903 724 377 [41]  λ1  −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='903 724 376  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='3)  First excited state  ε2,−  −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='145 974 048  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='9)  εref,2  −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='145 974 046 [79]  λ2  −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='145 974 045  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='5)  Second excited state  εref,3  −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='061 271 990 [79]  λ3  −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='061 271 989  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='4)  Variances  σ2  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='013 300 747  σ2  2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='129 998 619  σ2  3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='121 216 647  sis set so that the excited-state eigenvalues are not better  than the ground-state level, one may expect the method  to work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' This is demonstrated in Table IV, where us-  ing the known excited-state energy values, we can ensure  that the accuracy of all three levels is similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' However,  this does not answer the question as to how would it be  possible, without the knowledge of the numerically ex-  act values, to ensure that the PM equation leads to a  lower bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' Fortunately, for larger atoms, the problem  does not exist, and as we showed, it is straightforward  to obtain high-quality lower bounds for the Li and Be  atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='  SUMMARY AND DISCUSSION  A multi-state optimization strategy is developed to  systematically converge the Pollak–Martinazzo energy  lower bound with an explicitly correlated Gaussian basis  set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' Lower bounds to the ground- and first-excited state  energies of the lithium and beryllium atoms are com-  puted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' The resulting lower bounds are the most precise  to date, and their relative precision is comparable to that  of the energy upper bound in the same basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='  In view of the performance of the multi-state optimiza-  tion and the PM lower bound theory, the following con-  clusions can be drawn:  The multi-state optimization of ECG bases pro-  vides a systematic and robust improvement of the  low-lying eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='  The optimization of higher lying states does not  affect the already converged states adversely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='  The optimization of the energy of the (n+1)th state  improves the quality of the lower bound to the nth  state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='  The PM theory is able to provide lower bounds  with ppb relative precision for the energy levels of  few-electron systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='  The presented computational procedure and numeri-  cal results are for non-relativistic energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' Relativistic  and leading-order quantum electrodynamic effects have  been traditionally accounted for as perturbative correc-  tions to the non-relativistic energy, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=', [70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' The iden-  tification of a many-particle relativistic wave equation  based on relativistic quantum electrodynamics (QED)  is more challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' Most recently, it became possible  (for two particles) to start out from the Bethe–Salpeter  QED wave equation, exploit that interactions in atoms  and molecules are dominantly instantaneous, and arrive  at an eigenvalue equation for a no-pair Dirac–Coulomb–  Breit Hamiltonian [80].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='  This Hamiltonian appears to  be bounded from below, and robust variational proce-  dures could be developed to compute its eigenvalues,  which have an α fine-structure constant dependence that  is in agreement with the known α orders of the well-  established perturbative procedures [71–74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='  This theoretical approach provides variational ensur-  erelativistic upper bounds (including also some of the  so-called “non-radiative” QED corrections of the per-  turbative framework), and (with further development to  many-particle systems), it will be relevant to ask for en-  surerelativistic lower bounds in a spirit similar to this  work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='  ACKNOWLEDGMENTS  Financial support of the European Research Council  through a Starting Grant (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' 851421) is gratefully ac-  knowledged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='  This work has also been graciously sup-  ported by the Ben May Center for Chemical Theory and  Computation at the Weizmann Institute of Science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
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+page_content=' Almayouf,  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='  Krause,  and  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
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+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
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+page_content=' Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
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+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
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+page_content=' Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' Numer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
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+page_content=' Rumble, Upper  and lower bounds to atomic and molecular properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
+page_content='  iv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQf_gJj/content/2301.02827v1.pdf'}
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diff --git a/19E0T4oBgHgl3EQfdgD3/content/tmp_files/2301.02379v1.pdf.txt b/19E0T4oBgHgl3EQfdgD3/content/tmp_files/2301.02379v1.pdf.txt
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+CodeTalker: Speech-Driven 3D Facial Animation with Discrete Motion Prior
+Jinbo Xing* 1 Menghan Xia2 Yuechen Zhang1 Xiaodong Cun2 Jue Wang2 Tien-Tsin Wong1
+1The Chinese University of Hong Kong
+2Tencent AI Lab
+Abstract
+Speech-driven 3D facial animation has been widely stud-
+ied, yet there is still a gap to achieving realism and vividness
+due to the highly ill-posed nature and scarcity of audio-
+visual data. Existing works typically formulate the cross-
+modal mapping into a regression task, which suffers from
+the regression-to-mean problem leading to over-smoothed
+facial motions. In this paper, we propose to cast speech-
+driven facial animation as a code query task in a finite
+proxy space of the learned codebook, which effectively pro-
+motes the vividness of the generated motions by reducing
+the cross-modal mapping uncertainty.
+The codebook is
+learned by self-reconstruction over real facial motions and
+thus embedded with realistic facial motion priors. Over the
+discrete motion space, a temporal autoregressive model is
+employed to sequentially synthesize facial motions from the
+input speech signal, which guarantees lip-sync as well as
+plausible facial expressions. We demonstrate that our ap-
+proach outperforms current state-of-the-art methods both
+qualitatively and quantitatively. Also, a user study further
+justifies our superiority in perceptual quality. The source
+code will be publicly available.
+1. Introduction
+3D facial animation has been an active research topic for
+decades, as attributed to its broad applications in virtual re-
+ality, film production, and games. The high correlation be-
+tween speech and facial gestures (especially lip movements)
+makes it possible to drive the facial animation with a speech
+signal. Early attempts are mainly made to build the complex
+mapping rules between phonemes and their visual counter-
+part, which usually have limited performance [54,66]. With
+the advances in deep learning, recent speech-driven facial
+animation techniques push forward the state-of-the-art sig-
+nificantly. However, it still remains challenging to generate
+human-like motions.
+As an ill-posed problem, speech-driven facial animation
+generally has multiple plausible outputs for every input.
+* Work done during an internship at Tencent AI Lab.
+Such ambiguity tends to cause over-smoothed results. Any-
+how, person-specific approaches [31, 50] can usually ob-
+tain decent facial motions because of the relatively consis-
+tent talking style, but have low scalability to general ap-
+plications. Recently, VOCA [10] extends these methods
+to generalize across different identities, however, they gen-
+erally exhibit mild or static upper face expressions. This
+is because VOCA formulates the speech-to-motion map-
+ping as a regression task, which encourages averaged mo-
+tions, especially in the upper face that is only weakly or
+even uncorrelated to the speech signal. To reduce the un-
+certainty, FaceFormer [16] utilizes long-term audio context
+through a transformer-based model and synthesizes the se-
+quential motions in an autoregressive manner. Although
+it gains important performance promotion, it still inherits
+the weakness of one-to-one mapping formulation and suf-
+fers from a lack of subtle high-frequency motions.
+Dif-
+ferently, MeshTalk [51] models a categorical latent space
+for facial animation that disentangles audio-correlated and
+audio-uncorrelated information so that both aspects could
+be well handled. Anyway, the categorical latent space is
+represented by some proxy numbers instead of an explicit
+codebook with feature vectors, which makes the training
+tricky and hence impedes its performance.
+We get inspiration from 3D Face Morphable Model
+(3DMM) [37], where general facial expressions are rep-
+resented in a low-dimensional space.
+Accordingly, we
+propose to formulate speech-driven facial animation as a
+code query task in a finite proxy space of the learned dis-
+crete codebook prior.
+The codebook is learned by self-
+reconstruction over real facial motions using a vector-
+quantized autoencoder (VQ-VAE) [60], which along with
+the decoder stores the realistic facial motion priors.
+In
+contrast to the continuous linear space of 3DMM, com-
+binations of codebook items form a discrete prior space
+with only finite cardinality. Still, in the context of the de-
+coder, the code representation possesses high expressive-
+ness. Through mapping the speech to the finite proxy space,
+the uncertainty of the speech-to-motion mapping is signif-
+icantly attenuated and hence promotes the quality of mo-
+tion synthesis. Conceptually, the proxy space approximates
+the facial motion space, where the learned codebook items
+1
+arXiv:2301.02379v1  [cs.CV]  6 Jan 2023
+
+serve as discrete motion primitives.
+Based on the learned discrete codebook, we pro-
+pose a code-query-based temporal autoregressive model
+for speech-conditioned facial motion synthesis, called
+CodeTalker. Specifically, taking a speech signal as input,
+our model predicts the motion feature tokens in a temporal
+causal manner. Then, the feature tokens are used to query
+the code sequence in the discrete space, followed by facial
+motion reconstruction. Thanks to the contextual modeling
+over history motions and cross-modal alignment, the pro-
+posed CodeTalker shows the advantages of achieving accu-
+rate lip motions and natural expressions. Extensive experi-
+ments show that the proposed CodeTalker demonstrates su-
+perior performance on existing datasets. Systematic studies
+and experiments are conducted to demonstrate the merits of
+our method over previous works. The contributions of our
+work are as follows:
+• We make the first attempt to model the facial mo-
+tion space with discrete primitives, which offers ad-
+vantages to promote motion synthesis realism against
+cross-modal uncertainty.
+• We propose a discrete motion prior based temporal au-
+toregressive model for speech-driven facial animation,
+which outperforms existing state-of-the-art methods.
+• We will make our code fully available, including the
+data processing, metrics, trained models, etc., to stan-
+dardize the speech-driven 3D facial animation task for
+follow-ups.
+2. Related Works
+2.1. Speech-driven 3D Facial Animation
+Computer facial animation is a long-standing task [45]
+and has attracted rapidly increased interest over the past
+decades [5, 20, 32, 34, 36, 55, 65, 72]. As a branch, speech-
+driven facial animation is to reenact a person in sync with
+input speech sequences. While extensive literature in this
+field works on 2D talking heads [1,7–9,11,23,28,29,38,39,
+48, 52, 62, 64, 68, 69], we focus on facial animation on 3D
+models in this work, which can be roughly categorized into
+linguistics-based and learning-based methods.
+Linguistics-based methods.
+Typically, linguistics-
+based methods [12, 42, 54, 66] establish a set of complex
+mapping rules between phonemes and their visual counter-
+parts, i.e., visemes [19,35,43]. For example, the dominance
+function [42] is to determine the influence of phonemes on
+the respective facial animation control parameters. Xu et
+al. [66] defines animation curves for a constructed canonical
+set of visemes to generate synchronized mouth movements.
+There are also some methods considering the many-to-
+many mapping between phonemes and visemes, as demon-
+strated in the dynamic visemes model [54] and, more re-
+cently, the JALI [12]. Based on psycholinguistic consid-
+erations and built upon the Facial Action Coding System
+(FACS) [13], JALI factors mouth movements into lip and
+jaw rig animation and generate compelling co-articulation
+results. Although these methods have explicit control over
+the animation, they have complex procedures and lack a
+principled way to animate the entire face.
+Learning-based methods. Learning-based methods [6,
+10,16,17,24,31,40,47,53,63] resort to a data-driven frame-
+work.
+Cao et al. [6] achieve emotional lip sync by the
+proposed constrained search and Anime Graph structure.
+Recently, Taylor et al. [53] propose a deep-learning-based
+model utilizing a sliding window approach on the tran-
+scribed phoneme sequences input. Karras et al. [31] pro-
+pose a convolution-based network with a learnable emotion
+database to animate a speech-driven 3D mesh. More re-
+cently, VisemeNet [71] employ a three-stage Long Short-
+Term Memory (LSTM) network to predict the animation
+curve for a lower face lip model.
+We review the most related works more concretely here
+as they have the same setting as this work, i.e., training
+on high-resolution paired audio-mesh data and speaker-
+independently animating entire face meshes in vertex space.
+MeshTalk [51] successfully disentangles audio-correlated
+and uncorrelated facial information with a categorical latent
+space. However, the latent space adopted is not optimal with
+limited expressiveness, thus the animation quality is not sta-
+ble when applied in a data-scarcity setting. VOCA [10]
+employs powerful audio feature extraction models and can
+generate facial animation with different speaking styles.
+Furthermore, FaceFormer [16] considers long-term audio
+context with transformer [61] rendering temporally stable
+animations. Despite the appealing animations, both suffer
+from the over-smoothing problem, as they directly regress
+the facial motion in the highly ill-posed audio-visual map-
+ping with large uncertainty and ambiguity.
+2.2. Discrete Prior Learning
+In the last decades, discrete prior representation with
+learned dictionaries has demonstrated its superiority in im-
+age restoration tasks [14, 22, 30, 56, 57], since clear im-
+age details are well-preserved in the dictionaries.
+This
+line of techniques further inspires the high-capacity and
+high-compressed discrete prior learning. VQ-VAE [60] first
+presents to learn discrete representations (codebook) of im-
+ages and autoregressively model their distribution for image
+synthesis. The follow-up works, VQ-VAE2 [49] and VQ-
+GAN [15] further improve the quality of high-resolution
+image synthesis. Recently, discrete prior learning and its
+variants have been exploited for image colorization [26],
+inpainting [46], blind face restoration [70], text-to-image
+synthesis [21], etc.
+2
+
+In addition to the image modality, most recent works
+also explore the power of discrete prior learning in tasks
+with other modalities, such as dyadic face motion genera-
+tion [44], co-speech gesture synthesis [2], speech enhance-
+ment [67]. Inspired by codebook learning, this work inves-
+tigates to learn discrete motion prior for speech-driven 3D
+facial animation. Different from [44], we exploit the dis-
+crete motion primitives for facial motion representation in a
+context-rich manner, which is more effective to learn gen-
+eral priors.
+3. Method
+We aim to synthesize sequential 3D facial motions from
+a speech signal, so that any neutral face mesh could be an-
+imated as a lip-synchronized talking face. However, this is
+an ill-posed problem since one speech could be matched by
+multiple potential facial animations. Such ambiguity tends
+to make cross-modal learning suffer from averaged motions
+and lack of subtle variations. To bypass this barrier, we pro-
+pose to first model the facial motion space with the learned
+discrete motion prior, and then learn a speech-conditioned
+temporal autoregressive model over this space, which pro-
+motes robustness against the cross-modal uncertainty.
+Formulation.
+Let M1:T = (m1, ..., mT ) be a sequence
+of facial motions, where each frame mt ∈ RV ×3 denotes
+the 3D movement of V vertices over a neutral-face mesh
+template h ∈ RV ×3. Let further A1:T = (a1, ..., aT ) be
+a sequence of speech snippets, each of which at ∈ Rd has
+d samples to align with the corresponding (visual) frame
+mt. Then, our goal is to sequentially synthesize M1:T from
+A1:T so that an arbitrary neutral facial template f could be
+animated as H1:T = {m1 + h, ..., mT + h}.
+3.1. Discrete Facial Motion Space
+Visual realistic facial animations should present accu-
+rate lip motions and natural expressions. To achieve this
+from speech signals, extra motion priors are required to re-
+duce the uncertainty and complement realistic motion com-
+ponents.
+As witnessed by the recent image restoration
+task [70], discrete codebook prior [60] demonstrates ad-
+vantages in guaranteeing high-fidelity results even from a
+severely degraded input. Inspired by this, we propose to
+model the facial motion space as a discrete codebook by
+learning from tracked real-world facial motions.
+Codebook of motion primitives.
+We manage to learn a
+codebook Z = {zk ∈ RC}N
+k=1 that allows any facial mo-
+tion mt to be represented by a group of allocated items
+{zk}k∈S, where S denotes the selected index set through
+Eq. 1. Conceptually, the codebook items serve as the motion
+primitives of a facial motion space. To this end, we pre-train
+a transformer-based VQ-VAE that consists of an encoder
+E, a decoder D, and a context-rich codebook Z, under the
+𝑡
+𝑠
+𝐙q
+0
+1
+…
+N
+Codebook 𝒵
+Facial motion space
+෠𝐙
+arg min 𝑧𝑖∈𝒵 ||෠𝐙(𝑡,𝑠) − 𝒛𝑖||
+Reconstructed motions ෡𝐌1:𝑇
+Facial motions 𝐌1:𝑇
+Predicted  Motion 
+Feature ෝ𝑚1:𝑡
+Figure 1. Learning framework of facial motion space. The learned
+motion primitives, as embedded in the codebook, serve to repre-
+sent the facial motions in a spatial and temporal manner.
+self-reconstruction of realistic facial motions. As shown in
+Figure 1, the facial motions M1:T is first embedded as a
+temporal feature ˆZ = E(M1:T ) ∈ RT ′×H×C, where H is
+the number of face components and T ′ denotes the num-
+ber of encoded temporal units (P =
+T
+T ′ frames). Then, we
+obtain the quantized motion sequence Zq ∈ RT ′×H×C via
+an element-wise quantization function Q(·) that maps each
+item in ˆZ to its nearest entry in codebook Z:
+Zq = Q(ˆZ) := arg min
+zk∈Z
+∥ˆzt − zk∥2.
+(1)
+Then, the self-reconstruction is given by:
+ˆM1:T = D(Zq) = D(Q(E(M1:T ))).
+(2)
+Note that, the discrete facial motion space reduces the map-
+ping ambiguity with the finite cardinality, but never sacri-
+fices its expressiveness thanks to its context-rich represen-
+tation as a latent space.
+Training objectives.
+To supervise the quantized autoen-
+coder training, we adopt a motion-level loss and two inter-
+mediate code-level losses:
+LVQ =∥M1:T − ˆM1:T ∥1
++ ∥sg(ˆZ) − Zq∥2
+2 + β∥ˆZ − sg(Zq)∥2
+2,
+(3)
+where the first term is a reconstruction loss, the latter two
+are adopted to update the codebook items by reducing the
+distance between the codebook Z and embedded features
+ˆZ.
+sg(·) stands for a stop-gradient operation and β is a
+weighting factor controlling the update rate of the codebook
+and encoder. Since the quantization function (Eq. 1) is not
+differentiable, the straight-through gradient estimator [4,60]
+is employed to copy the gradients from the decoder input to
+the encoder output.
+Discussion.
+Recently, Learn2Listen [44] has applied VQ-
+VAE for facial expression synthesis in response to a given
+talking head harnessing 2D monocular videos to obtain
+3DMM coefficients.
+In addition to distinct applications,
+3
+
+Transformer
+decoder
+Speech 𝐀1:𝑇
+TCN
+Embedding
+0
+1
+…
+N
+Codebook 𝒵
+Style vector 𝒔
+quantization
+Past motions ෡𝐌1:𝑡−1
+Predicted motion ෝ𝒎𝑡
+Transformer
+encoder
+෠𝐙1:𝑡
+Speech encoder
+Cross-modal decoder
+𝐙𝐪1:𝑡
+Figure 2. Diagram of our speech-driven motion synthesis model. Given the speech A1:T and style vector s as input, the model learn
+to recursively generate a sequence of facial motions by predicting the motion codes. As embedded with well-learned motion priors, the
+pre-trained codebook and decoder are frozen during training.
+3dmm 
+coeffs
+𝑡
+𝑠
+𝐙q
+0
+1
+…
+N
+Codebook 𝒵
+Facial motion space
+෠𝐙
+arg min 𝑧𝑖∈𝒵 ||෠𝐙(𝑡,𝑠) − 𝒛𝑖||
+Reconstructed motions ෡𝐌1:𝑇
+Facial motions 𝐌1:𝑇
+Predicted  Motion 
+Feature ෝ𝑚1:𝑡
+𝒵1
+𝐸1
+𝐷1
+Speaker 1:
+Speaker N:
+. . .
+𝐸𝑁
+𝐷𝑁
+𝐷𝑁
+𝐸
+𝐷
+Generic 
+motion:
+(a)
+(b)
+𝒛1
+𝑖
+𝒛𝑁
+𝑖
+𝒛𝑖, … , 𝒛𝑘
+𝒵𝑁
+𝒵
+3dmm 
+coeffs
+3dmm 
+coeffs
+3dmm 
+coeffs
+3dmm
+𝐷1
+3dmm
+𝐷
+Figure 3.
+Concept comparison with Learn2Listen [44].
+(Top)
+The speaker-specific facial expression coefficient prior [44], in
+which each code represents a sequence of facial expression co-
+efficients. (Bottom) Our speaker-agnostic generic motion prior, in
+which each code represents the motion primitive of face compo-
+nents. The blue dotted boxes indicate what information each code
+may represent conceptually.
+here we would like to emphasize our major differences.
+First, Learn2Listen constructs speaker-specific codebooks
+while ours uses a generic codebook that is feasible to rep-
+resent arbitrary facial motions. Since cross-character mo-
+tions are absorbed, our codebook is naturally embedded
+with more plentiful priors. Second, Learn2Listen utilizes
+the codebook to represent common sequences of facial ex-
+pressions by the way of 3DMM coefficients, i.e., each code
+represents a sequence (8 frames) of facial expressions. Dif-
+ferently, our codebook is formulated to represent the vertex-
+based facial motion space, where the codes are embedded
+with per-vertex motions of facial components and repre-
+sent the facial motion (within a temporal unit) in a context-
+rich manner. As compared in Figure 3, the codebook of
+Learn2Listen is learned to memorize typical sequential fa-
+cial expressions of a specific speaker within 3DMM space,
+which cannot synthesize realistic facial motions with sub-
+tle details due to the limited expressiveness of 3DMM and
+is bounded by the accuracy of 3D reconstruction tech-
+niques [44]. As the first attempt, our codebook is learned to
+represent the generic facial motion space with motion prim-
+itives for captured facial mesh data, which is more effective
+to embed general priors preserving vivid facial details.
+We further discuss the hyper-parameters of the code-
+book.
+First, the length of the temporal unit P and the
+number of face components H determine the complexity
+of the motion primitives in temporal and spatial aspects
+respectively. Generally, complex motion primitives cause
+low flexibility and reusability and thus hinder representa-
+tion effectiveness. On the opposite, overly simple motion
+primitives challenge motion prediction due to the lack of
+semantics. Besides, the codebook size N and the feature
+dimension C determine the representation capability, which
+should be defined according to the complexity of the dataset
+and in cooperation with P and H. In our experiment, we
+set N = 256, P = 1, H = 8 or H = 16, and C = 64
+or C = 128 depending on the dataset, which lead to high-
+quality results as justified by the ablation studies in Sec-
+tion 4.5. More details can be found in the Supplement.
+3.2. Speech-Driven Motion Synthesis
+With the learned discrete motion prior, we can build a
+cross-modal mapping from the input speech to the target
+motion codes that could be further decoded into realistic
+facial motions. Along with the speech, we further adopt
+a control on the talking styles as input, i.e., a style vector
+s ∈ RM
++ ∪ {0}, where M is the dimension of the learned
+style space (see Eq. 4). Conditioning on the speech A1:T
+and the style vector s, a temporal autoregressive model,
+composed of a speech encoder Espeech and a cross-modal
+decoder Dcross-modal, is employed to learn over the facial mo-
+tion space, as depicted in Figure 2.
+Following FaceFormer [16], our speech encoder adopts
+the architecture of the state-of-the-art self-supervised pre-
+trained speech model, wav2vec 2.0 [3], which consists of
+an audio feature extractor and a multi-layer transformer en-
+coder. The audio feature extractor converts the speech of
+raw waveform into feature vectors through a temporal con-
+4
+
+volutions network (TCN). Benefiting from the effective at-
+tention scheme, the transformer encoder converts the audio
+features into contextualized speech representations. Apart
+from the pre-trained codebook and VQ-VAE decoder, our
+cross-modal decoder contains an embedding block and a
+multi-layer transformer decoder with causal self-attention.
+The embedding block combines the past facial motions and
+the style embedding via:
+F1:t−1
+emb
+= Pθ( ˆM1:t−1) + B ·
+s
+∥s∥1
+,
+(4)
+where Pθ
+is a linear projection layer,
+and B
+=
+[b1, ..., bM] ∈ RC×M denotes the M learnable basis vec-
+tors that span the style space linearly.
+Alike to Face-
+Former [16], we equip the transformer decoder with causal
+self-attention to learn the dependencies between each frame
+in the context of the past facial motion sequence, and with
+cross-modal attention to align the audio and motion modal-
+ities. The output features ˆZ1:t is further quantized into Z1:t
+q
+via Eq. 1 and decoded by the pre-trained VQ-VAE decoder.
+The newly predicted motion ˆmt is used to update the past
+motions as ˆM1:t, in preparation for the next prediction. For-
+mally, this recursive process can be written as:
+ˆmt = Dcross-modal(Espeech(A1:T ), s, ˆM1:t−1).
+(5)
+Training objectives.
+We train the transformer encoder,
+decoder and the embedding block for cross-modality map-
+ping, while keeping the codebook Z and motion decoder D
+frozen. To benefit from the speech representation learning
+from large-scale corpora, we initialize the TCN and trans-
+former encoder with the pre-trained wav2vec 2.0 weights.
+Overall, the autoregressive model is trained in a teaching-
+forcing scheme, under the constraint of two loss terms: (i)
+feature regularity loss Lreg measuring the deviation between
+the predicted motion feature ˆZ1:T and the quantized feature
+Z1:T
+q
+from codebook, and (ii) motion loss Lmotion measuring
+the difference between the predicted motions ˆM1:T and the
+ground-truth motions M1:T , which plays an important role
+to stabilize the training process. The final loss function is:
+Lsyn = Lreg + Lmotion
+= ∥ˆZ1:T − sg(Z1:T
+q )∥2
+2 + ∥ ˆM1:T − M1:T ∥2
+2.
+(6)
+3.3. Training Details
+At stage one, we train the VQ-VAE model (Figure 1)
+on a single NVIDIA V100 for 200 epochs (∼2 hours) with
+the AdamW [41] optimizer (β1 = 0.9, β2 = 0.999 and
+ϵ = 1e − 8), where the learning rate is initialized as 10−4,
+and the mini-batch size is set as 1.
+At stage two, we
+train the temporal autoregressive model with the Adam op-
+timizer [33]. The training duration is 150 epochs (∼5 hours)
+and other hyper-parameters remain unchanged as stage one.
+Style embedding space.
+The style embedding space is
+linearly spanned by M learned basis vectors, where each
+style is represented by a style vector s that serves as the lin-
+ear combination coefficients or a coordinate. During train-
+ing, we assign each speaker (e.g. no. i) with a standard unit
+vector ei as a style vector, under the assumption that each
+speaker is associated with a unique and consistent style.
+Anyway, arbitrary style vectors are allowed to interpolate
+new talking styles during inference.
+4. Experiments
+4.1. Datasets and Implementations
+We employ two widely used datasets, BIWI [18] and
+VOCASET [10], to train and test different methods in our
+experiments. Both datasets contain 4D face scans together
+with utterances spoken in English. BIWI contains 40 unique
+sentences shared across all speakers in the dataset, while
+VOCASET contains 255 unique sentences, which are par-
+tially shared among different speakers.
+BIWI dataset. BIWI is a 3D audio-visual corpus of af-
+fective speech and facial expression in the form of dense
+dynamic 3D face geometries, which is originally proposed
+to study affective communication among humans. There is
+a total of 40 sentences uttered by 14 subjects, eight females
+and six males. Each sentence was recorded twice: with and
+without emotion. On average, each sentence is 4.67 seconds
+long. The 3D face dynamics are captured at 25fps, each
+with 23370 vertices and registered topology. We follow the
+data splits from FaceFormer [16] and only use the emotional
+subset. Specifically, the training set (BIWI-Train) contains
+192 sentences, while the validation set (BIWI-Val) contains
+24 sentences. There are two testing sets, in which BIWI-
+Test-A includes 24 sentences spoken by six seen subjects
+and BIWI-Test-B contains 32 sentences spoken by eight un-
+seen subjects. BIWI-Test-A can be used for both quantita-
+tive and qualitative evaluation due to the seen subjects dur-
+ing training, while BIWI-Test-B is more suitable for quali-
+tative evaluation.
+VOCASET dataset. VOCASET is comprised of 480
+paired audio-visual sequences recorded from 12 subjects.
+The facial motion is captured at 60fps and is about 4 sec-
+onds long. Different from BIWI, each 3D face mesh is reg-
+istered to the FLAME [37] topology with 5023 vertices. We
+adopt the same training (VOCA-Train), validation (VOCA-
+Val), and testing (VOCA-Test) splits as VOCA [10] and
+FaceFormer for fair comparisons.
+Implementations.
+We compare our work with three
+state-of-the-art methods: VOCA [10], MeshTalk [51] and
+FaceFormer [16]. We train and test VOCA on BIWI us-
+ing the official codebase, while directly testing the released
+model that was trained on VOCASET. For MeshTalk, we
+train and test it using the official implementation on the two
+5
+
+Table 1. Quantitative evaluation on BIWI-Test-A. Lower means
+better for both metrics.
+Method
+Lip Vertex Error
+FDD
+(×10−4 mm)
+(×10−5 mm)
+VOCA
+6.5563
+8.1816
+MeshTalk
+5.9181
+5.1025
+FaceFormer
+5.3077
+4.6408
+CodeTalker (Ours)
+4.7914
+4.1170
+datasets. To compare with FaceFormer, we conduct test-
+ing directly using the pre-trained weights. Among the four
+methods, VOCA, FaceFormer and our CodeTalker require
+conditioning on a training speaking style during testing. For
+unseen subjects, we generate facial animations conditioned
+on all training styles. More details about the implementa-
+tion details can be found in the Supplementary Material.
+4.2. Quantitative Evaluation
+Following MeshTalk [51] and FaceFormer [16], we
+adopt the lip vertex error to measure the lip synchroniza-
+tion, which is the only publicly proposed metric for speech-
+driven facial animation evaluation, to our best knowledge.
+As a complement, we introduce a new quantitative measure-
+ment, i.e., upper-face motion statistics, to evaluate the over-
+all facial dynamics.
+Lip vertex error. It measures the lip deviation of a se-
+quence with respect to the ground truth, i.e., calculating the
+maximal L2 error of all lip vertices for each frame and takes
+the average over all frames.
+Upper-face dynamics deviation.
+The upper-face ex-
+pression is just loosely correlated with the speech, depend-
+ing on personal talking styles and the semantics of speech
+content. With this belief, we propose to measure the varia-
+tion of facial dynamics for a motion sequence in comparison
+with that of the ground truth. Specifically, the upper-face
+dynamics deviation (FDD) is calculated by:
+FDD(M1:T , ˆM1:T ) =
+�
+v∈SU (dyn(Mv
+1:T ) − dyn( ˆMv
+1:T ))
+|SU|
+,
+(7)
+where Mv
+1:T ∈ R3×T denotes the motions of the v-th ver-
+tex, and SU is the index set of upper-face vertices. dyn(·)
+denotes the standard deviation of the element-wise L2 norm
+along the temporal axis.
+We calculate the lip vertex error and upper-face dynam-
+ics deviation (FDD) over all sequences in BIWI-Test-A and
+take the average for comparison. According to Table 1, the
+proposed CodeTalker achieves lower error than the exist-
+ing state-of-the-arts, suggesting that it produces more accu-
+rate lip-synchronized movements. Besides, Table 1 shows
+that our CodeTalker achieves the best performance in terms
+of FDD. It indicates the high consistency between the pre-
+dicted upper-face expressions together with the trend of fa-
+cial dynamics (conditioned on the speech and talking styles)
+and those of the ground truth.
+4.3. Qualitative Evaluation
+We visually compare our method with other competitors
+in Figure 4. For fair comparison, we assign the same talking
+style to VOCA, FaceFormer and our CodeTalker as con-
+ditional input, which is sampled at random. To check the
+lip synchronization performance, we illustrate three typical
+frames of synthesized facial animations that speak at spe-
+cific syllables, as compared in the upper partition in Fig-
+ure 4. We can observe that compared with the competi-
+tors, the lip movements produced by our CodeTalker are
+more accurately articulated with the speech signals and also
+more consistent with those of the reference. For example,
+CodeTalker produces better lip sync with proper mouth clo-
+sures when pronouncing bilabial consonant /b/ (i.e., “bed-
+side” in the upper-right case of Figure 4), compared to
+VOCA and MeshTalk; for the even challenging speech parts
+“waterproof” and “shaving” that need to pout, CodeTalker
+can produce accurate lip shapes while other methods suffer
+from the over-smoothing problem and fail to lip-sync cor-
+rectly (Zoom in for better inspection).
+Different from lip movements, facial expressions only
+have weak correlations with the speech signal, which tends
+to be static in front of cross-modal mapping ambiguity. To
+visualize the facial motion dynamics, we calculate the tem-
+poral statistics of adjacent-frame facial motions within a
+sequence.
+Specifically, we first calculate the inter-frame
+motion L2 distance and then compute the mean and stan-
+dard deviation (std) across the sequence at each vertex.
+The higher mean value indicates stronger facial movements,
+while the higher std value suggests richer variations of facial
+dynamics. Two examples are visualized in the last two rows
+of Figure 4, evidencing that our method outperforms others
+in achieving both stronger facial movements and a broader
+range of dynamics. It is mainly attributed to the superior-
+ity of the discrete facial motion space, which promotes the
+robustness to cross-modal uncertainty effectively. Readers
+are recommended to watch the animation comparisons in
+the Supplemental Video.
+Talking style interpolation.
+Our model can synthe-
+size new speaking styles from the learned style embedding
+space. To inspect the effects on VOCA-Test, we select two
+speaking style vectors, i.e., ei and ej, which correspond to
+large and slight lip articulations respectively, and interpolate
+new talking style vectors snew = B · [ωei + (1 − ω)ej] with
+a linear coefficient ω. For the synthesized 3D animations of
+a sampled sequence, we plot the lower-upper lip distances
+across frames for each style in Figure 5, from which we
+observe the smooth transition of mouth amplitudes between
+the two typical styles. It is not only useful to synthesize new
+talking styles without additional constraints, but also prac-
+6
+
+“shaving”
+“diagnosis”
+“waterproof”
+“on”
+“sharing”
+“bedside”
+Mean
+Std
+0
+× 10−3 mm
+2.1
+Mean
+Std
+0
+× 10−3 mm
+7.4
+Reference
+VOCA
+MeshTalk
+FaceFormer
+Ours
+Reference
+VOCA
+MeshTalk FaceFormer
+Ours
+Figure 4. Visual comparisons of sampled facial motions animated by different methods on VOCA-Test (left) and BIWI-Test-B (right). The
+upper partition shows the facial animation conditioned on different speech parts, while the lower depicts the temporal statistics (mean and
+standard deviation) of adjacent-frame motion variations within a sequence.
+0.180
+0.182
+0.184
+0.186
+0.188
+0.190
+0.192
+0
+20
+40
+60
+80
+100
+Frame index
+Lip distance (mm)
+0               1
+Interpolation weight 𝜔:
+Figure 5. Distance between lower and upper lip for our predic-
+tions within a sequence conditioned on different weighted linear
+combinations of two style vectors.
+tical to match a specific speaking performance of an unseen
+subject during training.
+4.4. User Study
+The human perception system has been evolutionarily
+adapted to understanding subtle facial motions and captur-
+ing lip synchronization. Thus, it is still the most reliable
+measure in the speech-driven facial animation task. We con-
+duct a user study to evaluate the quality of animated faces in
+perceptual lip synchronization and realism, compared with
+VOCA, MeshTalk, FaceFormer and the ground truth. We
+adopt A/B tests for each comparison, i.e., ours vs. competi-
+tor, in terms of realistic facial animation and lip sync. For
+Table 2. User study results on BIWI-Test-B and VOCA-Test. We
+adopt A/B testing and report the percentage of answers where A is
+preferred over B.
+Competitors
+BIWI-Test-B
+VOCA-Test
+Lip Sync Realism Lip Sync Realism
+Ours vs. VOCA
+92.47
+89.25
+86.02
+84.95
+Ours vs. MeshTalk
+80.65
+82.80
+95.70
+92.47
+Ours vs. FaceFormer
+53.76
+56.99
+70.97
+69.89
+Ours vs. GT
+43.01
+49.46
+43.01
+43.01
+BIWI, we obtain the results of four kinds of comparisons by
+randomly selecting 30 samples from BIWI-Test-B, respec-
+tively. To achieve the most variations in terms of speaking
+styles, we ensure the sampling results can fairly cover all
+conditioning styles. Thus, 120 A vs. B pairs (30 samples
+× 4 comparisons) are created for BIWI-Test-B. Each pair is
+judged by at least 3 different participants separately, and fi-
+nally, 372 entries are collected in total. For the user study on
+VOCASET, we apply the same setting as that on the BIWI
+dataset, i.e., another 120 A vs. B pairs from VOCA-Test
+set, finally yielding 372 entries as well. In this study, 31
+participants with good vision and hearing ability complete
+the evaluation successfully. Moreover, each participant is
+involved in all 8 kinds of comparisons to make better expo-
+sure and cover the diversity of favorability.
+The percentage of A/B testing in terms of lip sync and re-
+7
+
+230ssTable 3. Ablation study on the representation space of codebook.
+The performance is measured by the reconstruction error (i.e., L2
+error) and lip vertex error on VOCA-Test and BIWI-Test-A.
+Variants
+VOCA-Test
+BIWI-Test-A
+Rec. Error
+Rec. Error
+Lip Vertex Error
+(×10−5 mm)
+(×10−5 mm)
+(×10−4 mm)
+Shape-ent. codebook
+2.75
+4.07
+6.41
+Motion codebook (Ours)
+0.08
+2.83
+4.79
+Reconstruction
+Speech-driven synthesis
+Ref./GT
+Shape-ent.
+codebook
+“finding Nan's cameo”
+“an excellent room”
+Motion 
+codebook
+(Ours)
+GT
+Shape-ent.
+codebook
+Motion 
+codebook
+(Ours)
+Figure 6. Visual comparisons of reconstruction and speech-driven
+motion synthesis results with different representation spaces on
+VOCA-Test (left) and BIWI-Test-A (right).
+alism on BIWI-Test-B is tabulated in Table 2, which shows
+that participants favor CodeTalker over competitors. Based
+on the visual analysis in Section 4.3, we attribute this to the
+facial animation synthesized by CodeTalker having more
+expressive facial motions, accurate lip shape, and well-
+synchronized mouth movements. For VOCA-Test, which
+has a nature of fewer upper-face motions, a similar favora-
+bility can still be observed in Table 2. We believe the rea-
+sons are at least three-fold: subtle motions around the eyes,
+more accurate lip movements and expressive motions in the
+lower face. Although be aware of a gap between our pre-
+dictions and the recorded performance (ground truth), we
+surprisingly get over 40% preference when compared to the
+ground truth. Overall, the user study justifies that the facial
+animations produced by CodeTalker have superior percep-
+tual quality.
+4.5. Ablation Studies
+We study several key designs of our proposed method
+in this section, including the representation space and the
+hyper-parameters of the codebook construction.
+Representation space. To study the superiority of our
+motion-based representation, we construct a baseline that
+learns a shape-entangled codebook, i.e., the codes repre-
+sent shapes instead of motions.
+As shown in Table 3,
+the baseline decreases the reconstruction accuracy signif-
+icantly, which is evidenced by the visualized examples in
+Figure 6. It is mainly because the shape-entangled sequence
+Face components 𝐻
+Temporal unit 𝑃
+(a) Reconstruction Error
+Face components 𝐻
+Temporal unit 𝑃
+(b) Lip Vertex Error
+mm
+mm
+Figure 7. Model performance comparisons with different hyper-
+parameters of the codebook, i.e., the length of temporal unit P and
+the number of face components H. We measure the reconstruction
+error and lip vertex error on BIWI-Test-A.
+contains more speaker-specific information that hinders the
+reusability of the codes. A direct weakness is the poor gen-
+eralization for self-reconstruction, which further impedes
+cross-modal mapping correctness.
+In contrast, our pro-
+posed speaker-agnostic motion representation is more ef-
+fective to represent generic motion priors shared across in-
+dividuals, and hence promotes the quality of both the self-
+reconstruction and the speech-driven motion synthesis.
+Codebook construction. We further study the hyper-
+parameters used for codebook construction. We evaluate
+the performance of different settings ⟨P, H⟩ by measuring
+their reconstruction accuracy and cross-modal mapping ac-
+curacy (namely lip vertex error).
+First, we evaluate the
+reconstruction accuracy as shown in Figure 7(a). On one
+hand, increasing P degrades the reconstruction accuracy,
+which could be explained by the increased complexity of
+the motion to be represented. On the other hand, increas-
+ing H eases the reconstruction but risks over-fitting, which
+explains the general benefits (H < 8) but inferior perfor-
+mance when H ≥ 8. Notably, a similar trend could be
+found in the cross-modal mapping performance, as shown
+in Figure 7(b). We conjecture that complex motion primi-
+tives cause lower reusability and higher redundancy, result-
+ing in ambiguity in the cross-modal code query process.
+5. Discussion and Conclusion
+We demonstrated the advantages of casting speech-
+driven facial animation as a code query task in the discrete
+space, which notably promotes the motion synthesis quality
+against cross-modal ambiguity. By comparing to the exist-
+ing state-of-the-arts, our proposed method shows superior-
+ity in achieving accurate lip sync and vivid facial expres-
+sions. However, we still follow the assumption that facial
+motions are independent of shapes, whose rationality may
+deserve further studies. Also, the overall perceptual qual-
+ity still lags behind the ground truth, mainly because of the
+limited paired audio-visual data. As a future work, it is in-
+teresting to guide the 3D facial animation by utilizing priors
+from large-scale available talking head videos.
+8
+
+32
+2.96
+3.84
+3.92
+16
+2.88
+3.84
+4.29
+8
+2.83
+3.13
+3.61
+4
+3.65
+3.72
+4.01
+1
+4.88
+5.40
+5.65
+232
+5.31
+5.48
+5.24
+16
+4.99
+5.14
+5.31
+8
+4.79
+4.97
+5.52
+4
+5.21
+4.90
+5.58
+1
+5.33
+5.95
+5.94
+2X10-55.8
+5.6
+5.4
+5.2
+5.0
+4.85.5
+5.0
+4.5
+4.0
+3.5
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+
+Appendix
+This supplemental document contains four sections:
+Section A shows implementation details of our CodeTalker;
+Section B presents more discussions on the proposed
+method; Section C presents details of the user study; and
+Section D presents short descriptions of the supplemental
+video. The source code and trained model will also be re-
+leased upon publication.
+A. Implementation Details
+A.1. Hyper-parameters of Codebook
+We have explored and discussed the important hyper-
+parameters of our motion codebook in Section 4.5 “Code-
+book construction” on the BIWI dataset in the main pa-
+per. Here we provide more specific parameters adopted for
+CodeTalker trained on the two datasets. For BIWI, we have
+the ground truth for quantitative evaluation on the testing
+set BIWI-Test-A to determine a group of parameters P = 1
+and H = 8 for high-quality results (i.e., Section 4.5 “Code-
+book construction” in the main paper). Additionally, we
+set the codebook item number N = 256 and the dimen-
+sion of items C = 128. Although more codebook items
+and dimensions could ease reconstruction, the redundant el-
+ements may cause ambiguity in speech-driven motion syn-
+thesis. Hence, we did not heavily tune these parameters
+and just empirically set them for good visual quality. For
+VOCASET, since there is no ground truth for us to obtain
+the quantitative results, we empirically select a group of pa-
+rameters (i.e., N = 256, P = 1, H = 16, C = 64), which
+could produce visually plausible facial animations in our
+experiments.
+A.2. Network Architecture
+To improve the reproducibility of our CodeTalker, we
+further illustrate the detailed network architectures for the
+facial motion space learning and the speech-driven motion
+synthesis (Section 3.1 and 3.2 in the main paper, respec-
+tively), which are shown Table 4.
+B. More Discussions on CodeTalker
+B.1. Instance Normalization in Self-reconstruction
+Learning
+Instance Normalization [58] (IN) has been widely used
+in the filed of style transfer [25,59], which is defined as:
+IN(x) = γ(x − µ(x)
+σ(x)
+) + β.
+(8)
+Different from BN [27] layers, here µ(x) and σ(x) are com-
+puted across temporal dimensions independently for each
+0.182
+0.183
+0.184
+0.185
+0.186
+0.187
+0.188
+0.189
+0.190
+0.191
+0
+20
+40
+60
+80
+100
+  Ours
+  Ours (w/o IN)
+  GT
+Lip distance (mm)
+Frame index
+(a)
+(b)
+0.183
+0.184
+0.185
+0.186
+0.187
+0.188
+0.189
+0.190
+0.191
+0.192
+0
+20
+40
+60
+80
+100
+  Ours
+  Ours (w/o IN)
+  Reference
+Lip distance (mm)
+Frame index
+Figure 8. Distance between lower and upper lip within a sampled
+sequence from VOCA-Test of (a) reconstruction and (b) speech-
+driven motion synthesis results produced by different variants.
+channel within each sample:
+µnc(x) = 1
+T
+T
+�
+t=1
+xnct
+(9)
+σnc(x) =
+�
+�
+�
+� 1
+T
+T
+�
+t=1
+(xnct − µnc(x))2 + ϵ
+(10)
+Interestingly, we empirically find that normalizing feature
+statistics (i.e., mean and variance) with IN (not BN due
+to small mini-batch size) can boost the performance of our
+CodeTalker in self-reconstruction learning, as shown in Ta-
+ble 5. In addition, it can also make self-reconstruction train-
+ing more stable. To better show the gain of normalization,
+we also visualize the lip distance of a sampled sequence
+of reconstruction results from VOCA-Test in Figure 8(a).
+The visualization result indicates that the predicted lip am-
+plitudes are closer to those of the ground truth by equip-
+ping with IN, while the ablated variant (i.e., Ours (w/o IN))
+cannot reconstruct lip movements with accurate amplitudes.
+The speech-driven facial motion synthesis (stage two) can
+also benefit from the facial motion codebook learned in self-
+reconstruction with IN, as shown in Figure 8(b). Note that
+we synthesize facial motions conditioned on a randomly
+sampled speaking style. We conjecture that facial motions
+with different magnitudes could be well encapsulated into
+12
+
+Table 4. Parameter illustration of network architectures. C(k,s,p,n) denotes a 1D Convolutional layer with kernel size k, stride size s,
+padding size p, and output channels of n. Tenc(d1,d2,h,l) denotes a transformer encoder layer with basic channel number of d1, forward
+channel number of d2, self-attention head number of h, and layer number l, while similarly, Tdec represents a transformer decoder layer.
+L(n) denotes a linear layer with output channels of n. CA[·] stands for the additional cross-attention input for transformer decoders.
+lCM = 12 for BIWI, while lCM = 6 for VOCASET. n · T stands for the interpolated audio feature length in order to align with visual
+frames, where n = 2 for BIWI and n = 1 for VOCASET. ‘+’ denotes the channel-wise addition. “Drop” means the dropout operation.
+Stage
+Module
+Input → Output
+Layer Operation
+I
+Encoder
+M(T, V, 3) → M(T, V · 3)
+Reshape
+M(T, V · 3) → Z1
+e(T, 1024)
+L(1024) → LReLU → C(5,1,2,1024) → LReLU → IN
+Z1
+e(T, 1024) → Z2
+e(T, H · C)
+L(1024) → Tenc(1024,1536,8,6) → L(H · C)
+Z2
+e(T, H · C) → Zq(T, H, C)
+Reshape → Quantize
+Decoder
+Zq(T, H, C) → Zq(T, H · C)
+Reshape
+Zq(T, H · C) → Z1
+d(T, 1024)
+L(1024) → C(5,1,2,1024) → LReLU → IN
+Z1
+d(T, 1024) → ˆ
+M(T, V · 3)
+L(1024) → Tenc(1024,1536,8,6) → L(V · 3)
+II
+A(T, d) → F1
+e(T ′, 512)
+C(10,5,0,512) → GN → GeLU → C(3,2,0,512) → GN → GeLU
+→ C(3,2,0,512) → GN → GeLU → C(3,2,0,512) → GN → GeLU
+Speech
+→ C(3,2,0,512) → GN → GeLU → C(3,2,0,512) → GN → GeLU
+Encoder
+→ C(2,2,0,512) → GN → GeLU → C(2,2,0,512) → GN → GeLU
+F1
+e(T ′, 512) → F2
+e(n · T, 768)
+Interpolate → LN → L(768) → Drop
+F2
+e(n · T, 768) → F3
+e(n · T, 1024)
+Tenc(768,3072,12,12) → L(1024)
+ˆ
+Mpast(T, V · 3) → Fpast
+emb(T, 1024)
+L(1024) → +StyleVector
+Fpast
+emb(T, 1024) → ˆZ1
+d(T, 1024)
+Tdec(1024,2048,4,lCM) with CA[F3
+e] → L(H · C)
+Cross-modal
+ˆZ1
+d(T, H · C) → ˆZq(T, H, C)
+Reshape → Quantize
+Decoder
+ˆZq(T, H, C) → ˆZq(T, H · C)
+Reshape
+ˆZq(T, H · C) → ˆZ2
+d(T, 1024)
+L(1024) → C(5,1,2,1024) → LReLU → IN
+ˆZ2
+d(T, 1024) → ˆ
+M(T, V · 3)
+L(1024) → Tenc(1024,1536,8,6) → L(V · 3)
+Table 5. Ablation study on the Instance Normalization (IN) for
+self-reconstruction learning. The performance is measured by the
+reconstruction error on VOCA-Test and BIWI-Test-A.
+Variants
+Reconstruction Error
+VOCA-Test (×10−5 mm)
+BIWI-Test-A (×10−5 mm)
+Ours (w/o IN)
+0.12
+3.27
+Ours
+0.08
+2.83
+the discrete motion prior by normalizing temporal elements
+within each channel. The rationality and effect of IN de-
+serve further studies as our potential direction.
+B.2. Alternative Data Flow and Supervision
+As we have summarized in Section 2.2 of the main paper,
+recent works explore the power of discrete prior learning in
+a large variety of tasks, among which most existing Vector
+Quantization (VQ)-based works [44, 70] adopt categorical
+cross-entropy (CE) loss to supervise their token predictions.
+Table 6. Comparison of lip-sync errors. We compare different
+methods on BIWI-Test-A. Lower means better. λ is the weighting
+factor.
+Method
+Lip Vertex Error (×10−4 mm)
+Alter. (Lce)
+9.6356
+Alter. (λLce+Lreg)
+5.1138
+Alter. (λLce+Lreg+Lmotion)
+5.0254
+CodeTalker (Ours)
+4.7914
+Hence, we also explore some alternative data flow and su-
+pervision frameworks as our cross-modal decoder, which is
+shown in Figure 9. It is worth noting that the style vector
+and audio features are omitted for simplicity.
+Different from our cross-modal decoder in the main pa-
+per, the alternative takes past motion code as input and then
+autoregressively predicts code sequences in form of n-way
+classification. The predicted code sequence then retrieves
+the respective code items from the learned codebook Z, and
+13
+
+Transformer
+decoder
+Codebook 𝒵
+Past motion code 
+𝐂1:𝑡−1
+Predicted motion 
+෡𝐌1:𝑡
+𝐙𝐪1:𝑡
+8
+5
+12
+68
+74
+54 𝑡
+𝑠
+10
+12
+9
+Predicted motion code
+𝐂𝑡
+Cross-modal decoder
+lookup
+Figure 9. Alternative data flow and supervision framework of our
+cross-modal decoder. Note that we omit the style vector and au-
+dio features input for simplicity. Given the past motion code as
+input, the alternative cross-modal decoder first autoregressively
+predict motion code and then decode them into motions with the
+pre-trained codebook and decoder.
+further produces facial motion sequences through the fixed
+decoder D. A CE loss is adopted to penalize error between
+the predicted code sequence ˆc ∈ {0, . . . , |N| − 1}T ′·H and
+the ground truth c generated by the pre-trained encoder E:
+Lce =
+T ′·H
+�
+i=0
+−ci log(ˆci).
+(11)
+We train the alternatives with the same settings as those in
+the main paper (Section 3.3). The lip-sync evaluation result
+is tabulated in Table 6. Alternative model with Lce alone
+cannot converge well due to the difficult cross-modality
+mapping of token prediction.
+While adding more con-
+straints (i.e., Lreg and Lmotion in the main paper Eq. 6) can
+ease the difficulty of token prediction learning, the per-
+formance is still limited with this token prediction frame-
+work. Overall, the lower average lip error achieved by our
+CodeTalker suggests its framework superiority in terms of
+the accuracy of lip movements.
+C. User Study
+The designed user study interface is shown in Figure 10.
+A user study is expected to be completed with 5–10 min-
+utes (24 video pairs × 5 seconds × 3 times watching).
+To remove the impact of random selection, we filter out
+those comparison results completed in less than two min-
+utes. For each participant, the user study interface shows
+24 video pairs and the participant is instructed to judge the
+videos twice with the following two questions, respectively:
+“Comparing the lips of two faces, which one is more in sync
+with the audio?” and “Comparing the two full faces, which
+one looks more realistic?”.
+D. Video Comparison
+To better evaluate the qualitative results produced by
+competitors [10, 16, 31, 51] and our CodeTalker, we pro-
+vide a supplemental video for demonstration and compar-
+ison. Specifically, we test our model using various audio
+clips, including the audio clips extracted from TED and
+TEDx videos, audio sequences from the VOCASET and
+BIWI datasets, and the speech from supplementary videos
+of previous methods.
+The video shows that CodeTalker
+can synthesize natural and plausible facial animations with
+well-synchronized lip movements. It is worth noting that,
+compared to the competitors (i.e., VOCA, MeshTalk and
+FaceFormer) suffering from the over-smoothing problem,
+our CodeTalker can produce more vivid and realistic facial
+motions and better lip sync. Besides, we also show the talk-
+ing style interpolation results and facial animations of talk-
+ing in different languages.
+14
+
+Figure 10. Designed user study interface. Each participant need to answer 24 video pairs and here only one video pair is shown due to the
+page limit.
+15
+
+Instructions:
+Please watch the short videos (duration ~5s) of two animated talking heads. You need to choose the talking head (a or
+b) that moves more naturally in terms of the full face and the lips (two guestions for each video). The total
+duration for this survey is about 5-10mins.
+Reminder: Please turn on the sound on your computer when watching.
+1.1 Comparing the lips of two faces, which one is more in sync with the audio?
+Oa
+Ob
+1.2 Comparing the two full faces, which one looks more realistic?
+a
+Ob
\ No newline at end of file
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@@ -0,0 +1,887 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf,len=886
+page_content='CodeTalker: Speech-Driven 3D Facial Animation with Discrete Motion Prior  Jinbo Xing* 1 Menghan Xia2 Yuechen Zhang1 Xiaodong Cun2 Jue Wang2 Tien-Tsin Wong1  1The Chinese University of Hong Kong  2Tencent AI Lab  Abstract  Speech-driven 3D facial animation has been widely stud-  ied, yet there is still a gap to achieving realism and vividness  due to the highly ill-posed nature and scarcity of audio-  visual data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' Existing works typically formulate the cross-  modal mapping into a regression task, which suffers from  the regression-to-mean problem leading to over-smoothed  facial motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' In this paper, we propose to cast speech-  driven facial animation as a code query task in a finite  proxy space of the learned codebook, which effectively pro-  motes the vividness of the generated motions by reducing  the cross-modal mapping uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  The codebook is  learned by self-reconstruction over real facial motions and  thus embedded with realistic facial motion priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' Over the  discrete motion space, a temporal autoregressive model is  employed to sequentially synthesize facial motions from the  input speech signal, which guarantees lip-sync as well as  plausible facial expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' We demonstrate that our ap-  proach outperforms current state-of-the-art methods both  qualitatively and quantitatively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' Also, a user study further  justifies our superiority in perceptual quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' The source  code will be publicly available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' Introduction  3D facial animation has been an active research topic for  decades, as attributed to its broad applications in virtual re-  ality, film production, and games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' The high correlation be-  tween speech and facial gestures (especially lip movements)  makes it possible to drive the facial animation with a speech  signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' Early attempts are mainly made to build the complex  mapping rules between phonemes and their visual counter-  part, which usually have limited performance [54,66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' With  the advances in deep learning, recent speech-driven facial  animation techniques push forward the state-of-the-art sig-  nificantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' However, it still remains challenging to generate  human-like motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  As an ill-posed problem, speech-driven facial animation  generally has multiple plausible outputs for every input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  Work done during an internship at Tencent AI Lab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  Such ambiguity tends to cause over-smoothed results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' Any-  how, person-specific approaches [31, 50] can usually ob-  tain decent facial motions because of the relatively consis-  tent talking style, but have low scalability to general ap-  plications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' Recently, VOCA [10] extends these methods  to generalize across different identities, however, they gen-  erally exhibit mild or static upper face expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' This  is because VOCA formulates the speech-to-motion map-  ping as a regression task, which encourages averaged mo-  tions, especially in the upper face that is only weakly or  even uncorrelated to the speech signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' To reduce the un-  certainty, FaceFormer [16] utilizes long-term audio context  through a transformer-based model and synthesizes the se-  quential motions in an autoregressive manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' Although  it gains important performance promotion, it still inherits  the weakness of one-to-one mapping formulation and suf-  fers from a lack of subtle high-frequency motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  Dif-  ferently, MeshTalk [51] models a categorical latent space  for facial animation that disentangles audio-correlated and  audio-uncorrelated information so that both aspects could  be well handled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' Anyway, the categorical latent space is  represented by some proxy numbers instead of an explicit  codebook with feature vectors, which makes the training  tricky and hence impedes its performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  We get inspiration from 3D Face Morphable Model  (3DMM) [37], where general facial expressions are rep-  resented in a low-dimensional space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  Accordingly, we  propose to formulate speech-driven facial animation as a  code query task in a finite proxy space of the learned dis-  crete codebook prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  The codebook is learned by self-  reconstruction over real facial motions using a vector-  quantized autoencoder (VQ-VAE) [60], which along with  the decoder stores the realistic facial motion priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  In  contrast to the continuous linear space of 3DMM, com-  binations of codebook items form a discrete prior space  with only finite cardinality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' Still, in the context of the de-  coder, the code representation possesses high expressive-  ness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' Through mapping the speech to the finite proxy space,  the uncertainty of the speech-to-motion mapping is signif-  icantly attenuated and hence promotes the quality of mo-  tion synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' Conceptually, the proxy space approximates  the facial motion space, where the learned codebook items  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='02379v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='CV]  6 Jan 2023  serve as discrete motion primitives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  Based on the learned discrete codebook, we pro-  pose a code-query-based temporal autoregressive model  for speech-conditioned facial motion synthesis, called  CodeTalker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' Specifically, taking a speech signal as input,  our model predicts the motion feature tokens in a temporal  causal manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' Then, the feature tokens are used to query  the code sequence in the discrete space, followed by facial  motion reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' Thanks to the contextual modeling  over history motions and cross-modal alignment, the pro-  posed CodeTalker shows the advantages of achieving accu-  rate lip motions and natural expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' Extensive experi-  ments show that the proposed CodeTalker demonstrates su-  perior performance on existing datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' Systematic studies  and experiments are conducted to demonstrate the merits of  our method over previous works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' The contributions of our  work are as follows:  We make the first attempt to model the facial mo-  tion space with discrete primitives, which offers ad-  vantages to promote motion synthesis realism against  cross-modal uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  We propose a discrete motion prior based temporal au-  toregressive model for speech-driven facial animation,  which outperforms existing state-of-the-art methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  We will make our code fully available, including the  data processing, metrics, trained models, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=', to stan-  dardize the speech-driven 3D facial animation task for  follow-ups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' Related Works  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' Speech-driven 3D Facial Animation  Computer facial animation is a long-standing task [45]  and has attracted rapidly increased interest over the past  decades [5, 20, 32, 34, 36, 55, 65, 72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' As a branch, speech-  driven facial animation is to reenact a person in sync with  input speech sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' While extensive literature in this  field works on 2D talking heads [1,7–9,11,23,28,29,38,39,  48, 52, 62, 64, 68, 69], we focus on facial animation on 3D  models in this work, which can be roughly categorized into  linguistics-based and learning-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  Linguistics-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  Typically, linguistics-  based methods [12, 42, 54, 66] establish a set of complex  mapping rules between phonemes and their visual counter-  parts, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=', visemes [19,35,43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' For example, the dominance  function [42] is to determine the influence of phonemes on  the respective facial animation control parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' Xu et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' [66] defines animation curves for a constructed canonical  set of visemes to generate synchronized mouth movements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  There are also some methods considering the many-to-  many mapping between phonemes and visemes, as demon-  strated in the dynamic visemes model [54] and, more re-  cently, the JALI [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' Based on psycholinguistic consid-  erations and built upon the Facial Action Coding System  (FACS) [13], JALI factors mouth movements into lip and  jaw rig animation and generate compelling co-articulation  results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' Although these methods have explicit control over  the animation, they have complex procedures and lack a  principled way to animate the entire face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  Learning-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' Learning-based methods [6,  10,16,17,24,31,40,47,53,63] resort to a data-driven frame-  work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' [6] achieve emotional lip sync by the  proposed constrained search and Anime Graph structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  Recently, Taylor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' [53] propose a deep-learning-based  model utilizing a sliding window approach on the tran-  scribed phoneme sequences input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' Karras et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' [31] pro-  pose a convolution-based network with a learnable emotion  database to animate a speech-driven 3D mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' More re-  cently, VisemeNet [71] employ a three-stage Long Short-  Term Memory (LSTM) network to predict the animation  curve for a lower face lip model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  We review the most related works more concretely here  as they have the same setting as this work, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=', training  on high-resolution paired audio-mesh data and speaker-  independently animating entire face meshes in vertex space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  MeshTalk [51] successfully disentangles audio-correlated  and uncorrelated facial information with a categorical latent  space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' However, the latent space adopted is not optimal with  limited expressiveness, thus the animation quality is not sta-  ble when applied in a data-scarcity setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' VOCA [10]  employs powerful audio feature extraction models and can  generate facial animation with different speaking styles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  Furthermore, FaceFormer [16] considers long-term audio  context with transformer [61] rendering temporally stable  animations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' Despite the appealing animations, both suffer  from the over-smoothing problem, as they directly regress  the facial motion in the highly ill-posed audio-visual map-  ping with large uncertainty and ambiguity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' Discrete Prior Learning  In the last decades, discrete prior representation with  learned dictionaries has demonstrated its superiority in im-  age restoration tasks [14, 22, 30, 56, 57], since clear im-  age details are well-preserved in the dictionaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  This  line of techniques further inspires the high-capacity and  high-compressed discrete prior learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' VQ-VAE [60] first  presents to learn discrete representations (codebook) of im-  ages and autoregressively model their distribution for image  synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' The follow-up works, VQ-VAE2 [49] and VQ-  GAN [15] further improve the quality of high-resolution  image synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' Recently, discrete prior learning and its  variants have been exploited for image colorization [26],  inpainting [46], blind face restoration [70], text-to-image  synthesis [21], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  2  In addition to the image modality, most recent works  also explore the power of discrete prior learning in tasks  with other modalities, such as dyadic face motion genera-  tion [44], co-speech gesture synthesis [2], speech enhance-  ment [67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' Inspired by codebook learning, this work inves-  tigates to learn discrete motion prior for speech-driven 3D  facial animation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' Different from [44], we exploit the dis-  crete motion primitives for facial motion representation in a  context-rich manner, which is more effective to learn gen-  eral priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' Method  We aim to synthesize sequential 3D facial motions from  a speech signal, so that any neutral face mesh could be an-  imated as a lip-synchronized talking face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' However, this is  an ill-posed problem since one speech could be matched by  multiple potential facial animations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' Such ambiguity tends  to make cross-modal learning suffer from averaged motions  and lack of subtle variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' To bypass this barrier, we pro-  pose to first model the facial motion space with the learned  discrete motion prior, and then learn a speech-conditioned  temporal autoregressive model over this space, which pro-  motes robustness against the cross-modal uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  Formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  Let M1:T = (m1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=', mT ) be a sequence  of facial motions, where each frame mt ∈ RV ×3 denotes  the 3D movement of V vertices over a neutral-face mesh  template h ∈ RV ×3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' Let further A1:T = (a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=', aT ) be  a sequence of speech snippets, each of which at ∈ Rd has  d samples to align with the corresponding (visual) frame  mt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' Then, our goal is to sequentially synthesize M1:T from  A1:T so that an arbitrary neutral facial template f could be  animated as H1:T = {m1 + h, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=', mT + h}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' Discrete Facial Motion Space  Visual realistic facial animations should present accu-  rate lip motions and natural expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' To achieve this  from speech signals, extra motion priors are required to re-  duce the uncertainty and complement realistic motion com-  ponents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  As witnessed by the recent image restoration  task [70], discrete codebook prior [60] demonstrates ad-  vantages in guaranteeing high-fidelity results even from a  severely degraded input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' Inspired by this, we propose to  model the facial motion space as a discrete codebook by  learning from tracked real-world facial motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  Codebook of motion primitives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  We manage to learn a  codebook Z = {zk ∈ RC}N  k=1 that allows any facial mo-  tion mt to be represented by a group of allocated items  {zk}k∈S, where S denotes the selected index set through  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' Conceptually, the codebook items serve as the motion  primitives of a facial motion space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' To this end, we pre-train  a transformer-based VQ-VAE that consists of an encoder  E, a decoder D, and a context-rich codebook Z, under the  𝑡  𝑠  𝐙q  0  1  …  N  Codebook 𝒵  Facial motion space  \u0de0𝐙  arg min 𝑧𝑖∈𝒵 ||\u0de0𝐙(𝑡,𝑠) − 𝒛𝑖||  Reconstructed motions \u0de1𝐌1:𝑇  Facial motions 𝐌1:𝑇  Predicted  Motion  Feature ෝ𝑚1:𝑡  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' Learning framework of facial motion space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' The learned  motion primitives, as embedded in the codebook, serve to repre-  sent the facial motions in a spatial and temporal manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  self-reconstruction of realistic facial motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' As shown in  Figure 1, the facial motions M1:T is first embedded as a  temporal feature ˆZ = E(M1:T ) ∈ RT ′×H×C, where H is  the number of face components and T ′ denotes the num-  ber of encoded temporal units (P =  T  T ′ frames).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' Then, we  obtain the quantized motion sequence Zq ∈ RT ′×H×C via  an element-wise quantization function Q(·) that maps each  item in ˆZ to its nearest entry in codebook Z:  Zq = Q(ˆZ) := arg min  zk∈Z  ∥ˆzt − zk∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  (1)  Then, the self-reconstruction is given by:  ˆM1:T = D(Zq) = D(Q(E(M1:T ))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  (2)  Note that, the discrete facial motion space reduces the map-  ping ambiguity with the finite cardinality, but never sacri-  fices its expressiveness thanks to its context-rich represen-  tation as a latent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  Training objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  To supervise the quantized autoen-  coder training, we adopt a motion-level loss and two inter-  mediate code-level losses:  LVQ =∥M1:T − ˆM1:T ∥1  + ∥sg(ˆZ) − Zq∥2  2 + β∥ˆZ − sg(Zq)∥2  2,  (3)  where the first term is a reconstruction loss, the latter two  are adopted to update the codebook items by reducing the  distance between the codebook Z and embedded features  ˆZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  sg(·) stands for a stop-gradient operation and β is a  weighting factor controlling the update rate of the codebook  and encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' Since the quantization function (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' 1) is not  differentiable, the straight-through gradient estimator [4,60]  is employed to copy the gradients from the decoder input to  the encoder output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  Discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  Recently, Learn2Listen [44] has applied VQ-  VAE for facial expression synthesis in response to a given  talking head harnessing 2D monocular videos to obtain  3DMM coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  In addition to distinct applications,  3  Transformer  decoder  Speech 𝐀1:𝑇  TCN  Embedding  0  1  …  N  Codebook 𝒵  Style vector 𝒔  quantization  Past motions \u0de1𝐌1:𝑡−1  Predicted motion ෝ𝒎𝑡  Transformer  encoder  \u0de0𝐙1:𝑡  Speech encoder  Cross-modal decoder  𝐙𝐪1:𝑡  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' Diagram of our speech-driven motion synthesis model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' Given the speech A1:T and style vector s as input, the model learn  to recursively generate a sequence of facial motions by predicting the motion codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' As embedded with well-learned motion priors, the  pre-trained codebook and decoder are frozen during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  3dmm  coeffs  𝑡  𝑠  𝐙q  0  1  …  N  Codebook 𝒵  Facial motion space  \u0de0𝐙  arg min 𝑧𝑖∈𝒵 ||\u0de0𝐙(𝑡,𝑠) − 𝒛𝑖||  Reconstructed motions \u0de1𝐌1:𝑇  Facial motions 𝐌1:𝑇  Predicted  Motion  Feature ෝ𝑚1:𝑡  𝒵1  𝐸1  𝐷1  Speaker 1:  Speaker N:  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  𝐸𝑁  𝐷𝑁  𝐷𝑁  𝐸  𝐷  Generic  motion:  (a)  (b)  𝒛1  𝑖  𝒛𝑁  𝑖  𝒛𝑖, … , 𝒛𝑘  𝒵𝑁  𝒵  3dmm  coeffs  3dmm  coeffs  3dmm  coeffs  3dmm  𝐷1  3dmm  𝐷  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  Concept comparison with Learn2Listen [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  (Top)  The speaker-specific facial expression coefficient prior [44], in  which each code represents a sequence of facial expression co-  efficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' (Bottom) Our speaker-agnostic generic motion prior, in  which each code represents the motion primitive of face compo-  nents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' The blue dotted boxes indicate what information each code  may represent conceptually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  here we would like to emphasize our major differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  First, Learn2Listen constructs speaker-specific codebooks  while ours uses a generic codebook that is feasible to rep-  resent arbitrary facial motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' Since cross-character mo-  tions are absorbed, our codebook is naturally embedded  with more plentiful priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' Second, Learn2Listen utilizes  the codebook to represent common sequences of facial ex-  pressions by the way of 3DMM coefficients, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=', each code  represents a sequence (8 frames) of facial expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' Dif-  ferently, our codebook is formulated to represent the vertex-  based facial motion space, where the codes are embedded  with per-vertex motions of facial components and repre-  sent the facial motion (within a temporal unit) in a context-  rich manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' As compared in Figure 3, the codebook of  Learn2Listen is learned to memorize typical sequential fa-  cial expressions of a specific speaker within 3DMM space,  which cannot synthesize realistic facial motions with sub-  tle details due to the limited expressiveness of 3DMM and  is bounded by the accuracy of 3D reconstruction tech-  niques [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' As the first attempt, our codebook is learned to  represent the generic facial motion space with motion prim-  itives for captured facial mesh data, which is more effective  to embed general priors preserving vivid facial details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  We further discuss the hyper-parameters of the code-  book.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  First, the length of the temporal unit P and the  number of face components H determine the complexity  of the motion primitives in temporal and spatial aspects  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' Generally, complex motion primitives cause  low flexibility and reusability and thus hinder representa-  tion effectiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' On the opposite, overly simple motion  primitives challenge motion prediction due to the lack of  semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' Besides, the codebook size N and the feature  dimension C determine the representation capability, which  should be defined according to the complexity of the dataset  and in cooperation with P and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' In our experiment, we  set N = 256, P = 1, H = 8 or H = 16, and C = 64  or C = 128 depending on the dataset, which lead to high-  quality results as justified by the ablation studies in Sec-  tion 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' More details can be found in the Supplement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' Speech-Driven Motion Synthesis  With the learned discrete motion prior, we can build a  cross-modal mapping from the input speech to the target  motion codes that could be further decoded into realistic  facial motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' Along with the speech, we further adopt  a control on the talking styles as input, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=', a style vector  s ∈ RM  + ∪ {0}, where M is the dimension of the learned  style space (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' Conditioning on the speech A1:T  and the style vector s, a temporal autoregressive model,  composed of a speech encoder Espeech and a cross-modal  decoder Dcross-modal, is employed to learn over the facial mo-  tion space, as depicted in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  Following FaceFormer [16], our speech encoder adopts  the architecture of the state-of-the-art self-supervised pre-  trained speech model, wav2vec 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='0 [3], which consists of  an audio feature extractor and a multi-layer transformer en-  coder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' The audio feature extractor converts the speech of  raw waveform into feature vectors through a temporal con-  4  volutions network (TCN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' Benefiting from the effective at-  tention scheme, the transformer encoder converts the audio  features into contextualized speech representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' Apart  from the pre-trained codebook and VQ-VAE decoder, our  cross-modal decoder contains an embedding block and a  multi-layer transformer decoder with causal self-attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  The embedding block combines the past facial motions and  the style embedding via:  F1:t−1  emb  = Pθ( ˆM1:t−1) + B ·  s  ∥s∥1  ,  (4)  where Pθ  is a linear projection layer,  and B  =  [b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=', bM] ∈ RC×M denotes the M learnable basis vec-  tors that span the style space linearly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  Alike to Face-  Former [16], we equip the transformer decoder with causal  self-attention to learn the dependencies between each frame  in the context of the past facial motion sequence, and with  cross-modal attention to align the audio and motion modal-  ities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' The output features ˆZ1:t is further quantized into Z1:t  q  via Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' 1 and decoded by the pre-trained VQ-VAE decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  The newly predicted motion ˆmt is used to update the past  motions as ˆM1:t, in preparation for the next prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' For-  mally, this recursive process can be written as:  ˆmt = Dcross-modal(Espeech(A1:T ), s, ˆM1:t−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  (5)  Training objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  We train the transformer encoder,  decoder and the embedding block for cross-modality map-  ping, while keeping the codebook Z and motion decoder D  frozen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' To benefit from the speech representation learning  from large-scale corpora, we initialize the TCN and trans-  former encoder with the pre-trained wav2vec 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='0 weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  Overall, the autoregressive model is trained in a teaching-  forcing scheme, under the constraint of two loss terms: (i)  feature regularity loss Lreg measuring the deviation between  the predicted motion feature ˆZ1:T and the quantized feature  Z1:T  q  from codebook, and (ii) motion loss Lmotion measuring  the difference between the predicted motions ˆM1:T and the  ground-truth motions M1:T , which plays an important role  to stabilize the training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' The final loss function is:  Lsyn = Lreg + Lmotion  = ∥ˆZ1:T − sg(Z1:T  q )∥2  2 + ∥ ˆM1:T − M1:T ∥2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  (6)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' Training Details  At stage one, we train the VQ-VAE model (Figure 1)  on a single NVIDIA V100 for 200 epochs (∼2 hours) with  the AdamW [41] optimizer (β1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='9, β2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='999 and  ϵ = 1e − 8), where the learning rate is initialized as 10−4,  and the mini-batch size is set as 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  At stage two, we  train the temporal autoregressive model with the Adam op-  timizer [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' The training duration is 150 epochs (∼5 hours)  and other hyper-parameters remain unchanged as stage one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  Style embedding space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  The style embedding space is  linearly spanned by M learned basis vectors, where each  style is represented by a style vector s that serves as the lin-  ear combination coefficients or a coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' During train-  ing, we assign each speaker (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' i) with a standard unit  vector ei as a style vector, under the assumption that each  speaker is associated with a unique and consistent style.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  Anyway, arbitrary style vectors are allowed to interpolate  new talking styles during inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' Experiments  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' Datasets and Implementations  We employ two widely used datasets, BIWI [18] and  VOCASET [10], to train and test different methods in our  experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' Both datasets contain 4D face scans together  with utterances spoken in English.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' BIWI contains 40 unique  sentences shared across all speakers in the dataset, while  VOCASET contains 255 unique sentences, which are par-  tially shared among different speakers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  BIWI dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' BIWI is a 3D audio-visual corpus of af-  fective speech and facial expression in the form of dense  dynamic 3D face geometries, which is originally proposed  to study affective communication among humans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' There is  a total of 40 sentences uttered by 14 subjects, eight females  and six males.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' Each sentence was recorded twice: with and  without emotion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' On average, each sentence is 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='67 seconds  long.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' The 3D face dynamics are captured at 25fps, each  with 23370 vertices and registered topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' We follow the  data splits from FaceFormer [16] and only use the emotional  subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' Specifically, the training set (BIWI-Train) contains  192 sentences, while the validation set (BIWI-Val) contains  24 sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' There are two testing sets, in which BIWI-  Test-A includes 24 sentences spoken by six seen subjects  and BIWI-Test-B contains 32 sentences spoken by eight un-  seen subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' BIWI-Test-A can be used for both quantita-  tive and qualitative evaluation due to the seen subjects dur-  ing training, while BIWI-Test-B is more suitable for quali-  tative evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  VOCASET dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' VOCASET is comprised of 480  paired audio-visual sequences recorded from 12 subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  The facial motion is captured at 60fps and is about 4 sec-  onds long.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' Different from BIWI, each 3D face mesh is reg-  istered to the FLAME [37] topology with 5023 vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' We  adopt the same training (VOCA-Train), validation (VOCA-  Val), and testing (VOCA-Test) splits as VOCA [10] and  FaceFormer for fair comparisons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  Implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  We compare our work with three  state-of-the-art methods: VOCA [10], MeshTalk [51] and  FaceFormer [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' We train and test VOCA on BIWI us-  ing the official codebase, while directly testing the released  model that was trained on VOCASET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' For MeshTalk, we  train and test it using the official implementation on the two  5  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' Quantitative evaluation on BIWI-Test-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' Lower means  better for both metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  Method  Lip Vertex Error  FDD  (×10−4 mm)  (×10−5 mm)  VOCA  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='5563  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='1816  MeshTalk  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='9181  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='1025  FaceFormer  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='3077  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='6408  CodeTalker (Ours)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='7914  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='1170  datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' To compare with FaceFormer, we conduct test-  ing directly using the pre-trained weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' Among the four  methods, VOCA, FaceFormer and our CodeTalker require  conditioning on a training speaking style during testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' For  unseen subjects, we generate facial animations conditioned  on all training styles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' More details about the implementa-  tion details can be found in the Supplementary Material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' Quantitative Evaluation  Following MeshTalk [51] and FaceFormer [16], we  adopt the lip vertex error to measure the lip synchroniza-  tion, which is the only publicly proposed metric for speech-  driven facial animation evaluation, to our best knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  As a complement, we introduce a new quantitative measure-  ment, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=', upper-face motion statistics, to evaluate the over-  all facial dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  Lip vertex error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' It measures the lip deviation of a se-  quence with respect to the ground truth, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=', calculating the  maximal L2 error of all lip vertices for each frame and takes  the average over all frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  Upper-face dynamics deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  The upper-face ex-  pression is just loosely correlated with the speech, depend-  ing on personal talking styles and the semantics of speech  content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' With this belief, we propose to measure the varia-  tion of facial dynamics for a motion sequence in comparison  with that of the ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' Specifically, the upper-face  dynamics deviation (FDD) is calculated by:  FDD(M1:T , ˆM1:T ) =  �  v∈SU (dyn(Mv  1:T ) − dyn( ˆMv  1:T ))  |SU|  ,  (7)  where Mv  1:T ∈ R3×T denotes the motions of the v-th ver-  tex, and SU is the index set of upper-face vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' dyn(·)  denotes the standard deviation of the element-wise L2 norm  along the temporal axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  We calculate the lip vertex error and upper-face dynam-  ics deviation (FDD) over all sequences in BIWI-Test-A and  take the average for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' According to Table 1, the  proposed CodeTalker achieves lower error than the exist-  ing state-of-the-arts, suggesting that it produces more accu-  rate lip-synchronized movements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' Besides, Table 1 shows  that our CodeTalker achieves the best performance in terms  of FDD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' It indicates the high consistency between the pre-  dicted upper-face expressions together with the trend of fa-  cial dynamics (conditioned on the speech and talking styles)  and those of the ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' Qualitative Evaluation  We visually compare our method with other competitors  in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' For fair comparison, we assign the same talking  style to VOCA, FaceFormer and our CodeTalker as con-  ditional input, which is sampled at random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' To check the  lip synchronization performance, we illustrate three typical  frames of synthesized facial animations that speak at spe-  cific syllables, as compared in the upper partition in Fig-  ure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' We can observe that compared with the competi-  tors, the lip movements produced by our CodeTalker are  more accurately articulated with the speech signals and also  more consistent with those of the reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' For example,  CodeTalker produces better lip sync with proper mouth clo-  sures when pronouncing bilabial consonant /b/ (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=', “bed-  side” in the upper-right case of Figure 4), compared to  VOCA and MeshTalk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' for the even challenging speech parts  “waterproof” and “shaving” that need to pout, CodeTalker  can produce accurate lip shapes while other methods suffer  from the over-smoothing problem and fail to lip-sync cor-  rectly (Zoom in for better inspection).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  Different from lip movements, facial expressions only  have weak correlations with the speech signal, which tends  to be static in front of cross-modal mapping ambiguity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' To  visualize the facial motion dynamics, we calculate the tem-  poral statistics of adjacent-frame facial motions within a  sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  Specifically, we first calculate the inter-frame  motion L2 distance and then compute the mean and stan-  dard deviation (std) across the sequence at each vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  The higher mean value indicates stronger facial movements,  while the higher std value suggests richer variations of facial  dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' Two examples are visualized in the last two rows  of Figure 4, evidencing that our method outperforms others  in achieving both stronger facial movements and a broader  range of dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' It is mainly attributed to the superior-  ity of the discrete facial motion space, which promotes the  robustness to cross-modal uncertainty effectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' Readers  are recommended to watch the animation comparisons in  the Supplemental Video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  Talking style interpolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  Our model can synthe-  size new speaking styles from the learned style embedding  space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' To inspect the effects on VOCA-Test, we select two  speaking style vectors, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=', ei and ej, which correspond to  large and slight lip articulations respectively, and interpolate  new talking style vectors snew = B · [ωei + (1 − ω)ej] with  a linear coefficient ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' For the synthesized 3D animations of  a sampled sequence, we plot the lower-upper lip distances  across frames for each style in Figure 5, from which we  observe the smooth transition of mouth amplitudes between  the two typical styles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' It is not only useful to synthesize new  talking styles without additional constraints, but also prac-  6  “shaving”  “diagnosis”  “waterproof”  “on”  “sharing”  “bedside”  Mean  Std  0  × 10−3 mm  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='1  Mean  Std  0  × 10−3 mm  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='4  Reference  VOCA  MeshTalk  FaceFormer  Ours  Reference  VOCA  MeshTalk FaceFormer  Ours  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' Visual comparisons of sampled facial motions animated by different methods on VOCA-Test (left) and BIWI-Test-B (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' The  upper partition shows the facial animation conditioned on different speech parts, while the lower depicts the temporal statistics (mean and  standard deviation) of adjacent-frame motion variations within a sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='180  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='182  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='184  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='186  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='188  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='190  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='192  0  20  40  60  80  100  Frame index  Lip distance (mm)  0               1  Interpolation weight 𝜔:  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' Distance between lower and upper lip for our predic-  tions within a sequence conditioned on different weighted linear  combinations of two style vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  tical to match a specific speaking performance of an unseen  subject during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' User Study  The human perception system has been evolutionarily  adapted to understanding subtle facial motions and captur-  ing lip synchronization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' Thus, it is still the most reliable  measure in the speech-driven facial animation task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' We con-  duct a user study to evaluate the quality of animated faces in  perceptual lip synchronization and realism, compared with  VOCA, MeshTalk, FaceFormer and the ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' We  adopt A/B tests for each comparison, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=', ours vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' competi-  tor, in terms of realistic facial animation and lip sync.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' For  Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' User study results on BIWI-Test-B and VOCA-Test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' We  adopt A/B testing and report the percentage of answers where A is  preferred over B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  Competitors  BIWI-Test-B  VOCA-Test  Lip Sync Realism Lip Sync Realism  Ours vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' VOCA  92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='47  89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='25  86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='02  84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='95  Ours vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' MeshTalk  80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='65  82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='80  95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='70  92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='47  Ours vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' FaceFormer  53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='76  56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='99  70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='97  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='89  Ours vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' GT  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='01  49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='46  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='01  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='01  BIWI, we obtain the results of four kinds of comparisons by  randomly selecting 30 samples from BIWI-Test-B, respec-  tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' To achieve the most variations in terms of speaking  styles, we ensure the sampling results can fairly cover all  conditioning styles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' Thus, 120 A vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' B pairs (30 samples  × 4 comparisons) are created for BIWI-Test-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' Each pair is  judged by at least 3 different participants separately, and fi-  nally, 372 entries are collected in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' For the user study on  VOCASET, we apply the same setting as that on the BIWI  dataset, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=', another 120 A vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' B pairs from VOCA-Test  set, finally yielding 372 entries as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' In this study, 31  participants with good vision and hearing ability complete  the evaluation successfully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' Moreover, each participant is  involved in all 8 kinds of comparisons to make better expo-  sure and cover the diversity of favorability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  The percentage of A/B testing in terms of lip sync and re-  7  230ssTable 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' Ablation study on the representation space of codebook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  The performance is measured by the reconstruction error (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=', L2  error) and lip vertex error on VOCA-Test and BIWI-Test-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  Variants  VOCA-Test  BIWI-Test-A  Rec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' Error  Rec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' Error  Lip Vertex Error  (×10−5 mm)  (×10−5 mm)  (×10−4 mm)  Shape-ent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' codebook  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='75  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='07  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='41  Motion codebook (Ours)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='08  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='83  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='79  Reconstruction  Speech-driven synthesis  Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='/GT  Shape-ent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content="  codebook  “finding Nan's cameo”  “an excellent room”  Motion  codebook  (Ours)  GT  Shape-ent." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  codebook  Motion  codebook  (Ours)  Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' Visual comparisons of reconstruction and speech-driven  motion synthesis results with different representation spaces on  VOCA-Test (left) and BIWI-Test-A (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  alism on BIWI-Test-B is tabulated in Table 2, which shows  that participants favor CodeTalker over competitors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' Based  on the visual analysis in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='3, we attribute this to the  facial animation synthesized by CodeTalker having more  expressive facial motions, accurate lip shape, and well-  synchronized mouth movements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' For VOCA-Test, which  has a nature of fewer upper-face motions, a similar favora-  bility can still be observed in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' We believe the rea-  sons are at least three-fold: subtle motions around the eyes,  more accurate lip movements and expressive motions in the  lower face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' Although be aware of a gap between our pre-  dictions and the recorded performance (ground truth), we  surprisingly get over 40% preference when compared to the  ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' Overall, the user study justifies that the facial  animations produced by CodeTalker have superior percep-  tual quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' Ablation Studies  We study several key designs of our proposed method  in this section, including the representation space and the  hyper-parameters of the codebook construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  Representation space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' To study the superiority of our  motion-based representation, we construct a baseline that  learns a shape-entangled codebook, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=', the codes repre-  sent shapes instead of motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  As shown in Table 3,  the baseline decreases the reconstruction accuracy signif-  icantly, which is evidenced by the visualized examples in  Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' It is mainly because the shape-entangled sequence  Face components 𝐻  Temporal unit 𝑃  (a) Reconstruction Error  Face components 𝐻  Temporal unit 𝑃  (b) Lip Vertex Error  mm  mm  Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' Model performance comparisons with different hyper-  parameters of the codebook, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=', the length of temporal unit P and  the number of face components H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' We measure the reconstruction  error and lip vertex error on BIWI-Test-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  contains more speaker-specific information that hinders the  reusability of the codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' A direct weakness is the poor gen-  eralization for self-reconstruction, which further impedes  cross-modal mapping correctness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  In contrast, our pro-  posed speaker-agnostic motion representation is more ef-  fective to represent generic motion priors shared across in-  dividuals, and hence promotes the quality of both the self-  reconstruction and the speech-driven motion synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  Codebook construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' We further study the hyper-  parameters used for codebook construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' We evaluate  the performance of different settings ⟨P, H⟩ by measuring  their reconstruction accuracy and cross-modal mapping ac-  curacy (namely lip vertex error).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  First, we evaluate the  reconstruction accuracy as shown in Figure 7(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' On one  hand, increasing P degrades the reconstruction accuracy,  which could be explained by the increased complexity of  the motion to be represented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' On the other hand, increas-  ing H eases the reconstruction but risks over-fitting, which  explains the general benefits (H < 8) but inferior perfor-  mance when H ≥ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' Notably, a similar trend could be  found in the cross-modal mapping performance, as shown  in Figure 7(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' We conjecture that complex motion primi-  tives cause lower reusability and higher redundancy, result-  ing in ambiguity in the cross-modal code query process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' Discussion and Conclusion  We demonstrated the advantages of casting speech-  driven facial animation as a code query task in the discrete  space, which notably promotes the motion synthesis quality  against cross-modal ambiguity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' By comparing to the exist-  ing state-of-the-arts, our proposed method shows superior-  ity in achieving accurate lip sync and vivid facial expres-  sions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' However, we still follow the assumption that facial  motions are independent of shapes, whose rationality may  deserve further studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' Also, the overall perceptual qual-  ity still lags behind the ground truth, mainly because of the  limited paired audio-visual data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' As a future work, it is in-  teresting to guide the 3D facial animation by utilizing priors  from large-scale available talking head videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  8  32  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
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+page_content=' 2  [65] Thibaut Weise, Sofien Bouaziz, Hao Li, and Mark Pauly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' Re-  altime performance-based facial animation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' ACM Transac-  tions on Graphics (TOG), 30(4):1–10, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' 2  [66] Yuyu Xu, Andrew W Feng, Stacy Marsella, and Ari Shapiro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  A practical and configurable lip sync method for games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' In  Proceedings of Motion on Games, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' 1, 2  [67] Karren Yang, Dejan Markovi´c, Steven Krenn, Vasu Agrawal,  and Alexander Richard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' Audio-visual speech codecs: Re-  thinking audio-visual speech enhancement by re-synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  In IEEE/CVF Conference on Computer Vision and Pattern  Recognition (CVPR), 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' 3  [68] Ran Yi, Zipeng Ye, Juyong Zhang, Hujun Bao, and Yong-  Jin Liu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  Audio-driven talking face video generation with  learning-based personalized head pose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  arXiv preprint  arXiv:2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='10137, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' 2  [69] Hang Zhou, Yasheng Sun, Wayne Wu, Chen Change Loy,  Xiaogang Wang, and Ziwei Liu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' Pose-controllable talking  face generation by implicitly modularized audio-visual rep-  resentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' In IEEE/CVF Conference on Computer Vision  and Pattern Recognition (CVPR), 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' 2  [70] Shangchen Zhou, Kelvin CK Chan, Chongyi Li, and  Chen Change Loy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  Towards robust blind face restoration  with codebook lookup transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' Conference on Neural  Information Processing Systems (NeurIPS), 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' 2, 3, 13  [71] Yang Zhou, Zhan Xu, Chris Landreth, Evangelos Kaloger-  akis, Subhransu Maji, and Karan Singh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' Visemenet: Audio-  driven animator-centric speech animation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  ACM Transac-  tions on Graphics (TOG), 37(4):1–10, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' 2  [72] Michael Zollh¨ofer, Justus Thies, Pablo Garrido, Derek  Bradley, Thabo Beeler, Patrick P´erez, Marc Stamminger,  Matthias Nießner, and Christian Theobalt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' State of the art on  monocular 3d face reconstruction, tracking, and applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  Computer Graphics Forum (CGF), 37(2):523–550, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' 2  11  Appendix  This supplemental document contains four sections:  Section A shows implementation details of our CodeTalker;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  Section B presents more discussions on the proposed  method;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' Section C presents details of the user study;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' and  Section D presents short descriptions of the supplemental  video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' The source code and trained model will also be re-  leased upon publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' Implementation Details  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' Hyper-parameters of Codebook  We have explored and discussed the important hyper-  parameters of our motion codebook in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='5 “Code-  book construction” on the BIWI dataset in the main pa-  per.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' Here we provide more specific parameters adopted for  CodeTalker trained on the two datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' For BIWI, we have  the ground truth for quantitative evaluation on the testing  set BIWI-Test-A to determine a group of parameters P = 1  and H = 8 for high-quality results (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=', Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='5 “Code-  book construction” in the main paper).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' Additionally, we  set the codebook item number N = 256 and the dimen-  sion of items C = 128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' Although more codebook items  and dimensions could ease reconstruction, the redundant el-  ements may cause ambiguity in speech-driven motion syn-  thesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' Hence, we did not heavily tune these parameters  and just empirically set them for good visual quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' For  VOCASET, since there is no ground truth for us to obtain  the quantitative results, we empirically select a group of pa-  rameters (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=', N = 256, P = 1, H = 16, C = 64), which  could produce visually plausible facial animations in our  experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' Network Architecture  To improve the reproducibility of our CodeTalker, we  further illustrate the detailed network architectures for the  facial motion space learning and the speech-driven motion  synthesis (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='2 in the main paper, respec-  tively), which are shown Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' More Discussions on CodeTalker  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' Instance Normalization in Self-reconstruction  Learning  Instance Normalization [58] (IN) has been widely used  in the filed of style transfer [25,59], which is defined as:  IN(x) = γ(x − µ(x)  σ(x)  ) + β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  (8)  Different from BN [27] layers, here µ(x) and σ(x) are com-  puted across temporal dimensions independently for each  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='182  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
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+page_content='184  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='185  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='186  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='187  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='188  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='189  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='190  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='191  0  20  40  60  80  100  Ours  Ours (w/o IN)  GT  Lip distance (mm)  Frame index  (a)  (b)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
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+page_content='185  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='186  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='187  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='188  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='189  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='190  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='191  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='192  0  20  40  60  80  100  Ours  Ours (w/o IN)  Reference  Lip distance (mm)  Frame index  Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' Distance between lower and upper lip within a sampled  sequence from VOCA-Test of (a) reconstruction and (b) speech-  driven motion synthesis results produced by different variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  channel within each sample:  µnc(x) = 1  T  T  �  t=1  xnct  (9)  σnc(x) =  �  �  �  � 1  T  T  �  t=1  (xnct − µnc(x))2 + ϵ  (10)  Interestingly, we empirically find that normalizing feature  statistics (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=', mean and variance) with IN (not BN due  to small mini-batch size) can boost the performance of our  CodeTalker in self-reconstruction learning, as shown in Ta-  ble 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' In addition, it can also make self-reconstruction train-  ing more stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' To better show the gain of normalization,  we also visualize the lip distance of a sampled sequence  of reconstruction results from VOCA-Test in Figure 8(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  The visualization result indicates that the predicted lip am-  plitudes are closer to those of the ground truth by equip-  ping with IN, while the ablated variant (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=', Ours (w/o IN))  cannot reconstruct lip movements with accurate amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  The speech-driven facial motion synthesis (stage two) can  also benefit from the facial motion codebook learned in self-  reconstruction with IN, as shown in Figure 8(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' Note that  we synthesize facial motions conditioned on a randomly  sampled speaking style.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' We conjecture that facial motions  with different magnitudes could be well encapsulated into  12  Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' Parameter illustration of network architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' C(k,s,p,n) denotes a 1D Convolutional layer with kernel size k, stride size s,  padding size p, and output channels of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' Tenc(d1,d2,h,l) denotes a transformer encoder layer with basic channel number of d1, forward  channel number of d2, self-attention head number of h, and layer number l, while similarly, Tdec represents a transformer decoder layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  L(n) denotes a linear layer with output channels of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' CA[·] stands for the additional cross-attention input for transformer decoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  lCM = 12 for BIWI, while lCM = 6 for VOCASET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' n · T stands for the interpolated audio feature length in order to align with visual  frames, where n = 2 for BIWI and n = 1 for VOCASET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' ‘+’ denotes the channel-wise addition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' “Drop” means the dropout operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  Stage  Module  Input → Output  Layer Operation  I  Encoder  M(T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' 3) → M(T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' V · 3)  Reshape  M(T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' V · 3) → Z1  e(T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' 1024)  L(1024) → LReLU → C(5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='1024) → LReLU → IN  Z1  e(T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' 1024) → Z2  e(T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' H · C)  L(1024) → Tenc(1024,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='1536,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='6) → L(H · C)  Z2  e(T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' H · C) → Zq(T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' H,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' C)  Reshape → Quantize  Decoder  Zq(T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' H,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' C) → Zq(T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' H · C)  Reshape  Zq(T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' H · C) → Z1  d(T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' 1024)  L(1024) → C(5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='1024) → LReLU → IN  Z1  d(T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' 1024) → ˆ  M(T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' V · 3)  L(1024) → Tenc(1024,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='1536,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='6) → L(V · 3)  II  A(T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' d) → F1  e(T ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' 512)  C(10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='512) → GN → GeLU → C(3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='512) → GN → GeLU  → C(3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='512) → GN → GeLU → C(3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='512) → GN → GeLU  Speech  → C(3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='512) → GN → GeLU → C(3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='512) → GN → GeLU  Encoder  → C(2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='512) → GN → GeLU → C(2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='512) → GN → GeLU  F1  e(T ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' 512) → F2  e(n · T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' 768)  Interpolate → LN → L(768) → Drop  F2  e(n · T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' 768) → F3  e(n · T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' 1024)  Tenc(768,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='3072,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='12) → L(1024)  ˆ  Mpast(T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' V · 3) → Fpast  emb(T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' 1024)  L(1024) → +StyleVector  Fpast  emb(T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' 1024) → ˆZ1  d(T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' 1024)  Tdec(1024,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='2048,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='lCM) with CA[F3  e] → L(H · C)  Cross-modal  ˆZ1  d(T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' H · C) → ˆZq(T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' H,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' C)  Reshape → Quantize  Decoder  ˆZq(T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' H,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' C) → ˆZq(T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' H · C)  Reshape  ˆZq(T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' H · C) → ˆZ2  d(T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' 1024)  L(1024) → C(5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='1024) → LReLU → IN  ˆZ2  d(T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' 1024) → ˆ  M(T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' V · 3)  L(1024) → Tenc(1024,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='1536,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='6) → L(V · 3)  Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' Ablation study on the Instance Normalization (IN) for  self-reconstruction learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' The performance is measured by the  reconstruction error on VOCA-Test and BIWI-Test-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  Variants  Reconstruction Error  VOCA-Test (×10−5 mm)  BIWI-Test-A (×10−5 mm)  Ours (w/o IN)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='12  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='27  Ours  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='08  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='83  the discrete motion prior by normalizing temporal elements  within each channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' The rationality and effect of IN de-  serve further studies as our potential direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' Alternative Data Flow and Supervision  As we have summarized in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='2 of the main paper,  recent works explore the power of discrete prior learning in  a large variety of tasks, among which most existing Vector  Quantization (VQ)-based works [44, 70] adopt categorical  cross-entropy (CE) loss to supervise their token predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' Comparison of lip-sync errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' We compare different  methods on BIWI-Test-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' Lower means better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' λ is the weighting  factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  Method  Lip Vertex Error (×10−4 mm)  Alter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' (Lce)  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='6356  Alter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' (λLce+Lreg)  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='1138  Alter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' (λLce+Lreg+Lmotion)  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='0254  CodeTalker (Ours)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='7914  Hence, we also explore some alternative data flow and su-  pervision frameworks as our cross-modal decoder, which is  shown in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' It is worth noting that the style vector  and audio features are omitted for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  Different from our cross-modal decoder in the main pa-  per, the alternative takes past motion code as input and then  autoregressively predicts code sequences in form of n-way  classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' The predicted code sequence then retrieves  the respective code items from the learned codebook Z, and  13  Transformer  decoder  Codebook 𝒵  Past motion code  𝐂1:𝑡−1  Predicted motion  \u0de1𝐌1:𝑡  𝐙𝐪1:𝑡  8  5  12  68  74  54 𝑡  𝑠  10  12  9  Predicted motion code  𝐂𝑡  Cross-modal decoder  lookup  Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' Alternative data flow and supervision framework of our  cross-modal decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' Note that we omit the style vector and au-  dio features input for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' Given the past motion code as  input, the alternative cross-modal decoder first autoregressively  predict motion code and then decode them into motions with the  pre-trained codebook and decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  further produces facial motion sequences through the fixed  decoder D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' A CE loss is adopted to penalize error between  the predicted code sequence ˆc ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' , |N| − 1}T ′·H and  the ground truth c generated by the pre-trained encoder E:  Lce =  T ′·H  �  i=0  −ci log(ˆci).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  (11)  We train the alternatives with the same settings as those in  the main paper (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' The lip-sync evaluation result  is tabulated in Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' Alternative model with Lce alone  cannot converge well due to the difficult cross-modality  mapping of token prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  While adding more con-  straints (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=', Lreg and Lmotion in the main paper Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' 6) can  ease the difficulty of token prediction learning, the per-  formance is still limited with this token prediction frame-  work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' Overall, the lower average lip error achieved by our  CodeTalker suggests its framework superiority in terms of  the accuracy of lip movements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' User Study  The designed user study interface is shown in Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  A user study is expected to be completed with 5–10 min-  utes (24 video pairs × 5 seconds × 3 times watching).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  To remove the impact of random selection, we filter out  those comparison results completed in less than two min-  utes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' For each participant, the user study interface shows  24 video pairs and the participant is instructed to judge the  videos twice with the following two questions, respectively:  “Comparing the lips of two faces, which one is more in sync  with the audio?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' and “Comparing the two full faces, which  one looks more realistic?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' Video Comparison  To better evaluate the qualitative results produced by  competitors [10, 16, 31, 51] and our CodeTalker, we pro-  vide a supplemental video for demonstration and compar-  ison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' Specifically, we test our model using various audio  clips, including the audio clips extracted from TED and  TEDx videos, audio sequences from the VOCASET and  BIWI datasets, and the speech from supplementary videos  of previous methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  The video shows that CodeTalker  can synthesize natural and plausible facial animations with  well-synchronized lip movements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' It is worth noting that,  compared to the competitors (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=', VOCA, MeshTalk and  FaceFormer) suffering from the over-smoothing problem,  our CodeTalker can produce more vivid and realistic facial  motions and better lip sync.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' Besides, we also show the talk-  ing style interpolation results and facial animations of talk-  ing in different languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  14  Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' Designed user study interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' Each participant need to answer 24 video pairs and here only one video pair is shown due to the  page limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  15  Instructions:  Please watch the short videos (duration ~5s) of two animated talking heads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' You need to choose the talking head (a or  b) that moves more naturally in terms of the full face and the lips (two guestions for each video).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content=' The total  duration for this survey is about 5-10mins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  Reminder: Please turn on the sound on your computer when watching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='1 Comparing the lips of two faces, which one is more in sync with the audio?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  Oa  Ob  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='2 Comparing the two full faces, which one looks more realistic?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
+page_content='  a  Ob' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfdgD3/content/2301.02379v1.pdf'}
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+arXiv:2301.03736v1  [math.AP]  10 Jan 2023
+HYPERBOLIC SYSTEMS OF QUASILINEAR EQUATIONS IN COMPRESSIBLE
+FLUID DYNAMICS WITH AN OBJECTIVE CATTANEO-TYPE EXTENSION FOR
+THE HEAT FLUX
+FELIPE ANGELES
+Abstract. We consider the coupling between the equations of motion of an inviscid compressible fluid in
+space with an objective Cattaneo-type extension for the heat flux. These equations are written in quasilinear
+form and we determine which of the given formulations for the heat flux allows for the hyperbolicity of the
+system. This feature is necessary for a physically acceptable sense of well-posedness for the Cauchy problem
+of such system of equations.
+1. Introduction
+One of the best known substitutes for Fourier’s law of heat conduction in continuum thermodynamics is
+the Maxwell-Cattaneo law [3]
+τqt + q = −κ∇θ,
+(1.1)
+where q is the heat flux, θ the temperature field and κ stands as the thermal conductivity. Although the
+Maxwell-Cattaneo law accounts for finite speed of heat conduction, this model is not Galilean invariant [5].
+By replacing the partial time derivative in (1.1) with a material derivative (see (6.1)) Christov and Jordan
+showed that a Galilean invariant formulation of the Maxwell-Cattaneo law is obtained, one that predicts the
+finite speed of propagation of heat [5]. However, they also showed that the heat flux in this model cannot
+be eliminated in order to obtain a single equation for the temperature field. In [4], Christov argues the
+importance of replacing the partial time derivative in (1.1) with an objective derivative and proposes
+τ [qt + v · ∇q − q · ∇v + (∇ · v)q] + q = −κ∇θ.
+(1.2)
+Moreover, Christov shows that, when this evolution equation is combined with the material invariant form of
+the balance of internal energy, it allows for the heat flux to be eliminated in order to obtain a single hyperbolic
+equation for the temperature. By considering the coupling between the local form of the conservation of
+mass (ρ), the balance of linear momentum (ρv) and the balance of total energy (E =
+1
+2ρ|u|2 + e) for a
+compressible inviscid fluid in space, that is,
+ρt + ∇ · (ρv)
+=
+0,
+(1.3)
+(ρv)t + ∇ · (ρv ⊗ v)
+=
+−∇ · p,
+(1.4)
+(ρE)t + ∇ · (ρEv)
+=
+−∇ · (pv) − ∇ · q,
+(1.5)
+together with (1.2), Straughan showed that, for a given wavenumber ξ0, there are values of q for which
+an acoustic wave propagates together with a thermal wave and completely determined the wavespeed for
+the purely thermal and purely mechanical cases [22]. However, it was recently shown that this system of
+equations, also known as the Cattaneo-Christov system, is not hyperbolic [1]. The notion of hyperbolicity
+Key words and phrases. Hyperbolic quasilinear systems, objective derivatives, hyperbolic heat conduction, Cattaneo-type
+extensions.
+1
+
+2
+F. ANGELES
+for a quasilinear systems of N equations is related with the possibility of finding N different (linearly
+independent) waves propagating in any spatial direction (cf. [7, Chapter III]). In [1, Section 5], it is shown
+that, for the Cattaneo-Christov system, we can always find particular directions ξ and values of q for which
+the N different waves doesn’t exist. In particular, since the characteristic speeds of this system are real,
+it can be classified as weakly hyperbolic. Although the Cauchy problem for weakly hyperbolic systems of
+equations can be well-posed in Grevey spaces, it is not a sufficient condition for the C∞-well-posedness (see
+[6] and the references therein). Moreover, it is well known that the hyperbolicity (cf. [2], [7], [20]) of a first
+order quasilinear system of equations is a necessary condition for the existence of L2-energy estimates (cf.
+[20, Theorem 3.1.2 and Lemma 3.1.3], [10] and [11]).
+For these reasons, the present work considers the coupling between (1.3)-(1.5) with
+τ
+�
+∂tq + v · ∇q − 1
+2
+�
+∇v − ∇v⊤�
+q + λ
+2
+�
+∇v + ∇v⊤�
+q + ν(∇ · v)q
+�
++ q = −κ∇θ,
+(1.6)
+In [17], Morro shows that (1.6) is objective for any pairs of invariant scalars λ, ν. Using an objective derivative
+for the heat flux and questioning if such formulation is compatible with thermodynamics is an issue that has
+been addressed in detail by Morro and Giorgi (see [15], [17], [16], [18] for example). In contrast, this work
+determines which of the frame indifferent formulations for the heat flux in (1.6), yields a hyperbolic system
+of quasilinear equations when it is coupled with (1.3)-(1.5). We show that there is only one heat flux model
+in (1.6) with such feature, namely, when (λ, ν) = (1, −1).
+2. Thermodynamical assumptions
+Throughout this paper we make the following thermodynamical and constitutive assumptions:
+(C1) The independent thermodynamical variables are the mass density field ρ(x, t) > 0 and the absolute
+temperature field θ(x, t) > 0. They vary within the domain D :=
+�
+(ρ, θ) ∈ R2 : ρ > 0, θ > 0
+�
+. The
+thermodynamic pressure p, the internal energy e and the thermal conductivity κ are given smooth
+functions of (ρ, θ) ∈ D.
+(C2) The fluid satisfies the following conditions p, pρ, pθ, eθ, κ > 0 for all (ρ, θ) ∈ D.
+In particular, assumption (C2) refers to compressible fluids satisfying the standard assumptions of Weyl
+[23]. Also, we assume that,
+(C3) λ and ν are real valued functions of (ρ, θ) ∈ D.
+3. Hyperbolic quasilinear systems of equations
+Let λ and ν be given functions of (ρ, θ) ∈ D. Consider the state variable U = (ρ, v, θ, q)⊤ ∈ O ⊂ RN, where
+O :=
+�
+(ρ, v, θ, q) ∈ RN : ρ > 0, θ > 0
+�
+, N = 2d + 2 and d is the spatial dimension. Then, the quasilinear
+form of (1.3)-(1.6) is,
+A0(U)Ut + Ai
+λν(U)∂iU + Q(U) = 0,
+(3.1)
+where repeated index notation has been used in the space derivatives ∂i and i = 1, .., d. Here, once U ∈ O
+is given, each coefficient A0(U), Ai(U) is a matrix of order N × N and Q(U) is a vector in RN. In the
+subsequence, we will refer to (3.1) as the induced (λ, ν)-quasilinear system of equations by the (λ, ν)-objective
+heat flux model (1.6). For ξ = (ξ1, .., ξd) ∈ Sd−1 and U ∈ O we define the symbol
+Aλν(ξ; U) :=
+3
+�
+i=1
+Ai
+λ,ν(U)ξi.
+(3.2)
+Let us recall the definition of hyperbolic quasilinear system of equations (cf. [2], [7], [20]).
+
+3
+Definition 3.1 (Hyperbolicity). A quasilinear system of the form (3.1) is called hyperbolic if for any fixed
+state U ∈ O and ξ ∈ Sd−1 the matrix A0(U) is non singular and the eigenvalue problem
+�
+Aλν(ξ; U) − ηA0(U)
+�
+V = 0
+(3.3)
+has real eigenvalues (η ∈ R) and N linearly independent eigenvectors. In particular, the eigenvalues of (3.3),
+also known as the characteristic speeds of (3.1), satisfy the equation
+det
+�
+Aλν(ξ; U) − ηA0�
+= 0,
+(3.4)
+for each U ∈ O and ξ ∈ Sd−1. Therefore, η = η(ξ; U). When d = 1 we simply write η = η(U).
+4. One dimensional system
+In the one dimensional case, the matrices defining system (3.1) are
+A0(U) :=
+
+
+
+
+1
+0
+0
+0
+0
+ρ
+0
+0
+0
+0
+ρeθ
+0
+0
+0
+0
+τ
+
+
+
+,
+A1(U) :=
+
+
+
+
+v
+ρ
+0
+0
+pρ
+ρv
+pθ
+0
+0
+θpθ
+ρveθ
+1
+0
+(λ + ν)q
+κ
+τv
+
+
+
+ ,
+Q(U) := (0, 0, 0, q)T .
+First, we seek the roots of (3.4). We use the formula,
+det
+� L
+M
+N
+P
+�
+= (det L) det
+�
+P − NL−1M
+�
+,
+(4.1)
+whenever L is invertible (see [24] for example). In this case,
+L =
+� v − η
+ρ
+Pρ
+ρ(v − η)
+�
+,
+M =
+� 0
+0
+pθ
+0
+�
+,
+N =
+� 0
+θpθ
+0
+(λ + ν)q
+�
+,
+P =
+� ρeθ(v − η)
+1
+κ
+τ(v − η)
+�
+,
+and det L = ρ(v − η)2 − ρpρ. Then, according with (4.1), (3.4) is equivalent to
+ρeθτ(v − η)4 −
+�
+ρeθpρτ + θp2
+θτ
+ρ
++ κ
+�
+(v − η)2 + (λ + ν)pθq
+ρ (v − η) + κpρ = 0.
+(4.2)
+If we set z := v − η, and multiply by (ρeθτ)−1 in (4.2), we obtain
+z4 + a2z2 + a1z + a0 = 0,
+(4.3)
+where
+a2 = −
+1
+ρeθτ
+�
+ρeθpρτ + θp2
+θτ
+ρ
++ κ
+�
+,
+a1 = (λ + ν)
+pθ
+ρ2eθτ q,
+a0 = κpρ
+ρeθτ .
+(4.4)
+First, we show that, if λ + ν ̸= 0, the one dimensional case of (3.1), is not a hyperbolic system of equations.
+We use the following result stated without a proof (see [8, Theorem 7] and also [19], for example).
+Lemma 1. A quartic equation of the form (4.3) with a0, a1, a2 real, a1 ̸= 0, and with discriminant ∆, has
+2 distinct real roots and 2 imaginary roots if ∆ < 0.
+□
+The discriminant, ∆ = ∆(a0, a1, a2), of (4.3) is given as
+∆(a0, a1, a2) = 256a3
+0 − 128a2
+2a2
+0 + 16a0a4
+2 + 144a0a2
+1a2 − 4a2
+1a3
+2 − 27a4
+1,
+(4.5)
+see [8, page 41] and [9, page 405] for example.
+
+4
+F. ANGELES
+Theorem 4.1. Let (ρ∗, θ∗) ∈ D be such that λ(ρ∗, θ∗) + ν(ρ∗, θ∗) ̸= 0. Then, there are values of q ∈ R for
+which the characteristic polynomial (4.2) has complex roots and thus, the (λ, ν)-quasilinear system (3.1) is
+not hyperbolic.
+Proof. Set γ∗ := λ(ρ∗, θ∗) + ν(ρ∗, θ∗). Observe that (4.5) can be rewritten as a fourth order polynomial in
+the variable a1, namely
+P(a1) := Aa4
+1 + Ba2
+1 + C,
+where A := −27, B := 144a0a2 − 4a3
+2 > 0, C := 16a0
+�
+a2
+2 − 4a0
+�2 ≥ 0 and, according with (4.4), B =
+B(ρ∗, θ∗), C = C(ρ∗, θ∗) and a1 = g(ρ∗, θ∗)q for g(ρ∗, θ∗) = γ∗
+pθ(ρ∗,θ∗)
+(ρ∗)2eθ(ρ∗,θ∗)τ ̸= 0. By taking,
+q2 > max
+�−C(ρ∗, θ∗) − B(ρ∗, θ∗)
+A(ρ∗, θ∗)g2(ρ∗, θ∗)
+,
+2
+g2(ρ∗, θ∗)
+�
+(4.6)
+and using that A < 0, it follows that Aa2
+1 + B < −C, and so
+P(a1) = a2
+1
+�
+Aa2
+1 + B
+�
++ C < −C.
+Hence, (4.5) is negative and according with Lemma 1, there are two complex roots of (4.3).
+Now, let
+v ∈ R, take q satisfying (4.6) and set U ∗ := (ρ∗, v, θ∗, q) ∈ O ⊂ R4. Then, by the previous argument, the
+characteristic polynomial (4.2) has complex roots at U ∗. This proves the result.
+□
+5. Three dimensional system
+Set d = 3 and observe that N = 8. In this case, A0(U) is a diagonal matrix given as
+A0(U) =
+
+
+
+
+1
+ρI3
+ρeθ
+τI3
+
+
+
+ ,
+(5.1)
+where I3 denotes the identity matrix of order 3 × 3 and all the empty spaces refer to zero block matrices of
+the appropriate sizes. We have that
+Aλ,ν(ξ; U) =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+ξ · v
+ξ1ρ
+ξ2ρ
+ξ3ρ
+0
+0
+0
+0
+ξ1pρ
+ρξ · v
+0
+0
+ξ1pθ
+0
+0
+0
+ξ2pρ
+0
+ρξ · v
+0
+ξ2pθ
+0
+0
+0
+ξ3pρ
+0
+0
+ρξ · v
+ξ3pθ
+0
+0
+0
+0
+ξ1θpθ
+ξ2θpθ
+ξ3θpθ
+ρeθξ · v
+ξ1
+ξ2
+ξ3
+0
+ξ1κ
+τξ · v
+0
+0
+0
+τQλ,ν(ξ; q)
+ξ2κ
+0
+τξ · v
+0
+0
+ξ3κ
+0
+0
+τξ · v
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+,
+(5.2)
+where, for each ξ ∈ S2, the sub-block matrix Qλ,ν(q; ξ) is of order 3 × 3 and given as
+Qλ,ν(q; ξ) =
+�
+�
+γξ1q1 + λ− (ξ2q2 + ξ3q3)
+λ+ξ1q2 + νξ2q1
+λ+ξ1q3 + νξ3q1
+λ+ξ2q1 + νξ1q2
+γξ2q2 + λ− (ξ1q1 + ξ3q3)
+λ+ξ2q3 + νξ3q2
+λ+ξ3q1 + νξ1q3
+λ+ξ3q2 + νξ2q3
+γξ3q3 + λ− (ξ1q1 + ξ2q2)
+�
+� ,
+(5.3)
+where γ := λ + ν, λ+ := λ
+2 + 1
+2 and λ− := λ
+2 − 1
+2, and
+Q(U) = (0, 0, 0, 0, 0, q1, q2, q3)⊤ .
+(5.4)
+By using formula (4.1), we can compute the characteristic polynomial for (3.1), yielding
+det
+�
+Aλν(ξ; U) − ηA0(ξ; U)
+�
+= ρ3τ 2(ξ · v − η)4Pλ,ν(ξ, U; η),
+(5.5)
+
+5
+where
+Pλ,ν(ξ, U; η) = ρeθτ(ξ · v − η)4 −
+�
+τρeθpρ + τθp2
+θ
+ρ
++ κ
+�
+(ξ · v − η)2 + pθ
+ρ hλ,ν(ξ; q)(ξ · v − η) + κpρ
+(5.6)
+and hλ,ν(ξ; q) := (Qλ,ν(ξ; q)ξ) · ξ.
+Lemma 2. Fix λ, ν ∈ R. The mapping, S2 × R3 ∋ (ξ; q) �→ Qλ,ν(ξ; q) ∈ M3×3, is non-trivial.
+Proof. Assume otherwise, that is, for all q ∈ R3 and all ξ ∈ S2, Qλ,ν(ξ; q) = 0. Take ξ = (1, 0, 0) and
+q = (q1, 0, 0) with q1 ̸= 0. Then, by (5.3), γq1 = 0 and λ−q1 = 0. This means that λ = 1 and ν = −1.
+On the other hand, take ξ = (1, 0, 0) and q = (0, q2, 0) with q2 ̸= 0. Then, (5.3) implies that νq2 = 0, a
+contradiction.
+□
+Lemma 3. Fix λ, ν ∈ R. The mapping, S2 × R3 ∋ (ξ, q) �→ hλ,ν(ξ; q) ∈ R, is null if and only if λ + ν = 0.
+Proof. First observe that for any q ∈ R3 and ξ ∈ S2 it holds that
+Qλ,ν(ξ; q)ξ · ξ = γ
+�
+ξ3
+1q1 + ξ3
+2q2 + ξ3
+3q3 + ξ2
+1ξ2q2 + ξ2
+1ξ3q3 + ξ1ξ2
+2q1 + ξ1ξ2
+3q1 + ξ3ξ2
+2q3 + ξ2ξ2
+3q2
+�
+,
+which yields the formula
+hλ,ν(q; ξ) = γ|ξ|2ξ · q
+∀
+q ∈ R3,
+ξ ∈ S2.
+(5.7)
+Then, if γ = λ + ν = 0, (5.7) implies that hλ,ν(q; ξ) = 0 for any q ∈ R3 and ξ ∈ S2. On the other hand,
+assume that hλ,ν(ξ; q) = 0 for all ξ ∈ S2 and q ∈ R3. If so, it must be true for q0 ∈ R3 and ξ0 ∈ S2 such that
+ξ0 · q0 ̸= 0. Then, by (5.7), we have that γ|ξ0|2ξ0 · q0 = 0, which implies that γ = λ + ν = 0.
+□
+Theorem 5.1. Set d = 3 and consider the (λ, ν)-quasilinear system defined by (5.1), (5.2), (5.3) and (5.4).
+If there is (ρ∗, θ∗) ∈ D such that γ∗ := λ(ρ∗, θ∗) + ν(ρ∗, θ∗) ̸= 0 then, the (λ, ν)-quasilinear system is not
+hyperbolic.
+Proof. For any q ∈ R3 \ {0} set ξq =
+q
+|q| ∈ S2. Then, by (5.7) and Lemma 3, hλ,ν(ξq; q) = γ|q| ̸= 0. Take
+v ∈ R3 and define the state U ∗ := (ρ∗, v, θ∗, q). Set p∗
+ρ = pρ(ρ∗, θ∗), p∗
+θ = pθ(ρ∗, θ∗), e∗
+θ = eθ(ρ∗, θ∗) and
+κ∗ = κ(ρ∗, θ∗). Consider Pλ,ν(ξq, U ∗; η) = 0, which according with (5.6) is equivalent to
+ρ∗e∗
+θτ(ξq · v − η)4 −
+�
+τρ∗e∗
+θp∗
+ρ + τθ∗(p∗
+θ)2
+ρ∗
++ κ∗
+�
+(ξq · v − η)2 + p∗
+θ
+ρ∗ γ∗|q|(ξq · v − η) + κ∗p∗
+ρ = 0.
+(5.8)
+Notice that, (5.8) is the same as the polynomial given in (4.2) but with |q| and v · ξq replacing q and v,
+respectively. Thus, by (4.6), with |q|2 instead of q2, it holds that (5.8) has complex roots. Therefore, the
+(λ, ν)-quasilinear system is not hyperbolic if λ + ν ̸= 0 on D.
+□
+6. The Cattaneo-Christov-Jordan system
+Consider the coupling between (1.3)-(1.5) and the material formulation of the Cattaneo heat flux model,
+namely,
+τ (qt + u · ∇q) + q = −κ∇θ,
+(6.1)
+
+6
+F. ANGELES
+This system can be written in the quasilinear form (3.1) where for each U ∈ O, A0(U) and Q(U) are given
+by (5.1) and (5.4) respectively, however, instead of Aλ,ν(ξ; U) we have the following symbol
+A(ξ; U) =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+ξ · v
+ξ1ρ
+ξ2ρ
+ξ3ρ
+0
+0
+0
+0
+ξ1pρ
+ρξ · v
+0
+0
+ξ1pθ
+0
+0
+0
+ξ2pρ
+0
+ρξ · v
+0
+ξ2pθ
+0
+0
+0
+ξ3pρ
+0
+0
+ρξ · v
+ξ3pθ
+0
+0
+0
+0
+ξ1θpθ
+ξ2θpθ
+ξ3θpθ
+ρeθξ · v
+ξ1
+ξ2
+ξ3
+0
+ξ1κ
+τξ · v
+0
+0
+0
+O3×3
+ξ2κ
+0
+τξ · v
+0
+0
+ξ3κ
+0
+0
+τξ · v
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+,
+(6.2)
+where, O3×3 denotes the zero matrix of order 3 × 3. This system was first proposed by Christov and Jordan
+[5]. They showed that (6.1) is Galilean invariant. We refer to the quasilinear system defined by (5.1), (5.4)
+and (6.2) as the Cattaneo-Christov-Jordan system. Morro points out that (6.1) is not objective and therefore
+for no value of λ and ν this equation can be deduced from (1.6). This coincides with Lemma 2. Nonetheless,
+formally, we can understand the Cattaneo-Christov-Jordan system as a quasilinear system with O3×3 instead
+of Qλ,ν(ξ; q). The characteristic polynomial of the Cattaneo-Christov-Jordan system has the form
+det
+�
+Aλν(ξ; U) − ηA0(ξ; U)
+�
+= ρ3τ 2(ξ · v − η)4P0(ξ, U; η),
+(6.3)
+where
+P0(ξ, U; η) = ρeθτ(ξ · v − η)4 −
+�
+τρeθpρ + τθp2
+θ
+ρ
++ κ
+�
+(ξ · v − η)2 + κpρ.
+(6.4)
+The characteristic roots of the three dimensional Cattaneo-Christov-Jordan system are real and given by
+η0(ξ; U) = ξ · v,
+η1(ξ; U) = ξ · v + 1
+√
+2
+�
+r(ρ, θ) + m(ρ, θ),
+η2(ξ; U) = ξ · v + 1
+√
+2
+�
+r(ρ, θ) − m(ρ, θ),
+(6.5)
+η3(ξ; U) = ξ · v − 1
+√
+2
+�
+r(ρ, θ) + m(ρ, θ),
+η4(ξ; U) = ξ · v − 1
+√
+2
+�
+r(ρ, θ) − m(ρ, θ),
+where, for each ρ, θ > 0 we have set
+r(ρ, θ) :=
+�
+pρ + θp2
+θ
+ρ2eθ
++
+κ
+ρeθτ
+�
+and
+m(ρ, θ) :=
+��
+pρ + θp2
+θ
+ρ2eθ
++
+κ
+ρeθτ
+�2
+− 4pρκ
+ρeθτ .
+Observe that, by (6.3), η0(ξ; U) is a root of algebraic multiplicity four, and it is easy to see that η3 < η4 <
+η2 < η1 (see [1, Sections 3 and 4.1]). Moreover, this system is Friedrichs symmetrizable [1, Section 4], which
+in turn implies that it is hyperbolic and thus, its Cauchy problem is locally well-posed in L2 (see, [12], [13]
+and [14], for example).
+Theorem 6.1. Consider a (λ, ν)-quasilinear system for which λ + ν = 0 on D. Then, the characteristic
+speeds of such system are real and coincide with the characteristic speeds of the Cattaneo-Christov-Jordan
+system given in (6.5).
+Proof. Since λ + ν = 0, Lemma 3 assures that hλ,ν(ξ; q) = 0 for all ξ ∈ S2 and q ∈ R3. Therefore,
+Pλ,ν(ξ, U; η) = P0(ξ, U; η)
+for all
+ξ ∈ S2, U ∈ R8, η ∈ R,
+and the result follows by comparing (5.5) and (6.3).
+□
+Remark 1. As it was pointed out in [1], a quasilinear system of the form (3.1), with real characteristic
+speeds, is not necessarily hyperbolic. Such is the case of the Cattaneo-Christov system, since in this case,
+(λ, ν) = (−1, 1) and so λ + ν = 0 (see [1, Theorem 3.5]).
+
+7
+Theorem 6.2. Let assumptions (C1)-(C3) be satisfied. Consider a (λ, ν)-quasilinear system (3.1) with the
+property that λ + ν = 0 on D. This system is hyperbolic if and only if (λ, ν) = (1, −1).
+Proof. Assume that λ + ν = 0 on D. According with the definition 3.1, we have to show that the only
+values of (λ, ν) for which the eigenvalue problem (3.3) has a complete set of eigenvectors are (λ, ν) = (−1, 1).
+Since λ+ ν = 0, Theorem 6.1 assures that the eigenvalues of the (λ, ν)-quasilinear system are given by (6.5).
+Let {Vj}4
+j=1 be the eigenvectors associated with the eigenvalues {ηj}4
+j=1, respectively. By (C1)-(C2), these
+eigenvalues are different and so, the set of eigenvectors {Vj}4
+j=1 is linearly independent. Since η0(ξ; U) is
+different from the other roots, the (λ, ν)-quasilinear system will be hyperbolic if and only if, for every ξ ∈ S2
+and U ∈ O, the geometric multiplicity of η0(ξ; U) equals four. Let η = η0 and A(ξ; U) = Aλ,ν(ξ; U) in (3.3)
+and set V = (V1, V2, .., V8)⊤ to obtain the algebraic system of equations,
+ξ · V ′ = 0,
+(6.6)
+pρV1 + pθV5 = 0,
+(6.7)
+ξ · V ′′ = 0,
+(6.8)
+τQλ,ν(q; ξ)V ′ + κV5ξ = 0,
+(6.9)
+where V ′ = (V2, V3, V4)⊤ and V ′′ = (V6, V7, V8)⊤. From (6.8), it follows the existence of exactly two linearly
+independent solutions, say (V 1
+6 , V 1
+7 , V 1
+8 )⊤ and (V 2
+6 , V 2
+7 , V 2
+7 )⊤. Then, we can take the vectors
+(0, 0, 0, 0, 0, V i
+6, V i
+7 , V i
+8 )⊤,
+for i = 1, 2,
+(6.10)
+as two linearly independent solutions of (6.6)-(6.9). Therefore, in order to comply with the hyperbolicity,
+equations (6.6), (6.7) and (6.9), must have two more linearly independent solutions for each ξ ∈ S2 and
+U = (ρ, v, θ, q) ∈ O. Observe that, for each ξ ∈ S2, (6.6) implies that V ′ ∈ {ξ}⊥, a two dimensional space.
+Hence, if {Vξ, Wξ} is a basis of {ξ}⊥, V ′ is of the form
+V ′ = α1Vξ + α2Wξ,
+for some α1, α2 ∈ R.
+(6.11)
+If we multiply (6.9) by ξ and use (6.11) we obtain that
+V5 = −τ
+κ {α1 (Qλ,ν(q; ξ)Vξ) · ξ + α2 (Qλ,ν(q; ξ)Wξ) · ξ}
+for each ξ ∈ S2.
+(6.12)
+Let ξ = (ξ1, ξ2, ξ3) ∈ S2 and assume that ξ1 ̸= 0. Then, the vectors
+Vξ = (−ξ2, ξ1, 0)⊤
+and
+Wξ = (−ξ3, 0, ξ1)⊤
+are linearly independent and form a basis of {ξ}⊥. Hence, by (6.12)
+V5 = −τ
+κλ+ {α1 (ξ1q2 − ξ2q1) + α2 (ξ1q3 − ξ3q1)}
+for any q = (q1, q2, q3) ∈ R3.
+(6.13)
+By using (6.11) and (6.13) into (6.9) we obtain the equations
+α1
+�
+−λ−(ξ2
+2q2 + ξ2ξ3q3) + (ν + λ+)ξ1ξ2q1
+�
++ α2
+�
+−λ−(ξ2ξ3q2 + ξ2
+3q3) + (ν + λ+)ξ1ξ3q1
+�
+= 0,
+(6.14)
+α1
+�
+−(ν + λ+)ξ1ξ2q2 + λ−(ξ2
+1q1 + ξ1ξ3q3)
+�
+= 0,
+(6.15)
+α2
+�
+−(ν + λ+)ξ1ξ3q3 + λ−(ξ2
+1q1 + ξ1ξ2q2)
+�
+= 0,
+(6.16)
+valid for any ξ ∈ S2 with ξ1 ̸= 0. Take ξ = (ξ1, ξ2, ξ3) ∈ S2 such that ξi ̸= 0 for all i = 1, 2, 3 and q ∈ R3
+with the property that ξ ·q ̸= 0. Assume the existence of (ρ′, θ′) ∈ D such that (λ, ν)(ρ′, θ′) ̸= (1, −1). Then,
+(ν + λ+) (ρ′, θ′) = −λ−(ρ′, θ′) ̸= 0 and (6.14)-(6.16) can be rewritten as
+α1ξ2
+�
+ξ · q
+�
+= −α2ξ3
+�
+ξ · q
+�
+,
+α1ξ1
+�
+ξ · q
+�
+= 0,
+α2ξ1
+�
+ξ · q
+�
+= 0.
+
+8
+F. ANGELES
+By the particular choices of ξ and q, it follows that α1 = α2 = 0 and thus, V ′ = 0. This implies that
+V1 = V5 = 0. Therefore, if (λ, ν) ̸= (1, −1) we can find wavenumbers (ξ) and states (U = (ρ′, v, θ′, q)⊤ ∈ O)
+such that the geometric multiplicity of the eigenvalue η0(ξ; U) equals two.
+This means that the (λ, ν)-
+quasilinear system is not hyperbolic if (λ, ν) ̸= (1, −1).
+Now assume that (λ, ν) = (1, −1) on D. If ξ = (ξ1, ξ2, ξ3) ∈ S2 is such that ξ1 ̸= 0, V5 is given by (6.13) and
+(6.9) is equivalent to equations (6.14)-(6.16). Moreover, since λ+ = 1 and λ− = 0, equations (6.14)-(6.16)
+are trivially satisfied for any choice of α1, α2 ∈ R. In particular, this means that Vξ and Wξ are linearly
+independent solutions of equations (6.6) and (6.9), where V5 is given by (6.13). Consequently, the vectors
+V(ξ) =
+�
+−pθ
+pρ
+V5, Vξ, V5, 0, 0, 0
+�⊤
+,
+W(ξ) =
+�
+−pθ
+pρ
+V5, Wξ, V5, 0, 0, 0
+�⊤
+,
+(6.17)
+are solutions of equations (6.6)-(6.9) and together with (6.10) form a linearly independent set. Therefore,
+the geometric multiplicity of η0(ξ; U) equals four, for any ξ ∈ S2 with ξ1 ̸= 0 and U ∈ O. If, on the other
+hand, ξ ∈ S2 is such that ξ2 ̸= 0, we take the linearly independent vectors,
+Vξ = (ξ2, −ξ1, 0)⊤
+and
+Wξ = (0, −ξ3, ξ2)⊤ ,
+as solutions of (6.6). In this case,
+V5 = −τ
+κ {α1 (ξ2q1 − ξ1q2) + α2 (ξ2q3 − ξ3q2)}
+for any q = (q1, q2, q3) ∈ R3.
+By setting V(ξ) and W(ξ) as in (6.17), the conclusion follows. Finally, if ξ = (0, 0, 1) take the canonical
+vectors ˆe1 and ˆe2 as solutions of (6.9), that is, Vξ = ˆe1 and Wξ = ˆe2. Then, V5 = − τ
+κ (α1q1 + α2q2) and we
+proceed as before. Therefore, if (λ, ν) = (1, −1), the geometric multiplicity of the eigenvalue η0(ξ, U) equals
+four for any choice of ξ ∈ S2 and U ∈ O and thus, the (1, −1)-quasilinear system is hyperbolic.
+□
+7. Final comments and conclusions
+In this work we have shown that the only case in which the coupling between (1.3)-(1.5) and the frame
+indifferent formulation for the heat flux (1.6) yields a hyperbolic system of equations is when we take (λ, ν)
+as the constant value functions (1, −1) on D. In particular, this implies that for a constant state Vc ∈ O,
+the Cauchy problem for the linear equation
+A0(Vc)Ut + Ai
+1,−1(Vc)∂iU + Q(Vc) = 0,
+is well posed in L2, as consequence of the existence of energy estimates [20, Theorem 3.1.2]. Moreover, if
+Vc ∈ O is taken as constant or constant outside a compact set and since the characteristic speeds of this
+model (see Theorem 6.1) have constant algebraic multiplicity with respect to ξ ∈ S2, the finite speed of
+propagation holds true [2, Theorem 2.11]. Now, for the general (1, −1)-quasilinear system (3.1), although
+its hyperbolicity is necessary for the existence of L2 energy estimates, it is not sufficient [21]. Typically,
+when the quasilinear system (3.1) is derived from a system of conservation laws, the existence of a convex
+entropy is equivalent to the existence of a symmetrizer (as the Hessian of the entropy) (cf. [7], [20]). Given
+the non-conservative structure of the (1, −1)-heat flux model, it is unknown for the author if a Friedrichs
+symmetrizer exists for this quasilinear system of equations in more than one space dimensions. Due to the
+presence of the skew matrix Q1,−1(ξ; q), a diagonal symmetrizer as in the Cattaneo-Christov-Jordan system,
+doesn’t work for this case when d = 3.
+By proceeding as in [4, Section 3.1], it is easy to show that the (1, −1)-heat flux model is irreducible in several
+space dimensions, in the sense that, together with the material invariant form of the internal energy equation,
+it is impossible to derive a single equation for the temperature field θ. Observe that the one dimensional
+version of this model coincides with the one dimensional Cattaneo-Christov-Jordan system. Therefore, it is
+locally well-posed in L2 and satisfies the finite speed of propagation property (see [12] and [2, Theorem 2.11].
+
+9
+Acknowledgments
+Thanks to Angelo Morro for clarifying some aspects of objective rates. This work was supported by
+CONACyT (Mexico) through a postdoctoral fellowship under grant A1-S-10457.
+Bibliography
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+Email address: teojkd@ciencias.unam.mx
+
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+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='03736v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='AP]  10 Jan 2023  HYPERBOLIC SYSTEMS OF QUASILINEAR EQUATIONS IN COMPRESSIBLE  FLUID DYNAMICS WITH AN OBJECTIVE CATTANEO-TYPE EXTENSION FOR  THE HEAT FLUX  FELIPE ANGELES  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' We consider the coupling between the equations of motion of an inviscid compressible fluid in  space with an objective Cattaneo-type extension for the heat flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' These equations are written in quasilinear  form and we determine which of the given formulations for the heat flux allows for the hyperbolicity of the  system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' This feature is necessary for a physically acceptable sense of well-posedness for the Cauchy problem  of such system of equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' Introduction  One of the best known substitutes for Fourier’s law of heat conduction in continuum thermodynamics is  the Maxwell-Cattaneo law [3]  τqt + q = −κ∇θ,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='1)  where q is the heat flux, θ the temperature field and κ stands as the thermal conductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' Although the  Maxwell-Cattaneo law accounts for finite speed of heat conduction, this model is not Galilean invariant [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='  By replacing the partial time derivative in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='1) with a material derivative (see (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='1)) Christov and Jordan  showed that a Galilean invariant formulation of the Maxwell-Cattaneo law is obtained, one that predicts the  finite speed of propagation of heat [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' However, they also showed that the heat flux in this model cannot  be eliminated in order to obtain a single equation for the temperature field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' In [4], Christov argues the  importance of replacing the partial time derivative in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='1) with an objective derivative and proposes  τ [qt + v · ∇q − q · ∇v + (∇ · v)q] + q = −κ∇θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='2)  Moreover, Christov shows that, when this evolution equation is combined with the material invariant form of  the balance of internal energy, it allows for the heat flux to be eliminated in order to obtain a single hyperbolic  equation for the temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' By considering the coupling between the local form of the conservation of  mass (ρ), the balance of linear momentum (ρv) and the balance of total energy (E =  1  2ρ|u|2 + e) for a  compressible inviscid fluid in space, that is,  ρt + ∇ · (ρv)  =  0,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='3)  (ρv)t + ∇ · (ρv ⊗ v)  =  −∇ · p,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='4)  (ρE)t + ∇ · (ρEv)  =  −∇ · (pv) − ∇ · q,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='5)  together with (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='2), Straughan showed that, for a given wavenumber ξ0, there are values of q for which  an acoustic wave propagates together with a thermal wave and completely determined the wavespeed for  the purely thermal and purely mechanical cases [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' However, it was recently shown that this system of  equations, also known as the Cattaneo-Christov system, is not hyperbolic [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' The notion of hyperbolicity  Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' Hyperbolic quasilinear systems, objective derivatives, hyperbolic heat conduction, Cattaneo-type  extensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='  1  2  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' ANGELES  for a quasilinear systems of N equations is related with the possibility of finding N different (linearly  independent) waves propagating in any spatial direction (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' [7, Chapter III]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' In [1, Section 5], it is shown  that, for the Cattaneo-Christov system, we can always find particular directions ξ and values of q for which  the N different waves doesn’t exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' In particular, since the characteristic speeds of this system are real,  it can be classified as weakly hyperbolic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' Although the Cauchy problem for weakly hyperbolic systems of  equations can be well-posed in Grevey spaces, it is not a sufficient condition for the C∞-well-posedness (see  [6] and the references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' Moreover, it is well known that the hyperbolicity (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' [2], [7], [20]) of a first  order quasilinear system of equations is a necessary condition for the existence of L2-energy estimates (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='  [20, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='2 and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='3], [10] and [11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='  For these reasons, the present work considers the coupling between (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='3)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='5) with  τ  �  ∂tq + v · ∇q − 1  2  �  ∇v − ∇v⊤�  q + λ  2  �  ∇v + ∇v⊤�  q + ν(∇ · v)q  �  + q = −κ∇θ,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='6)  In [17], Morro shows that (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='6) is objective for any pairs of invariant scalars λ, ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' Using an objective derivative  for the heat flux and questioning if such formulation is compatible with thermodynamics is an issue that has  been addressed in detail by Morro and Giorgi (see [15], [17], [16], [18] for example).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' In contrast, this work  determines which of the frame indifferent formulations for the heat flux in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='6), yields a hyperbolic system  of quasilinear equations when it is coupled with (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='3)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' We show that there is only one heat flux model  in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='6) with such feature, namely, when (λ, ν) = (1, −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' Thermodynamical assumptions  Throughout this paper we make the following thermodynamical and constitutive assumptions:  (C1) The independent thermodynamical variables are the mass density field ρ(x, t) > 0 and the absolute  temperature field θ(x, t) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' They vary within the domain D :=  �  (ρ, θ) ∈ R2 : ρ > 0, θ > 0  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' The  thermodynamic pressure p, the internal energy e and the thermal conductivity κ are given smooth  functions of (ρ, θ) ∈ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='  (C2) The fluid satisfies the following conditions p, pρ, pθ, eθ, κ > 0 for all (ρ, θ) ∈ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='  In particular, assumption (C2) refers to compressible fluids satisfying the standard assumptions of Weyl  [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' Also, we assume that,  (C3) λ and ν are real valued functions of (ρ, θ) ∈ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' Hyperbolic quasilinear systems of equations  Let λ and ν be given functions of (ρ, θ) ∈ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' Consider the state variable U = (ρ, v, θ, q)⊤ ∈ O ⊂ RN, where  O :=  �  (ρ, v, θ, q) ∈ RN : ρ > 0, θ > 0  �  , N = 2d + 2 and d is the spatial dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' Then, the quasilinear  form of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='3)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='6) is,  A0(U)Ut + Ai  λν(U)∂iU + Q(U) = 0,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='1)  where repeated index notation has been used in the space derivatives ∂i and i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='., d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' Here, once U ∈ O  is given, each coefficient A0(U), Ai(U) is a matrix of order N × N and Q(U) is a vector in RN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' In the  subsequence, we will refer to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='1) as the induced (λ, ν)-quasilinear system of equations by the (λ, ν)-objective  heat flux model (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' For ξ = (ξ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='., ξd) ∈ Sd−1 and U ∈ O we define the symbol  Aλν(ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' U) :=  3  �  i=1  Ai  λ,ν(U)ξi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='2)  Let us recall the definition of hyperbolic quasilinear system of equations (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' [2], [7], [20]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='  3  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='1 (Hyperbolicity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' A quasilinear system of the form (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='1) is called hyperbolic if for any fixed  state U ∈ O and ξ ∈ Sd−1 the matrix A0(U) is non singular and the eigenvalue problem  �  Aλν(ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' U) − ηA0(U)  �  V = 0  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='3)  has real eigenvalues (η ∈ R) and N linearly independent eigenvectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' In particular, the eigenvalues of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='3),  also known as the characteristic speeds of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='1), satisfy the equation  det  �  Aλν(ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' U) − ηA0�  = 0,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='4)  for each U ∈ O and ξ ∈ Sd−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' Therefore, η = η(ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' When d = 1 we simply write η = η(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' One dimensional system  In the one dimensional case, the matrices defining system (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='1) are  A0(U) :=  \uf8eb  \uf8ec  \uf8ec  \uf8ed  1  0  0  0  0  ρ  0  0  0  0  ρeθ  0  0  0  0  τ  \uf8f6  \uf8f7  \uf8f7  \uf8f8,  A1(U) :=  \uf8eb  \uf8ec  \uf8ec  \uf8ed  v  ρ  0  0  pρ  ρv  pθ  0  0  θpθ  ρveθ  1  0  (λ + ν)q  κ  τv  \uf8f6  \uf8f7  \uf8f7  \uf8f8 ,  Q(U) := (0, 0, 0, q)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='  First, we seek the roots of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' We use the formula,  det  � L  M  N  P  �  = (det L) det  �  P − NL−1M  �  ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='1)  whenever L is invertible (see [24] for example).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' In this case,  L =  � v − η  ρ  Pρ  ρ(v − η)  �  ,  M =  � 0  0  pθ  0  �  ,  N =  � 0  θpθ  0  (λ + ν)q  �  ,  P =  � ρeθ(v − η)  1  κ  τ(v − η)  �  ,  and det L = ρ(v − η)2 − ρpρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' Then, according with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='1), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='4) is equivalent to  ρeθτ(v − η)4 −  �  ρeθpρτ + θp2  θτ  ρ  + κ  �  (v − η)2 + (λ + ν)pθq  ρ (v − η) + κpρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='2)  If we set z := v − η, and multiply by (ρeθτ)−1 in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='2), we obtain  z4 + a2z2 + a1z + a0 = 0,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='3)  where  a2 = −  1  ρeθτ  �  ρeθpρτ + θp2  θτ  ρ  + κ  �  ,  a1 = (λ + ν)  pθ  ρ2eθτ q,  a0 = κpρ  ρeθτ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='4)  First, we show that, if λ + ν ̸= 0, the one dimensional case of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='1), is not a hyperbolic system of equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='  We use the following result stated without a proof (see [8, Theorem 7] and also [19], for example).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' A quartic equation of the form (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='3) with a0, a1, a2 real, a1 ̸= 0, and with discriminant ∆, has  2 distinct real roots and 2 imaginary roots if ∆ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='  □  The discriminant, ∆ = ∆(a0, a1, a2), of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='3) is given as  ∆(a0, a1, a2) = 256a3  0 − 128a2  2a2  0 + 16a0a4  2 + 144a0a2  1a2 − 4a2  1a3  2 − 27a4  1,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='5)  see [8, page 41] and [9, page 405] for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='  4  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' ANGELES  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' Let (ρ∗, θ∗) ∈ D be such that λ(ρ∗, θ∗) + ν(ρ∗, θ∗) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' Then, there are values of q ∈ R for  which the characteristic polynomial (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='2) has complex roots and thus, the (λ, ν)-quasilinear system (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='1) is  not hyperbolic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' Set γ∗ := λ(ρ∗, θ∗) + ν(ρ∗, θ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' Observe that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='5) can be rewritten as a fourth order polynomial in  the variable a1, namely  P(a1) := Aa4  1 + Ba2  1 + C,  where A := −27, B := 144a0a2 − 4a3  2 > 0, C := 16a0  �  a2  2 − 4a0  �2 ≥ 0 and, according with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='4), B =  B(ρ∗, θ∗), C = C(ρ∗, θ∗) and a1 = g(ρ∗, θ∗)q for g(ρ∗, θ∗) = γ∗  pθ(ρ∗,θ∗)  (ρ∗)2eθ(ρ∗,θ∗)τ ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' By taking,  q2 > max  �−C(ρ∗, θ∗) − B(ρ∗, θ∗)  A(ρ∗, θ∗)g2(ρ∗, θ∗)  ,  2  g2(ρ∗, θ∗)  �  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='6)  and using that A < 0, it follows that Aa2  1 + B < −C, and so  P(a1) = a2  1  �  Aa2  1 + B  �  + C < −C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='  Hence, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='5) is negative and according with Lemma 1, there are two complex roots of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='  Now, let  v ∈ R, take q satisfying (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='6) and set U ∗ := (ρ∗, v, θ∗, q) ∈ O ⊂ R4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' Then, by the previous argument, the  characteristic polynomial (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='2) has complex roots at U ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' This proves the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='  □  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' Three dimensional system  Set d = 3 and observe that N = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' In this case, A0(U) is a diagonal matrix given as  A0(U) =  \uf8eb  \uf8ec  \uf8ec  \uf8ed  1  ρI3  ρeθ  τI3  \uf8f6  \uf8f7  \uf8f7  \uf8f8 ,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='1)  where I3 denotes the identity matrix of order 3 × 3 and all the empty spaces refer to zero block matrices of  the appropriate sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' We have that  Aλ,ν(ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' U) =  �  �  �  �  �  �  �  �  �  �  �  �  ξ · v  ξ1ρ  ξ2ρ  ξ3ρ  0  0  0  0  ξ1pρ  ρξ · v  0  0  ξ1pθ  0  0  0  ξ2pρ  0  ρξ · v  0  ξ2pθ  0  0  0  ξ3pρ  0  0  ρξ · v  ξ3pθ  0  0  0  0  ξ1θpθ  ξ2θpθ  ξ3θpθ  ρeθξ · v  ξ1  ξ2  ξ3  0  ξ1κ  τξ · v  0  0  0  τQλ,ν(ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' q)  ξ2κ  0  τξ · v  0  0  ξ3κ  0  0  τξ · v  �  �  �  �  �  �  �  �  �  �  �  �  ,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='2)  where, for each ξ ∈ S2, the sub-block matrix Qλ,ν(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' ξ) is of order 3 × 3 and given as  Qλ,ν(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' ξ) =  �  �  γξ1q1 + λ− (ξ2q2 + ξ3q3)  λ+ξ1q2 + νξ2q1  λ+ξ1q3 + νξ3q1  λ+ξ2q1 + νξ1q2  γξ2q2 + λ− (ξ1q1 + ξ3q3)  λ+ξ2q3 + νξ3q2  λ+ξ3q1 + νξ1q3  λ+ξ3q2 + νξ2q3  γξ3q3 + λ− (ξ1q1 + ξ2q2)  �  � ,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='3)  where γ := λ + ν, λ+ := λ  2 + 1  2 and λ− := λ  2 − 1  2, and  Q(U) = (0, 0, 0, 0, 0, q1, q2, q3)⊤ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='4)  By using formula (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='1), we can compute the characteristic polynomial for (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='1), yielding  det  �  Aλν(ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' U) − ηA0(ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' U)  �  = ρ3τ 2(ξ · v − η)4Pλ,ν(ξ, U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' η),  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='5)  5  where  Pλ,ν(ξ, U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' η) = ρeθτ(ξ · v − η)4 −  �  τρeθpρ + τθp2  θ  ρ  + κ  �  (ξ · v − η)2 + pθ  ρ hλ,ν(ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' q)(ξ · v − η) + κpρ  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='6)  and hλ,ν(ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' q) := (Qλ,ν(ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' q)ξ) · ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' Fix λ, ν ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' The mapping, S2 × R3 ∋ (ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' q) �→ Qλ,ν(ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' q) ∈ M3×3, is non-trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' Assume otherwise, that is, for all q ∈ R3 and all ξ ∈ S2, Qλ,ν(ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' q) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' Take ξ = (1, 0, 0) and  q = (q1, 0, 0) with q1 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' Then, by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='3), γq1 = 0 and λ−q1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' This means that λ = 1 and ν = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='  On the other hand, take ξ = (1, 0, 0) and q = (0, q2, 0) with q2 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' Then, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='3) implies that νq2 = 0, a  contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='  □  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' Fix λ, ν ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' The mapping, S2 × R3 ∋ (ξ, q) �→ hλ,ν(ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' q) ∈ R, is null if and only if λ + ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' First observe that for any q ∈ R3 and ξ ∈ S2 it holds that  Qλ,ν(ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' q)ξ · ξ = γ  �  ξ3  1q1 + ξ3  2q2 + ξ3  3q3 + ξ2  1ξ2q2 + ξ2  1ξ3q3 + ξ1ξ2  2q1 + ξ1ξ2  3q1 + ξ3ξ2  2q3 + ξ2ξ2  3q2  �  ,  which yields the formula  hλ,ν(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' ξ) = γ|ξ|2ξ · q  ∀  q ∈ R3,  ξ ∈ S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='7)  Then, if γ = λ + ν = 0, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='7) implies that hλ,ν(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' ξ) = 0 for any q ∈ R3 and ξ ∈ S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' On the other hand,  assume that hλ,ν(ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' q) = 0 for all ξ ∈ S2 and q ∈ R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' If so, it must be true for q0 ∈ R3 and ξ0 ∈ S2 such that  ξ0 · q0 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' Then, by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='7), we have that γ|ξ0|2ξ0 · q0 = 0, which implies that γ = λ + ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='  □  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' Set d = 3 and consider the (λ, ν)-quasilinear system defined by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='1), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='2), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='3) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='  If there is (ρ∗, θ∗) ∈ D such that γ∗ := λ(ρ∗, θ∗) + ν(ρ∗, θ∗) ̸= 0 then, the (λ, ν)-quasilinear system is not  hyperbolic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' For any q ∈ R3 \\ {0} set ξq =  q  |q| ∈ S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' Then, by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='7) and Lemma 3, hλ,ν(ξq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' q) = γ|q| ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' Take  v ∈ R3 and define the state U ∗ := (ρ∗, v, θ∗, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' Set p∗  ρ = pρ(ρ∗, θ∗), p∗  θ = pθ(ρ∗, θ∗), e∗  θ = eθ(ρ∗, θ∗) and  κ∗ = κ(ρ∗, θ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' Consider Pλ,ν(ξq, U ∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' η) = 0, which according with (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='6) is equivalent to  ρ∗e∗  θτ(ξq · v − η)4 −  �  τρ∗e∗  θp∗  ρ + τθ∗(p∗  θ)2  ρ∗  + κ∗  �  (ξq · v − η)2 + p∗  θ  ρ∗ γ∗|q|(ξq · v − η) + κ∗p∗  ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='8)  Notice that, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='8) is the same as the polynomial given in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='2) but with |q| and v · ξq replacing q and v,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' Thus, by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='6), with |q|2 instead of q2, it holds that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='8) has complex roots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' Therefore, the  (λ, ν)-quasilinear system is not hyperbolic if λ + ν ̸= 0 on D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='  □  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' The Cattaneo-Christov-Jordan system  Consider the coupling between (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='3)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='5) and the material formulation of the Cattaneo heat flux model,  namely,  τ (qt + u · ∇q) + q = −κ∇θ,  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='1)  6  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' ANGELES  This system can be written in the quasilinear form (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='1) where for each U ∈ O, A0(U) and Q(U) are given  by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='1) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='4) respectively, however, instead of Aλ,ν(ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' U) we have the following symbol  A(ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' U) =  �  �  �  �  �  �  �  �  �  �  �  �  ξ · v  ξ1ρ  ξ2ρ  ξ3ρ  0  0  0  0  ξ1pρ  ρξ · v  0  0  ξ1pθ  0  0  0  ξ2pρ  0  ρξ · v  0  ξ2pθ  0  0  0  ξ3pρ  0  0  ρξ · v  ξ3pθ  0  0  0  0  ξ1θpθ  ξ2θpθ  ξ3θpθ  ρeθξ · v  ξ1  ξ2  ξ3  0  ξ1κ  τξ · v  0  0  0  O3×3  ξ2κ  0  τξ · v  0  0  ξ3κ  0  0  τξ · v  �  �  �  �  �  �  �  �  �  �  �  �  ,  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='2)  where, O3×3 denotes the zero matrix of order 3 × 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' This system was first proposed by Christov and Jordan  [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' They showed that (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='1) is Galilean invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' We refer to the quasilinear system defined by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='1), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='4)  and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='2) as the Cattaneo-Christov-Jordan system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' Morro points out that (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='1) is not objective and therefore  for no value of λ and ν this equation can be deduced from (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' This coincides with Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' Nonetheless,  formally, we can understand the Cattaneo-Christov-Jordan system as a quasilinear system with O3×3 instead  of Qλ,ν(ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' The characteristic polynomial of the Cattaneo-Christov-Jordan system has the form  det  �  Aλν(ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' U) − ηA0(ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' U)  �  = ρ3τ 2(ξ · v − η)4P0(ξ, U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' η),  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='3)  where  P0(ξ, U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' η) = ρeθτ(ξ · v − η)4 −  �  τρeθpρ + τθp2  θ  ρ  + κ  �  (ξ · v − η)2 + κpρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='4)  The characteristic roots of the three dimensional Cattaneo-Christov-Jordan system are real and given by  η0(ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' U) = ξ · v,  η1(ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' U) = ξ · v + 1  √  2  �  r(ρ, θ) + m(ρ, θ),  η2(ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' U) = ξ · v + 1  √  2  �  r(ρ, θ) − m(ρ, θ),  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='5)  η3(ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' U) = ξ · v − 1  √  2  �  r(ρ, θ) + m(ρ, θ),  η4(ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' U) = ξ · v − 1  √  2  �  r(ρ, θ) − m(ρ, θ),  where, for each ρ, θ > 0 we have set  r(ρ, θ) :=  �  pρ + θp2  θ  ρ2eθ  +  κ  ρeθτ  �  and  m(ρ, θ) :=  ��  pρ + θp2  θ  ρ2eθ  +  κ  ρeθτ  �2  − 4pρκ  ρeθτ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='  Observe that, by (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='3), η0(ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' U) is a root of algebraic multiplicity four, and it is easy to see that η3 < η4 <  η2 < η1 (see [1, Sections 3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' Moreover, this system is Friedrichs symmetrizable [1, Section 4], which  in turn implies that it is hyperbolic and thus, its Cauchy problem is locally well-posed in L2 (see, [12], [13]  and [14], for example).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='  Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' Consider a (λ, ν)-quasilinear system for which λ + ν = 0 on D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' Then, the characteristic  speeds of such system are real and coincide with the characteristic speeds of the Cattaneo-Christov-Jordan  system given in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' Since λ + ν = 0, Lemma 3 assures that hλ,ν(ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' q) = 0 for all ξ ∈ S2 and q ∈ R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' Therefore,  Pλ,ν(ξ, U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' η) = P0(ξ, U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' η)  for all  ξ ∈ S2, U ∈ R8, η ∈ R,  and the result follows by comparing (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='5) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='  □  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' As it was pointed out in [1], a quasilinear system of the form (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='1), with real characteristic  speeds, is not necessarily hyperbolic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' Such is the case of the Cattaneo-Christov system, since in this case,  (λ, ν) = (−1, 1) and so λ + ν = 0 (see [1, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='  7  Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' Let assumptions (C1)-(C3) be satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' Consider a (λ, ν)-quasilinear system (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='1) with the  property that λ + ν = 0 on D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' This system is hyperbolic if and only if (λ, ν) = (1, −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' Assume that λ + ν = 0 on D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' According with the definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='1, we have to show that the only  values of (λ, ν) for which the eigenvalue problem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='3) has a complete set of eigenvectors are (λ, ν) = (−1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='  Since λ+ ν = 0, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='1 assures that the eigenvalues of the (λ, ν)-quasilinear system are given by (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='  Let {Vj}4  j=1 be the eigenvectors associated with the eigenvalues {ηj}4  j=1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' By (C1)-(C2), these  eigenvalues are different and so, the set of eigenvectors {Vj}4  j=1 is linearly independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' Since η0(ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' U) is  different from the other roots, the (λ, ν)-quasilinear system will be hyperbolic if and only if, for every ξ ∈ S2  and U ∈ O, the geometric multiplicity of η0(ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' U) equals four.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' Let η = η0 and A(ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' U) = Aλ,ν(ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' U) in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='3)  and set V = (V1, V2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='., V8)⊤ to obtain the algebraic system of equations,  ξ · V ′ = 0,  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='6)  pρV1 + pθV5 = 0,  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='7)  ξ · V ′′ = 0,  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='8)  τQλ,ν(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' ξ)V ′ + κV5ξ = 0,  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='9)  where V ′ = (V2, V3, V4)⊤ and V ′′ = (V6, V7, V8)⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' From (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='8), it follows the existence of exactly two linearly  independent solutions, say (V 1  6 , V 1  7 , V 1  8 )⊤ and (V 2  6 , V 2  7 , V 2  7 )⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' Then, we can take the vectors  (0, 0, 0, 0, 0, V i  6, V i  7 , V i  8 )⊤,  for i = 1, 2,  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='10)  as two linearly independent solutions of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='6)-(6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' Therefore, in order to comply with the hyperbolicity,  equations (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='6), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='7) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='9), must have two more linearly independent solutions for each ξ ∈ S2 and  U = (ρ, v, θ, q) ∈ O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' Observe that, for each ξ ∈ S2, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='6) implies that V ′ ∈ {ξ}⊥, a two dimensional space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='  Hence, if {Vξ, Wξ} is a basis of {ξ}⊥, V ′ is of the form  V ′ = α1Vξ + α2Wξ,  for some α1, α2 ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='11)  If we multiply (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='9) by ξ and use (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='11) we obtain that  V5 = −τ  κ {α1 (Qλ,ν(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' ξ)Vξ) · ξ + α2 (Qλ,ν(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' ξ)Wξ) · ξ}  for each ξ ∈ S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='12)  Let ξ = (ξ1, ξ2, ξ3) ∈ S2 and assume that ξ1 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' Then, the vectors  Vξ = (−ξ2, ξ1, 0)⊤  and  Wξ = (−ξ3, 0, ξ1)⊤  are linearly independent and form a basis of {ξ}⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' Hence, by (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='12)  V5 = −τ  κλ+ {α1 (ξ1q2 − ξ2q1) + α2 (ξ1q3 − ξ3q1)}  for any q = (q1, q2, q3) ∈ R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='13)  By using (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='11) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='13) into (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='9) we obtain the equations  α1  �  −λ−(ξ2  2q2 + ξ2ξ3q3) + (ν + λ+)ξ1ξ2q1  �  + α2  �  −λ−(ξ2ξ3q2 + ξ2  3q3) + (ν + λ+)ξ1ξ3q1  �  = 0,  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='14)  α1  �  −(ν + λ+)ξ1ξ2q2 + λ−(ξ2  1q1 + ξ1ξ3q3)  �  = 0,  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='15)  α2  �  −(ν + λ+)ξ1ξ3q3 + λ−(ξ2  1q1 + ξ1ξ2q2)  �  = 0,  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='16)  valid for any ξ ∈ S2 with ξ1 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' Take ξ = (ξ1, ξ2, ξ3) ∈ S2 such that ξi ̸= 0 for all i = 1, 2, 3 and q ∈ R3  with the property that ξ ·q ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' Assume the existence of (ρ′, θ′) ∈ D such that (λ, ν)(ρ′, θ′) ̸= (1, −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' Then,  (ν + λ+) (ρ′, θ′) = −λ−(ρ′, θ′) ̸= 0 and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='14)-(6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='16) can be rewritten as  α1ξ2  �  ξ · q  �  = −α2ξ3  �  ξ · q  �  ,  α1ξ1  �  ξ · q  �  = 0,  α2ξ1  �  ξ · q  �  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='  8  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' ANGELES  By the particular choices of ξ and q, it follows that α1 = α2 = 0 and thus, V ′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' This implies that  V1 = V5 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' Therefore, if (λ, ν) ̸= (1, −1) we can find wavenumbers (ξ) and states (U = (ρ′, v, θ′, q)⊤ ∈ O)  such that the geometric multiplicity of the eigenvalue η0(ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' U) equals two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='  This means that the (λ, ν)-  quasilinear system is not hyperbolic if (λ, ν) ̸= (1, −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='  Now assume that (λ, ν) = (1, −1) on D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' If ξ = (ξ1, ξ2, ξ3) ∈ S2 is such that ξ1 ̸= 0, V5 is given by (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='13) and  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='9) is equivalent to equations (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='14)-(6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' Moreover, since λ+ = 1 and λ− = 0, equations (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='14)-(6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='16)  are trivially satisfied for any choice of α1, α2 ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' In particular, this means that Vξ and Wξ are linearly  independent solutions of equations (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='6) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='9), where V5 is given by (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' Consequently, the vectors  V(ξ) =  �  −pθ  pρ  V5, Vξ, V5, 0, 0, 0  �⊤  ,  W(ξ) =  �  −pθ  pρ  V5, Wξ, V5, 0, 0, 0  �⊤  ,  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='17)  are solutions of equations (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='6)-(6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='9) and together with (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='10) form a linearly independent set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' Therefore,  the geometric multiplicity of η0(ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' U) equals four, for any ξ ∈ S2 with ξ1 ̸= 0 and U ∈ O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' If, on the other  hand, ξ ∈ S2 is such that ξ2 ̸= 0, we take the linearly independent vectors,  Vξ = (ξ2, −ξ1, 0)⊤  and  Wξ = (0, −ξ3, ξ2)⊤ ,  as solutions of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' In this case,  V5 = −τ  κ {α1 (ξ2q1 − ξ1q2) + α2 (ξ2q3 − ξ3q2)}  for any q = (q1, q2, q3) ∈ R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='  By setting V(ξ) and W(ξ) as in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='17), the conclusion follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' Finally, if ξ = (0, 0, 1) take the canonical  vectors ˆe1 and ˆe2 as solutions of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='9), that is, Vξ = ˆe1 and Wξ = ˆe2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' Then, V5 = − τ  κ (α1q1 + α2q2) and we  proceed as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' Therefore, if (λ, ν) = (1, −1), the geometric multiplicity of the eigenvalue η0(ξ, U) equals  four for any choice of ξ ∈ S2 and U ∈ O and thus, the (1, −1)-quasilinear system is hyperbolic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='  □  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' Final comments and conclusions  In this work we have shown that the only case in which the coupling between (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='3)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='5) and the frame  indifferent formulation for the heat flux (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='6) yields a hyperbolic system of equations is when we take (λ, ν)  as the constant value functions (1, −1) on D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' In particular, this implies that for a constant state Vc ∈ O,  the Cauchy problem for the linear equation  A0(Vc)Ut + Ai  1,−1(Vc)∂iU + Q(Vc) = 0,  is well posed in L2, as consequence of the existence of energy estimates [20, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' Moreover, if  Vc ∈ O is taken as constant or constant outside a compact set and since the characteristic speeds of this  model (see Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='1) have constant algebraic multiplicity with respect to ξ ∈ S2, the finite speed of  propagation holds true [2, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' Now, for the general (1, −1)-quasilinear system (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='1), although  its hyperbolicity is necessary for the existence of L2 energy estimates, it is not sufficient [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' Typically,  when the quasilinear system (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='1) is derived from a system of conservation laws, the existence of a convex  entropy is equivalent to the existence of a symmetrizer (as the Hessian of the entropy) (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' [7], [20]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' Given  the non-conservative structure of the (1, −1)-heat flux model, it is unknown for the author if a Friedrichs  symmetrizer exists for this quasilinear system of equations in more than one space dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' Due to the  presence of the skew matrix Q1,−1(ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' q), a diagonal symmetrizer as in the Cattaneo-Christov-Jordan system,  doesn’t work for this case when d = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='  By proceeding as in [4, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='1], it is easy to show that the (1, −1)-heat flux model is irreducible in several  space dimensions, in the sense that, together with the material invariant form of the internal energy equation,  it is impossible to derive a single equation for the temperature field θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' Observe that the one dimensional  version of this model coincides with the one dimensional Cattaneo-Christov-Jordan system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' Therefore, it is  locally well-posed in L2 and satisfies the finite speed of propagation property (see [12] and [2, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='  9  Acknowledgments  Thanks to Angelo Morro for clarifying some aspects of objective rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' This work was supported by  CONACyT (Mexico) through a postdoctoral fellowship under grant A1-S-10457.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
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+page_content=' Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' SIAM, (Philadelphia  1973).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
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+page_content=' Morro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' A Thermodynamic Approach to Rate Equations in Continuum Physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
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+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
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+page_content='  [17] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' Morro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
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+page_content=' Contin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
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+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
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+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
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+page_content=' Equ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
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+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
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+page_content='  [23] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' Weyl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' Shock waves in arbitrary fluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
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+page_content=' Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
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+page_content=' Springer, (New York 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='  (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' Angeles) Instituto de Matem´aticas, Universidad Nacional Aut´onoma de M´exico, Circuito Exterior s/n, Ciu-  dad de M´exico C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content=' 04510 (Mexico)  Email address: teojkd@ciencias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='unam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
+page_content='mx' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfNQbL/content/2301.03736v1.pdf'}
diff --git a/1NAyT4oBgHgl3EQfPfZU/content/tmp_files/2301.00025v1.pdf.txt b/1NAyT4oBgHgl3EQfPfZU/content/tmp_files/2301.00025v1.pdf.txt
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@@ -0,0 +1,734 @@
+Search for echoes on the edge of quantum black
+holes
+Jahed Abedi1,2,3,*
+1Department of Mathematics and Physics, University of Stavanger, NO-4036 Stavanger, Norway
+2Albert-Einstein-Institut, Max-Planck-Institut f¨ur Gravitationsphysik, Callinstraße 38, 30167 Hannover, Germany
+3Leibniz Universit¨at Hannover, 30167 Hannover, Germany
+*e-mail: jahed.abedi@uis.no
+ABSTRACT
+I perform an unprecedented template-based search for stimulated emission of Hawking radiation (or Boltzmann echoes) by
+combining the gravitational wave data from 65 binary black hole merger events observed by the LIGO/Virgo collaboration. With
+a careful Bayesian inference approach, I found no statistically significant evidence for this signal in either of the 3 Gravitational
+Wave Transient Catalogs GWTC-1, GWTC-2 and GWTC-3. However, the data cannot yet conclusively rule out the presence of
+Boltzmann echoes either, with the Bayesian evidence ranging within 0.3-1.6 for most events, and a common (non-vanishing)
+echo amplitude for all mergers being disfavoured at only 2:5 odds. The only exception is GW190521, the most massive and
+confidently detected event ever observed, which shows a positive evidence of 9.2 for stimulated Hawking radiation. An optimal
+combination of posteriors yields an upper limit of A < 0.42 (at 90% confidence level) for a universal echo amplitude, whereas
+A ∼ 1 was predicted in the canonical model. The next generation of gravitational wave detectors such as LISA, Einstein
+Telescope, and Cosmic Explorer can draw a definitive conclusion on the quantum nature of black hole horizons.
+1 Introduction
+Post-merger gravitational wave (GW) echoes are our most
+direct observational windows into the quantum structure of
+black hole (BH) event horizons1–3, while their non-existence
+would rule out different hypotheses about the nature of these
+enigmatic objects. The best view of these horizons can be
+achieved by combining a large number of binary black hole
+(BBH) merger events. As such, the GW data release for BBH
+mergers during the first, second and third observing run of
+LIGO/Virgo observatories4–10 provides an unprecedented op-
+portunity to test classical general relativity (GR), as well as its
+alternatives, in the strong gravity regime. One can assume GR
+as the base model and contrast it to GR+phenomenological
+echo waveforms to see which one fits the data better. Nonethe-
+less, despite many attempts in searching for echoes11–26 using
+different models, we still lack a waveform as physical/accurate
+as waveforms in GR. Additionally, there is no consensus on
+the optimal procedure to combine the events, for best sensi-
+tivity to fundamental physics. Although, the reported GW
+detections have so far been consistent with predictions of
+GR8–10, the first search for echoes11, motivated by a resolu-
+tion to the BH quantum information paradox, was conducted
+for the first observing run of the Advanced LIGO detectors
+(O1), which then motivated further searches within different
+GW data analysis frameworks, and using more physical echo
+waveforms. Accordingly, several attempts with replication
+and extention were made with positive11–15, mixed16–18, and
+negative14,19–22 results. These searches lead to tentative ev-
+idence and detection found with different groups11–16,18,26
+at false alarm rates of 0.002% − 5% (but see3,16,18,27–30 for
+the ongoing discussion, comments, and rebuttals on statistical
+significance of these findings that motivate further investiga-
+Figure 1. Schematic diagram of GW echoes (stimulated
+Hawking radiation) from remnant of a BBH merger.
+tions). So far, the searches for echoes have employed three
+strategies that can be classified into:
+1. Waveform dependent11,14,16,17,19,21–23,25,26.
+2. Model-agnostic or coherent12,13,15,18,24,26.
+3. Electromagnetic confirmation by Gill et al.31.
+For more details, discussions and review please see3.
+A confirmed detection of echoes would imply that the BH
+horizon is not totally absorbing. This would lead to post-
+merger repeating signals which are produced in the cavity that
+traps GWs between the classical angular momentum barrier
+and the near-horizon membrane/firewall1,2,11,26,32. However,
+firewalls are not a necessary condition to have observable
+echoes33. Stimulated emission of Hawking radiation, caused
+by the GWs that excite the quantum BH microstructure has a
+similar effect26,34–37. As shown in Figs. 1 and 2, the trapped
+GWs slowly leak out, leading to repeating echoes within time
+intervals of:
+1
+arXiv:2301.00025v1  [gr-qc]  30 Dec 2022
+
+barriel
+momentum
+Stimulated Hawking radiation
+Gravitational waves
+Second echo
+△techo
+First echo
+techo
+LmergerFigure 2. Boltzmann GW echoes template for GW150914 like signal with amplitude A = 1.
+∆techo ≃ 4GMBH
+c3
+�
+1+
+1
+√
+1−a2
+�
+×ln
+�GMBH
+c2ℓQG
+�
+,
+(1)
+where MBH and a are the mass and the dimensionless spin of
+the final BH remnant. Here, ℓQG is the characteristic physi-
+cal length scale for quantum gravity effects where GWs are
+reflected near the (would-be) horizon. For ℓQG = ℓPlanck, the
+reflection happens at a Planck distance from the horizon. More
+generally, for ℓQG = ℓPlanck/Λ, deviations from GR happen
+sub-Planckian Λ > 1 or super-Planckian Λ < 1 scales. In this
+paper, we choose a conservative prior −13 ≤ log10 Λ ≤ 13.
+Here, in comparison to former attempts, I used a more phys-
+ical waveform, based on stimulated Hawking radiation34,35
+to test for the existence of echoes. Furthermore, I adopt the
+Bayesian methodoly and p, as in Abedi et al.26. Note that this
+search has been implemented before the search and release
+of26. The delay in release was due to large number of events
+and high computational costs to combine all 65 events. In this
+approach, I set our model and search pipeline from a rather
+novel point of view to look for echoes combining 65 LVK
+BBH merger events. I perform the search for echoes on BBH
+signals using the GWTC-14, GWTC-25 and GWTC-36. This
+search includes almost the bulk of all the confident observa-
+tions4–6. The missed events are either the marginal ones or
+needed a high computational cost (ones with very small mass).
+In this approach of combining events I assume echo model
+is the same for all the events. In particular, I assume all the
+events have same echo amplitude A. Although, this approach
+does not cover entire space of former searches, it makes a
+complementary search in overall.
+One such proposal to search for quantum black holes in
+GW data is given by phenomenological Boltzmann echoes
+waveform34,35, where the general relativistic prediction for
+GW signal from BBH mergers hGR(ω) in Fourier space is
+modified to:
+hGR+echoes(A,ω) = hGR(ω)
+�
+1+Aeiφ
+∞
+∑
+n=1
+Rn
+�
+,
+(2)
+R ≡ exp[−ℏ|ω −mΩH|
+2kTH
++iω∆techo],
+(3)
+where Aeiφ quantifies their overall amplitude, while the modu-
+lus and phase of R quantify their relative damping and tempo-
+ral separation, respectively. Generally, we expect 0 ≲ A ≲ 2
+and 0 < φ ≤ 2π due to GR non-linearities. Furthermore, I set
+the horizon mode frequency m×ΩH to m=2 for quadrupolar
+gravitational radiation (with the assumption that the energy in
+BBH ringdown and echoes are dominated by this mode) as
+main frequency of search pipeline.
+Although, there is no doubt that the hGR(ω) (main event
+GR part) exists, we want to answer whether the echoes part
+exists. Existence of hGR(ω) helps us to obtain physical prior
+for echo model i.e. improvements in priors for ∆techo in
+(1) or ΩH and TH variables. Here, the Boltzmann factor
+exp[− |ω−2ΩH|
+2TH
+] originates from Hawking tunnelling rate to
+fuzzy states of quantum BH (please see Fig 2 for this wave-
+form). Note that, repeating echoes time delay modifies this
+factor to exp[ ω−2ΩH
+2TH
++iω∆techo]34,35. Here, I only keep first
+two echoes of the waveform in this search pipeline. Indeed,
+this waveform is not as perfect as GR waveform, while it helps
+us in future research and establishment of better waveforms.
+Next section describes method and search pipeline. Then, I
+conclude with the search results and findings.
+2 Method and search pipeline and results
+In order to combine the events, the amplitude A for all the 65
+events is fixed to a universal value and the individual bayes
+factors of events BEvent(A) are combine as follows.
+Combined Bayes Factor = B(A) = ∏
+i=Events
+Bi(A)
+(4)
+The combined bayes factors is shown in Figs. 3a and 3b.
+I employed PyCBC inference38 pipeline using a dynamic
+nested sampling MCMC algorithm, dynesty139. It is based
+on sampling the likelihood function for a hypothesis that
+gives a measure of existence of a signal in the data. The
+likelihood function is supposed to be compatible with the
+natural assumption that the background is Gaussian. I have
+used two/three detector networks H1-L1/H1-L1-V1 (Han-
+ford, Livingston, Virgo) depending on the event and available
+1I used 25,000 live points in each run.
+2/6
+
+1e-21
+1.5
+1.0
+0.5
+0.0 -
+-0.5 
+-1.0
+1.5
+-0.20
+-0.10
+0.00
+0.25
+0.30
+0.50
+0.60
+t-tmerger (sec)0.00
+0.25
+0.50
+0.75
+1.00
+1.25
+1.50
+1.75
+2.00
+Amplitude of echoes (A)
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+Bayes factor
+(a)
+0.00
+0.25
+0.50
+0.75
+1.00
+1.25
+1.50
+1.75
+2.00
+Amplitude of echoes (A)
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+Posterior density
+All events
+GW150914
+GW190521
+(b)
+Figure 3. (a): Combined Bayes factor density in terms of amplitude for 65 events. Combined events give an overall value of
+BGR+echo
+GR
+≃ 0.4 for bayes factor. (b): Combined bayes factor posterior density where the bayes factor curves are normalized by
+� Amax
+Amin B(A)dA. Here, in order to compare the individual events (GW150914 and GW190521) and combined events we plot the
+posterior density of the bayes factors.
+data for this analyse. First I obtain the Bayes factor com-
+paring the log likelihood to the log likelihood of the Gaus-
+sian noise. Then the combined Bayes factors of alternative
+models with different amplitude A are compared (here they
+are hGR+echoes(A,ω) and hGR+echoes(A = 0,ω) = hGR(ω) in
+Eq. (2)). I used class of phenomenological IMR waveform
+family IMRPhenomPv240,41 which is freely available as part
+of LALSuite42. Although the main search in Abedi et al.26
+for GW190521 has been performed with NRSur7dq4 wave-
+form43, in order to make identical search with other events
+in this paper I employed IMRPhenomPv2 for this event. The
+slight change in reported bayes factor for this event in this
+paper is due to change in waveform. It is worth to mention
+that other waveforms/changes have shown consistent result
+for this event26.
+For each event I run for discrete set of amplitudes, where
+each run has different seed number. Since the bayes factor
+estimation in PyCBC has error, it would be hard to read the re-
+sult. Due to this error and in order to get smooth/stable result,
+for each amplitude A, the bayes factor density ¯B(A) which is
+the average of B(A) within (A−∆A/2,A+∆A/2) (see Figs.
+3a and 3b) evaluated. Although the lower ∆A gives a better
+resolution, it leads to more fluctuations and error. In order
+to improve the resolution one needs to increase the number
+of runs (increase the amplitude bins) as well, which leads to
+higher computational cost. So it requires a balance between
+computational cost and targeted resolution. In order to get
+a satisfying smoothness along with optimum computational
+cost ∆A = 0.2 is chosen. In order to satisfy the approxima-
+tion, three amplitude range is arranged. First range is near
+zero A ∼ 0 which is the place of GR model for comparison.
+This range needs to have as high as possible runs to estimate
+the bayes factor of GR accurately. I chose A = (10−4,2 ×
+10−4,3×10−4,5×10−4,10−3,2×10−3,3×10−3,5×10−3)
+for amplitude bins to accomplish this task. Here the bayes
+factor is re-normalized to B(A ∼ 0) = 1. The second range is
+where the combined bayes factor is B(A) ≥ 1. This range is
+between ∼ (0,0.5) with uniform intervals of dA = 5×10−3.
+In order to get satisfying result this range needs to have sec-
+ond priority in amplitude resolution. The last range where the
+bayes factor drops significantly from 1 doesn’t need to have
+high resolution as it already disfavoured by model when the
+events are combined. This range is between (0.5,2) with uni-
+form intervals of dA = 0.01. However, for individual events
+one may need high resolution in all the amplitudes as well.
+I set 257 runs for each event and 65 × 257 = 16705 runs in
+total.
+In order to obtain the overall bayes factor of individ-
+ual events and combined events we just do BGR+echo
+GR
+=
+1
+Amax−Amin
+� Amax
+Amin B(A)dA with (Amin,Amax) = (0,2). For com-
+bined events I got BGR+echo
+GR
+≃ 0.4. The result for individual
+events and their histogram are reported in Table 1 and Fig. 4
+respectively.
+3 Conclusion and discussions
+I presented the outcome which gives measure of possibility
+for preference of hGR+echoes over hGR based on Bayes factors
+comparison of these two models. The 65 analysed events in
+Table 1 and Fig. 4 for individual events show inconclusive
+result in preference for GR or GR+echo, although with slight
+preference for GR but not by much. The main scope and
+result of the paper is combination of echoes for large number
+of events. We see that the combined bayes factor which is
+BGR+echo
+GR
+≃ 0.4 is still inconclusive about GR+echo and GR.
+It is realised that this combining method gives five order of
+magnitude higher bayes factor compared to when we simply
+3/6
+
+GWTC-1
+log10 BGR+echo
+GR
+GWTC-1
+log10 BGR+echo
+GR
+GWTC-1
+log10 BGR+echo
+GR
+GW150914
+-0.53
+GW170608
+0.05
+GW170818
+-0.06
+GW151012
+0.05
+GW170729
+-0.12
+GW170823
+-0.25
+GW151226
+-0.09
+GW170809
+0.08
+GW170104
+0.13
+GW170814
+-0.30
+GWTC-2
+log10 BGR+echo
+GR
+GWTC-2
+log10 BGR+echo
+GR
+GWTC-2
+log10 BGR+echo
+GR
+GW190408_181802
+-0.16
+GW190521
+0.96
+GW190728_064510
+-0.01
+GW190412
+-0.09
+GW190521_074359
+-0.54
+GW190731_140936
+-0.15
+GW190413_052954
+0.03
+GW190527_092055
+0.01
+GW190814
+-0.42
+GW190413_134308
+-0.10
+GW190602_175927
+-0.22
+GW190828_063405
+0.04
+GW190421_213856
+0.21
+GW190620_030421
+-0.16
+GW190828_065509
+-0.14
+GW190424_180648
+-0.17
+GW190630_185205
+-0.17
+GW190910_112807
+-0.30
+GW190503_185404
+-0.02
+GW190706_222641
+-0.06
+GW190915_235702
+-0.09
+GW190512_180714
+-0.06
+GW190707_093326
+-0.02
+GW190924_021846
+0.00
+GW190513_205428
+-0.15
+GW190708_232457
+-0.01
+GW190925_232845
+-0.03
+GW190514_065416
+-0.03
+GW190719_215514
+-0.01
+GW190929_012149
+-0.13
+GW190517_055101
+0.07
+GW190720_000836
+-0.07
+GW190519_153544
+-0.35
+GW190727_060333
+-0.30
+GWTC-3
+log10 BGR+echo
+GR
+GWTC-3
+log10 BGR+echo
+GR
+GWTC-3
+log10 BGR+echo
+GR
+GW191109_010717
+-0.36
+GW200112_155838
+-0.28
+GW200219_094415
+-0.07
+GW191129_134029
+0.01
+GW200128_022011
+-0.2
+GW200220_061928
+0.21
+GW191204_171526
+0.01
+GW200129_065458
+-0.43
+GW200224_222234
+-0.34
+GW191215_223052
+0.2
+GW200202_154313
+0.21
+GW200225_060421
+-0.01
+GW191216_213338
+0.03
+GW200208_130117
+0.08
+GW200302_015811
+-0.12
+GW191222_033537
+-0.32
+GW200209_085452
+0.14
+GW200311_115853
+-0.37
+GW191230_180458
+-0.21
+GW200216_220804
+-0.15
+GW200316_215756
+-0.01
+Table 1. Results of bayes factor for GW echoes in GWTC-1, GWTC-2, and GWTC-3 events. Positive value of the log10 bayes
+factor indicates a preference for the GR+echoes model over GR model, while the negative value suggests instead a preference
+for the GR model over the GR+echoes model. Here GW190521 shows loudest echo. Here based on44 all the individual events
+appear as inconclusive to both GR or GR+echoes with GW190521 as exception! (see Fig. 4).
+combine the individual events bayes factor via multiplica-
+tion ∏
+i=Events
+BiGR+echo
+GR
+= 2.2 × 10−6. In another words the
+fact that the combined bayes factor for preference to GR has
+dropped from ∼ 4.6 × 105 to ∼ 2.5 indicates that there are
+still much to do in method improvement. Additionally, the
+large number of events and computational costs is a guarantee
+against bayes factor hack making the result robust. The only
+event that has shown evidence for preference of GR+echo
+model is GW190521 with BGR+echo
+GR
+= 9.2 (see Fig. 4). This
+is the most massive and confidently detected BBH merger
+event observed to date5. I refer the detailed interpretation and
+investigation about this event to Abedi et al.26. Presuming
+a simple speculation that we can compare all the events as
+same (echo model remain same for all the 65 events and their
+echo amplitudes compare to main event amplitude doesn’t
+change by much despite the change in initial condition of the
+progenitor BBH mergers) and all the 65 BBH events should
+show evidence for echo signals in this model and the space of
+parameters considered in this search, I found an upper bound
+amplitude A < 0.42 (at 90% confidence level) for echoes. I
+remind the reader that bounds from our search only relate to
+the family of echo waveforms considered here.
+It is worth to note that I didn’t see any evidence for echoes
+in O1 in contrast to11,16,17, possibly because the model I
+used here is different and has much suppressed amplitudes in
+contrast to ADA model in11.
+In order to do a better search for quieter echoes, we might
+need to have a more physical echo waveforms. In another
+words, concrete models from alternatives to GR are needed to
+use in PyCBC pipeline. Without better models, we might wait
+for O4. Observations will improve in number. LISA, Einstein
+Telescope, and Cosmic Explorer will make a big breakthrough
+in sensitivity in search for alternatives to GR.
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+funded by the French Centre National de Recherche Scien-
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+Nucleare (INFN) and the Dutch Nikhef, with contributions by
+Polish and Hungarian institutes.
+6/6
+
diff --git a/1NAyT4oBgHgl3EQfPfZU/content/tmp_files/load_file.txt b/1NAyT4oBgHgl3EQfPfZU/content/tmp_files/load_file.txt
new file mode 100644
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@@ -0,0 +1,832 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf,len=831
+page_content='Search for echoes on the edge of quantum black  holes  Jahed Abedi1,2,3,*  1Department of Mathematics and Physics, University of Stavanger, NO-4036 Stavanger, Norway  2Albert-Einstein-Institut, Max-Planck-Institut f¨ur Gravitationsphysik, Callinstraße 38, 30167 Hannover, Germany  3Leibniz Universit¨at Hannover, 30167 Hannover, Germany  e-mail: jahed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='abedi@uis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='no  ABSTRACT  I perform an unprecedented template-based search for stimulated emission of Hawking radiation (or Boltzmann echoes) by  combining the gravitational wave data from 65 binary black hole merger events observed by the LIGO/Virgo collaboration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' With  a careful Bayesian inference approach, I found no statistically significant evidence for this signal in either of the 3 Gravitational  Wave Transient Catalogs GWTC-1, GWTC-2 and GWTC-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' However, the data cannot yet conclusively rule out the presence of  Boltzmann echoes either, with the Bayesian evidence ranging within 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='3-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='6 for most events, and a common (non-vanishing)  echo amplitude for all mergers being disfavoured at only 2:5 odds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' The only exception is GW190521, the most massive and  confidently detected event ever observed, which shows a positive evidence of 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='2 for stimulated Hawking radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' An optimal  combination of posteriors yields an upper limit of A < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='42 (at 90% confidence level) for a universal echo amplitude, whereas  A ∼ 1 was predicted in the canonical model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' The next generation of gravitational wave detectors such as LISA, Einstein  Telescope, and Cosmic Explorer can draw a definitive conclusion on the quantum nature of black hole horizons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='  1 Introduction  Post-merger gravitational wave (GW) echoes are our most  direct observational windows into the quantum structure of  black hole (BH) event horizons1–3, while their non-existence  would rule out different hypotheses about the nature of these  enigmatic objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' The best view of these horizons can be  achieved by combining a large number of binary black hole  (BBH) merger events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' As such, the GW data release for BBH  mergers during the first, second and third observing run of  LIGO/Virgo observatories4–10 provides an unprecedented op-  portunity to test classical general relativity (GR), as well as its  alternatives, in the strong gravity regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' One can assume GR  as the base model and contrast it to GR+phenomenological  echo waveforms to see which one fits the data better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' Nonethe-  less, despite many attempts in searching for echoes11–26 using  different models, we still lack a waveform as physical/accurate  as waveforms in GR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' Additionally, there is no consensus on  the optimal procedure to combine the events, for best sensi-  tivity to fundamental physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' Although, the reported GW  detections have so far been consistent with predictions of  GR8–10, the first search for echoes11, motivated by a resolu-  tion to the BH quantum information paradox, was conducted  for the first observing run of the Advanced LIGO detectors  (O1), which then motivated further searches within different  GW data analysis frameworks, and using more physical echo  waveforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' Accordingly, several attempts with replication  and extention were made with positive11–15, mixed16–18, and  negative14,19–22 results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' These searches lead to tentative ev-  idence and detection found with different groups11–16,18,26  at false alarm rates of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='002% − 5% (but see3,16,18,27–30 for  the ongoing discussion, comments, and rebuttals on statistical  significance of these findings that motivate further investiga-  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' Schematic diagram of GW echoes (stimulated  Hawking radiation) from remnant of a BBH merger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='  tions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' So far, the searches for echoes have employed three  strategies that can be classified into:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' Waveform dependent11,14,16,17,19,21–23,25,26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' Model-agnostic or coherent12,13,15,18,24,26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' Electromagnetic confirmation by Gill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='  For more details, discussions and review please see3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='  A confirmed detection of echoes would imply that the BH  horizon is not totally absorbing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' This would lead to post-  merger repeating signals which are produced in the cavity that  traps GWs between the classical angular momentum barrier  and the near-horizon membrane/firewall1,2,11,26,32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' However,  firewalls are not a necessary condition to have observable  echoes33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' Stimulated emission of Hawking radiation, caused  by the GWs that excite the quantum BH microstructure has a  similar effect26,34–37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' As shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' 1 and 2, the trapped  GWs slowly leak out, leading to repeating echoes within time  intervals of:  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='00025v1  [gr-qc]  30 Dec 2022  barriel  momentum  Stimulated Hawking radiation  Gravitational waves  Second echo  △techo  First echo  techo  LmergerFigure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' Boltzmann GW echoes template for GW150914 like signal with amplitude A = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='  ∆techo ≃ 4GMBH  c3  �  1+  1  √  1−a2  �  ×ln  �GMBH  c2ℓQG  �  ,  (1)  where MBH and a are the mass and the dimensionless spin of  the final BH remnant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' Here, ℓQG is the characteristic physi-  cal length scale for quantum gravity effects where GWs are  reflected near the (would-be) horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' For ℓQG = ℓPlanck, the  reflection happens at a Planck distance from the horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' More  generally, for ℓQG = ℓPlanck/Λ, deviations from GR happen  sub-Planckian Λ > 1 or super-Planckian Λ < 1 scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' In this  paper, we choose a conservative prior −13 ≤ log10 Λ ≤ 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='  Here, in comparison to former attempts, I used a more phys-  ical waveform, based on stimulated Hawking radiation34,35  to test for the existence of echoes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' Furthermore, I adopt the  Bayesian methodoly and p, as in Abedi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' Note that this  search has been implemented before the search and release  of26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' The delay in release was due to large number of events  and high computational costs to combine all 65 events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' In this  approach, I set our model and search pipeline from a rather  novel point of view to look for echoes combining 65 LVK  BBH merger events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' I perform the search for echoes on BBH  signals using the GWTC-14, GWTC-25 and GWTC-36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' This  search includes almost the bulk of all the confident observa-  tions4–6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' The missed events are either the marginal ones or  needed a high computational cost (ones with very small mass).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='  In this approach of combining events I assume echo model  is the same for all the events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' In particular, I assume all the  events have same echo amplitude A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' Although, this approach  does not cover entire space of former searches, it makes a  complementary search in overall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='  One such proposal to search for quantum black holes in  GW data is given by phenomenological Boltzmann echoes  waveform34,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='35,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' where the general relativistic prediction for  GW signal from BBH mergers hGR(ω) in Fourier space is  modified to:  hGR+echoes(A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='ω) = hGR(ω)  �  1+Aeiφ  ∞  ∑  n=1  Rn  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='  (2)  R ≡ exp[−ℏ|ω −mΩH|  2kTH  +iω∆techo],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='  (3)  where Aeiφ quantifies their overall amplitude,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' while the modu-  lus and phase of R quantify their relative damping and tempo-  ral separation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' Generally, we expect 0 ≲ A ≲ 2  and 0 < φ ≤ 2π due to GR non-linearities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' Furthermore, I set  the horizon mode frequency m×ΩH to m=2 for quadrupolar  gravitational radiation (with the assumption that the energy in  BBH ringdown and echoes are dominated by this mode) as  main frequency of search pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='  Although, there is no doubt that the hGR(ω) (main event  GR part) exists, we want to answer whether the echoes part  exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' Existence of hGR(ω) helps us to obtain physical prior  for echo model i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' improvements in priors for ∆techo in  (1) or ΩH and TH variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' Here, the Boltzmann factor  exp[− |ω−2ΩH|  2TH  ] originates from Hawking tunnelling rate to  fuzzy states of quantum BH (please see Fig 2 for this wave-  form).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' Note that, repeating echoes time delay modifies this  factor to exp[ ω−2ΩH  2TH  +iω∆techo]34,35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' Here, I only keep first  two echoes of the waveform in this search pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' Indeed,  this waveform is not as perfect as GR waveform, while it helps  us in future research and establishment of better waveforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='  Next section describes method and search pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' Then, I  conclude with the search results and findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='  2 Method and search pipeline and results  In order to combine the events, the amplitude A for all the 65  events is fixed to a universal value and the individual bayes  factors of events BEvent(A) are combine as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='  Combined Bayes Factor = B(A) = ∏  i=Events  Bi(A)  (4)  The combined bayes factors is shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' 3a and 3b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='  I employed PyCBC inference38 pipeline using a dynamic  nested sampling MCMC algorithm, dynesty139.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' It is based  on sampling the likelihood function for a hypothesis that  gives a measure of existence of a signal in the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' The  likelihood function is supposed to be compatible with the  natural assumption that the background is Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' I have  used two/three detector networks H1-L1/H1-L1-V1 (Han-  ford, Livingston, Virgo) depending on the event and available  1I used 25,000 live points in each run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='  2/6  1e-21  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
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+page_content='60  t-tmerger (sec)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
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+page_content='00  Amplitude of echoes (A)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='0  Bayes factor  (a)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='75  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='00  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='25  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='50  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='75  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='00  Amplitude of echoes (A)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='0  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='5  Posterior density  All events  GW150914  GW190521  (b)  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' (a): Combined Bayes factor density in terms of amplitude for 65 events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' Combined events give an overall value of  BGR+echo  GR  ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='4 for bayes factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' (b): Combined bayes factor posterior density where the bayes factor curves are normalized by  � Amax  Amin B(A)dA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' Here, in order to compare the individual events (GW150914 and GW190521) and combined events we plot the  posterior density of the bayes factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='  data for this analyse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' First I obtain the Bayes factor com-  paring the log likelihood to the log likelihood of the Gaus-  sian noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' Then the combined Bayes factors of alternative  models with different amplitude A are compared (here they  are hGR+echoes(A,ω) and hGR+echoes(A = 0,ω) = hGR(ω) in  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' (2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' I used class of phenomenological IMR waveform  family IMRPhenomPv240,41 which is freely available as part  of LALSuite42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' Although the main search in Abedi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='26  for GW190521 has been performed with NRSur7dq4 wave-  form43, in order to make identical search with other events  in this paper I employed IMRPhenomPv2 for this event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' The  slight change in reported bayes factor for this event in this  paper is due to change in waveform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' It is worth to mention  that other waveforms/changes have shown consistent result  for this event26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='  For each event I run for discrete set of amplitudes, where  each run has different seed number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' Since the bayes factor  estimation in PyCBC has error, it would be hard to read the re-  sult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' Due to this error and in order to get smooth/stable result,  for each amplitude A, the bayes factor density ¯B(A) which is  the average of B(A) within (A−∆A/2,A+∆A/2) (see Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='  3a and 3b) evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' Although the lower ∆A gives a better  resolution, it leads to more fluctuations and error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' In order  to improve the resolution one needs to increase the number  of runs (increase the amplitude bins) as well, which leads to  higher computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' So it requires a balance between  computational cost and targeted resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' In order to get  a satisfying smoothness along with optimum computational  cost ∆A = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='2 is chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' In order to satisfy the approxima-  tion, three amplitude range is arranged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' First range is near  zero A ∼ 0 which is the place of GR model for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='  This range needs to have as high as possible runs to estimate  the bayes factor of GR accurately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' I chose A = (10−4,2 ×  10−4,3×10−4,5×10−4,10−3,2×10−3,3×10−3,5×10−3)  for amplitude bins to accomplish this task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' Here the bayes  factor is re-normalized to B(A ∼ 0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' The second range is  where the combined bayes factor is B(A) ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' This range is  between ∼ (0,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='5) with uniform intervals of dA = 5×10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='  In order to get satisfying result this range needs to have sec-  ond priority in amplitude resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' The last range where the  bayes factor drops significantly from 1 doesn’t need to have  high resolution as it already disfavoured by model when the  events are combined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' This range is between (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='5,2) with uni-  form intervals of dA = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' However, for individual events  one may need high resolution in all the amplitudes as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='  I set 257 runs for each event and 65 × 257 = 16705 runs in  total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='  In order to obtain the overall bayes factor of individ-  ual events and combined events we just do BGR+echo  GR  =  1  Amax−Amin  � Amax  Amin B(A)dA with (Amin,Amax) = (0,2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' For com-  bined events I got BGR+echo  GR  ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' The result for individual  events and their histogram are reported in Table 1 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' 4  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='  3 Conclusion and discussions  I presented the outcome which gives measure of possibility  for preference of hGR+echoes over hGR based on Bayes factors  comparison of these two models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' The 65 analysed events in  Table 1 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' 4 for individual events show inconclusive  result in preference for GR or GR+echo, although with slight  preference for GR but not by much.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' The main scope and  result of the paper is combination of echoes for large number  of events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' We see that the combined bayes factor which is  BGR+echo  GR  ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='4 is still inconclusive about GR+echo and GR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='  It is realised that this combining method gives five order of  magnitude higher bayes factor compared to when we simply  3/6  GWTC-1  log10 BGR+echo  GR  GWTC-1  log10 BGR+echo  GR  GWTC-1  log10 BGR+echo  GR  GW150914  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='53  GW170608  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='05  GW170818  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='06  GW151012  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='05  GW170729  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='12  GW170823  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='25  GW151226  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='09  GW170809  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='08  GW170104  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='13  GW170814  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='30  GWTC-2  log10 BGR+echo  GR  GWTC-2  log10 BGR+echo  GR  GWTC-2  log10 BGR+echo  GR  GW190408_181802  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='16  GW190521  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='96  GW190728_064510  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='01  GW190412  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='09  GW190521_074359  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='54  GW190731_140936  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='15  GW190413_052954  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='03  GW190527_092055  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='01  GW190814  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='42  GW190413_134308  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='10  GW190602_175927  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='22  GW190828_063405  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='04  GW190421_213856  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='21  GW190620_030421  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='16  GW190828_065509  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='14  GW190424_180648  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='17  GW190630_185205  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='17  GW190910_112807  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='30  GW190503_185404  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='02  GW190706_222641  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='06  GW190915_235702  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='09  GW190512_180714  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='06  GW190707_093326  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='02  GW190924_021846  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='00  GW190513_205428  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='15  GW190708_232457  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='01  GW190925_232845  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='03  GW190514_065416  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='03  GW190719_215514  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='01  GW190929_012149  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='13  GW190517_055101  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='07  GW190720_000836  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='07  GW190519_153544  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='35  GW190727_060333  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='30  GWTC-3  log10 BGR+echo  GR  GWTC-3  log10 BGR+echo  GR  GWTC-3  log10 BGR+echo  GR  GW191109_010717  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='36  GW200112_155838  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='28  GW200219_094415  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='07  GW191129_134029  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='01  GW200128_022011  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='2  GW200220_061928  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='21  GW191204_171526  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='01  GW200129_065458  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='43  GW200224_222234  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='34  GW191215_223052  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='2  GW200202_154313  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='21  GW200225_060421  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='01  GW191216_213338  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='03  GW200208_130117  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='08  GW200302_015811  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='12  GW191222_033537  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='32  GW200209_085452  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='14  GW200311_115853  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='37  GW191230_180458  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='21  GW200216_220804  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='15  GW200316_215756  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='01  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' Results of bayes factor for GW echoes in GWTC-1, GWTC-2, and GWTC-3 events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' Positive value of the log10 bayes  factor indicates a preference for the GR+echoes model over GR model, while the negative value suggests instead a preference  for the GR model over the GR+echoes model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' Here GW190521 shows loudest echo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' Here based on44 all the individual events  appear as inconclusive to both GR or GR+echoes with GW190521 as exception!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='  combine the individual events bayes factor via multiplica-  tion ∏  i=Events  BiGR+echo  GR  = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='2 × 10−6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' In another words the  fact that the combined bayes factor for preference to GR has  dropped from ∼ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='6 × 105 to ∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='5 indicates that there are  still much to do in method improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' Additionally, the  large number of events and computational costs is a guarantee  against bayes factor hack making the result robust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' The only  event that has shown evidence for preference of GR+echo  model is GW190521 with BGR+echo  GR  = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='2 (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' This  is the most massive and confidently detected BBH merger  event observed to date5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' I refer the detailed interpretation and  investigation about this event to Abedi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' Presuming  a simple speculation that we can compare all the events as  same (echo model remain same for all the 65 events and their  echo amplitudes compare to main event amplitude doesn’t  change by much despite the change in initial condition of the  progenitor BBH mergers) and all the 65 BBH events should  show evidence for echo signals in this model and the space of  parameters considered in this search, I found an upper bound  amplitude A < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='42 (at 90% confidence level) for echoes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' I  remind the reader that bounds from our search only relate to  the family of echo waveforms considered here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='  It is worth to note that I didn’t see any evidence for echoes  in O1 in contrast to11,16,17, possibly because the model I  used here is different and has much suppressed amplitudes in  contrast to ADA model in11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='  In order to do a better search for quieter echoes, we might  need to have a more physical echo waveforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' In another  words, concrete models from alternatives to GR are needed to  use in PyCBC pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' Without better models, we might wait  for O4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' Observations will improve in number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' LISA, Einstein  Telescope, and Cosmic Explorer will make a big breakthrough  in sensitivity in search for alternatives to GR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='  References  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' Cardoso, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' Franzin and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' Pani, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' 116 (2016)  no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='17, 171101 [erratum: Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' 117 (2016) no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='8, 089902]  doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='1103/PhysRevLett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='116.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='171101 [arXiv:1602.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='07309 [gr-qc]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='  4/6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='75  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='75  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='00  log10[Evidence for echoes]  0  2  4  6  8  10  12  14  16  18  Number of events  Combined with same amplitude  GW190521  Barely worth mentioning  Substantial  Substantial  Histogram of events  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' Histogram of log10 bayes factors of 65 events in  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' Vertical regions identify Jeffreys scale for  interpretation of bayes factor44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' Cardoso, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' Hopper, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' Macedo, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' Palenzuela and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' Pani,  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' D 94 (2016) no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='8, 084031 doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='1103/PhysRevD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='084031  [arXiv:1608.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='08637 [gr-qc]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' Abedi, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' Afshordi, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' Oshita and Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' Wang, Universe 6 (2020) no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='3,  43 doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
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+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
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+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
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+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
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+page_content=' Afshordi, [arXiv:2201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
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+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
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+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
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+page_content='07363 [hep-th]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
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+page_content='  36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
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+page_content=' Chen, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
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+page_content='  Acknowledgements  I would like to thank Niayesh Afshordi and Alex B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' Nielsen  for helpful comments and discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' I also thank Conner  Dailey for suggestion about Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' I thank the Max Planck  Gesellschaft and the Atlas cluster computing team at AEI Han-  nover for support and computational help.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' I was supported by  ROMFORSK grant Project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' 302640.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' This research has  made use of data, software and/or web tools obtained from the  Gravitational Wave Open Science Center (https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='gw-  openscience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='org), a service of LIGO Laboratory, the LIGO  Scientific Collaboration and the Virgo Collaboration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' LIGO  is funded by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' National Science Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content=' Virgo is  funded by the French Centre National de Recherche Scien-  tifique (CNRS), the Italian Instituto Nazionale della Fisica  Nucleare (INFN) and the Dutch Nikhef, with contributions by  Polish and Hungarian institutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
+page_content='  6/6' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfPfZU/content/2301.00025v1.pdf'}
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+IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. XX, NO. XX, XXXX 2022
+1
+GraVIS: Grouping Augmented Views from
+Independent Sources for Dermatology Analysis
+Hong-Yu Zhou, Member, IEEE, Chixiang Lu, Liansheng Wang, Member, IEEE,
+and Yizhou Yu, Fellow, IEEE
+Abstract— Self-supervised representation learning has
+been extremely successful in medical image analysis, as it
+requires no human annotations to provide transferable rep-
+resentations for downstream tasks. Recent self-supervised
+learning methods are dominated by noise-contrastive es-
+timation (NCE, also known as contrastive learning), which
+aims to learn invariant visual representations by contrast-
+ing one homogeneous image pair with a large number of
+heterogeneous image pairs in each training step. Nonethe-
+less, NCE-based approaches still suffer from one major
+problem that is one homogeneous pair is not enough to
+extract robust and invariant semantic information. Inspired
+by the archetypical triplet loss, we propose GraVIS, which is
+specifically optimized for learning self-supervised features
+from dermatology images, to group homogeneous derma-
+tology images while separating heterogeneous ones. In ad-
+dition, a hardness-aware attention is introduced and incor-
+porated to address the importance of homogeneous image
+views with similar appearance instead of those dissimilar
+homogeneous ones. GraVIS significantly outperforms its
+transfer learning and self-supervised learning counterparts
+in both lesion segmentation and disease classification
+tasks, sometimes by 5 percents under extremely limited
+supervision. More importantly, when equipped with the pre-
+trained weights provided by GraVIS, a single model could
+achieve better results than winners that heavily rely on
+ensemble strategies in the well-known ISIC 2017 challenge.
+Code is available at https://bit.ly/3xiFyjx.
+Index Terms— self-supervised learning, dermatology di-
+agnosis, triplet loss, self-supervised pre-training
+I. INTRODUCTION
+Considering the protection of patients’ privacy and limi-
+tation of accessible expertise, self-supervised learning (SSL)
+[1]–[7] has been widely adopted in medical image anal-
+ysis to learn transferable feature representations in the
+stage of pre-training. Recent state-of-the-art SSL approaches
+are often based on noise-contrastive estimation (NCE) [8]
+(i.e., contrastive loss or contrastive learning), which learn
+transformation-invariant features by contrasting a few ho-
+mogeneous image pairs with quite a large number (usually
+(Corresponding author: Yizhou Yu.)
+Hong-Yu Zhou, Chixiang Lu and Yizhou Yu are with the Department
+of Computer Science, The University of Hong Kong, Pokfulam, Hong
+Kong
+(e-mail:
+whuzhouhongyu@gmail.com,
+luchixiang@gmail.com,
+yizhouy@acm.org).
+Liansheng Wang is with the Department of Computer Science, Xi-
+amen University, Siming District, Xiamen, Fujian Province, P.R. China
+(e-mail: lswang@xmu.edu.cn).
+First two authors contributed equally.
+a. triplet loss
+b. NCE
+c. our GraVIS
+anchor view
+positive view
+negative view
+Fig. 1.
+Comparison with the archetypical triplet loss (a) and noise-
+contrastive estimation (b. NCE). The hollow arrow denotes the direction
+to be pushed, where positive views are pushed towards the anchor
+view and negative views are pushed away. Here, one anchor view and
+one positive view construct one homogeneous pair. Accordingly, one
+heterogeneous pair consists of one anchor view and one negative view.
+To generate positive views, we randomly augment the same image as
+used to generate the anchor view, while negative views are produced by
+augmenting different images.
+tens of thousands) of heterogeneous pairs [1, 4, 5] in each
+training step. However, the above characteristic may cause
+one intuitive problem in practice, which is the limited number
+of homogeneous pairs in contrastive analysis may prevent
+the model from capturing the most representative semantic
+features.
+On the other hand, in dermatology analysis, there are about
+1,500 skin diseases and many variants that can be located in
+different body parts1. Although the medical imaging commu-
+nity has proposed several publicly available large-scale derma-
+tology datasets, such as HAM10000 [9] and BCN20000 [10],
+to address the limitation of data and labels for training neural
+networks, there still exists a great need of labeled dermatology
+images, especially when we consider the domain shift problem
+that often emerges in small-scale multi-domain, multi-center
+and multi-device datasets. From a different perspective, SSL
+provides a solution to utilize unlabeled dermatology data for
+pre-training, from which transferable images representations
+can be learned and serve as the feature basis for downstream
+1https://dermnetnz.org/cme/principles/
+an-overview-of-dermatology/
+arXiv:2301.04410v1  [cs.CV]  11 Jan 2023
+
+EMB
+NPS
+UFFC
+SignalProcessing Society
+0222
+IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. XX, NO. XX, XXXX 2022
+small-scale datasets.
+In this paper, we introduce Grouping augmented Views
+from Independent Sources (GraVIS) to learn self-supervised
+dermatology image representations. The core idea behind
+GraVIS is intuitive, which is good representations should be
+able to aggregate homogeneous image views while separating
+heterogeneous views. Here, the term homogeneous views in-
+clude one anchor view and at least one positive view, which
+are all generated by randomly augmenting the same source
+image. The heterogeneous views involve one anchor view and
+at least one negative view, where each negative image view
+is generated by augmenting an image that is different from
+the one used for homogeneous views. Specifically, we use
+homogeneous pair to refer to an image pair that consists
+of one anchor view and one positive view. Similarly, each
+heterogeneous pair contains one anchor view and one negative
+view.
+In Fig. 1, we contrast our GraVIS with the canonical triplet
+loss [11] and NCE [8]. In practice, GraVIS addresses the
+aforementioned major problem of NCE (i.e., few homoge-
+neous pairs) by applying N-times augmentation to a given
+source dermatology image in order to generate one anchor
+view and N-1 positive views (i.e., N-1 homogeneous pairs).
+Meanwhile, GraVIS reduces the number of negative views to
+hundreds by taking into account one specific characteristic
+of dermatology images, which is heterogeneous dermatology
+image pairs are easy to recognize and thus we do not need a
+large number of heterogeneous pairs for contrast. Compared
+to the triplet loss, our method aims to aggregate multiple
+homogeneous views and separating heterogeneous views, si-
+multaneously. Compared to NCE, GraVIS addresses more on
+grouping multiple homogeneous views for capturing the most
+general and transferable information hidden in the data. To
+achieve these goals, GraVIS introduce View Grouping Loss
+(VGL) to adaptively aggregate different types of views, where
+the hardness-aware attention is proposed to help the model
+focus more on homogeneous views with similar appearance
+by assigning larger penalty to them. To summarize, our
+contributions can be organized as follows:
+• A novel self-supervised learning methodology, GraVIS,
+is proposed for conduct unsupervised pre-training on
+dermatology images. GraVIS is inspired by the archetyp-
+ical triplet loss and utilizes a similarity loss instead of
+the recent NCE to learn transferable dermatology image
+representations.
+• In comparison to contrastive SSL approaches, GraVIS
+brings improvements by laying emphasis on homoge-
+neous pairs instead of heterogeneous ones. We introduce
+a hardness-aware attention to adaptively assign different
+degrees of penalty to homogeneous pairs based on each
+pair’s similarity score.
+• Extensive experiments are conducted on two tasks: lesion
+segmentation and disease classification. Our GraVIS dis-
+plays substantial and significant improvements over other
+SSL methods. More importantly, by fine-tuning the pre-
+trained weights acquired using GraVIS, we achieve quite
+competitive results using a SINGLE model on each task,
+even compared to the winners of ISIC 2017 challenges.
+II. RELATED WORK
+1) Contrastive and non-contrastive self-supervised learning:
+Contrastive learning aims to learn invariant representations
+of images via noise-contrastive estimation (NCE) [8]. Doso-
+vitskiy et al. [12] considered each image as its own class
+and used a linear classifier to classify each image. Wu et al.
+[13] replaced the classifier with a memory bank that stores
+previously-computed representations as the former approach
+is unpractical in big datasets. Based on the memory bank,
+Misra et al. [14] proposed to learn invariant representations
+based on pretext tasks. Similarly, Tian et al. [15] tried to
+maximize mutual information between different views of the
+same scene. CPC [16] and CPCv2 [17] combined predicting
+future observations (predictive coding) with a probabilistic
+contrastive loss. Chen et al. [4] introduced SimCLR, which
+shows that as long as the batch size for pre-training is
+large enough, well transferable image representations can be
+directly learned from the image batch without the memory
+bank. He et al. [18] introduced MoCo, which improves the
+training of contrastive methods by comparing representations
+from the momentum network and the ordinary network. To
+replace the memory bank and solve the memory issue of
+SimCLR (caused by large batch sizes), MoCo utilizes a queue
+to store past representations. Caron et al. [19] incorporated
+deep clustering strategies to facilitate contrastive learning by
+enforcing consistency between cluster assignments produced
+for different augmentations. Hjelm et al. [20] further showed
+the necessity of employing a large number of negative samples
+in the training batch to achieve satisfactory performance.
+Different from the above contrastive approaches that need a
+large number (tens of thousands) of heterogeneous pairs for
+contrast, our GraVIS is specifically designed for dermatology
+images, which lays more emphasis on homogeneous pairs
+and only requires hundreds of heterogeneous pairs to learn
+invariant and transferable image representations. On the other
+hand, there are some recent works [21]–[25] trying to replace
+the contrastive procedure with the siamese learning. Different
+from them, our GraVIS is inspired by the archetypical triplet
+loss.
+2) Triplet loss and triplet mining: The objective of the triplet
+loss is to minimize the distance between the anchor and posi-
+tive sample while maximizing the distance between the anchor
+and negative sample. Schroff et al. [26] first applied the triplet
+loss to face recognition and verification using deep neural
+network. Cheng et al. [27] introduced an improved version of
+the triplet loss to pull the instances of the same person closer,
+and at the same time push the instances belonging to different
+persons farther from each other in the learned feature space.
+Xie et al. [28] used the triplet loss to learn representations
+for nuclei segmentation. Puch et al. [29] implemented the
+triplet loss in the setting of few-shot learning to recognize
+brain imaging modality. When using the triplet loss to train
+neural networks, triplet mining is a widely adopted strategy to
+select hard [30] or semi-hard [26, 31] triplets for reducing the
+computational cost and accelerating training process. Oh Song
+et al. [32] incorporated batch-wise hard triplet mining into the
+training process of deep neural networks. In our GraVIS, we
+
+AUTHOR zhou et al.: GRAVIS: GROUPING AUGMENTED VIEWS FROM INDEPENDENT SOURCES FOR DERMATOLOGY ANALYSIS
+3
+Input
+batch
+Augmentation
+Augmented
+batch
+Backbone
+Representation
+batch ������������
+…
+…
+…
+…
+…
+…
+…
+…
+������������
+������������
+View
+grouping
+. . .
+…
+…
+…
+…
+…
+…
+…
+������������
+������������
+ℒ1
+VGL
+ℒ������������
+VGL
+× ������������
+. . .
+Backbone
+Backbone
+Backbone
+1st random
+augmentation
+2nd random
+augmentation
+N-th random
+augmentation
+������������-1 positive views
+. . .
+Backbone
+Backbone
+Backbone
+1st random
+augmentation
+2nd random
+augmentation
+N-th random
+augmentation
+������������ negative views
+Anchor view
+. . .
+1st image
+������������-th image
+Different source images in the input batch
+Fig. 2.
+Overview of our proposed GraVIS. By augmenting the input
+image batch by N times, we can acquire the augmented image batch,
+which is then passed to the backbone network to get a batch of
+representations, denoted as Q. Next, we apply view grouping loss to
+Q, where we randomly select a view representation as the anchor view
+(marked using red). We treat those representations that share the same
+source image with the anchor view as positive views (marked using
+green). Accordingly, representations that come from different source
+images are defined as negative views (marked using blue).
+introduce an attention mechanism to help the network focus
+on the most similar homogeneous pairs while avoiding being
+dominated by dissimilar homogeneous pairs. This is different
+from the traditional triplet mining methods that often address
+the negative samples (i.e., heterogeneous pairs). It should be
+mentioned that the idea of addressing the importance of similar
+homogeneous pairs has been introduced by BagLoss [33]
+to improve the performance of image retrieval models. The
+technical difference between BagLoss and GraVIS lies in
+the way to generate and utilize negative samples. BagLoss
+picks the hardest negative sample from the dataset, while
+GraVIS dynamically constructs negative samples using various
+augmentation strategies and uses all of them to learn invariant
+representations.
+3) Self-supervised learning in medical image analysis: Be-
+sides NCE-based SSL approaches, restoring/reconstructing the
+input images is the mostly adopted way to provide intrinsic su-
+pervision for learning self-supervised medical image represen-
+tations. Chen et al. [34] used context restoration to learn self-
+supervised representations on 2D ultrasound image, 3D ab-
+dominal Computerized Tomography (CT) scans and 3D brain
+Magnetic Resonance Imaging (MRI) scans. Zhou et al. [1]
+introduced Model Genesis that applies various augmentation
+strategies to build diverse reconstruction targets for both 2D
+and 3D medical images. Haghighi et al. [35] modified Model
+Genesis by adding a classification branch to classify high-level
+features. Zhuang et al. [36] introduced a jigsaw problem on
+reconstructing 3D brain MRI scans, which is then improved by
+[37] with a generative adversarial network to recover shuffled
+3D patches. For NCE-based methods, Zhou et al. [2] pro-
+posed C2L which first applied NCE to 2D radiographs. C2L
+learns more invariant radiograph representations by contrasting
+homogeneous with heterogeneous pairs in both image- and
+feature-space. Taleb et al. [38] investigated the performance of
+both NCE and self-reconstruction based approaches and found
+contrastive SSL methods are more advantageous. Besides self-
+supervised learning methodologies, Ke et al. [39] made a
+range of experiments to study the influence of ImageNet-
+�� ������ ����
+       �� positive ����� 
+          �� positive �����
+�� negative �����
+�� negative �����    �� negative ����3
+Fig. 3.
+An example of the formation of the anchor view, positive views
+and negative views. Note that views a, b and c come from the same
+source image while views d, e and f are from a different source image.
+Thus, view pairs (a, b) and (a, c) are homogeneous. Pairs (a, d), (a, e)
+and (a, f) are heterogeneous. All views are generated via different types
+and degrees of augmentation.
+based pre-training on chest X-ray models. Different from the
+above works, we introduce a new SSL methodology which
+replies neither on NCE nor context restoration but surpasses
+both of them by substantial and significant margins. Recently,
+Azizi et al. [3] introduced a modified version of SimCLR (i.e.,
+SimCLR-Derm) and a multi-instance version of SimCLR (i.e.,
+MiCLE) to dermatology images. The second version utilizes
+the consistency of multiple images from a single patient. Vu
+et al. [40] also employed a similar idea on X-ray data to
+make use of multiple views from the same patient to construct
+homogeneous pairs. For implementation, these two approaches
+require the access to the patient meta data in order to know
+which images come from the same patient. In contrast, our
+GraVIS does not have this constraint.
+III. METHODOLOGY
+A. Overview
+Grouping augmented Views from Independent Sources
+(GraVIS) aims at conducting pre-training using a large number
+of unlabeled dermatology images, after which we fine-tune the
+pre-trained model on different downstream tasks to verify the
+transferable ability of learned representations. We provide an
+overview of our GraVIS in Fig. 2. In each training step, we
+first forward a batch of dermatology images to the backbone
+network in order to acquire their representations, on top of
+which the view grouping loss (VGL) is applied to aggregate
+homogeneous samples while separating heterogeneous ones.
+Compared to recent contrastive learning based approaches
+[2, 3], GraVIS does not need any contrastive operations
+but still displays state-of-the-art performance on a variety of
+downstream tasks.
+B. Positive and negative sets
+Given an input image batch, we first randomly augment
+each image in this batch by N times. The augmentation
+strategies mainly include random crop, random horizontal flip,
+random rotation and color distortion. We provide an example
+in Fig. 3 where we display one anchor view, two positive
+
+4
+IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. XX, NO. XX, XXXX 2022
+a. anchor view
+b. view1
+d. view3
+Hard
+Easy
+c. view2
+Easy
+Fig. 4.
+Hardness of recognizing homogeneous image pairs based
+on appearance. It is observable that the anchor view (a), view1 (b)
+and view2 (c) are from the same source image as they look similar
+in appearance. However, although view3 (d) is randomly cropped from
+the same source image as that of the anchor view, we can hardly tell
+whether they come from the same source without expertise as view3
+has quite different appearance and there is no clear overlap between
+them. Thus, it may not be reasonable to directly ask the representation
+of view3 to be as close to the representation of anchor view as other
+positive features.
+views and three negative views. We apply random crop and
+random rotation (a small degree) to the source dermatology
+image to acquire the anchor view (a). To generate the positive
+view1 (b), we include random crop and random rotation with
+a relatively large degree. For positive view2 (c), we include
+random crop, horizontal flip, random rotation and gaussian
+blur. Similar combinations of augmentation strategies can also
+be observed in negative views (d, e and f). Supposing the
+size of the input image batch is B (we ignore the height,
+width and channel number of images for simplicity), after
+N-times random augmentation, we obtain an enlarged image
+batch whose size is B*N. During the pre-training stage,
+in each training step, we pass this enlarged batch to the
+backbone network to acquire a batch of corresponding visual
+representations (denoted as batch Q in the following), from
+which we randomly pick a representation and regard it as
+the anchor representation. Given this anchor view, its positive
+set contains the rest N − 1 views (i.e., representations) that
+are derived from the same source image. Intuitively, the rest
+(B−1)*N views construct the negative set as they correspond
+to different source dermatology images. Based on positive
+and negative sets, we calculate VGL to minimize the distance
+between the anchor view and its positive set while keeping the
+anchor away from those views in the negative set.
+C. View grouping loss
+In GraVIS, we define the view grouping loss (VGL) to be:
+LVGL
+q
+= 1 −
+1
+∥Sps∥
+�
+i∈Sps
+1
+γq,i
+�
+j∈Sns
+σ(cq,j − cq,i) + 1
+,
+γq,i =
+1
+�
+j∈Sps,j̸=i
+σ(cq,j − cq,i)
+.
+(1)
+LVGL
+q
+stands for VGL with q-th sample in batch Q as the
+anchor view. Accordingly, Sps denotes the positive set and
+Sns stands for the negative set. ∥Sps∥ refers to the number
+of views in Sps. ci,j measures the similarity between the i-th
+Algorithm 1 GraVIS:
+1: Notations: NET denotes the backbone network. K stands
+for the number of training steps. ITER is the index of
+iteration.
+2: for ITER ← 0 to K do
+3:
+Randomly sample a batch of images Xo;
+4:
+Augment Xo by N times to acquire an enlarged image
+batch Xe;
+5:
+Pass Xe to NET to get Q;
+6:
+Initialize LQ to 0;
+7:
+for q ← 0 to B ∗ N do
+8:
+LQ += LVGL
+q
+.;
+9:
+end for
+10:
+Backward (LQ) to update NET;
+11: end for
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+Similarity Gap (cq, 3
+cq, 6)
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+0.35
+Gradient Gap (|
+Q
+cq, 3|
+|
+Q
+cq, 6|)
+Impact of  (gradient)
+w/ 
+w/o 
+(a) Impact of γ on the gradient
+0.605
+0.610
+0.615
+0.620
+0.625
+0.630
+Similarity Gap (cq, 3
+cq, 6)
+0.1
+0.2
+0.3
+0.4
+0.5
+Loss Value
+Impact of  (loss)
+w/ 
+w/o 
+(b) Impact of γ on the loss value
+Fig. 5.
+Impact of the hardness-aware attention γ. The left figure
+(a) shows that the introduced attention γ focuses more on the similar
+homogeneous pair (vq, vps
+3 ) (the similarity score is cq,3) instead of the
+dissimilar pair (vq, vps
+6 ) (the similarity score is cq,6). The right figure
+(b) demonstrates that adding γ makes the view grouping loss more
+sensitive to the similarity gap between two homogenenous pairs.
+sample and the j-th one, which is calculated using the cosine
+similarity:
+ci,j =
+vT
+i vj
+∥vi∥∥vj∥,
+(2)
+where vi and vj correspond to the i-th and j-th feature vectors
+in batch Q, respectively. In Equation 1, σ(·) stands for the
+sigmoid function:
+σ(x) =
+1
+1 + e
+−x
+τ
+(3)
+with the temperature hyper-parameter τ controlling how much
+GraVIS addresses similar image pairs.
+Term
+�
+j∈Sns
+σ(cq,j − cq,i) evaluates cq,i among the negative
+set (Sns). By aggregating the anchor view (q) and positive
+view (i) and separating the anchor view from negative views
+(∀j ∈ Sns),
+�
+j∈Sns
+σ(cq,j − cq,i) becomes smaller and LVGL
+q
+is
+accordingly minimized.
+The hardness-aware attention γq,i evaluates cq,i among the
+positive set (Sps). The intuition behind is that we believe
+more attention should be allocated to homogeneous pairs of
+similar views (in appearance) instead of dissimilar (though
+homogeneous) pairs. We give an example to illustrate our
+intuition in Fig. 4, where one anchor view and three positive
+views are displayed. It is easy to recognize the relationship
+
+AUTHOR zhou et al.: GRAVIS: GROUPING AUGMENTED VIEWS FROM INDEPENDENT SOURCES FOR DERMATOLOGY ANALYSIS
+5
+between the anchor view and view1 (b) as both of them are
+overlapped after being cropped from the same source image.
+Similar phenomenon can also be observed between the anchor
+view and view2 (c). However, when we compare the anchor
+view with view3 (d), we can hardly tell whether they two
+share the same origin without relying on expertise because
+they have quite dissimilar appearance. In fact, it may not
+be suitable to require the representation of view3 to be as
+close to the anchor representation as other homogeneous pairs
+(i.e., forcing ca,3 ≈ ca,1 ≈ ca,2 is questionable). To address
+this issue, the hardness-aware attention γq,i tends to assign
+a large penalty to similar homogeneous pairs by taking into
+account their similarity (cq,i) in all homogeneous pairs (cq,j,
+∀j ∈ Sps, j ̸= i). In practice, the hardness-aware mechanism
+helps GraVIS to focus more on recognizing potentially “easy”
+relationships instead of regularizing meaningless dissimilar
+homogeneous pairs.
+We provide an example to illustrate the impact of γ on
+VGL. Considering three positive homogeneous view vectors
+(i.e., Sps={vps
+1 , vps
+3 , vps
+6 }) and three negative view vectors (i.e.,
+Sns={vns
+2 , vns
+4 , vns
+5 }), all of which are ordered based on the
+similar score with the anchor view in a descending order, i.e.,
+{vps
+1 , vns
+2 , vps
+3 , vns
+4 , vns
+5 , vps
+6 },
+s.t. ∀i ∈ {1, 2, 3, 4, 5}, cq,i − cq,i+1 = C,
+(4)
+where cq,i stands for the similarity score between the i-th
+view and the anchor view. For simplicity, we define the score
+interval between two adjacent views to be a positive constant
+C. As aforementioned, the proposed hardness-aware attention
+γ helps VGL to lay more emphasis on shifting easy samples.
+In Fig. 5(a), we can see that as the similarity gap between
+cq,3 and cq,6 becomes larger, VGL (with the attention factor
+γ) assigns larger gradients (denoted as | ∂Q
+∂cq,3 | − | ∂Q
+∂cq,6 |) to the
+more similar homogeneous pair (vq, vps
+3 ) instead of the less
+similar homogeneous pair (vq, vps
+6 ). Moreover, in Fig. 5(b), we
+observe that equipped with γ, VGL becomes more sensitive
+to the similarity gap between two homogenenous pairs, which
+again verifies the impact of the hardness-aware attention on
+handling dissimilar homogeneous pairs.
+Finally, for each batch Q, we calculate the overall loss by
+summing up all LVGL
+q
+, which can be presented as follows:
+LQ =
+B∗N
+�
+q=1
+LVGL
+q
+.
+(5)
+1) Comparison
+with
+noise-contrastive
+estimation:
+Here
+we would like to clarify the advantages of proposed VGL
+over the noise-contrastive estimation (NCE) [8], which has
+been widely used in self-supervised visual representation
+learning [2, 4, 5]. Similar to VGL, NCE contrasts the
+homogeneous pair with multiple heterogeneous pairs to
+learn invariant representations; however, it suffers from two
+issues when dealing with dermatology images: i) the locality
+problem and ii) the equality problem. For issue i), given
+an anchor view in the input image batch, the archetypical
+NCE compares it with only one homogeneous view (i.e.,
+one homogeneous pair), which cannot guarantee the learned
+semantic information are robustness enough towards a variety
+of transformations (i.e., augmentation strategies). For issue
+ii), NCE treats all homogeneous pairs equally regardless of
+the degree of applied augmentations. As we have mentioned
+in Fig. 4, for dissimilar homogeneous views (e.g., the anchor
+view and view3), it may not be appropriate to force their
+representations to be close to each other. In contrast, our
+GraVIS addresses issue i) by introducing more homogeneous
+pairs
+to
+learn
+more
+invariant
+semantic
+representations.
+To tackle issue ii), VGL comprises the hardness-aware
+attention to assign adaptive penalty on top of the hardness of
+recognizing the relationship between two homogeneous views.
+2) Comparison with the triplet loss: Each time, the triplet
+loss contrasts one homogeneous pair with one heterogeneous
+pair as follows:
+Ltriplet(vq, vps, vns) = max(0, m + d(vq, vps) − d(vq, vns))
+(6)
+where vq, vps and vns denote the query representation (i.e.,
+the anchor view), positive and negative representations, respec-
+tively. m is a pre-defined margin. d stands for the distance
+function, where euclidean distance is mostly used and cosine
+similarity is also applicable. Compared to GraVIS, the triplet
+loss does not require to apply N-times augmentation, which
+lead to only one homogeneous pair and one heterogeneous
+pair in each computation. Thus, the learned representations
+are much less invariant than those learned with VGL or
+NCE, which may be the reason why the triplet loss has not
+been widely used in self-supervised representation learning.
+Another drawback of the triplet loss is that it linearly combines
+each term, ignoring the difference and importance of different
+homogeneous or heterogeneous pairs. As a result, the training
+process may be less effective as it is dominated by the trivial
+information from a few meaningless yet hard samples. For
+instance, it is possible that vq and vps originate from two
+augmented images that share the same source image but have
+dramatically different appearance (such as view1 and view3 in
+Fig. 4). In this case, minimizing the distance between vq and
+vps becomes meaningless, especially when vns comes from a
+dermatology image that suffers from the same skin disease as
+that of vq and vps. In contrast, our GraVIS is able to reduce
+the importance of such dissimilar homogeneous pairs using
+the hardness-aware attention.
+IV. EXPERIMENT
+A. Dataset
+We evaluate GraVIS on ISIC 2017 dataset [41]. The num-
+bers of images in the training, validation and test sets are 2000,
+150 and 600, respectively. Lesions in dermatology images
+are all paired with a gold standard (definitive) diagnosis,
+i.e. melanoma, nevus, and seborrheic keratosis. There are
+three major tasks in ISIC 2017 Challenge and we evaluate
+our framework on lesion segmentation (task1) and disease
+classification (task2). In disease classification, we conduct
+two binary classification sub-tasks: i) distinguishing between
+(a) melanoma and (b) nevus and seborrheic keratosis; ii)
+distinguishing (a) seborrheic keratosis from (b) nevus and
+melanoma. The evaluation metrics are jaccard index and mean
+
+6
+IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. XX, NO. XX, XXXX 2022
+TABLE I
+EXPERIMENTS IN LESION SEGMENTATION AND DISEASE CLASSIFICATION TASKS. DIFFERENT RATIOS STAND FOR DIFFERENT AMOUNTS OF DATA
+FOR FINE-TUNING. WE REPORT THE JACCARD INDEX FOR LESION SEGMENTATION AND MEAN ACCURACY FOR DISEASE CLASSIFICATION.
+Lesion segmentation
+Disease classification
+Methodology
+10%
+20%
+30%
+40%
+50%
+100%
+10%
+20%
+30%
+40%
+50%
+100%
+TS
+58.9
+61.3
+63.4
+69.5
+70.5
+72.3
+78.3
+79.7
+80.4
+81.8
+82.5
+85.9
+IN
+68.4
+70.1
+71.8
+72.1
+72.5
+74.1
+80.1
+83.5
+84.1
+84.3
+85.1
+87.0
+NPTL
+66.7
+68.9
+70.4
+70.8
+71.1
+73.9
+79.2
+81.7
+83.4
+83.8
+84.2
+86.6
+SimCLR-Derm
+67.7
+69.6
+70.8
+71.4
+71.5
+73.9
+80.5
+83.8
+84.3
+84.6
+85.3
+86.5
+MoCov2
+68.0
+70.0
+71.9
+72.3
+72.6
+74.5
+83.1
+83.8
+85.0
+85.7
+85.8
+87.8
+BYOL
+68.3
+69.6
+72.2
+72.7
+73.0
+74.9
+81.4
+83.3
+85.3
+85.8
+86.0
+86.7
+BagLoss
+63.7
+68.8
+69.5
+70.8
+71.2
+73.3
+76.8
+80.3
+81.2
+81.7
+82.4
+83.5
+C2L
+68.9
+70.7
+71.9
+72.4
+72.5
+74.4
+81.3
+84.7
+85.2
+85.7
+85.8
+87.6
+MG
+66.2
+68.9
+70.3
+70.7
+71.3
+72.8
+77.1
+78.7
+79.2
+80.8
+81.7
+85.9
+Our GraVIS
+70.9
+72.6
+73.0
+73.3
+74.0
+75.7
+85.9
+86.4
+86.7
+86.9
+87.0
+89.1
+accuracy for task1 and task2, respectively, which are also used
+in the released official leaderboard of ISIC 2017.
+B. Implementation details
+1) Details of pre-training: We first remove the labels of all
+dermatology images in the training set to pre-train the back-
+bone network using GraVIS. Specifically, the size of the input
+image batch is 32 and the number of random augmentation
+N is 20. We employ SGD with momentum as the default
+optimizer whose initial learning rate is set to 1e-3 while the
+momentum value is set to 0.9. We employ the cosine annealing
+strategy for learning rate decay and train the network for 240
+epochs. We use ResNet50 [18] with two fully-connected layers
+as the backbone network whose output dimension is 1×128.
+The pre-training procedure is finished on 4 NIVIDA TITAN
+V GPUs within 8 hours. In each training step, the augmented
+image batch contain B*N images, where all positive images
+are continuous and thus the model is likely to find a short
+cut by directly making use of the location information (i.e.,
+the model knows which images are from the same source and
+which are not). So we shuffle the batch before we pass it to the
+backbone network, which prevents the network from finding
+a trivial solution that may lead to trivial representations.
+2) Details of fine-tuning: In the fine-tuning stage, we remove
+two fully-connected layers used in the pre-training stage and
+load the weight parameters of convolutional layers. In task1,
+we use ResU-Net [42] for lesion segmentation, where we load
+pre-trained ResNet as the encoder and randomly initialize the
+corresponding decoder. The batch size for segmentation task is
+32. The loss function includes both dice loss and cross entropy
+loss whose coefficients are 1 and 0.2, respectively. In task2,
+we add a linear classification head with a dropout (p = 0.2)
+layer before it. The batch size of classification task is 64 and
+the loss function is binary cross entropy loss. For all tasks, we
+unfreeze the weights of convolutional layers and fine-tune the
+entire network. Validation score (accuracy or jaccard index)
+is obtained after each training epoch and then we anneal the
+learning rate by a factor of 0.5 if the validation score is not
+improved after 3 epochs. Different from the pre-training stage,
+we employ Adam as the optimizer and the initial learning
+rate is set to 1e-4. We stop the fine-tuning process when the
+validation score does not decrease for 10 epochs, and we save
+the checkpoint with the lowest validation loss value for testing.
+To evaluate the ability of GraVIS under different amounts of
+labeled data, we randomly sample 10% to 50% labeled data
+from the whole training set to fine-tune the model.
+C. Baselines
+Training from scratch (TS) and ImageNet-based initializa-
+tion (IN) are involved as two basic baselines. For contrastive
+approaches, we include SimCLR-Derm [3], MoCov2 [43],
+BYOL [21] and C2L [2] for comparison. As for traditional
+self-supervised learning methods, we compare GraVIS against
+Model Genesis (MG) [1], N-pairs triplet loss with the triplet
+mining strategy (NPTL) [44] and BagLoss [33], where the last
+two methods are based on metric learning.
+D. Boosting performance under limited supervision
+1) Lesion segmentation: We present experimental results
+of fine-tuning under different labeling ratios in Table I.
+Firstly, it is obvious that all representation learning (i.e.,
+transfer learning and self-supervised learning) methods can
+distinctly boost the performance compared to TS, verifying
+the necessity of learning transferable representations during
+the pre-training stage. Secondly, we can easily find that self-
+supervised approaches based on NCE are advantageous over
+the predictive method (i.e., MG) under different ratios, which
+implies that NCE does perform better than the traditional
+predictive representation learning for dermatology images.
+Somewhat surprisingly, we found that an improved multi-pair
+version of triplet loss (i.e., NPTL) can already achieve
+comparable results with MG. Considering MG needs various
+pre-defined restoration tasks, such phenomenon suggests
+the potential of applying the triplet loss to self-supervised
+representation learning. Among all NCE-based baselines,
+BYOL achieves the best performance in large labeling ratios
+while C2L performs better in small ratios. When comparing
+our GraVIS against other baselines, it is observable that
+GraVIS has the ability to outperform other baselines in
+
+AUTHOR zhou et al.: GRAVIS: GROUPING AUGMENTED VIEWS FROM INDEPENDENT SOURCES FOR DERMATOLOGY ANALYSIS
+7
+various ratios by obvious margins. More importantly, GraVIS
+surpasses the best performing self-supervised baselines C2L
+and BYOL by 2 percents under extremely limited supervision
+(10% and 20%) without explicitly using any contrastive
+functions. Compared to IN using 100% labeled data (the
+most widely used method for tackling limited annotations),
+GraVIS achieves comparable results using only 50% labeled
+dermatology images. It is worth noting that similar to the
+introduced hardness-aware attention, BagLoss addresses the
+importance of aggregating multiple homogeneous pairs in
+image retrieval, which is the reason why we include it as
+one of our baselines. From Table I, we can see that the
+performance of BagLoss are not satisfactory. The reason
+behind is that BagLoss only assigns one hard negative sample
+to each anchor view, which may prevent the model from
+learning invariant and discriminative representations from a
+number of negative views.
+Statistical significance. A t-test validation is conducted
+in all labeling ratios. The p-values between GraVIS and
+C2L/BYOL are smaller than 0.01, which indicates that
+GraVIS is statistically better than C2L/BYOL at the 1%
+significance level. In addition, the p-values between GraVIS
+(using 50% annotations) and IN (using 100% labeled data)
+are much larger than 0.05, which shows that GraVIS produces
+competitive results compared to the most widely used fully-
+supervised baseline (IN-100%).
+2) Disease classification: The best performing baselines in
+this task are MoCov2 and C2L, where MoCov2 maintains
+relatively obvious advantages when the labeling ratio is 10%.
+In comparison to the performance on lesion segmentation, our
+GraVIS maintains relatively larger advantages over different
+baselines in the problem disease classification. For instance,
+GraVIS outperforms MoCov2 by nearly 3 percents when
+using 10% labeled data. We believe these large improvements
+brought by GraVIS can be attributed to the strong demand of
+annotations in distinguishing three diseases (i.e., melanoma,
+nevus and seborrheic keratosis), where GraVIS can make
+use of limited annotations more effectively. Compared to the
+most widely adopted IN-100%, our approach again produces
+quite comparable performance by using only half labeled
+data. These phenomena suggest the potential of GraVIS to
+replace canonical self-supervised and transfer learning based
+methods to learn more effective pre-trained representations
+for dermatology images.
+Statistical significance. The p-values between GraVIS and
+the best performing baselines C2L/MoCov2 are smaller than
+0.01 under different labeling ratios. Similar to the results of
+lesion segmentation, we believe these p-values help verify the
+statistical effectiveness of GraVIS over C2L/MoCov2. Again,
+the p-values between GraVIS-50% and IN-100% are much
+larger than 0.05, suggesting that GraVIS can greatly reduce the
+amount of human annotations that are necessary for training
+a fully-supervised diagnosis model that is based on ImageNet
+pre-training.
+TABLE II
+COMPARISON WITH TOP-5 APPROACHES IN THE LEADERBOARD OF
+ISIC 2017 AND RECENT STATE-OF-THE-ART METHODS. NOTE THAT WE
+ONLY USE SINGLE MODELS FOR TESTING ON BOTH TASKS. 50% AND
+100% STAND FOR THE LABELING RATIOS OF ANNOTATED DATA USED IN
+THE FINE-TUNING STAGE.
+Rank
+Lesion segmentation
+Disease classification
+1
+Mt.Sinai [48]
+76.5
+Casio and Shinshu [49]
+91.1
+2
+NLP LOGIX [50]
+76.2
+Multimedia [51]
+91.0
+3
+USYD-BMIT [52]
+76.0
+RECOD Titans [53]
+90.8
+4
+USYD-BMIT [52]
+75.8
+USYD-BMIT [52]
+89.6
+5
+RECOD Titans [53]
+75.4
+A*STAR [54]
+88.6
+Xie et al. [46]
+80.4
+Bisa et al. [55]
+91.5
+GraVIS (50%)
+80.2
+Zhang et al. [45]
+91.7
+GraVIS (100%)
+81.0
+GraVIS (50%)
+91.4
+GraVIS (100%)
+92.1
+E. Advancing fully-supervised fine-tuning
+To demonstrate that our GraVIS can boost the performance
+of fully-supervised fine-tuning (denoted as GraVIS (100%)
+in Table II), we present a comparison of our method with
+top-5 results in the official leaderboard2 and recent state-of-
+the-art approaches [45, 46]. Like [45, 46], we also collected
+1320 additional dermoscopy images, including 466 melanoma,
+822 nevus images, and 32 seborrheic keratosis images, from
+the ISIC Archive3 to enlarge the training dataset. We also
+incorporate advanced pre- and post-processing (PP) techniques
+to improve the model performance. For both tasks, we apply
+aggressive rotation augmentation to each image, where we
+rotate each image 15 degrees more on the basis of the previous
+one. Thus, we have 24 (i.e.,
+360
+15 ) rotated versions given a
+source image. Note that the accompanying segmentation mask
+is also rotated and aligned with the rotated input. Specifically,
+for lesion segmentation, we add L (lightness) channel in
+CIELAB space [47] for better capturing the lesion boundary.
+We then resize each image to 192×256 (height×width), where
+a similar aspect ratio (3:4) is mostly used in the training
+set. To produce consistent predictions, we first use a high-
+confidence threshold (0.8) to determine the center of lesion,
+and then a middle-threshold (0.5) is used to highlight the lesion
+area. After filling holes, we ask the lesion area to contain the
+detected lesion center and take the maximum connected area
+as the final prediction. In practice, our GraVIS firstly pre-
+trains the ResU-Net (segmentation)/ResNet50 (classification)
+using the whole ISIC 2017 training set and the external data
+(without labels). Then, we fine-tune the pre-trained models
+on the labeled training set and external data.
+As shown in
+Table II, GraVIS achieves the highest jaccard index in lesion
+segmentation and ranks 1st in the disease classification task,
+outperforming approaches in the leaderboard and recent state-
+of-the-art methods. We believe these observations show that
+GraVIS has the potential for advancing the performance of
+fully-supervised models.
+To demonstrate the effectiveness of GraVIS in reducing the
+2https://challenge.isic-archive.com/leaderboards/2017
+3https://www.isic-archive.com/
+
+8
+IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. XX, NO. XX, XXXX 2022
+TABLE III
+INFLUENCE OF INTRODUCED HARDNESS-AWARE ATTENTION γ. W/ AND
+W/O DENOTES WITH AND WITHOUT, RESPECTIVELY.
+Method
+w/ γ
+w/o γ
+Jaccard index (%)
+75.7
+73.4
+amount of labeled data, we also present the results of fine-
+tuning GraVIS-based pre-trained models with 50% annotated
+data in Table II (denoted as GraVIS (50%)). We see that
+GraVIS trained with only 50% annotated data performs com-
+parably with the fully-supervised state-of-arts in both lesion
+segmentation (i.e., [46]) and disease classification (i.e., [45])
+tasks. We believe these results show the potential of GraVIS
+for serving as an annotation-efficient learning methodology.
+F. Ablation study
+In this section, we conduct ablative experiments for different
+modules and hyper-parameters to investigate their significance
+towards the performance of GraVIS. Specifically, the study
+targets consist of the hardness-aware attention, two factors that
+affect the number of positive and negative views: the number
+of augmentation N and the input batch size B. We also display
+the influence of the slope controller τ in Eq. 3. Last but not
+least, we demonstrate the effects of pre-training with respect
+to the number of pre-training epochs. Note that all ablation
+studies are fine-tuned with 100% labeled data in the task of
+lesion segmentation.
+1) Influence of hardness-aware attention γ: In Table III, we
+report the results equipped with γ and without γ. It is apparent
+that the proposed hardness-aware attention can bring sub-
+stantial improvements in the setting of fully-supervised fine-
+tuning. We believe the underlying reason is that γ prevents the
+model from laying too much emphasis on hard homogeneous
+pairs with quite different appearance that would cover up
+the effects of similar homogeneous pairs and dominate the
+training process. Without γ, GraVIS performs slightly worse
+than SimCLR-Derm but still better than MG, which indicates
+that GraVIS could become a useful self-supervised learning
+method even without the proposed attention mechanism.
+2) Investigation of N-times augmentation: One of the core
+ideas of GraVIS is to contrast N − 1 homogeneous pairs
+(excluding the anchor view) with (B − 1)*N heterogeneous
+image pairs. Intuitively, a larger N would bring more in-
+variance to learned representations but inevitably reduces the
+training efficiency. In Table IV, we study the influence of N by
+gradually increasing it. It is worth mentioning that N=0 means
+the anchor view is the same as the positive view, which leads to
+the worst result in Table IV. The reason behind is GraVIS can
+easily distinguish positive views from negative ones, reducing
+the discriminative ability of learned representations. When N
+is equal to 2, GraVIS degenerates into a similar format to the
+archetypical triplet loss but has more heterogeneous pairs. If
+we increase N to 10, the learned invariant features provide
+1.3 percents improvements over N=2. Obviously, as we add
+more augmented views, the overall segmentation performance
+continue to increase. However, we can find that N=40 is only
+TABLE IV
+INVESTIGATION OF N-TIMES AUGMENTATION.
+N
+0
+2
+10
+20
+40
+Jaccard index(%)
+52.3
+73.5
+74.8
+75.7
+75.8
+marginally better than N=20, implying that the performance
+becomes saturated in large values of N. Considering the
+need of training efficiency (i.e., N=40 costs much more time
+and GPU memory), we set N as the default value in all
+experiments.
+3) Influence of the batch size B: Different from N, the
+batch size B only influences the number of heterogeneous
+pairs in each loss computation. From Table V where we
+gradually increase the batch size from 16 to 128, we observe
+an obvious trend that is increasing the input batch size would
+not affect as much as increasing N. Thus, we can draw a
+conclusion that the number of heterogeneous pairs play a less
+important role in GraVIS compared to that of homogeneous
+pairs. These phenomena imply that B=16 already provides
+enough negative views (i.e., 16×(20-1)=304) for contrasting
+with the positive ones. In addition, the performance become
+saturated as we increase B to 128. Considering large batch
+sizes would consume too much more GPU memory for loss
+computation, we set B to 32 in all experiments.
+TABLE V
+INVESTIGATION OF THE BATCH SIZE B. N IS SET TO 20.
+B
+16
+32
+64
+128
+Jaccard index(%)
+75.4
+75.7
+75.7
+75.8
+TABLE VI
+COMPUTATIONAL OVERHEAD OF GRAVIS UNDER DIFFERENT N. WE
+REPORT THE AVERAGE TRAINING TIME (IN SECOND) PER EPOCH.
+N
+0
+2
+10
+20
+SimCLR-Derm
+-
+32
+-
+-
+-
+GraVIS (w/o γ)
+25
+47
+76
+115
+GraVIS (w/ γ)
+27
+51
+83
+125
+TABLE VII
+INVESTIGATION OF THE SENSITIVITY TO DIFFERENT AUGMENTATION
+STRATEGIES. RANDCROP, RANDFLIP, RANDROT, COLORDIST STAND
+FOR RANDOM CROP, RANDOM HORIZONTAL FLIP, RANDOM ROTATION
+AND COLOR DISTORTION, RESPECTIVELY.
+w/o RandCrop
+w/o RandFlip
+w/o RandRot
+w/o ColorDist
+Optimal
+GraVIS
+70.5
+74.5
+74.8
+72.8
+75.7
+4) Computational overhead: In Table VI, we present the
+computational overhead of GraVIS. Besides, we also inves-
+tigate the impact of adding the hardness-aware attention γ.
+Compared to SimCLR-Derm, our GraVIS naturally takes more
+training time because it includes more homogeneous and
+heterogeneous pairs in the view grouping loss as N increases.
+When N becomes 20 (the optimal choice), it costs about
+2 minutes to finish each epoch, which is approximate 4×
+
+AUTHOR zhou et al.: GRAVIS: GROUPING AUGMENTED VIEWS FROM INDEPENDENT SOURCES FOR DERMATOLOGY ANALYSIS
+9
+TABLE VIII
+INVESTIGATION OF THE INFLUENCE OF τ .
+τ
+0.01
+0.1
+0.2
+0.5
+1.0
+Jaccard index(%)
+73.5
+75.4
+75.7
+75.5
+74.3
+−8
+−6
+−4
+−2
+0
+2
+4
+6
+8
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+(a) τ = 0.01
+−8
+−6
+−4
+−2
+0
+2
+4
+6
+8
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+(b) τ = 0.2
+−8
+−6
+−4
+−2
+0
+2
+4
+6
+8
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+(c) τ = 0.5
+Fig. 6. Sigmoid functions with different temperature values (τ).
+of SimCLR-Derm. On the other hand, we found that the
+hardness-aware attention γ has little influence on the training
+time per epoch. Considering the obvious performance gains
+provided by γ (as shown in Table III), the helpfulness and
+efficiency of adding γ to the loss function can be verified.
+5) Sensitivity to different augmentation strategies: We study
+the sensitivity to different augmentation strategies in Table VII.
+We found that removing the random crop operation has the
+largest influence on the overall performance, where the jaccard
+index falls by over 5 percents. The reason behind is that
+applying random crop can dramatically increase the diversity
+of both homogeneous and heterogeneous image pairs. Besides
+random crop, it seems that color distortion becomes the most
+influential augmentation strategy. Adding the color distortion
+operation to GraVIS brings about 3-percent gains in lesion
+segmentation. The underlying reason is that lesions usually
+vary in colors, where applying color distortion helps learn
+color-invariant representations.
+6) Slope controller τ: The temperature hyper-parameter τ
+in Eq. 3 controls the slope of the sigmoid function, which
+is closely related to how much GraVIS addresses similar
+image pairs. To achieve satisfactory performance, the sigmoid
+function should appropriately address image pairs with dif-
+ferent degrees of similarity. From Fig. 6, we can see that τ
+determines the range of scope (centered around zero in the
+horizontal axis) with large slopes. When we set τ to 0.01, the
+scope is narrowed. If we set τ to 0.5, the scope becomes
+wider. As shown in Eq. 3, such scope corresponds to the
+similarity between two image pairs. In practice, large slopes
+within the scope usually lead to large gradient updates, which
+means GraVIS focuses more on image pairs within a specific
+range of similarity. Inspired by these phenomena, we study
+the influence of τ and report the experimental results in Table
+VIII. It is obvious that either a too large or too small τ would
+adversely affect the performance of GraVIS. A too large τ
+(such as 0.5) cannot well address similar image pairs as the
+corresponding gradient updates are not prominent enough. In
+contrast, a too small τ (such as 0.01) only addresses very
+similar pairs but ignores less similar pairs. Consider τ=0.2
+produces the best segmentation result, we make it the default
+choice for τ in all experiments.
+To help explain these phenomena, we study the impact of
+τ on the gradient update in Fig. 7(a). Here, we still refer to
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+Similarity Gap (cq, 3
+cq, 6)
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+0.35
+Gradient Gap (|
+Q
+cq, 3|
+|
+Q
+cq, 6|)
+Impact of  (gradient)
+=0.2
+=0.5
+=0.01
+(a) Impact of τ on the gradient
+0.4
+0.2
+0.0
+-0.2
+-0.4
+-0.6
+-0.8
+Similarity Score (cq, 3)
+0.000
+0.025
+0.050
+0.075
+0.100
+0.125
+0.150
+0.175
+0.200
+Loss Value
+Impact of  (loss)
+(b) Influence of using σ in VGL
+Fig. 7.
+Impact of τ on the gradient update and investigation of the
+influence of using the sigmoid function σ in VGL.
+0.77
+0.54
+0.36
+0.64
+0.56
+0.60
+0.56
+0.63
+0.47
+0.82
+0.63
+0.70
+0.66
+0.67
+0.77
+0.69
+0.73
+0.66
+0.76
+0.63
+0.44
+0.43
+0.65
+0.64
+0.58
+0.71
+0.62
+a. IN
+b. NPTL c. SimCLR-Derm d. MoCov2
+Images
+e. BYOL
+f. BagLoss
+g. C2L
+h. MG
+i. GraVIS
+GT
+Fig. 8.
+Lesion segmentation results of different approaches. Each row
+includes one randomly selected test case. GT denotes the ground truth.
+the example in Eq. 4. We can see that when τ is equal to
+0.2, | ∂Q
+∂cq,3 | becomes much larger than | ∂Q
+∂cq,6 | as the similarity
+gap increases. This observation verifies that VGL can appro-
+priately address more on the more similar homogenenous pair
+(vq, vps
+3 ) with a higher similarity score cq,3 than the dissimilar
+homogeneous pair (vq, vps
+6 ). Particularly, when τ is equal to
+0.01, we found that the gradient diminishes as the similarity
+gap becomes larger, which would lead to instability during
+training. In Fig. 7(b), we investigate the influence of using the
+sigmoid function σ in VGL. We found that as the similarity
+of a give image pair gets lower, its influence to the loss
+(i.e., VGL) becomes weaker. This phenomenon shows that
+the sigmoid function σ may serve as a soft margin that helps
+prevent the effect of null similarities.
+G. Visual analysis
+In Fig. 8, we visualize the segmentation results of different
+segmentation methods. The top two rows display the seg-
+mentation results of nevus, which is a non-specific medical
+term for a visible, circumscribed, chronic lesion of the skin or
+mucosa. Clinically, nevus in individuals are usually uniform
+in color and border. We can see that our GraVIS has the
+ability to detect most parts of nevus while delineating their
+boundary better than other approaches. In the third row,
+we present predicted masks of seborrheic keratosis, a non-
+cancerous (benign) skin tumour that originates from cells in
+the outer layer of the skin. Lesions of seborrheic keratosis
+often appear in various colors, sizes and shapes, which usually
+come in association with other skin conditions. Even so,
+GraVIS is still able to capture the majority of the main lesion
+site, which demonstrates the discrimination ability of learned
+invariant representations.
+
+10
+IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. XX, NO. XX, XXXX 2022
+TABLE IX
+LINEAR FINE-TUNING AUCS ACROSS THE TASKS OF ATELECTASIS,
+CARDIOMEGALY, CONSOLIDATION, AND EDEMA CLASSIFICATION ON
+CHEXPERT. NOTE THAT MICLE AND GRAVIS SHARE THE SAME
+AUGMENTATION STRATEGIES AS USED IN MEDAUG.
+Methods
+Average
+Atelectasis
+Cardiomegaly
+Consolidation
+Edema
+MedAug
+0.792
+0.721
+0.779
+0.801
+0.866
+MiCLE
+0.795
+0.728
+0.777
+0.794
+0.882
+GraVIS
+0.805
+0.735
+0.782
+0.824
+0.877
+H. Applying GraVIS to multiple views
+In medical image analysis, it is possible that each patient
+study contains multiple views. In this case, it is intuitive
+to treat these views and their augmented images as positive
+views. In this part, we apply GraVIS to multiple views in
+order to investigate whether GraVIS has the ability to handle
+this situation. We conducted experiments on CheXpert [56].
+The dataset consists of 224,316 images from 65,240 patients
+labeled for the presence or absence of 14 radiological ob-
+servations. We use these images for pre-training and random
+samples of 1% of these images for fine-tuning. We compare
+GraVIS with MiCLE [3] and MedAug [40], both of which
+were initially designed for dealing with multiple views.
+In this scenario, we randomly select a view from a given
+patient as the anchor view, while the rest views and their
+augmented versions are considered as positive views. In con-
+trast, negative views consist of chest X-rays from different
+patients and their augmented versions. We employed the same
+set of augmentation strategies as used in [40]. For other
+experimental settings, we simply followed our experiences
+on dermatology images and report the results in Table IX.
+From Table IX, we can see that GraVIS outperforms both
+MedAug and MiCLE across different tasks on X-ray images,
+which verify the effectiveness of GraVIS in the multi-instance
+scenario.
+V. DISCUSSION
+Compared to contrastive SSL approaches (i.e., NCE-based
+methods), GraVIS focuses more on extracting invariant se-
+mantic information given a number of homogeneous views.
+To address the importance of similar homogeneous pairs and
+prevent dissimilar homogeneous pairs from dominating the
+training process, a hardness-aware attention mechanism is
+introduced to amplify the influence of similar homogeneous
+pairs by assigning large penalty to them.
+The main idea of GraVIS, i.e., focusing on similar ho-
+mogeneous pairs to learn invariant representations, is also
+addressed in some recent works [21, 22] that attempt to utilize
+only homogeneous pairs for self-supervised representation
+learning. However, these approaches only display comparable
+results with NCE-based methods that are surpassed by our
+GraVIS. The improvements of GraVIS can be attributed to
+the fact that we focus more on homogeneous pairs while
+not completely ignoring heterogeneous ones (unlike [21, 22]).
+Thus, we believe how to appropriately address and balance
+the importance of both homogeneous and heterogeneous pairs
+can be an important direction for contrastive or non-contrastive
+SSL approaches.
+For future work, we will explore conducting more exper-
+iments using GraVIS on a larges number of dermatology
+images (without labels) and fine-tune the pre-trained models
+on some relatively small skin datasets. We will introduce the
+hard triplet mining into GraVIS to see if it could help the view
+grouping cost to utilize heterogeneous samples as effectively
+as homogeneous ones. Last but not least, efforts will be made
+to explore a more efficient training strategy to reduce the
+batch size in the pre-training stage. In fact, although the batch
+size may not be problem in 2D medical image processing,
+it will become an inescapable problem in 3D medical image
+segmentation, which often consumes a great amount of GPU
+memory.
+VI. CONCLUSION
+We propose GraVIS to learn transferable representations in
+the pre-training stage by aggregating multiple homogeneous
+views while separating heterogeneous ones. GraVIS does
+not need tens of thousands of heterogeneous samples but
+achieves better performance than state-of-the-art contrastive
+pre-training methods. We conduct extensive experiments and
+comprehensive ablation studies to demonstrate the effective-
+ness of GraVIS. We found that when pre-trained with GraVIS,
+a single model can already achieve better results compared
+to the winners and recent state-of-the-art approaches in ISIC
+2017 challenges. We believe these phenomena suggest that
+GraVIS has the potential to replace canonical NCE-based SSL
+approaches.
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+in Proc. of the AAAI Conf. on Art. Intell. (AAAI), pp. 590–597, 2019.
+
diff --git a/4dE3T4oBgHgl3EQfQQkz/content/tmp_files/load_file.txt b/4dE3T4oBgHgl3EQfQQkz/content/tmp_files/load_file.txt
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@@ -0,0 +1,1441 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf,len=1440
+page_content='IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' XX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' XX, XXXX 2022  1  GraVIS: Grouping Augmented Views from  Independent Sources for Dermatology Analysis  Hong-Yu Zhou, Member, IEEE, Chixiang Lu, Liansheng Wang, Member, IEEE,  and Yizhou Yu, Fellow, IEEE  Abstract— Self-supervised representation learning has  been extremely successful in medical image analysis, as it  requires no human annotations to provide transferable rep-  resentations for downstream tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' Recent self-supervised  learning methods are dominated by noise-contrastive es-  timation (NCE, also known as contrastive learning), which  aims to learn invariant visual representations by contrast-  ing one homogeneous image pair with a large number of  heterogeneous image pairs in each training step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' Nonethe-  less, NCE-based approaches still suffer from one major  problem that is one homogeneous pair is not enough to  extract robust and invariant semantic information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' Inspired  by the archetypical triplet loss, we propose GraVIS, which is  specifically optimized for learning self-supervised features  from dermatology images, to group homogeneous derma-  tology images while separating heterogeneous ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' In ad-  dition, a hardness-aware attention is introduced and incor-  porated to address the importance of homogeneous image  views with similar appearance instead of those dissimilar  homogeneous ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' GraVIS significantly outperforms its  transfer learning and self-supervised learning counterparts  in both lesion segmentation and disease classification  tasks, sometimes by 5 percents under extremely limited  supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' More importantly, when equipped with the pre-  trained weights provided by GraVIS, a single model could  achieve better results than winners that heavily rely on  ensemble strategies in the well-known ISIC 2017 challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='  Code is available at https://bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='ly/3xiFyjx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='  Index Terms— self-supervised learning, dermatology di-  agnosis, triplet loss, self-supervised pre-training  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' INTRODUCTION  Considering the protection of patients’ privacy and limi-  tation of accessible expertise, self-supervised learning (SSL)  [1]–[7] has been widely adopted in medical image anal-  ysis to learn transferable feature representations in the  stage of pre-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' Recent state-of-the-art SSL approaches  are often based on noise-contrastive estimation (NCE) [8]  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=', contrastive loss or contrastive learning), which learn  transformation-invariant features by contrasting a few ho-  mogeneous image pairs with quite a large number (usually  (Corresponding author: Yizhou Yu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=')  Hong-Yu Zhou, Chixiang Lu and Yizhou Yu are with the Department  of Computer Science, The University of Hong Kong, Pokfulam, Hong  Kong  (e-mail:  whuzhouhongyu@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='com,  luchixiang@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='com,  yizhouy@acm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='org).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='  Liansheng Wang is with the Department of Computer Science, Xi-  amen University, Siming District, Xiamen, Fujian Province, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' China  (e-mail: lswang@xmu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='  First two authors contributed equally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='  a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' triplet loss  b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' NCE  c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' our GraVIS  anchor view  positive view  negative view  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='  Comparison with the archetypical triplet loss (a) and noise-  contrastive estimation (b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' NCE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' The hollow arrow denotes the direction  to be pushed, where positive views are pushed towards the anchor  view and negative views are pushed away.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' Here, one anchor view and  one positive view construct one homogeneous pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' Accordingly, one  heterogeneous pair consists of one anchor view and one negative view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='  To generate positive views, we randomly augment the same image as  used to generate the anchor view, while negative views are produced by  augmenting different images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='  tens of thousands) of heterogeneous pairs [1, 4, 5] in each  training step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' However, the above characteristic may cause  one intuitive problem in practice, which is the limited number  of homogeneous pairs in contrastive analysis may prevent  the model from capturing the most representative semantic  features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='  On the other hand, in dermatology analysis, there are about  1,500 skin diseases and many variants that can be located in  different body parts1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' Although the medical imaging commu-  nity has proposed several publicly available large-scale derma-  tology datasets, such as HAM10000 [9] and BCN20000 [10],  to address the limitation of data and labels for training neural  networks, there still exists a great need of labeled dermatology  images, especially when we consider the domain shift problem  that often emerges in small-scale multi-domain, multi-center  and multi-device datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' From a different perspective, SSL  provides a solution to utilize unlabeled dermatology data for  pre-training, from which transferable images representations  can be learned and serve as the feature basis for downstream  1https://dermnetnz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='org/cme/principles/  an-overview-of-dermatology/  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='04410v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='CV]  11 Jan 2023  EMB  NPS  UFFC  SignalProcessing Society  0222  IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' XX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' XX, XXXX 2022  small-scale datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='  In this paper, we introduce Grouping augmented Views  from Independent Sources (GraVIS) to learn self-supervised  dermatology image representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' The core idea behind  GraVIS is intuitive, which is good representations should be  able to aggregate homogeneous image views while separating  heterogeneous views.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' Here, the term homogeneous views in-  clude one anchor view and at least one positive view, which  are all generated by randomly augmenting the same source  image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' The heterogeneous views involve one anchor view and  at least one negative view, where each negative image view  is generated by augmenting an image that is different from  the one used for homogeneous views.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' Specifically, we use  homogeneous pair to refer to an image pair that consists  of one anchor view and one positive view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' Similarly, each  heterogeneous pair contains one anchor view and one negative  view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' 1, we contrast our GraVIS with the canonical triplet  loss [11] and NCE [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' In practice, GraVIS addresses the  aforementioned major problem of NCE (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=', few homoge-  neous pairs) by applying N-times augmentation to a given  source dermatology image in order to generate one anchor  view and N-1 positive views (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=', N-1 homogeneous pairs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='  Meanwhile, GraVIS reduces the number of negative views to  hundreds by taking into account one specific characteristic  of dermatology images, which is heterogeneous dermatology  image pairs are easy to recognize and thus we do not need a  large number of heterogeneous pairs for contrast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' Compared  to the triplet loss, our method aims to aggregate multiple  homogeneous views and separating heterogeneous views, si-  multaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' Compared to NCE, GraVIS addresses more on  grouping multiple homogeneous views for capturing the most  general and transferable information hidden in the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' To  achieve these goals, GraVIS introduce View Grouping Loss  (VGL) to adaptively aggregate different types of views, where  the hardness-aware attention is proposed to help the model  focus more on homogeneous views with similar appearance  by assigning larger penalty to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' To summarize, our  contributions can be organized as follows:  A novel self-supervised learning methodology, GraVIS,  is proposed for conduct unsupervised pre-training on  dermatology images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' GraVIS is inspired by the archetyp-  ical triplet loss and utilizes a similarity loss instead of  the recent NCE to learn transferable dermatology image  representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='  In comparison to contrastive SSL approaches, GraVIS  brings improvements by laying emphasis on homoge-  neous pairs instead of heterogeneous ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' We introduce  a hardness-aware attention to adaptively assign different  degrees of penalty to homogeneous pairs based on each  pair’s similarity score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='  Extensive experiments are conducted on two tasks: lesion  segmentation and disease classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' Our GraVIS dis-  plays substantial and significant improvements over other  SSL methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' More importantly, by fine-tuning the pre-  trained weights acquired using GraVIS, we achieve quite  competitive results using a SINGLE model on each task,  even compared to the winners of ISIC 2017 challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' RELATED WORK  1) Contrastive and non-contrastive self-supervised learning:  Contrastive learning aims to learn invariant representations  of images via noise-contrastive estimation (NCE) [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' Doso-  vitskiy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' [12] considered each image as its own class  and used a linear classifier to classify each image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='  [13] replaced the classifier with a memory bank that stores  previously-computed representations as the former approach  is unpractical in big datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' Based on the memory bank,  Misra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' [14] proposed to learn invariant representations  based on pretext tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' Similarly, Tian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' [15] tried to  maximize mutual information between different views of the  same scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' CPC [16] and CPCv2 [17] combined predicting  future observations (predictive coding) with a probabilistic  contrastive loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' [4] introduced SimCLR, which  shows that as long as the batch size for pre-training is  large enough, well transferable image representations can be  directly learned from the image batch without the memory  bank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' [18] introduced MoCo, which improves the  training of contrastive methods by comparing representations  from the momentum network and the ordinary network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' To  replace the memory bank and solve the memory issue of  SimCLR (caused by large batch sizes), MoCo utilizes a queue  to store past representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' Caron et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' [19] incorporated  deep clustering strategies to facilitate contrastive learning by  enforcing consistency between cluster assignments produced  for different augmentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' Hjelm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' [20] further showed  the necessity of employing a large number of negative samples  in the training batch to achieve satisfactory performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='  Different from the above contrastive approaches that need a  large number (tens of thousands) of heterogeneous pairs for  contrast, our GraVIS is specifically designed for dermatology  images, which lays more emphasis on homogeneous pairs  and only requires hundreds of heterogeneous pairs to learn  invariant and transferable image representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' On the other  hand, there are some recent works [21]–[25] trying to replace  the contrastive procedure with the siamese learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' Different  from them, our GraVIS is inspired by the archetypical triplet  loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='  2) Triplet loss and triplet mining: The objective of the triplet  loss is to minimize the distance between the anchor and posi-  tive sample while maximizing the distance between the anchor  and negative sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' Schroff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' [26] first applied the triplet  loss to face recognition and verification using deep neural  network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' Cheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' [27] introduced an improved version of  the triplet loss to pull the instances of the same person closer,  and at the same time push the instances belonging to different  persons farther from each other in the learned feature space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='  Xie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' [28] used the triplet loss to learn representations  for nuclei segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' Puch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' [29] implemented the  triplet loss in the setting of few-shot learning to recognize  brain imaging modality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' When using the triplet loss to train  neural networks, triplet mining is a widely adopted strategy to  select hard [30] or semi-hard [26, 31] triplets for reducing the  computational cost and accelerating training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' Oh Song  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' [32] incorporated batch-wise hard triplet mining into the  training process of deep neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' In our GraVIS, we  AUTHOR zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' : GRAVIS: GROUPING AUGMENTED VIEWS FROM INDEPENDENT SOURCES FOR DERMATOLOGY ANALYSIS  3  Input  batch  Augmentation  Augmented  batch  Backbone  Representation  batch ������������  …  …  …  …  …  …  …  …  ������������  ������������  View  grouping  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='  …  …  …  …  …  …  …  ������������  ������������  ℒ1  VGL  ℒ������������  VGL  × ������������  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='  Backbone  Backbone  Backbone  1st random  augmentation  2nd random  augmentation  N-th random  augmentation  ������������-1 positive views  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='  Backbone  Backbone  Backbone  1st random  augmentation  2nd random  augmentation  N-th random  augmentation  ������������ negative views  Anchor view  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='  1st image  ������������-th image  Different source images in the input batch  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='  Overview of our proposed GraVIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' By augmenting the input  image batch by N times, we can acquire the augmented image batch,  which is then passed to the backbone network to get a batch of  representations, denoted as Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' Next, we apply view grouping loss to  Q, where we randomly select a view representation as the anchor view  (marked using red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' We treat those representations that share the same  source image with the anchor view as positive views (marked using  green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' Accordingly, representations that come from different source  images are defined as negative views (marked using blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='  introduce an attention mechanism to help the network focus  on the most similar homogeneous pairs while avoiding being  dominated by dissimilar homogeneous pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' This is different  from the traditional triplet mining methods that often address  the negative samples (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=', heterogeneous pairs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' It should be  mentioned that the idea of addressing the importance of similar  homogeneous pairs has been introduced by BagLoss [33]  to improve the performance of image retrieval models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' The  technical difference between BagLoss and GraVIS lies in  the way to generate and utilize negative samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' BagLoss  picks the hardest negative sample from the dataset, while  GraVIS dynamically constructs negative samples using various  augmentation strategies and uses all of them to learn invariant  representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='  3) Self-supervised learning in medical image analysis: Be-  sides NCE-based SSL approaches, restoring/reconstructing the  input images is the mostly adopted way to provide intrinsic su-  pervision for learning self-supervised medical image represen-  tations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' [34] used context restoration to learn self-  supervised representations on 2D ultrasound image, 3D ab-  dominal Computerized Tomography (CT) scans and 3D brain  Magnetic Resonance Imaging (MRI) scans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' [1]  introduced Model Genesis that applies various augmentation  strategies to build diverse reconstruction targets for both 2D  and 3D medical images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' Haghighi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' [35] modified Model  Genesis by adding a classification branch to classify high-level  features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' Zhuang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' [36] introduced a jigsaw problem on  reconstructing 3D brain MRI scans, which is then improved by  [37] with a generative adversarial network to recover shuffled  3D patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' For NCE-based methods, Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' [2] pro-  posed C2L which first applied NCE to 2D radiographs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' C2L  learns more invariant radiograph representations by contrasting  homogeneous with heterogeneous pairs in both image- and  feature-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' Taleb et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' [38] investigated the performance of  both NCE and self-reconstruction based approaches and found  contrastive SSL methods are more advantageous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' Besides self-  supervised learning methodologies, Ke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' [39] made a  range of experiments to study the influence of ImageNet-  �� ������ ����  �� positive �����  �� positive �����  �� negative �����  �� negative �����    �� negative ����3  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='  An example of the formation of the anchor view, positive views  and negative views.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' Note that views a, b and c come from the same  source image while views d, e and f are from a different source image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='  Thus, view pairs (a, b) and (a, c) are homogeneous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' Pairs (a, d), (a, e)  and (a, f) are heterogeneous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' All views are generated via different types  and degrees of augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='  based pre-training on chest X-ray models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' Different from the  above works, we introduce a new SSL methodology which  replies neither on NCE nor context restoration but surpasses  both of them by substantial and significant margins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' Recently,  Azizi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' [3] introduced a modified version of SimCLR (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=',  SimCLR-Derm) and a multi-instance version of SimCLR (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=',  MiCLE) to dermatology images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' The second version utilizes  the consistency of multiple images from a single patient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' Vu  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' [40] also employed a similar idea on X-ray data to  make use of multiple views from the same patient to construct  homogeneous pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' For implementation, these two approaches  require the access to the patient meta data in order to know  which images come from the same patient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' In contrast, our  GraVIS does not have this constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' METHODOLOGY  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' Overview  Grouping augmented Views from Independent Sources  (GraVIS) aims at conducting pre-training using a large number  of unlabeled dermatology images, after which we fine-tune the  pre-trained model on different downstream tasks to verify the  transferable ability of learned representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' We provide an  overview of our GraVIS in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' In each training step, we  first forward a batch of dermatology images to the backbone  network in order to acquire their representations, on top of  which the view grouping loss (VGL) is applied to aggregate  homogeneous samples while separating heterogeneous ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='  Compared to recent contrastive learning based approaches  [2, 3], GraVIS does not need any contrastive operations  but still displays state-of-the-art performance on a variety of  downstream tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' Positive and negative sets  Given an input image batch, we first randomly augment  each image in this batch by N times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' The augmentation  strategies mainly include random crop, random horizontal flip,  random rotation and color distortion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' We provide an example  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' 3 where we display one anchor view, two positive  4  IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' XX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' XX, XXXX 2022  a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' anchor view  b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' view1  d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' view3  Hard  Easy  c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' view2  Easy  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='  Hardness of recognizing homogeneous image pairs based  on appearance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' It is observable that the anchor view (a), view1 (b)  and view2 (c) are from the same source image as they look similar  in appearance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' However, although view3 (d) is randomly cropped from  the same source image as that of the anchor view, we can hardly tell  whether they come from the same source without expertise as view3  has quite different appearance and there is no clear overlap between  them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' Thus, it may not be reasonable to directly ask the representation  of view3 to be as close to the representation of anchor view as other  positive features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='  views and three negative views.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' We apply random crop and  random rotation (a small degree) to the source dermatology  image to acquire the anchor view (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' To generate the positive  view1 (b), we include random crop and random rotation with  a relatively large degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' For positive view2 (c), we include  random crop, horizontal flip, random rotation and gaussian  blur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' Similar combinations of augmentation strategies can also  be observed in negative views (d, e and f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' Supposing the  size of the input image batch is B (we ignore the height,  width and channel number of images for simplicity), after  N-times random augmentation, we obtain an enlarged image  batch whose size is B*N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' During the pre-training stage,  in each training step, we pass this enlarged batch to the  backbone network to acquire a batch of corresponding visual  representations (denoted as batch Q in the following), from  which we randomly pick a representation and regard it as  the anchor representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' Given this anchor view, its positive  set contains the rest N − 1 views (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=', representations) that  are derived from the same source image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' Intuitively, the rest  (B−1)*N views construct the negative set as they correspond  to different source dermatology images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' Based on positive  and negative sets, we calculate VGL to minimize the distance  between the anchor view and its positive set while keeping the  anchor away from those views in the negative set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' View grouping loss  In GraVIS, we define the view grouping loss (VGL) to be:  LVGL  q  = 1 −  1  ∥Sps∥  �  i∈Sps  1  γq,i  �  j∈Sns  σ(cq,j − cq,i) + 1  ,  γq,i =  1  �  j∈Sps,j̸=i  σ(cq,j − cq,i)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='  (1)  LVGL  q  stands for VGL with q-th sample in batch Q as the  anchor view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' Accordingly, Sps denotes the positive set and  Sns stands for the negative set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' ∥Sps∥ refers to the number  of views in Sps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' ci,j measures the similarity between the i-th  Algorithm 1 GraVIS:  1: Notations: NET denotes the backbone network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' K stands  for the number of training steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' ITER is the index of  iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='  2: for ITER ← 0 to K do  3:  Randomly sample a batch of images Xo;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='  4:  Augment Xo by N times to acquire an enlarged image  batch Xe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='  5:  Pass Xe to NET to get Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='  6:  Initialize LQ to 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='  7:  for q ← 0 to B ∗ N do  8:  LQ += LVGL  q  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='  9:  end for  10:  Backward (LQ) to update NET;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='  11: end for  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
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+page_content='35  Gradient Gap (|  Q  cq, 3|  |  Q  cq, 6|)  Impact of  (gradient)  w/  w/o  (a) Impact of γ on the gradient  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
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+page_content='630  Similarity Gap (cq, 3  cq, 6)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
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+page_content='5  Loss Value  Impact of  (loss)  w/  w/o  (b) Impact of γ on the loss value  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='  Impact of the hardness-aware attention γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' The left figure  (a) shows that the introduced attention γ focuses more on the similar  homogeneous pair (vq, vps  3 ) (the similarity score is cq,3) instead of the  dissimilar pair (vq, vps  6 ) (the similarity score is cq,6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' The right figure  (b) demonstrates that adding γ makes the view grouping loss more  sensitive to the similarity gap between two homogenenous pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='  sample and the j-th one, which is calculated using the cosine  similarity:  ci,j =  vT  i vj  ∥vi∥∥vj∥,  (2)  where vi and vj correspond to the i-th and j-th feature vectors  in batch Q, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' In Equation 1, σ(·) stands for the  sigmoid function:  σ(x) =  1  1 + e  −x  τ  (3)  with the temperature hyper-parameter τ controlling how much  GraVIS addresses similar image pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='  Term  �  j∈Sns  σ(cq,j − cq,i) evaluates cq,i among the negative  set (Sns).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' By aggregating the anchor view (q) and positive  view (i) and separating the anchor view from negative views  (∀j ∈ Sns),  �  j∈Sns  σ(cq,j − cq,i) becomes smaller and LVGL  q  is  accordingly minimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='  The hardness-aware attention γq,i evaluates cq,i among the  positive set (Sps).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' The intuition behind is that we believe  more attention should be allocated to homogeneous pairs of  similar views (in appearance) instead of dissimilar (though  homogeneous) pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' We give an example to illustrate our  intuition in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' 4, where one anchor view and three positive  views are displayed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' It is easy to recognize the relationship  AUTHOR zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' : GRAVIS: GROUPING AUGMENTED VIEWS FROM INDEPENDENT SOURCES FOR DERMATOLOGY ANALYSIS  5  between the anchor view and view1 (b) as both of them are  overlapped after being cropped from the same source image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='  Similar phenomenon can also be observed between the anchor  view and view2 (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' However, when we compare the anchor  view with view3 (d), we can hardly tell whether they two  share the same origin without relying on expertise because  they have quite dissimilar appearance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' In fact, it may not  be suitable to require the representation of view3 to be as  close to the anchor representation as other homogeneous pairs  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=', forcing ca,3 ≈ ca,1 ≈ ca,2 is questionable).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' To address  this issue, the hardness-aware attention γq,i tends to assign  a large penalty to similar homogeneous pairs by taking into  account their similarity (cq,i) in all homogeneous pairs (cq,j,  ∀j ∈ Sps, j ̸= i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' In practice, the hardness-aware mechanism  helps GraVIS to focus more on recognizing potentially “easy”  relationships instead of regularizing meaningless dissimilar  homogeneous pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='  We provide an example to illustrate the impact of γ on  VGL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' Considering three positive homogeneous view vectors  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=', Sps={vps  1 , vps  3 , vps  6 }) and three negative view vectors (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=',  Sns={vns  2 , vns  4 , vns  5 }), all of which are ordered based on the  similar score with the anchor view in a descending order, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=',  {vps  1 , vns  2 , vps  3 , vns  4 , vns  5 , vps  6 },  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' ∀i ∈ {1, 2, 3, 4, 5}, cq,i − cq,i+1 = C,  (4)  where cq,i stands for the similarity score between the i-th  view and the anchor view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' For simplicity, we define the score  interval between two adjacent views to be a positive constant  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' As aforementioned, the proposed hardness-aware attention  γ helps VGL to lay more emphasis on shifting easy samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' 5(a), we can see that as the similarity gap between  cq,3 and cq,6 becomes larger, VGL (with the attention factor  γ) assigns larger gradients (denoted as | ∂Q  ∂cq,3 | − | ∂Q  ∂cq,6 |) to the  more similar homogeneous pair (vq, vps  3 ) instead of the less  similar homogeneous pair (vq, vps  6 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' Moreover, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' 5(b), we  observe that equipped with γ, VGL becomes more sensitive  to the similarity gap between two homogenenous pairs, which  again verifies the impact of the hardness-aware attention on  handling dissimilar homogeneous pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='  Finally, for each batch Q, we calculate the overall loss by  summing up all LVGL  q  , which can be presented as follows:  LQ =  B∗N  �  q=1  LVGL  q  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='  (5)  1) Comparison  with  noise-contrastive  estimation:  Here  we would like to clarify the advantages of proposed VGL  over the noise-contrastive estimation (NCE) [8], which has  been widely used in self-supervised visual representation  learning [2, 4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' Similar to VGL, NCE contrasts the  homogeneous pair with multiple heterogeneous pairs to  learn invariant representations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' however, it suffers from two  issues when dealing with dermatology images: i) the locality  problem and ii) the equality problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' For issue i), given  an anchor view in the input image batch, the archetypical  NCE compares it with only one homogeneous view (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=',  one homogeneous pair), which cannot guarantee the learned  semantic information are robustness enough towards a variety  of transformations (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=', augmentation strategies).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' For issue  ii), NCE treats all homogeneous pairs equally regardless of  the degree of applied augmentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' As we have mentioned  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' 4, for dissimilar homogeneous views (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=', the anchor  view and view3), it may not be appropriate to force their  representations to be close to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' In contrast, our  GraVIS addresses issue i) by introducing more homogeneous  pairs  to  learn  more  invariant  semantic  representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='  To tackle issue ii), VGL comprises the hardness-aware  attention to assign adaptive penalty on top of the hardness of  recognizing the relationship between two homogeneous views.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='  2) Comparison with the triplet loss: Each time, the triplet  loss contrasts one homogeneous pair with one heterogeneous  pair as follows:  Ltriplet(vq, vps, vns) = max(0, m + d(vq, vps) − d(vq, vns))  (6)  where vq, vps and vns denote the query representation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=',  the anchor view), positive and negative representations, respec-  tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' m is a pre-defined margin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' d stands for the distance  function, where euclidean distance is mostly used and cosine  similarity is also applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' Compared to GraVIS, the triplet  loss does not require to apply N-times augmentation, which  lead to only one homogeneous pair and one heterogeneous  pair in each computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' Thus, the learned representations  are much less invariant than those learned with VGL or  NCE, which may be the reason why the triplet loss has not  been widely used in self-supervised representation learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='  Another drawback of the triplet loss is that it linearly combines  each term, ignoring the difference and importance of different  homogeneous or heterogeneous pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' As a result, the training  process may be less effective as it is dominated by the trivial  information from a few meaningless yet hard samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' For  instance, it is possible that vq and vps originate from two  augmented images that share the same source image but have  dramatically different appearance (such as view1 and view3 in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' In this case, minimizing the distance between vq and  vps becomes meaningless, especially when vns comes from a  dermatology image that suffers from the same skin disease as  that of vq and vps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' In contrast, our GraVIS is able to reduce  the importance of such dissimilar homogeneous pairs using  the hardness-aware attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' EXPERIMENT  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' Dataset  We evaluate GraVIS on ISIC 2017 dataset [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' The num-  bers of images in the training, validation and test sets are 2000,  150 and 600, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' Lesions in dermatology images  are all paired with a gold standard (definitive) diagnosis,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' melanoma, nevus, and seborrheic keratosis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' There are  three major tasks in ISIC 2017 Challenge and we evaluate  our framework on lesion segmentation (task1) and disease  classification (task2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' In disease classification, we conduct  two binary classification sub-tasks: i) distinguishing between  (a) melanoma and (b) nevus and seborrheic keratosis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' ii)  distinguishing (a) seborrheic keratosis from (b) nevus and  melanoma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' The evaluation metrics are jaccard index and mean  6  IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' XX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' XX, XXXX 2022  TABLE I  EXPERIMENTS IN LESION SEGMENTATION AND DISEASE CLASSIFICATION TASKS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' DIFFERENT RATIOS STAND FOR DIFFERENT AMOUNTS OF DATA  FOR FINE-TUNING.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' WE REPORT THE JACCARD INDEX FOR LESION SEGMENTATION AND MEAN ACCURACY FOR DISEASE CLASSIFICATION.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='  Lesion segmentation  Disease classification  Methodology  10%  20%  30%  40%  50%  100%  10%  20%  30%  40%  50%  100%  TS  58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='9  61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='3  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='4  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='5  70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='5  72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='3  78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='3  79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='7  80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
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+page_content='9  IN  68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='4  70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
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+page_content='0  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='9  81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='4  83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='3  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='3  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='8  86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='0  86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='7  BagLoss  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='7  68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='8  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='5  70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='8  71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='2  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='3  76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='8  80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='3  81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='2  81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='7  82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='4  83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='5  C2L  68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='9  70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='7  71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='9  72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='4  72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='5  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='4  81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='3  84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='7  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='2  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='7  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='8  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='6  MG  66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='2  68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='9  70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='3  70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='7  71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='3  72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='8  77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='1  78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='7  79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='2  80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='8  81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='7  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='9  Our GraVIS  70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='9  72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='6  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='0  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='3  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='0  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='7  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='9  86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='4  86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='7  86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='9  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='0  89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='1  accuracy for task1 and task2, respectively, which are also used  in the released official leaderboard of ISIC 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' Implementation details  1) Details of pre-training: We first remove the labels of all  dermatology images in the training set to pre-train the back-  bone network using GraVIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' Specifically, the size of the input  image batch is 32 and the number of random augmentation  N is 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' We employ SGD with momentum as the default  optimizer whose initial learning rate is set to 1e-3 while the  momentum value is set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' We employ the cosine annealing  strategy for learning rate decay and train the network for 240  epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' We use ResNet50 [18] with two fully-connected layers  as the backbone network whose output dimension is 1×128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='  The pre-training procedure is finished on 4 NIVIDA TITAN  V GPUs within 8 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' In each training step, the augmented  image batch contain B*N images, where all positive images  are continuous and thus the model is likely to find a short  cut by directly making use of the location information (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=',  the model knows which images are from the same source and  which are not).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' So we shuffle the batch before we pass it to the  backbone network, which prevents the network from finding  a trivial solution that may lead to trivial representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='  2) Details of fine-tuning: In the fine-tuning stage, we remove  two fully-connected layers used in the pre-training stage and  load the weight parameters of convolutional layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' In task1,  we use ResU-Net [42] for lesion segmentation, where we load  pre-trained ResNet as the encoder and randomly initialize the  corresponding decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' The batch size for segmentation task is  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' The loss function includes both dice loss and cross entropy  loss whose coefficients are 1 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' In task2,  we add a linear classification head with a dropout (p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='2)  layer before it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' The batch size of classification task is 64 and  the loss function is binary cross entropy loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' For all tasks, we  unfreeze the weights of convolutional layers and fine-tune the  entire network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' Validation score (accuracy or jaccard index)  is obtained after each training epoch and then we anneal the  learning rate by a factor of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='5 if the validation score is not  improved after 3 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' Different from the pre-training stage,  we employ Adam as the optimizer and the initial learning  rate is set to 1e-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' We stop the fine-tuning process when the  validation score does not decrease for 10 epochs, and we save  the checkpoint with the lowest validation loss value for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='  To evaluate the ability of GraVIS under different amounts of  labeled data, we randomly sample 10% to 50% labeled data  from the whole training set to fine-tune the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' Baselines  Training from scratch (TS) and ImageNet-based initializa-  tion (IN) are involved as two basic baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' For contrastive  approaches, we include SimCLR-Derm [3], MoCov2 [43],  BYOL [21] and C2L [2] for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' As for traditional  self-supervised learning methods, we compare GraVIS against  Model Genesis (MG) [1], N-pairs triplet loss with the triplet  mining strategy (NPTL) [44] and BagLoss [33], where the last  two methods are based on metric learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' Boosting performance under limited supervision  1) Lesion segmentation: We present experimental results  of fine-tuning under different labeling ratios in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='  Firstly, it is obvious that all representation learning (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=',  transfer learning and self-supervised learning) methods can  distinctly boost the performance compared to TS, verifying  the necessity of learning transferable representations during  the pre-training stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' Secondly, we can easily find that self-  supervised approaches based on NCE are advantageous over  the predictive method (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=', MG) under different ratios, which  implies that NCE does perform better than the traditional  predictive representation learning for dermatology images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='  Somewhat surprisingly, we found that an improved multi-pair  version of triplet loss (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=', NPTL) can already achieve  comparable results with MG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' Considering MG needs various  pre-defined restoration tasks, such phenomenon suggests  the potential of applying the triplet loss to self-supervised  representation learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' Among all NCE-based baselines,  BYOL achieves the best performance in large labeling ratios  while C2L performs better in small ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' When comparing  our GraVIS against other baselines, it is observable that  GraVIS has the ability to outperform other baselines in  AUTHOR zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' : GRAVIS: GROUPING AUGMENTED VIEWS FROM INDEPENDENT SOURCES FOR DERMATOLOGY ANALYSIS  7  various ratios by obvious margins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' More importantly, GraVIS  surpasses the best performing self-supervised baselines C2L  and BYOL by 2 percents under extremely limited supervision  (10% and 20%) without explicitly using any contrastive  functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' Compared to IN using 100% labeled data (the  most widely used method for tackling limited annotations),  GraVIS achieves comparable results using only 50% labeled  dermatology images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' It is worth noting that similar to the  introduced hardness-aware attention, BagLoss addresses the  importance of aggregating multiple homogeneous pairs in  image retrieval, which is the reason why we include it as  one of our baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' From Table I, we can see that the  performance of BagLoss are not satisfactory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' The reason  behind is that BagLoss only assigns one hard negative sample  to each anchor view, which may prevent the model from  learning invariant and discriminative representations from a  number of negative views.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='  Statistical significance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' A t-test validation is conducted  in all labeling ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' The p-values between GraVIS and  C2L/BYOL are smaller than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='01, which indicates that  GraVIS is statistically better than C2L/BYOL at the 1%  significance level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' In addition, the p-values between GraVIS  (using 50% annotations) and IN (using 100% labeled data)  are much larger than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='05, which shows that GraVIS produces  competitive results compared to the most widely used fully-  supervised baseline (IN-100%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='  2) Disease classification: The best performing baselines in  this task are MoCov2 and C2L, where MoCov2 maintains  relatively obvious advantages when the labeling ratio is 10%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='  In comparison to the performance on lesion segmentation, our  GraVIS maintains relatively larger advantages over different  baselines in the problem disease classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' For instance,  GraVIS outperforms MoCov2 by nearly 3 percents when  using 10% labeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' We believe these large improvements  brought by GraVIS can be attributed to the strong demand of  annotations in distinguishing three diseases (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=', melanoma,  nevus and seborrheic keratosis), where GraVIS can make  use of limited annotations more effectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' Compared to the  most widely adopted IN-100%, our approach again produces  quite comparable performance by using only half labeled  data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' These phenomena suggest the potential of GraVIS to  replace canonical self-supervised and transfer learning based  methods to learn more effective pre-trained representations  for dermatology images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='  Statistical significance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' The p-values between GraVIS and  the best performing baselines C2L/MoCov2 are smaller than  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='01 under different labeling ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' Similar to the results of  lesion segmentation, we believe these p-values help verify the  statistical effectiveness of GraVIS over C2L/MoCov2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' Again,  the p-values between GraVIS-50% and IN-100% are much  larger than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='05, suggesting that GraVIS can greatly reduce the  amount of human annotations that are necessary for training  a fully-supervised diagnosis model that is based on ImageNet  pre-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='  TABLE II  COMPARISON WITH TOP-5 APPROACHES IN THE LEADERBOARD OF  ISIC 2017 AND RECENT STATE-OF-THE-ART METHODS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' NOTE THAT WE  ONLY USE SINGLE MODELS FOR TESTING ON BOTH TASKS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' 50% AND  100% STAND FOR THE LABELING RATIOS OF ANNOTATED DATA USED IN  THE FINE-TUNING STAGE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='  Rank  Lesion segmentation  Disease classification  1  Mt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='Sinai [48]  76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='5  Casio and Shinshu [49]  91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='1  2  NLP LOGIX [50]  76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='2  Multimedia [51]  91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='0  3  USYD-BMIT [52]  76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='0  RECOD Titans [53]  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='8  4  USYD-BMIT [52]  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='8  USYD-BMIT [52]  89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='6  5  RECOD Titans [53]  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='4  A*STAR [54]  88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='6  Xie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' [46]  80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='4  Bisa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' [55]  91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='5  GraVIS (50%)  80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='2  Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' [45]  91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='7  GraVIS (100%)  81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='0  GraVIS (50%)  91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='4  GraVIS (100%)  92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='1  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' Advancing fully-supervised fine-tuning  To demonstrate that our GraVIS can boost the performance  of fully-supervised fine-tuning (denoted as GraVIS (100%)  in Table II), we present a comparison of our method with  top-5 results in the official leaderboard2 and recent state-of-  the-art approaches [45, 46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' Like [45, 46], we also collected  1320 additional dermoscopy images, including 466 melanoma,  822 nevus images, and 32 seborrheic keratosis images, from  the ISIC Archive3 to enlarge the training dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' We also  incorporate advanced pre- and post-processing (PP) techniques  to improve the model performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' For both tasks, we apply  aggressive rotation augmentation to each image, where we  rotate each image 15 degrees more on the basis of the previous  one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' Thus, we have 24 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=',  360  15 ) rotated versions given a  source image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' Note that the accompanying segmentation mask  is also rotated and aligned with the rotated input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' Specifically,  for lesion segmentation, we add L (lightness) channel in  CIELAB space [47] for better capturing the lesion boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='  We then resize each image to 192×256 (height×width), where  a similar aspect ratio (3:4) is mostly used in the training  set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' To produce consistent predictions, we first use a high-  confidence threshold (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='8) to determine the center of lesion,  and then a middle-threshold (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='5) is used to highlight the lesion  area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' After filling holes, we ask the lesion area to contain the  detected lesion center and take the maximum connected area  as the final prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' In practice, our GraVIS firstly pre-  trains the ResU-Net (segmentation)/ResNet50 (classification)  using the whole ISIC 2017 training set and the external data  (without labels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' Then, we fine-tune the pre-trained models  on the labeled training set and external data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='  As shown in  Table II, GraVIS achieves the highest jaccard index in lesion  segmentation and ranks 1st in the disease classification task,  outperforming approaches in the leaderboard and recent state-  of-the-art methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' We believe these observations show that  GraVIS has the potential for advancing the performance of  fully-supervised models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='  To demonstrate the effectiveness of GraVIS in reducing the  2https://challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='isic-archive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='com/leaderboards/2017  3https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='isic-archive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='com/  8  IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' XX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' XX, XXXX 2022  TABLE III  INFLUENCE OF INTRODUCED HARDNESS-AWARE ATTENTION γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' W/ AND  W/O DENOTES WITH AND WITHOUT, RESPECTIVELY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='  Method  w/ γ  w/o γ  Jaccard index (%)  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='7  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='4  amount of labeled data, we also present the results of fine-  tuning GraVIS-based pre-trained models with 50% annotated  data in Table II (denoted as GraVIS (50%)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' We see that  GraVIS trained with only 50% annotated data performs com-  parably with the fully-supervised state-of-arts in both lesion  segmentation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=', [46]) and disease classification (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=', [45])  tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' We believe these results show the potential of GraVIS  for serving as an annotation-efficient learning methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' Ablation study  In this section, we conduct ablative experiments for different  modules and hyper-parameters to investigate their significance  towards the performance of GraVIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' Specifically, the study  targets consist of the hardness-aware attention, two factors that  affect the number of positive and negative views: the number  of augmentation N and the input batch size B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' We also display  the influence of the slope controller τ in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' Last but not  least, we demonstrate the effects of pre-training with respect  to the number of pre-training epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' Note that all ablation  studies are fine-tuned with 100% labeled data in the task of  lesion segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='  1) Influence of hardness-aware attention γ: In Table III, we  report the results equipped with γ and without γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' It is apparent  that the proposed hardness-aware attention can bring sub-  stantial improvements in the setting of fully-supervised fine-  tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' We believe the underlying reason is that γ prevents the  model from laying too much emphasis on hard homogeneous  pairs with quite different appearance that would cover up  the effects of similar homogeneous pairs and dominate the  training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' Without γ, GraVIS performs slightly worse  than SimCLR-Derm but still better than MG, which indicates  that GraVIS could become a useful self-supervised learning  method even without the proposed attention mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='  2) Investigation of N-times augmentation: One of the core  ideas of GraVIS is to contrast N − 1 homogeneous pairs  (excluding the anchor view) with (B − 1)*N heterogeneous  image pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' Intuitively, a larger N would bring more in-  variance to learned representations but inevitably reduces the  training efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' In Table IV, we study the influence of N by  gradually increasing it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' It is worth mentioning that N=0 means  the anchor view is the same as the positive view, which leads to  the worst result in Table IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' The reason behind is GraVIS can  easily distinguish positive views from negative ones, reducing  the discriminative ability of learned representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' When N  is equal to 2, GraVIS degenerates into a similar format to the  archetypical triplet loss but has more heterogeneous pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' If  we increase N to 10, the learned invariant features provide  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='3 percents improvements over N=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' Obviously, as we add  more augmented views, the overall segmentation performance  continue to increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' However, we can find that N=40 is only  TABLE IV  INVESTIGATION OF N-TIMES AUGMENTATION.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='  N  0  2  10  20  40  Jaccard index(%)  52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='3  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='5  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='8  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='7  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='8  marginally better than N=20, implying that the performance  becomes saturated in large values of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' Considering the  need of training efficiency (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=', N=40 costs much more time  and GPU memory), we set N as the default value in all  experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='  3) Influence of the batch size B: Different from N, the  batch size B only influences the number of heterogeneous  pairs in each loss computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' From Table V where we  gradually increase the batch size from 16 to 128, we observe  an obvious trend that is increasing the input batch size would  not affect as much as increasing N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' Thus, we can draw a  conclusion that the number of heterogeneous pairs play a less  important role in GraVIS compared to that of homogeneous  pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' These phenomena imply that B=16 already provides  enough negative views (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=', 16×(20-1)=304) for contrasting  with the positive ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' In addition, the performance become  saturated as we increase B to 128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' Considering large batch  sizes would consume too much more GPU memory for loss  computation, we set B to 32 in all experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='  TABLE V  INVESTIGATION OF THE BATCH SIZE B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' N IS SET TO 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='  B  16  32  64  128  Jaccard index(%)  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='4  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='7  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='7  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='8  TABLE VI  COMPUTATIONAL OVERHEAD OF GRAVIS UNDER DIFFERENT N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' WE  REPORT THE AVERAGE TRAINING TIME (IN SECOND) PER EPOCH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='  N  0  2  10  20  SimCLR-Derm 32 GraVIS (w/o γ)  25  47  76  115  GraVIS (w/ γ)  27  51  83  125  TABLE VII  INVESTIGATION OF THE SENSITIVITY TO DIFFERENT AUGMENTATION  STRATEGIES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' RANDCROP, RANDFLIP, RANDROT, COLORDIST STAND  FOR RANDOM CROP, RANDOM HORIZONTAL FLIP, RANDOM ROTATION  AND COLOR DISTORTION, RESPECTIVELY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='  w/o RandCrop  w/o RandFlip  w/o RandRot  w/o ColorDist  Optimal  GraVIS  70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='5  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='5  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='8  72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='8  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='7  4) Computational overhead: In Table VI, we present the  computational overhead of GraVIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' Besides, we also inves-  tigate the impact of adding the hardness-aware attention γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='  Compared to SimCLR-Derm, our GraVIS naturally takes more  training time because it includes more homogeneous and  heterogeneous pairs in the view grouping loss as N increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='  When N becomes 20 (the optimal choice), it costs about  2 minutes to finish each epoch, which is approximate 4×  AUTHOR zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' : GRAVIS: GROUPING AUGMENTED VIEWS FROM INDEPENDENT SOURCES FOR DERMATOLOGY ANALYSIS  9  TABLE VIII  INVESTIGATION OF THE INFLUENCE OF τ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='  τ  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='0  Jaccard index(%)  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='5  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='4  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='7  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='5  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='3  −8  −6  −4  −2  0  2  4  6  8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='0  (a) τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='01  −8  −6  −4  −2  0  2  4  6  8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='0  (b) τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='2  −8  −6  −4  −2  0  2  4  6  8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='0  (c) τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='5  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' Sigmoid functions with different temperature values (τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='  of SimCLR-Derm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' On the other hand, we found that the  hardness-aware attention γ has little influence on the training  time per epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' Considering the obvious performance gains  provided by γ (as shown in Table III), the helpfulness and  efficiency of adding γ to the loss function can be verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='  5) Sensitivity to different augmentation strategies: We study  the sensitivity to different augmentation strategies in Table VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='  We found that removing the random crop operation has the  largest influence on the overall performance, where the jaccard  index falls by over 5 percents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' The reason behind is that  applying random crop can dramatically increase the diversity  of both homogeneous and heterogeneous image pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' Besides  random crop, it seems that color distortion becomes the most  influential augmentation strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' Adding the color distortion  operation to GraVIS brings about 3-percent gains in lesion  segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' The underlying reason is that lesions usually  vary in colors, where applying color distortion helps learn  color-invariant representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='  6) Slope controller τ: The temperature hyper-parameter τ  in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' 3 controls the slope of the sigmoid function, which  is closely related to how much GraVIS addresses similar  image pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' To achieve satisfactory performance, the sigmoid  function should appropriately address image pairs with dif-  ferent degrees of similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' 6, we can see that τ  determines the range of scope (centered around zero in the  horizontal axis) with large slopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' When we set τ to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='01, the  scope is narrowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' If we set τ to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='5, the scope becomes  wider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' As shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' 3, such scope corresponds to the  similarity between two image pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' In practice, large slopes  within the scope usually lead to large gradient updates, which  means GraVIS focuses more on image pairs within a specific  range of similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' Inspired by these phenomena, we study  the influence of τ and report the experimental results in Table  VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' It is obvious that either a too large or too small τ would  adversely affect the performance of GraVIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' A too large τ  (such as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='5) cannot well address similar image pairs as the  corresponding gradient updates are not prominent enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' In  contrast, a too small τ (such as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='01) only addresses very  similar pairs but ignores less similar pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' Consider τ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='2  produces the best segmentation result, we make it the default  choice for τ in all experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='  To help explain these phenomena, we study the impact of  τ on the gradient update in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' 7(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' Here, we still refer to  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
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+page_content='35  Gradient Gap (|  Q  cq, 3|  |  Q  cq, 6|)  Impact of  (gradient)  =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
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+page_content='62  a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' IN  b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' NPTL c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' SimCLR-Derm d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' MoCov2  Images  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' BYOL  f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' BagLoss  g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' C2L  h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' MG  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' GraVIS  GT  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='  Lesion segmentation results of different approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' Each row  includes one randomly selected test case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' GT denotes the ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='  the example in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' We can see that when τ is equal to  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='2, | ∂Q  ∂cq,3 | becomes much larger than | ∂Q  ∂cq,6 | as the similarity  gap increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' This observation verifies that VGL can appro-  priately address more on the more similar homogenenous pair  (vq, vps  3 ) with a higher similarity score cq,3 than the dissimilar  homogeneous pair (vq, vps  6 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' Particularly, when τ is equal to  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='01, we found that the gradient diminishes as the similarity  gap becomes larger, which would lead to instability during  training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' 7(b), we investigate the influence of using the  sigmoid function σ in VGL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' We found that as the similarity  of a give image pair gets lower, its influence to the loss  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=', VGL) becomes weaker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' This phenomenon shows that  the sigmoid function σ may serve as a soft margin that helps  prevent the effect of null similarities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' Visual analysis  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' 8, we visualize the segmentation results of different  segmentation methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' The top two rows display the seg-  mentation results of nevus, which is a non-specific medical  term for a visible, circumscribed, chronic lesion of the skin or  mucosa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' Clinically, nevus in individuals are usually uniform  in color and border.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' We can see that our GraVIS has the  ability to detect most parts of nevus while delineating their  boundary better than other approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' In the third row,  we present predicted masks of seborrheic keratosis, a non-  cancerous (benign) skin tumour that originates from cells in  the outer layer of the skin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' Lesions of seborrheic keratosis  often appear in various colors, sizes and shapes, which usually  come in association with other skin conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' Even so,  GraVIS is still able to capture the majority of the main lesion  site, which demonstrates the discrimination ability of learned  invariant representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='  10  IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' XX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' XX, XXXX 2022  TABLE IX  LINEAR FINE-TUNING AUCS ACROSS THE TASKS OF ATELECTASIS,  CARDIOMEGALY, CONSOLIDATION, AND EDEMA CLASSIFICATION ON  CHEXPERT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' NOTE THAT MICLE AND GRAVIS SHARE THE SAME  AUGMENTATION STRATEGIES AS USED IN MEDAUG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='  Methods  Average  Atelectasis  Cardiomegaly  Consolidation  Edema  MedAug  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
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+page_content='721  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
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+page_content='801  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='866  MiCLE  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
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+page_content='794  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='882  GraVIS  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='805  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='735  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='782  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='824  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='877  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' Applying GraVIS to multiple views  In medical image analysis, it is possible that each patient  study contains multiple views.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' In this case, it is intuitive  to treat these views and their augmented images as positive  views.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' In this part, we apply GraVIS to multiple views in  order to investigate whether GraVIS has the ability to handle  this situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' We conducted experiments on CheXpert [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='  The dataset consists of 224,316 images from 65,240 patients  labeled for the presence or absence of 14 radiological ob-  servations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' We use these images for pre-training and random  samples of 1% of these images for fine-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' We compare  GraVIS with MiCLE [3] and MedAug [40], both of which  were initially designed for dealing with multiple views.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='  In this scenario, we randomly select a view from a given  patient as the anchor view, while the rest views and their  augmented versions are considered as positive views.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' In con-  trast, negative views consist of chest X-rays from different  patients and their augmented versions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' We employed the same  set of augmentation strategies as used in [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' For other  experimental settings, we simply followed our experiences  on dermatology images and report the results in Table IX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='  From Table IX, we can see that GraVIS outperforms both  MedAug and MiCLE across different tasks on X-ray images,  which verify the effectiveness of GraVIS in the multi-instance  scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' DISCUSSION  Compared to contrastive SSL approaches (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=', NCE-based  methods), GraVIS focuses more on extracting invariant se-  mantic information given a number of homogeneous views.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='  To address the importance of similar homogeneous pairs and  prevent dissimilar homogeneous pairs from dominating the  training process, a hardness-aware attention mechanism is  introduced to amplify the influence of similar homogeneous  pairs by assigning large penalty to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='  The main idea of GraVIS, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=', focusing on similar ho-  mogeneous pairs to learn invariant representations, is also  addressed in some recent works [21, 22] that attempt to utilize  only homogeneous pairs for self-supervised representation  learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' However, these approaches only display comparable  results with NCE-based methods that are surpassed by our  GraVIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' The improvements of GraVIS can be attributed to  the fact that we focus more on homogeneous pairs while  not completely ignoring heterogeneous ones (unlike [21, 22]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='  Thus, we believe how to appropriately address and balance  the importance of both homogeneous and heterogeneous pairs  can be an important direction for contrastive or non-contrastive  SSL approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='  For future work, we will explore conducting more exper-  iments using GraVIS on a larges number of dermatology  images (without labels) and fine-tune the pre-trained models  on some relatively small skin datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' We will introduce the  hard triplet mining into GraVIS to see if it could help the view  grouping cost to utilize heterogeneous samples as effectively  as homogeneous ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' Last but not least, efforts will be made  to explore a more efficient training strategy to reduce the  batch size in the pre-training stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' In fact, although the batch  size may not be problem in 2D medical image processing,  it will become an inescapable problem in 3D medical image  segmentation, which often consumes a great amount of GPU  memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content='  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' CONCLUSION  We propose GraVIS to learn transferable representations in  the pre-training stage by aggregating multiple homogeneous  views while separating heterogeneous ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' GraVIS does  not need tens of thousands of heterogeneous samples but  achieves better performance than state-of-the-art contrastive  pre-training methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' We conduct extensive experiments and  comprehensive ablation studies to demonstrate the effective-  ness of GraVIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' We found that when pre-trained with GraVIS,  a single model can already achieve better results compared  to the winners and recent state-of-the-art approaches in ISIC  2017 challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
+page_content=' We believe these phenomena suggest that  GraVIS has the potential to replace canonical NCE-based SSL  approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
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+page_content=' (AAAI), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE3T4oBgHgl3EQfQQkz/content/2301.04410v1.pdf'}
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+arXiv:2301.02843v1  [cs.IT]  7 Jan 2023
+ON VECTORIAL FUNCTIONS WITH MAXIMAL NUMBER OF
+BENT COMPONENTS
+XIANHONG XIE1,2, YI OUYANG3,4
+Abstract. We study vectorial functions with maximal number of bent compo-
+nents in this paper. We first give a construction of such functions from known
+ones, thus obtain two new classes from the Niho class and the Maiorana-
+McFarland class. Our construction gives a partial answer to an open prob-
+lem proposed by Pott et al., and also solves an open problem proposed by
+Mesnager.
+We then show that the vectorial function F : F22m → F22m,
+x �→ x2m+1 + x2i+1 has maximal number of bent components if and only if
+i = 0.
+Keywords Vectorial bent functions, Vectorial functions, Bent components,
+Niho quadratic function, Maiorana-McFarland class.
+1. Introduction
+Bent functions, as a special class of Boolean functions, were introduced by
+Rothaus [1] and have been extensively studied (see [2]-[10]) due to their impor-
+tant applications in cryptography, coding theory and combinatorics.
+The bentness of Boolean functions can be extended to a general vectorial function
+F : F2n → F2k by requesting all component functions fc(x) = Tr2k/2(cF(x)) (c ∈
+F∗
+2k) of F to be bent. Nyberg [11] showed that vectorial bent functions can only
+exist if n is even and n ≥ 2k, and presented two different constructions of such
+functions from known classes of bent functions. The reader can refer to [12]-[19]
+for more constructions of vectorial bent functions. However, relatively little work
+was done to construct bent functions from known vectorial bent functions. In this
+direction, Mesnager [20] proved the following result:
+Theorem. If F : F2n → F2k is a vectorial bent function, and c1, c2, c3 ∈ F∗
+2k
+satisfying c1+c2+c3 ̸= 0 and f ∗
+c1+f ∗
+c2+f ∗
+c3 = f ∗
+c1+c2+c3, then fc1fc2+fc1fc3 +fc2fc3
+is bent and its dual is f ∗
+c1f ∗
+c2 + f ∗
+c1f ∗
+c3 + f ∗
+c2f ∗
+c3.
+She raised an open problem to find vectorial bent functions satisfying the above
+condition.
+Another interesting and important question in studying bentness of vectorial
+functions is that how large the number of bent components of a vectorial function
+could be. Suppose F : F2n → F2k is a vectorial function. Nyberg’s condition [11] is
+equivalent to that the possible maximum 2k − 1 can be attained only if n is even
+and k ≤ n
+2 . For k = n, Pott et al. [22] proved that this number is at most 2n − 2
+n
+2
+2020 Mathematics Subject Classification. 11T71, 94A60.
+Partially supported by Innovation Program for Quantum Science and Technology (Grant
+No.
+2021ZD0302904) and Anhui Initiative in Quantum Information Technologies (Grant No.
+AHY150200).
+1
+
+2
+XIANHONG XIE1,2, YI OUYANG3,4
+and presented a class of binomial functions attaining the upper bound. Further,
+Pott et al. [22] raised the following problem:
+Problem 1. Determine all linear mappings ℓ(x) over F2n such that the number of
+bent components of xℓ(x) is 2n − 2
+n
+2 .
+For k ≥ n
+2 , Zheng et al. [21] showed that the number of bent components is at
+most 2k − 2k− n
+2 and presented a class of vectorial functions with 2k − 2k− n
+2 bent
+components.
+Let n = k = 2m. Pott et al. [22] firstly gave the upper bound of the number of
+bent components, and then constructing vectorial Boolean functions attaining the
+upper bound has been studied. There are five known classes of vectorial functions
+from F2n to itself with 2n − 2m bent components:
+(a) F(x) = x2m+1 ([23]);
+(b) F(x) = x2i(x + x2m) = x2iTr2n/2m(x), 0 ≤ i ≤ n − 1 ([22]);
+(c) F(x) = x2i(Tr2n/2m(x) +
+ρ�
+j=1
+γ(j)Tr2n/2m(x)2tj ), where γ(j) ∈ F2m, ρ ≤ m
+satisfying �ρ
+j=1(γ(j))2m−tj z2k−tj −1+1 ̸= 0 and �ρ
+j=1(γ(j))2m−jz2tj −1+1 ̸=
+0 for any x ∈ F2m ([19]);
+(d) F(x) = xh(Tr2n/2m(x)), where h : F2m → F2m is a permutation ([21]).
+(e) F(x) = x2iTr2n/2m(Λ(x)), where Λ(x) is a linearized permutation polyno-
+mial of F2m[x] ([24]).
+We note that functions in (a), (b) and (e) are of the form xℓ(x).
+We shall work on vectorial functions from F2n to itself with maximal number of
+bent components in this paper. Our main contributions are:
+(A) We present two new classes of vectorial functions from F2n to itself with
+maximal number of bent components via the Niho quadratic function and
+the Maiorana–McFarland class. Moreover,
+(A.1) From the Niho quadratic function, we obtain a new class of quadratic
+vectorial functions of the form
+F(x) = x2m+1 + u1xTr2n/2(u2x) = xℓ(x),
+where u1, u2 ∈ F2n−F2m satisfying u2m
+1 u2 ∈ F2m and Tr2m/2(u2m
+1 u2) =
+0. This gives a partial answer to Problem 1.
+(A.2) From the Maiorana–McFarland class, we obtain a family of vectorial
+bent functions and find three distinct bent components Gc1, Gc2, Gc3
+such that G∗
+c1 + G∗
+c2 + G∗
+c3 = G∗
+c, where c1, c2, c3 ∈ F∗
+2m and c =
+c1 + c2 + c3 ̸= 0, so that the conditions of Mesnager [20] are satisfied.
+(B) We prove that the binomial vectorial function F(x) = x2m+1 + x2i+1 (0 ≤
+i ≤ m − 1) has 2n − 2m bent components if and only if i = 0.
+2. Preliminaries
+2.1. Basic Notations. Let n and k be positive integers.
+For k | n, the trace
+function from F2n to its subfield F2k is Tr2n/2k(x) =
+n
+k −1
+�
+i=0
+x2ki.
+
+VECTORIAL FUNCTIONS WITH MAXIMAL NUMBER OF BENT COMPONENTS
+3
+For a finite dimensional F2-vector space V , we always fix a non-degenerate inner
+product ⟨ , ⟩ = ⟨ , ⟩V on V . In particular, if V = Fn
+2, we let
+⟨(vi), (wi)⟩ =
+n
+�
+i=1
+viwi.
+If V = F2n, let
+⟨ω, x⟩ = Tr2n/2(ωx).
+For a subspace W of V , let W ⊥ = {v ∈ V, ⟨v, w⟩ = 0 for all w ∈ W} be the
+orthogonal complementary of W, then dim W ⊥ = dim V − dim W.
+For a vectorial function F : V → W, the component function of F at w ∈ W is
+the function Fw : v → F2, v �→ ⟨w, F(v)⟩W .
+2.2. Bent and vectorial bent functions. Let V be a finite dimensional F2-
+vector space, the function F : V → F2 is called a Boolean function. In particular, if
+V = Fn
+2, then F is represented by a unique reduced polynomial R(X1, X2, · · · , Xn)
+over F2 (see [4, 10]).
+Suppose F : V → F2, the Walsh transform of F is given by
+WF (w) =
+�
+v∈V
+(−1)F (v)+⟨w,v⟩, w ∈ V,
+and its inverse transform is
+(−1)F (v) =
+1
+2dim V
+�
+w∈V
+WF (w)(−1)⟨w,v⟩.
+In particular, if V = Fn
+2, then
+WF (w1, · · · , wn) =
+�
+(v1,··· ,vn)∈Fn
+2
+(−1)
+F (v1,··· ,vn)+
+n
+�
+i=1
+wivi,
+and its inverse transform is
+(−1)F (v1,··· ,vn) = 1
+2n
+�
+(w1,··· ,wn)∈Fn
+2
+WF (w1, · · · , wn)(−1)
+n
+�
+i=1
+wivi;
+if V = F2n, then
+WF (ω) =
+�
+v∈F2n
+(−1)F (v)+Tr2n/2(ωv),
+and its inverse transform is
+(−1)F (v) = 1
+2n
+�
+w∈F2n
+WF (w)(−1)Tr2n/2(wv).
+Definition 1. A Boolean function F : V → F2 is called bent if WF (w) = ±2
+dim V
+2
+for all w ∈ V .
+The dual of a bent function F, denoted as F ∗, is defined via the equality
+WF (w) = 2
+dim V
+2
+(−1)F ∗(w).
+Lemma 1. A Boolean function F : V → F2 is bent if and only if its first derivative
+DaF(v) = F(v + a) + F(v)
+in the direction of a is balanced for all 0 ̸= a ∈ V .
+
+4
+XIANHONG XIE1,2, YI OUYANG3,4
+For a vectorial function F : V → W, its Walsh transform is
+WF (a, ω) = WFa(ω) =
+�
+v∈V
+(−1)Fa(v)+⟨ω,v⟩,
+a ∈ W − {0}, ω ∈ V.
+In particular, if F : F2n → F2k, then WF (a, ω) can be written as
+WF (a, ω) = WFa(ω) =
+�
+v∈F2n
+(−1)Tr2k/2(aF (v))+Tr2n/2(ωv),
+a ∈ F∗
+2n, ω ∈ F2n.
+Definition 2. A vectorial Boolean function F : V → W is called bent if WF (a, ω) =
+WFa(ω) = ±2
+dim V
+2
+for any a ∈ W − {0} and ω ∈ V , i.e., its component functions
+Fa for all a ∈ W − {0} are bent.
+Lemma 2 ([11]). If F : V → W is a vectorial bent function, then dim V is even
+and dim W ≤ dim V
+2
+.
+2.3. Bent components.
+Proposition 1. Let dim V = n = 2m and F : V → V be a vectorial function. Set
+SF := {v ∈ V : Fv is not bent}.
+Then
+(1) (Pott et al. [22]) |SF | ≥ 2m, and |SF | = 2m if and only if SF is an m-
+dimensional F2-subspace of V .
+(2) (Hu et al. [23]) Moreover, if V = F2n and |SF | = 2m, then SF = F2m.
+3. Construction of Vectorial Functions With Maximum Number of
+Bent Components
+Assume n = 2m. The main goal of this section is to construct vectorial functions
+from V of dimension n to itself with maximal number of bent components.
+3.1. A Generic construction. This construction is inspired by recent work of
+Tang et al. [3] and Zheng et al. [28].
+Definition 3. Suppose f : V → F2 and {u1, u2, . . . , uk} ⊆ V for 2 ≤ k ≤ n. We
+say that (f; u1, · · · , uk) satisfies Condition A if f(x) is a bent function and
+DuiDujf ∗(x) = 0
+for all pairs
+1 ≤ i < j ≤ k.
+(1)
+Eq. (1) means that
+f ∗(x + ui + uj) =f ∗(x + ui) + f ∗(x + uj) + f ∗(x)
+=f ∗(x + ui) + f ∗(x) + f ∗(x + uj) + f ∗(x) + f ∗(x)
+=Duif ∗(x) + Dujf ∗(x) + f ∗(x).
+By induction, for any (w1, w2, · · · , wk) ∈ Fk
+2, one has
+f ∗(x +
+k
+�
+i=1
+wiui) = f ∗(x) +
+k
+�
+i=1
+wiDuif ∗(x).
+Theorem 1. Suppose G : V → V , 0 ̸= β ∈ V , 2 ≤ k ≤ n and {u1, u2, . . . , uk} ⊆ V
+such that (Gβ(x); u1, · · · , uk) satisfies Condition A. Then for any reduced polyno-
+mial H(X1, · · · , Xk) over F2, the function
+Fβ(x) := Gβ(x) + H(⟨u1, x⟩, · · · , ⟨uk, x⟩)
+
+VECTORIAL FUNCTIONS WITH MAXIMAL NUMBER OF BENT COMPONENTS
+5
+is a bent function, whose dual is
+F ∗
+β(x) = G∗
+β(x) + H(Du1G∗
+β(x), · · · , DukG∗
+β(x)).
+Our proof of this theorem is almost identical to that of [3, Theorem 8], which
+we include here for completeness.
+Proof. Applying the inverse Walsh transform to the Boolean function H : Fk
+2 → F2
+defined by H(X1, · · · , Xk), we get
+(−1)H(X1,X2,··· ,Xk) = 1
+2k
+�
+(w1,··· ,wk)∈Fk
+2
+WH(w1, · · · , wk)(−1)
+�k
+i=1 wiXi.
+(2)
+Take Xi = xi = ⟨ui, x⟩ for 1 ≤ i ≤ k and then multiply both sides of the above
+identity by (−1)Gβ(x)+⟨ω,x⟩, note that �k
+i=1 wixi = ⟨�k
+i=1 wiui, x⟩, we get
+(−1)Gβ(x)+H(x1,··· ,xk)+⟨ω,x⟩
+= 1
+2k
+�
+(w1,··· ,wk)∈Fk
+2
+WH(w1, · · · , wk)(−1)
+Gβ(x)+⟨ω+
+k
+�
+i=1
+wiui,x⟩
+,
+which leads to
+WFβ(ω) =
+�
+x∈V
+(−1)Gβ(x)+H(x1,··· ,xk)+⟨ω,x⟩
+= 1
+2k
+�
+x∈V
+�
+(w1,··· ,wk)∈Fk
+2
+WH(w1, · · · , wk)(−1)
+Gβ(x)+⟨ω+
+k
+�
+i=1
+wiui,x⟩
+= 1
+2k
+�
+(w1,··· ,wk)∈Fk
+2
+WH(w1, · · · , wk)WGβ(ω +
+k
+�
+i=1
+wiui).
+By definition of the dual of a bent function, then
+WFβ(ω) = 2m
+2k
+�
+(w1,··· ,wk)∈Fk
+2
+WH(w1, · · · , wk)(−1)
+G∗
+β(ω+
+k
+�
+i=1
+wiui)
+= 2m
+2k (−1)G∗
+β(ω)
+�
+(w1,··· ,wk)∈Fk
+2
+WH(w1, · · · , wk)(−1)
+k
+�
+i=1
+wiDui G∗
+β(ω)
+.
+This together with (2) yields
+WFβ(ω) = 2m(−1)G∗
+β(ω)+H(Du1 G∗
+β(ω),Du2 G∗
+β(ω),··· ,Duk G∗
+β(ω)).
+Hence Fβ is bent and
+F ∗
+β(x) = G∗
+β(x) + H(Du1G∗
+β(x), · · · , DukG∗
+β(x)).
+□
+3.2. Construction via the Niho quadratic function. Take V = F2n with the
+inner product given by the trace map. By Proposition 1 and Theorem 1, we have
+Theorem 2. Suppose G : F2n → F2n has 2n − 2m bent components.
+Suppose
+2 ≤ k ≤ n and {u1, · · · , uk} ⊆ F2n such that (Gβ(x); βu1, u2, · · · , uk) satisfies
+
+6
+XIANHONG XIE1,2, YI OUYANG3,4
+Condition A for all β ∈ F2n−F2m. Then for any reduced polynomial R(X2, · · · , Xk)
+over F2, the vectorial function
+F(x) := G(x) + u1xR(Tr2n/2(u2x), Tr2n/2(u3x), · · · , Tr2n/2(ukx))
+has 2n −2m bent components. Furthermore, the component Fβ(x) = Tr2n/2(βF(x))
+is bent and its dual
+F ∗
+β(x) = G∗
+β(x) + Dβu1G∗
+β(x)R(Du2G∗
+β(x), · · · , DukG∗
+β(x)).
+Remark 1. Comparing with the constructions in [21, 22], the vectorial function
+constructed by Theorem 2 is new and its dual is explicitly given.
+Moreover, the vectorial functions with maximal number of bent components by
+our construction can have high algebraic degrees if we choose the reduced polynomial
+R with high algebraic degree and u1, u2, . . . , uk linearly independent over F2, in this
+case the algebraic degree of R(Tr2n/2(u1x), . . . , Tr2n/2(ukx)) is equal to the algebraic
+degree of R(X1, . . . , Xk) (see [3]).
+From now on in this subsection, let G(x) = x2m+1. For β ∈ F2n − F2m, let
+γ = β + β2m ∈ F∗
+2m.
+The component function of G at β is the monomial Niho quadratic function
+Gβ : x ∈ F2n �→ Tr2n/2(βx2m+1).
+It is a bent function (see [20]) and its dual G∗
+β is given by
+G∗
+β(x) = Tr2m/2(γ−1x2m+1) + 1.
+(3)
+To apply Theorem 2, we first show that the function Gβ(x) satisfies Condition
+A when u1, u2, · · · , uk are appropriately chosen.
+Lemma 3. Suppose β ∈ F2n − F2m, k ≤ m and {u1, u2, · · · , uk} ⊆ F2n such that
+Tr2n/2(γ−1u2m
+j ui) = 0 for all 1 ≤ i < j ≤ k. Then (Gβ(x); u1, · · · , uk) satisfies
+Condition A and for 1 ≤ j ≤ k,
+DujG∗
+β(x) = Tr2n/2(γ−1xu2m
+j ) + Tr2m/2(γ−1u2m+1
+j
+).
+Proof. By Eq. (3), the derivative of G∗
+β(x) in the direction of uj ∈ F2n is
+DujG∗
+β(x) = Tr2m/2(γ−1x2m+1) + 1 + Tr2m/2(γ−1(x + uj)2m+1) + 1
+= Tr2n/2(γ−1u2m
+j x) + Tr2m/2(γ−1u2m+1
+j
+).
+Then the second order derivative in the direction of (ui, uj) is
+DuiDujG∗
+β(x) = DujG∗
+β(x + ui) + DujG∗
+β(x)
+= Tr2n/2(γ−1u2m
+j x) + Tr2n/2(γ−1u2m
+j (x + ui))
+= Tr2n/2(γ−1u2m
+j ui) = 0,
+with the last equality followed by our assumption.
+□
+By Lemma 3 and Theorem 1, then we have
+Theorem 3. Let β ∈ F2n − F2m and Gβ(x) = Tr2n/2(βx2m+1). If k ≤ m and
+{u1, u2, · · · , uk} ⊆ F2n satisfying Tr2n/2(γ−1u2m
+j ui) = 0 for any 1 ≤ i < j ≤ k,
+then the function
+Fβ(x) := Gβ(x) + H(Tr2n/2(u1x), Tr2n/2(u2x), · · · , Tr2n/2(ukx))
+
+VECTORIAL FUNCTIONS WITH MAXIMAL NUMBER OF BENT COMPONENTS
+7
+where H(X1, X2, · · · , Xk) is any reduced polynomial over F2, is bent and its dual is
+F ∗
+β(x) = G∗
+β(x) + H(Du1G∗
+β(x), Du2G∗
+β(x), · · · , DukG∗
+β(x)).
+Our first construction of vectorial functions with maximal number of bent com-
+ponents is the following result:
+Theorem 4. Let 3 ≤ k ≤ m and {u1, u2, · · · , uk} ⊆ F2m satisfy Tr2m/2(u1uj) = 0
+for j ≥ 2. Then for any reduced polynomial R(X2, · · · , Xk) over F2,
+F(x) = x2m+1 + u1xR(Tr2n/2(u2x), Tr2n/2(u3x), · · · , Tr2n/2(ukx)),
+has 2n − 2m bent components. More precisely, for β ∈ F2n − F2m, Fβ is bent and
+F ∗
+β(x) = G∗
+β(x) + Dβu1G∗
+β(x)R(Du2G∗
+β(x), · · · , DukG∗
+β(x)).
+Proof. Since ui ∈ F2m and Tr2m/2(u1uj) = 0, we have γ−1ujui ∈ F2m,
+DuiDujG∗
+β(x) = Tr2n/2(γ−1ujui) = 0, 2 ≤ i < j ≤ k ≤ m,
+and
+Dβu1DujG∗
+β(x) = Tr2n/2(γ−1ujβu1) = Tr2m/2(uju1) = 0, 2 ≤ j ≤ k.
+Therefore, (Gβ(x); βu1, u2, . . . , uk) satisfies Condition A for any β ∈ F2n −F2m.
+□
+In Theorem 4, if we take k = m and {ui : 1 ≤ i ≤ m} to be an orthogonal
+basis of F2m over F2 under the inner product ⟨x, y⟩ = Tr2m/2(xy), then Condition
+A holds, and what’s more, we have the following result.
+Corollary 1. Let R(X2, . . . , Xm) = X2 · · · Xm. Then the function
+F(x) = x2m+1 + u1x
+m
+�
+i=2
+Tr2n/2(uix)
+has maximal algebraic degree m and maximal number of bent components 2n − 2m.
+In particular, for k = 2, i.e., R(X2, . . . , Xm) = X2, we can take u1, u2 ∈ F2n
+such that (Gβ; βu1, u2) satisfies Condition A. Thus we have
+Theorem 5. Suppose u1, u2 ∈ F2n such that u1u2m
+2
+∈ F2m and Tr2m/2(u1u2m
+2 ) = 0.
+Then
+F(x) = x2m+1 + u1xTr2n/2(u2x)
+has 2n − 2m bent components: for β ∈ F2n − F2m, Fβ is bent and
+F ∗
+β(x) =Tr2m/2(λ−1x2m+1) + 1+
+�
+Tr2n/2(λ−1(βu1)2mx) + Tr2m/2(λ−1(βu1)2m+1)
+�
+×
+�
+Tr2n/2(λ−1u2m
+2 x) + Tr2m/2(λ−1u2m+1
+2
+)
+�
+.
+Proof. The proof is similar to Theorem 4, so we omit it here.
+□
+Note that the function F(x) given by Theorem 5 is of the form xℓ(x), thus is a
+solution of Problem 1. We now show it is not equivalent to the functions in cases
+(a), (b) and (e) in the Introduction. Recall for a vectorial function F and a, b ∈ V ,
+δF (a, b) := |{x ∈ F2n : F(x + a) + F(x) = b}|. The differential spectrum of F is
+{δF(a, b) : a ∈ F∗
+2n, b ∈ F2n}.
+It was shown in [26], [22] and [25] respectively that
+δx2m+1(a, b) ∈ {0, 2m},
+δx2i(x+x2m)(a, b) ∈ {0, 2gcd(i,m), 2m} (0 < i < m) and
+
+8
+XIANHONG XIE1,2, YI OUYANG3,4
+δx2i(Tr2n/2m(Λ(x)))(a, b) ∈ {0, 2s, 2m},
+where s is the dimension of the solution space (in F2m) of z2iΛ(Tr2n/2m(a)) +
+(Tr2n/2m(a))2iΛ(z) = 0. Then the inequivalence of our function in Theorem 5 to
+theirs follows from
+Theorem 6. Suppose u1, u2 ∈ F2n − F2m, u1u2m
+2
+∈ F2m and Tr2m/2(u1u2m
+2 ) = 0.
+Then the differential spectrum of F(x) = x2m+1 + u1xTr2n/2(u2x) is given by
+δF (a, b) ∈
+�
+{0, 2},
+if Tr2n/2(u2a) = 1,
+{0, 2m−1, 2m},
+if Tr2n/2(u2a) = 0.
+Proof. We have
+F(x + a) + F(x)
+= x2ma + xa2m + a2m+1 + u1xTr2n/2(u2a) + u1aTr2n/2(u2(x + a)).
+Notice that if x is a solution of F(x + a) + F(x) = b, so is x + a.
+(A) Assume Tr2n/2(u2a) = 1. The equation F(x + a) + F(x) = b is reduced to
+x2ma + xa2m + a2m+1 + u1x + u1aTr2n/2(u2x) + u1a = b,
+and then to one of the following two systems of equations:
+�
+x2ma + xa2m + u1x = b + a2m+1 + u1a,
+Tr2n/2(u2x) = 0;
+�
+x2ma + xa2m + u1x = b + a2m+1,
+Tr2n/2(u2x) = 1.
+We claim that x2ma + xa2m + u1x is a permutation over F2n. Then δF (a, b) ∈
+{0, 2} follows from the claim immediately.
+For x, y ∈ F2n, let
+x2ma + xa2m + u1x = y2ma + ya2m + u1y.
+Set z = Tr2n/2m(xa2m) − Tr2n/2m(ya2m) ∈ F2m, then y = x + u−1
+1 z and
+z = Tr2n/2m(xa2m − ya2m) = −Tr2n/2m(a2mu−1
+1 z) = −zTr2n/2m(a2mu−1
+1 )
+⇒ z(1 + Tr2n/2m(a2mu−1
+1 )) = 0.
+Suppose Tr2n/2m(a2mu−1
+1 ) = 1. Notice that u2m
+1 u2 ∈ F∗
+2m and au−2m
+1
+=
+au2
+u2m
+1
+u2 ,
+thus
+Tr2n/2m(a2mu−1
+1 ) = Tr2n/2m(au−2m
+1
+) = Tr2n/2m( au2
+u2m
+1 u2
+) = Tr2n/2m(au2)
+u2m
+1 u2
+= 1
+⇒ Tr2n/2m(au2) = u2m
+1 u2.
+Since Tr2n/2(u2a) = 1, we have Tr2n/2(u2a) = Tr2m/2(u2m
+1 u2) = 1, which is
+a contradiction to the assumption Tr2m/2(u2m
+1 u2) = 0. Thus z = 0 and x2ma +
+xa2m + u1x is a linear permutation over F2n.
+(B) Assume Tr2n/2(u2a) = 0. The equation F(x + a) + F(x) = b is reduced to
+x2ma + xa2m + a2m+1 + u1aTr2n/2(u2x) = b.
+(4)
+Assume that x, y are two solutions of (4). Then
+x2ma + xa2m + a2m+1 + u1aTr2n/2(u2x) = b,
+y2ma + ya2m + a2m+1 + u1aTr2n/2(u2y) = b,
+
+VECTORIAL FUNCTIONS WITH MAXIMAL NUMBER OF BENT COMPONENTS
+9
+which means that z = x + y is a solution of
+z2ma + za2m + u1aTr2n/2(u2z) = 0.
+or equivalently,
+�
+z2ma + za2m = 0,
+Tr2n/2(u2z) = 0;
+or
+�
+z2ma + za2m = u1a,
+Tr2n/2(u2z) = 1.
+Thus δF (a, b) = 0 or the number of solutions of these two systems of equations.
+Let Xu = {x ∈ F2n : Tr2n/2(ux) = 0}.
+The zero set of the first system of
+equations is the F2-vector space a−2mF2m ∩ Xu2. Note that dimF2 a−2mF2m = m
+and dimF2 Xu2 = n − 1, a−2mF2m ∩ Xu2 must be of dimension either m − 1 or m.
+For the second system, note that z2ma + za2m ∈ F2m, we must have u1a ∈ F2m.
+Hence u2a−2m =
+u2m
+1
+u2
+(u1a)2m ∈ F2m. The solution of z2ma+za2m = u1a is z =
+u1a
+a2m(1+ξ)
+with ξ2m+1 = 1. Then
+Tr2n/2(u2z) = Tr2n/2(
+u2u1a
+a2m(1 + ξ)) = Tr2n/2(u2a−2mu1a
+1 + ξ
+)
+= Tr2m/2(u2a−2mu1a) = Tr2m/2(u2u2m
+1 ) = 0.
+Hence the second system has no zeros at all.
+□
+3.3. Construction via the Maiorana-MacFarland class. In this case, we let
+V = F2m × F2m and the corresponding inner product be
+⟨(y1, z1), (y2, z2)⟩ = Tr2m/2(y1y2) + Tr2m/2(z1z2).
+Let φ be a permutation of F2m, and G be the associated map defined by
+G : F2m × F2m −→ F2m × F2m
+(y, z) �−→ (yφ(z), z).
+Then G has maximal number of bent components: for (a, b) ∈ F∗
+2m × F2m, the
+component function
+Ga,b(y, z) = Tr2m/2(ayφ(z) + bz),
+(5)
+at (a, b) is a bent function in the Maiorana- MacFarland class, whose dual
+G∗
+a,b(y, z) = Tr2m/2((z + b)φ−1(a−1y)).
+(6)
+We assume that φ is an automorphism of F2m from now on in this subsection.
+Theorem 7. Let 2 ≤ k ≤ m, (a, b) ∈ F∗
+2m × F2m, φ and G be given as above. If
+the set {ui = (ui,1, ui,2) ∈ V : 1 ≤ i ≤ k} satisfies
+Tr2m/2
+�
+uj,2φ−1(a−1ui,1) + ui,2φ−1(a−1uj,1)
+�
+= 0 for all 1 ≤ i < j ≤ k,
+then (Ga,b(y, z); u1, · · · , uk) satisfies Condition A and for 1 ≤ i ≤ k,
+DuiG∗
+a,b(y, z) = Tr2m/2
+�
+(z + b)φ−1(a−1ui,1) + ui,2φ−1(a−1(y + ui,1))
+�
+.
+As a consequence, for any reduced polynomial H(X1, · · · Xk) over F2, the function
+Fa,b(y, z) = Ga,b(y, z) + H(Tr2m/2(u1,1y + u1,2z), . . . , Tr2m/2(uk,1y + uk,2z))
+is bent and its dual is
+F ∗
+a,b(y, z) = G∗
+a,b(y, z) + H(Du1G∗
+a,b(y, z), . . . , DukG∗
+a,b(y, z)).
+
+10
+XIANHONG XIE1,2, YI OUYANG3,4
+Proof. It suffices to check that (Ga,b(y, z); u1, · · · , uk) satisfies Condition A. By
+Eq. (6), the derivative of G∗
+a,b(y, z) in the direction of ui is
+DuiG∗
+a,b(y, z) = G∗
+a,b(y + ui,1, z + ui,2) + G∗
+a,b(y, z)
+= Tr2m/2
+�
+(z + b)φ−1(a−1ui,1) + ui,2φ−1(a−1(y + ui,1))
+�
+.
+Then the second order derivative in the direction (ui, uj) is
+DujDuiG∗
+a,b(y, z)
+= DujG∗
+a,b(y + ui,1, z + ui,2) + DujG∗
+a,b(y, z)
+= Tr2m/2
+�
+φ−1(a−1)(uj,2φ−1(ui,1) + ui,2φ−1(uj,1))
+�
+= 0.
+□
+Our second construction of vectorial functions with maximal number of bent
+components is the following result:
+Theorem 8. Let 2 ≤ k ≤ m, φ and G be given as above. Suppose u = (u1,1, 0)
+and choose ui = (ui,1, ui,2) for 2 ≤ i ≤ k such that
+Tr2m/2(φ−1(u1,1)ui,2) = 0 and ui,1 = φ(ui,2).
+Then for any reduced polynomial R(X2, · · · Xk) over F2, the vectorial function
+F(y, z) = (yφ(z), z) + (u1,1y, 0)R(Tr2m/2(u2,1y + u2,2z), . . . , Tr2m/2(uk,1y + uk,2z))
+has 2n − 2m bent components: for any (a, b) ∈ F∗
+2m × F2m,
+Fa,b(y, z) = ⟨(a, b), F(y, z)⟩ = Tr2m/2(ayφ(z) + bz)
++ Tr2m/2(au1,1y)R(Tr2m/2(u2,1y + u2,2z), . . . , Tr2m/2(uk,1y + uk,2z))
+is bent and
+F ∗
+a,b(y, z) = G∗
+a,b(y, z) + Du1,1a,0G∗
+a,b(y, z)R(Du2G∗
+a,b(y, z), . . . , DukG∗
+a,b(y, z)).
+Proof. Since Tr2m/2(φ−1(u1,1)ui,2) = 0 and ui,1 = φ(ui,2), then for 2 ≤ i ≤ k,
+DuiD(au1,1,0)G∗
+a,b(y, z) = Tr2m/2(φ−1(a−1)ui,2φ−1(au1,1))
+= Tr2m/2(ui,2φ−1(u1,1)) = 0,
+and for 2 ≤ i < j ≤ k,
+DujDuiG∗
+a,b(y, z) = Tr2m/2
+�
+φ−1(a−1)(uj,2φ−1(ui,1) + ui,2φ−1(uj,1))
+�
+= 0.
+Therefore, (Ga,b; (au1,1, 0), u2, . . . , uk) satisfies Condition A for (a, b) ∈ F∗
+2m × F2m.
+□
+Suppose m′ is a divisor of m, then (5) can be written as
+Ga,b(y, z) = Tr2m/2(ayφ(z) + bz) = Tr2m′ /2(G′
+a,b(y, z)),
+where G′
+a,b(y, z) = Tr2m/2m′ (ayφ(z) + bz).
+Theorem 9. Suppose m′ is a divisor of m, G′
+a,b(y, z) = Tr2m/2m′ (ayφ(z) + bz).
+Then G′
+a,b(y, z) is a vectorial bent function for (a, b) ∈ F∗
+2m × F2m. Furthermore,
+for c ∈ F∗
+2m′, Gca,cb(y, z) = Tr2m′/2(cG′
+a,b(y, z)) is a bent function and its dual is
+G∗
+ca,cb(y, z) = Tr2m/2
+�
+zφ−1((ac)−1y) + bcφ−1((ac)−1y)
+�
+.
+Proof. Note that for any c ∈ F∗
+2m′, one has (ca, cb) ∈ F∗
+2m × F2m, thus Gca,cb(y, z)
+is bent function in the Maiorana-MacFarland class and the dual can be obtained
+directly from (6).
+□
+
+VECTORIAL FUNCTIONS WITH MAXIMAL NUMBER OF BENT COMPONENTS
+11
+Remark 2. Take a = b = 1. Let c1, c2, c3 be three pairwise distinct elements in
+F∗
+2m′ such that c := c1 + c2 + c3 ̸= 0. For s ∈ {c, c1, c2, c3}, let
+Gs(y, z) := Gs,s(y, z) = Tr2m/2(syφ(z) + sz).
+Then Gc and Gci are all bent functions and
+Gc1(y, z) + Gc2(y, z) + Gc3(y, z) = Gc(y, z).
+To have the equality
+G∗
+c1(y, z) + G∗
+c2(y, z) + G∗
+c3(y, z) = G∗
+c(y, z).
+(7)
+It suffices to find an φ of F2m such that
+(C1) φ−1(c−1y) = φ−1(c−1
+1 y) + φ−1(c−1
+2 y) + φ−1(c−1
+3 y);
+(C2) cφ−1(c−1y) = c1φ−1(c−1
+1 y) + c2φ−1(c−1
+2 y) + c3φ−1(c−1
+3 y).
+Let φ be an involution and automorphism of F2m, then (C1) and (C2) are satisfied,
+so is Eq.(7). This gives a solution of the open problem proposed by Mesnager ([20,
+Open Problem 2]). Specially, the case of φ = φ−1 : z �→ z−1 was presented by [27].
+4. Binomial vectorial functions with maximal number of bent
+components
+Let n = 2m and v2(i) be the 2-adic valuation of i ∈ Z. The main result of this
+section is
+Theorem 10. The binomial vectorial function F(x) = x2m+1 + x2i+1 for 0 ≤ i ≤
+m − 1 on F2n has 2n − 2m bent components if and only if i = 0, i.e., F(x) is affine
+equivalent to x2m+1.
+Remark 3. The special case of odd m was proved by Zheng et al. [21].
+From now on, fix i such that 0 ≤ i < m, and let
+d = gcd(m + i, 2m) = gcd(m + i, 2i).
+Let F(x) = x2m+1 + x2i+1.
+For a ∈ F2n, the component function Fa(x) =
+Tr2n/2(ax2m+1 + ax2i+1). Let
+La(y) := a2iy22i + (a + a2m)2iy2m+i + ay.
+(8)
+If a ∈ F2m, then Fa(x) = Tr2n/2(ax2i+1) and (8) is reduced to
+La(y) := a2iy22i + ay.
+For any y ∈ F∗
+2n, the derivative of Fa(x) at direction y is
+DyFa(x) = Tr2n/2(a((x + y)2m+1 + (x + y)2i+1)) + Tr2n/2(a(x2m+1 + x2i+1))
+= Tr2n/2(x(ay2i + (a + a2m)y2m + a2n−iy2n−i)) = Tr2n/2(xLa(y)−2i).
+The root set of La(y) in F2n forms an F2d-vector space, hence the number of the
+roots of La(y) in F2n is either 1 or a power of 2d.
+Lemma 4. Assume v2(i) = v2(m). For ξ ∈ F2d such that ξ2d/2+1 = 1, let a =
+1
+1+ξ.
+Then a /∈ F2m and La(y) = 0 for any y ∈ F2d.
+
+12
+XIANHONG XIE1,2, YI OUYANG3,4
+Proof. By v2(i) = v2(m), d is even, m = d
+2 · m′ and i = d
+2 · i′ with m′ and i′ both
+odd. Then
+ξ2m = ξ2i = ξ−1 =⇒ a2m = a2i = ξa.
+This means that a /∈ F2m and
+La(y) = a2iy22i + (a + a2m)2iy2m+i + ay = a(ξy22i + (1 + ξ)y2m+i + y).
+For any y0 ∈ F2d = F22i ∩ F2m+i, one has y22i
+0
+= y2m+i
+0
+, hence La(y0) = 0.
+□
+We need the following two general results.
+Lemma 5. [29, Theorem 5.30] Let χ′ be a multiplicative character of F∗
+2m of order
+2d − 1. Then for any (a, b) ∈ F∗
+2m × F2m,
+�
+x∈F2m
+(−1)Tr2m/2(ax2d−1+b) = (−1)Tr2m/2(b)
+2d−2
+�
+j=1
+χ′j(a)G(χ′j),
+where χ and G(χ) are the conjugate and the Gauss sum of χ.
+Lemma 6. Suppose d < m is a factor of m. Let gcd(2d − 1, m
+d ) = t. Then the set
+N = {y ∈ F∗
+2m : Tr2m/2d(y2d−1) = 0} has order
+|N| =
+
+
+
+
+
+
+
+2m − 2d
+2d
++ (2d − 1)(−1)
+m
+d −1
+2d
+�
+χ∈(�F∗
+2d)
+2d−1
+t
+\{χ0}
+G(χ)
+m
+d ,
+if t ̸= 1;
+2m−d − 1,
+if t = 1.
+where �F∗
+2d is the set of the multiplicative characters of F∗
+2d and χ0 is the trivial
+character. In particular, N is non-empty.
+Proof. We have
+|N| = 1
+2d
+�
+v∈F2d
+�
+y∈F∗
+2m
+(−1)Tr2d/2(vTr2m/2d(y2d−1))
+= 2m − 1
+2d
++ 1
+2d
+�
+v∈F∗
+2d
+�
+y∈F∗
+2m
+(−1)Tr2m/2(vy2d−1).
+(9)
+Suppose F∗
+2m = ⟨β⟩, then F∗
+2m = � 2m−1
+2d−1 −1
+i=0
+βiF∗
+2d. Note that gcd( 2m−1
+2d−1 , 2d − 1) =
+gcd( m
+d , 2d − 1) = t. If t = 1, one has
+|N| = 2m − 1
+2d
++ 2d − 1
+2d
+�
+v∈F∗
+2d
+2m−1
+2d−1 −1
+�
+i=0
+(−1)Tr2m/2(vβi(2d−1))
+= 2m − 1
+2d
++ 2d − 1
+2d
+�
+v∈F∗
+2d
+2m−1
+2d−1 −1
+�
+i=0
+(−1)Tr2m/2(vβi)
+= 2m − 1
+2d
++ 2d − 1
+2d
+�
+v∈F∗
+2m
+(−1)Tr2m/2(v) = 2m − 2d
+2d
+≥ 1.
+
+VECTORIAL FUNCTIONS WITH MAXIMAL NUMBER OF BENT COMPONENTS
+13
+If t ̸= 1, suppose χ′ is a multiplicative character of F∗
+2m of order 2d − 1, then by
+Lemma 5 and Eq. (9),
+|N| = 2m − 1
+2d
++ 1
+2d
+�
+v∈F∗
+2d
+� �
+y∈F2m
+(−1)Tr2m/2(vy2d−1) − 1
+�
+= 2m − 1
+2d
++ 1
+2d
+�
+v∈F∗
+2d
+�2d−2
+�
+j=1
+χ′j(v)G(χ′j) − 1
+�
+= 2m − 1
+2d
++ 1
+2d
+�
+v∈F∗
+2d
+2d−2
+�
+j=0
+χ′j(v)G(χ′j).
+(10)
+Suppose N is the norm mapping from F2m to F2d. For χ ∈ �F∗
+2d, it can be lifted
+from F2d to F2m by χ′ = χ ◦ N (see [29, Theorem 5.28]). Furthermore, χ is of order
+2d − 1 if and only if χ′ is of order 2d − 1. Then
+2d−2
+�
+j=0
+χ′j(v)G(χ′j) =
+�
+χ∈�F∗
+2d
+χ(N(v))G(χ ◦ N) = (−1)
+m
+d −1 �
+χ∈�F∗
+2d
+χ(v
+2m−1
+2d−1 )G(χ)
+m
+d .
+Suppose δ = β
+2m−1
+2d−1 ∈ F∗
+2d, then F∗
+2d =
+2d−1
+t
+−1
+�
+j=0
+δj⟨δ
+2d−1
+t
+⟩. By Eq. (10), we get
+|N| = 2m − 1
+2d
++ (−1)
+m
+d −1
+2d
+�
+v∈F∗
+2d
+�
+χ∈�F∗
+2d
+χ(v
+2m−1
+2d−1 )G(χ)
+m
+d
+= 2m − 1
+2d
++ (−1)
+m
+d −1
+2d
+�
+χ∈�F∗
+2d
+G(χ)
+m
+d
+2d−1
+t
+−1
+�
+j=0
+�
+v∈δj⟨δ
+2d−1
+t
+⟩
+χ(v
+2m−1
+2d−1 )
+= 2m − 1
+2d
++ (−1)
+m
+d −1t
+2d
+�
+χ∈�F∗
+2d
+G(χ)
+m
+d
+2d−1
+t
+−1
+�
+j=0
+χ(δ
+j 2m−1
+2d−1 ).
+Note that gcd( 2m−1
+t(2d−1), 2d−1
+t
+) = 1, then
+2d−1
+t
+−1
+�
+i=0
+χ(δi 2m−1
+2d−1 ) =
+2d−1
+t
+−1
+�
+i=0
+χ(δit) =
+�
+x∈⟨δt⟩
+χ(x) =
+�
+0,
+if χ ̸= χ0;
+2d−1
+t
+,
+if χ = χ0.
+Hence we have
+|N| = 2m − 2d
+2d
++ (2d − 1)(−1)
+m
+d −1
+2d
+�
+χ∈(�F∗
+2d)
+2d−1
+t
+\{χ0}
+G(χ)
+m
+d .
+Note that |G(χ)| = 2
+d
+2 for χ ̸= χ0 (see [29, Theorem 5.11]), then we have
+|N| ≥ 2m − 2d − (2d − 1)(t − 1)2
+m
+2
+2d
+.
+
+14
+XIANHONG XIE1,2, YI OUYANG3,4
+Note that t ̸= 1 and
+2m − 2d − (2d − 1)(t − 1)2
+m
+2 = 2m − t2
+m
+2 +d + (t − 1)2
+m
+2 − 2d > 2m − t2
+m
+2 +d.
+Since t = gcd( m
+d , 2d − 1) ≤ 2d − 1, then 2m − t2
+m
+2 +d > 2m − 2
+m
+2 +2d ≥ 0 if m
+d ≥ 4.
+If m = 3d, then t = gcd(3, 2d − 1) = 3 and d is even. One has
+|N| = 23d − 2d − (2d − 1)2
+3d
+2 +1 > 23d − 2
+5d
+2 +1 ≥ 0.
+If m = 2d, then t = gcd(2, 2d − 1) = 1, which contradicts to t ̸= 1. Thus we
+complete the proof.
+□
+Back to our situation, we have the following result.
+Lemma 7. Suppose v2(m) < v2(i), then there exists a ∈ F2n −F2m such that La(y)
+has roots in F∗
+2n.
+Proof. For v2(m) < v2(i), note that d = gcd(m, i) = gcd(m + i, n). It suffices to
+show that there exists a ∈ F2n − F2m such that La(y) has a root y0 ∈ F∗
+2m. Note
+that for y ∈ F∗
+2m,
+La(y) = a2iy22i + (a + a2m)2iy2m+i + ay = a2iy22i + (a + a2m)2iy2i + ay.
+(11)
+Then we just need to find (a, y) ∈ F2n × F∗
+2m such that
+�
+a + a2m = y−2i−1v,
+(ay2i+1)2i + ay2i+1 = y2i−22iv,
+(12)
+for some v ∈ F∗
+2d (here a /∈ F2m is automatic). Let z = av−1y2i+1, then we just
+need to find (z, y) ∈ F2n × F∗
+2m such that
+�
+z + z2m = 1,
+(13)
+z2i + z = y2i−22i.
+(14)
+We consider Eq. (14).
+Note that the F2d-linear maps ϕi : z �→ z2i + z and
+ϕd : z �→ z2d + z from F2m to itself have the same kernel F2d and ϕi(z) = ϕd(z +
+z2d + · · · + z2( i
+d −1)d), then Im(ϕi) ⊆ Im(ϕd) and hence Im(ϕd) = Im(ϕi). Note also
+that the group homomorphisms y �→ y2i(1−2i) and y �→ y2d−1 from F∗
+2m to itself
+have the same kernel and image. Then there is an one-to-one correspondents of
+solutions (z, y) ∈ F2m × F∗
+2m of Eq. (14) and of
+z2d + z = y2d−1.
+(15)
+Eq. (15) is soluble if and only if there exists y ∈ F∗
+2m such that Tr2m/2d(y2d−1) = 0,
+which is guaranteed by Lemma 6 as d < m in this case. Thus there exists (z0, y0) ∈
+F∗
+2m × F∗
+2m such that z2i
+0 + z0 = y2i−22i
+0
+.
+Let w ∈ F22d \ F2d satisfy w2d + w = v0, then w2m−d = w2i = w and z =
+z0 + w ∈ F2n \ F2m is a solution of Eqs. (13) and (14).
+Thus, y0 ∈ F∗
+2m and
+a = (z0 + w)y−2i−1
+0
+v satisfy the equation La(y) = 0.
+□
+Lemma 8. For 0 ≤ i ≤ m − 1, if F(x) = x2m+1 + x2i+1 has 2n − 2m bent
+components, then v2(m) ≤ v2(i).
+
+VECTORIAL FUNCTIONS WITH MAXIMAL NUMBER OF BENT COMPONENTS
+15
+Proof. Assume v2(m) > v2(i). In this case d = gcd(m, i) = gcd(n, i), and 2d =
+gcd(2i, m). This means 2d−1 = gcd(2m−1, 2i−1) and 22d−1 = gcd(2m−1, 22i−1),
+which then implies that 2d + 1 is a factor of 2m − 1 and thus prime to 2m + 1.
+Let α be a primitive element of F2n. Let a = αk(2m+1) ∈ F∗
+2m such that a2i−1 =
+α(2m+1)(2d−1). By Proposition 1, for this a, Fa(x) = Tr2n/2(ax2i+1) is not bent. By
+Lemma 1, DyFa(x) = Tr2n/2(x(ay2i + (ay)2n−i)) is not balanced for some y ∈ F2n,
+i.e., a2i−1y22i−1 + 1 = 0 is soluble. Let a2i−1 = α(2m+1)(2d−1) = y1−22i
+0
+and let
+y1 ∈ F∗
+2n such that y1−22i
+0
+= y22d−1
+1
+. Then the congruent equation
+(22d − 1)x ≡ (2d − 1)(2m + 1) mod (2n − 1)
+is soluble, equivalently, the equation
+(2d + 1)x ≡ 2m + 1 mod (2d + 1) · 2n − 1
+22d − 1
+is soluble. This is not possible since 2d + 1 is prime to 2m + 1.
+□
+Proof of Theorem 10. If i = 0, the result is trivial. We now assume i ̸= 0 and
+v2(m) ≤ v2(i).
+If F(x) has maximal number of bent components, by Lemma 1, Fa(x) is bent
+function for all a ∈ F2n −F2m and hence Dyfa(x) is balanced for any y ∈ F∗
+2n. This
+implies La(y) ̸= 0 for all y ∈ F∗
+2n. Hence to show F(x) does not have maximal
+number of bent components, it suffices to show there exists a ∈ F2n − F2m, such
+that La(y) has a root in F∗
+2n:
+(i) If v2(m) = v2(i), this is implied by Lemma 4.
+(ii) If v2(m) < v2(i), this is implied by Lemma 7.
+Thus for i ̸= 0, F(x) cannot have 2n − 2m bent components.
+□
+Remark 4. For a general binomial vectorial function F(x) = xd1 +xd2, our exper-
+imental result indicates that F(x) is affine equivalent to x2m+1 or x2i(x + x2m) if
+F(x) has maximal number of bent components, but so far we do not have a proof.
+We leave this as an open problem for future study.
+5. Conclusion
+We firstly give a generic construction of vectorial functions with maximal number
+of bent components, and obtain two new classes of such vectorial functions based
+on the Niho quadratic function and the Maiorana-MacFarland class. Moreover,
+we solve the open problem proposed by Mesnager, and partially answer the open
+problem proposed by Pott et al. We then show that the binomial function F(x) =
+x2m+1 + x2i+1 : F22m → F22m has maximal number of bent components if and only
+if i = 0.
+References
+[1] O. Rothaus, “On ’bent’ functions.” J. Combinat. Theory, Ser. A, 20(3), 300-305 (1976).
+[2] C. Carlet, S. Mesnager, “ Four decades of research on bent functions.” Des. Codes Cryptogr.,
+78(1), 5-50 (2016).
+[3] C. Tang, Z. Zhou, Y. Qi, X. Zhang, C. Fan and T. Helleseth, “Generic construction of
+bent functions and bent idempotents with any possible algebraic degrees.” IEEE Trans. Inf.
+Theory, 63(10), 6149-6157 (2017).
+[4] S. Mesnager, Bent Functions: Fundamentals and Results. Springer, Cham (2016).
+[5] G. Leander, “Monomial bent functions.” IEEE Trans. Inf. Theory, 52(2), 738-743 (2006).
+
+16
+XIANHONG XIE1,2, YI OUYANG3,4
+[6] G. Leander and A. Kholosha, “Bent functions with 2r Niho exponents.” IEEE Trans. Inf.
+Theory, 52(12), 5529-5532 (2006).
+[7] N. Li, T. Helleseth, X. Tang and A. Kholosha, “Several new classes of bent functions from
+Dillon exponents.” IEEE Trans. Inf. Theory, 59(3), 1818-1831 (2013).
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+Scientia Sinica Math., 47(9), 995-1010 (2017).
+[9] C. Carlet, “On bent and highly nonlinear balanced/resilient functions and their algebraic
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+Imai, S. Lin, A. Poli, Eds, New York, NY, USA: Springer-Verlag, 1-28 (2006).
+[10] C. Carlet, Boolean Functions for Cryptography and Coding Theory. Cambridge, U.K.: Cam-
+bridge Univ. Press (2021).
+[11] K. Nyberg, “Perfect nonlinear S-boxs,” in Advance in Cryptology-EUROCRYPT. Berlin,
+Germany: Springer-Verlag, 547, 378-385 (1991).
+[12] A. C¸e¸smelio˘glu, W. Meidl and A. Pott, “Vectorial bent functions and their duals,” Linear
+Algebra Appl., 548, 305-320 (2018).
+[13] K. Feng, J. Yang, “Vectorial Boolean functions with good cryptographic properties,” Int. J.
+Found. Comput. Sci., 22(6), 1271-1282 (2011).
+[14] S. Mesnager, “Bent functions from spreads,” J. Amer. Math. Soc., to be published.
+[15] E. Pasalic and W. Zhang, “On multiple output bent functions.” Inform. Process. Lett.,
+112(21), 811-815 (2012).
+[16] S. Mesnager, “Bent vectorial functions and linear codes from o-polynomials.” Des. Codes
+Cryptogr., 77(1), 99-116 (2015).
+[17] A. Muratovi´c-Ribi´c, E. Pasalic and S. Bajri´c, “Vectorial bent functions from multiple terms
+trace functions.” IEEE Trans. Inf. Theory, 60(2), 1337-1347 (2014).
+[18] Y. Xu, C. Carlet, S. Mesnager and C. Wu, “Classification of bent monomials, constructions
+of bent multinomials and upper bounds on the nonlinearity of vectorial functions.” IEEE
+Trans. Inf. Theory, 64(1), 367-383 (2018).
+[19] S. Mesnager, F. Zhang, C. Tang and Y. Zhou, “Further study on the maximum number of
+bent components of vectorial functions.” Des., Codes Cryptogr., 87, 2597-2610 (2019).
+[20] S. Mesnager, “Several new infinite families of bent functions and their duals.” IEEE Trans.
+Inf. Theory, 60(7), 4397-4407 (2014).
+[21] L. Zheng, J. Peng, H. Kan, Y. Li and J. Luo, “On constructions and properties of (n, m)-
+functions with maximal number of bent components.” Des. Codes Cryptogr., 88(9), 2171-2186
+(2020).
+[22] A. Pott, E. Pasalic, A. Muratovic and S. Bajric, “On the maximum number of bent compo-
+nents of vectorial functions.” IEEE Trans. Inf. Theory, 64(1), 403-411 (2018).
+[23] H. Hu, B. Wang, X. Xie and Y. Luo, “Two problems about monomial bent functions,”
+arXiv:2102.12304v1.
+[24] N. Anbar, T. Kalayci, W. Meidl, “Analysis of (n, n)-functions obtained from the Maiorana-
+McFarland class.” IEEE Trans. Inf. Theory, 67(7), 4891-4901 (2021).
+[25] N. Anbar, T. Kalayci, W. Meidl, L. M´erai, “On a class of functions with the maximal number
+of Bent components.” IEEE Trans. Inf. Theory, 68(9), 6174-6186 (2022).
+[26] K. Nyberg, “Differentially uniform mappings for cryptography,” in Advance in Cryptology-
+EUROCRYPT. Berlin, Germany: Springer-Verlag, 765, 55-64 (1993).
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+Crypto., 89, 75-89 (2021).
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+second order derivatives.” Cryptogr. Commun., 12(1), 1143-1160 (2020).
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+1School of Information and Computer, Anhui Agricultural University, Hefei 230036,
+China
+2School of Cyber Science and Technology, University of Science and Technology
+of China, Hefei 230027, China
+Email address: xianhxie@ahau.edu.cn
+3School of Mathematical Sciences, CAS Wu Wen-Tsun Key Laboratory of Mathe-
+matics, University of Science and Technology of China, Hefei 230026, China
+
+VECTORIAL FUNCTIONS WITH MAXIMAL NUMBER OF BENT COMPONENTS
+17
+4Hefei National Laboratory, Hefei 230088, China
+Email address: yiouyang@ustc.edu.cn
+
diff --git a/4tE1T4oBgHgl3EQfAwJw/content/tmp_files/load_file.txt b/4tE1T4oBgHgl3EQfAwJw/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..c281beae5d4caeeab13e3e5435af550bd0bf1e7a
--- /dev/null
+++ b/4tE1T4oBgHgl3EQfAwJw/content/tmp_files/load_file.txt
@@ -0,0 +1,608 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf,len=607
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='02843v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='IT]  7 Jan 2023  ON VECTORIAL FUNCTIONS WITH MAXIMAL NUMBER OF  BENT COMPONENTS  XIANHONG XIE1,2, YI OUYANG3,4  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' We study vectorial functions with maximal number of bent compo-  nents in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' We first give a construction of such functions from known  ones, thus obtain two new classes from the Niho class and the Maiorana-  McFarland class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' Our construction gives a partial answer to an open prob-  lem proposed by Pott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=', and also solves an open problem proposed by  Mesnager.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  We then show that the vectorial function F : F22m → F22m,  x �→ x2m+1 + x2i+1 has maximal number of bent components if and only if  i = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  Keywords Vectorial bent functions, Vectorial functions, Bent components,  Niho quadratic function, Maiorana-McFarland class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' Introduction  Bent functions, as a special class of Boolean functions, were introduced by  Rothaus [1] and have been extensively studied (see [2]-[10]) due to their impor-  tant applications in cryptography, coding theory and combinatorics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  The bentness of Boolean functions can be extended to a general vectorial function  F : F2n → F2k by requesting all component functions fc(x) = Tr2k/2(cF(x)) (c ∈  F∗  2k) of F to be bent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' Nyberg [11] showed that vectorial bent functions can only  exist if n is even and n ≥ 2k, and presented two different constructions of such  functions from known classes of bent functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' The reader can refer to [12]-[19]  for more constructions of vectorial bent functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' However, relatively little work  was done to construct bent functions from known vectorial bent functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' In this  direction, Mesnager [20] proved the following result:  Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' If F : F2n → F2k is a vectorial bent function, and c1, c2, c3 ∈ F∗  2k  satisfying c1+c2+c3 ̸= 0 and f ∗  c1+f ∗  c2+f ∗  c3 = f ∗  c1+c2+c3, then fc1fc2+fc1fc3 +fc2fc3  is bent and its dual is f ∗  c1f ∗  c2 + f ∗  c1f ∗  c3 + f ∗  c2f ∗  c3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  She raised an open problem to find vectorial bent functions satisfying the above  condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  Another interesting and important question in studying bentness of vectorial  functions is that how large the number of bent components of a vectorial function  could be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' Suppose F : F2n → F2k is a vectorial function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' Nyberg’s condition [11] is  equivalent to that the possible maximum 2k − 1 can be attained only if n is even  and k ≤ n  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' For k = n, Pott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' [22] proved that this number is at most 2n − 2  n  2  2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' 11T71, 94A60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  Partially supported by Innovation Program for Quantum Science and Technology (Grant  No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  2021ZD0302904) and Anhui Initiative in Quantum Information Technologies (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  AHY150200).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  1  2  XIANHONG XIE1,2, YI OUYANG3,4  and presented a class of binomial functions attaining the upper bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' Further,  Pott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' [22] raised the following problem:  Problem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' Determine all linear mappings ℓ(x) over F2n such that the number of  bent components of xℓ(x) is 2n − 2  n  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  For k ≥ n  2 , Zheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' [21] showed that the number of bent components is at  most 2k − 2k− n  2 and presented a class of vectorial functions with 2k − 2k− n  2 bent  components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  Let n = k = 2m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' Pott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' [22] firstly gave the upper bound of the number of  bent components, and then constructing vectorial Boolean functions attaining the  upper bound has been studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' There are five known classes of vectorial functions  from F2n to itself with 2n − 2m bent components:  (a) F(x) = x2m+1 ([23]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  (b) F(x) = x2i(x + x2m) = x2iTr2n/2m(x), 0 ≤ i ≤ n − 1 ([22]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  (c) F(x) = x2i(Tr2n/2m(x) +  ρ�  j=1  γ(j)Tr2n/2m(x)2tj ), where γ(j) ∈ F2m, ρ ≤ m  satisfying �ρ  j=1(γ(j))2m−tj z2k−tj −1+1 ̸= 0 and �ρ  j=1(γ(j))2m−jz2tj −1+1 ̸=  0 for any x ∈ F2m ([19]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  (d) F(x) = xh(Tr2n/2m(x)), where h : F2m → F2m is a permutation ([21]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  (e) F(x) = x2iTr2n/2m(Λ(x)), where Λ(x) is a linearized permutation polyno-  mial of F2m[x] ([24]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  We note that functions in (a), (b) and (e) are of the form xℓ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  We shall work on vectorial functions from F2n to itself with maximal number of  bent components in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' Our main contributions are:  (A) We present two new classes of vectorial functions from F2n to itself with  maximal number of bent components via the Niho quadratic function and  the Maiorana–McFarland class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' Moreover,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='1) From the Niho quadratic function, we obtain a new class of quadratic  vectorial functions of the form  F(x) = x2m+1 + u1xTr2n/2(u2x) = xℓ(x),  where u1, u2 ∈ F2n−F2m satisfying u2m  1 u2 ∈ F2m and Tr2m/2(u2m  1 u2) =  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' This gives a partial answer to Problem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='2) From the Maiorana–McFarland class, we obtain a family of vectorial  bent functions and find three distinct bent components Gc1, Gc2, Gc3  such that G∗  c1 + G∗  c2 + G∗  c3 = G∗  c, where c1, c2, c3 ∈ F∗  2m and c =  c1 + c2 + c3 ̸= 0, so that the conditions of Mesnager [20] are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  (B) We prove that the binomial vectorial function F(x) = x2m+1 + x2i+1 (0 ≤  i ≤ m − 1) has 2n − 2m bent components if and only if i = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' Preliminaries  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' Basic Notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' Let n and k be positive integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  For k | n, the trace  function from F2n to its subfield F2k is Tr2n/2k(x) =  n  k −1  �  i=0  x2ki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  VECTORIAL FUNCTIONS WITH MAXIMAL NUMBER OF BENT COMPONENTS  3  For a finite dimensional F2-vector space V , we always fix a non-degenerate inner  product ⟨ , ⟩ = ⟨ , ⟩V on V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' In particular, if V = Fn  2, we let  ⟨(vi), (wi)⟩ =  n  �  i=1  viwi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  If V = F2n, let  ⟨ω, x⟩ = Tr2n/2(ωx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  For a subspace W of V , let W ⊥ = {v ∈ V, ⟨v, w⟩ = 0 for all w ∈ W} be the  orthogonal complementary of W, then dim W ⊥ = dim V − dim W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  For a vectorial function F : V → W, the component function of F at w ∈ W is  the function Fw : v → F2, v �→ ⟨w, F(v)⟩W .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' Bent and vectorial bent functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' Let V be a finite dimensional F2-  vector space, the function F : V → F2 is called a Boolean function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' In particular, if  V = Fn  2, then F is represented by a unique reduced polynomial R(X1, X2, · · · , Xn)  over F2 (see [4, 10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  Suppose F : V → F2, the Walsh transform of F is given by  WF (w) =  �  v∈V  (−1)F (v)+⟨w,v⟩, w ∈ V,  and its inverse transform is  (−1)F (v) =  1  2dim V  �  w∈V  WF (w)(−1)⟨w,v⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  In particular, if V = Fn  2, then  WF (w1, · · · , wn) =  �  (v1,··· ,vn)∈Fn  2  (−1)  F (v1,··· ,vn)+  n  �  i=1  wivi,  and its inverse transform is  (−1)F (v1,··· ,vn) = 1  2n  �  (w1,··· ,wn)∈Fn  2  WF (w1, · · · , wn)(−1)  n  �  i=1  wivi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  if V = F2n, then  WF (ω) =  �  v∈F2n  (−1)F (v)+Tr2n/2(ωv),  and its inverse transform is  (−1)F (v) = 1  2n  �  w∈F2n  WF (w)(−1)Tr2n/2(wv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' A Boolean function F : V → F2 is called bent if WF (w) = ±2  dim V  2  for all w ∈ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  The dual of a bent function F, denoted as F ∗, is defined via the equality  WF (w) = 2  dim V  2  (−1)F ∗(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' A Boolean function F : V → F2 is bent if and only if its first derivative  DaF(v) = F(v + a) + F(v)  in the direction of a is balanced for all 0 ̸= a ∈ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  4  XIANHONG XIE1,2, YI OUYANG3,4  For a vectorial function F : V → W, its Walsh transform is  WF (a, ω) = WFa(ω) =  �  v∈V  (−1)Fa(v)+⟨ω,v⟩,  a ∈ W − {0}, ω ∈ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  In particular, if F : F2n → F2k, then WF (a, ω) can be written as  WF (a, ω) = WFa(ω) =  �  v∈F2n  (−1)Tr2k/2(aF (v))+Tr2n/2(ωv),  a ∈ F∗  2n, ω ∈ F2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' A vectorial Boolean function F : V → W is called bent if WF (a, ω) =  WFa(ω) = ±2  dim V  2  for any a ∈ W − {0} and ω ∈ V , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=', its component functions  Fa for all a ∈ W − {0} are bent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  Lemma 2 ([11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' If F : V → W is a vectorial bent function, then dim V is even  and dim W ≤ dim V  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' Bent components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' Let dim V = n = 2m and F : V → V be a vectorial function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' Set  SF := {v ∈ V : Fv is not bent}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  Then  (1) (Pott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' [22]) |SF | ≥ 2m, and |SF | = 2m if and only if SF is an m-  dimensional F2-subspace of V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  (2) (Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' [23]) Moreover, if V = F2n and |SF | = 2m, then SF = F2m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' Construction of Vectorial Functions With Maximum Number of  Bent Components  Assume n = 2m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' The main goal of this section is to construct vectorial functions  from V of dimension n to itself with maximal number of bent components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' A Generic construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' This construction is inspired by recent work of  Tang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' [3] and Zheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' Suppose f : V → F2 and {u1, u2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' , uk} ⊆ V for 2 ≤ k ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' We  say that (f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' u1, · · · , uk) satisfies Condition A if f(x) is a bent function and  DuiDujf ∗(x) = 0  for all pairs  1 ≤ i < j ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  (1)  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' (1) means that  f ∗(x + ui + uj) =f ∗(x + ui) + f ∗(x + uj) + f ∗(x)  =f ∗(x + ui) + f ∗(x) + f ∗(x + uj) + f ∗(x) + f ∗(x)  =Duif ∗(x) + Dujf ∗(x) + f ∗(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  By induction, for any (w1, w2, · · · , wk) ∈ Fk  2, one has  f ∗(x +  k  �  i=1  wiui) = f ∗(x) +  k  �  i=1  wiDuif ∗(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' Suppose G : V → V , 0 ̸= β ∈ V , 2 ≤ k ≤ n and {u1, u2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' , uk} ⊆ V  such that (Gβ(x);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' u1, · · · , uk) satisfies Condition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' Then for any reduced polyno-  mial H(X1, · · · , Xk) over F2, the function  Fβ(x) := Gβ(x) + H(⟨u1, x⟩, · · · , ⟨uk, x⟩)  VECTORIAL FUNCTIONS WITH MAXIMAL NUMBER OF BENT COMPONENTS  5  is a bent function, whose dual is  F ∗  β(x) = G∗  β(x) + H(Du1G∗  β(x), · · · , DukG∗  β(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  Our proof of this theorem is almost identical to that of [3, Theorem 8], which  we include here for completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' Applying the inverse Walsh transform to the Boolean function H : Fk  2 → F2  defined by H(X1, · · · , Xk), we get  (−1)H(X1,X2,··· ,Xk) = 1  2k  �  (w1,··· ,wk)∈Fk  2  WH(w1, · · · , wk)(−1)  �k  i=1 wiXi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  (2)  Take Xi = xi = ⟨ui,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' x⟩ for 1 ≤ i ≤ k and then multiply both sides of the above  identity by (−1)Gβ(x)+⟨ω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='x⟩,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' note that �k  i=1 wixi = ⟨�k  i=1 wiui,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' x⟩,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' we get  (−1)Gβ(x)+H(x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='xk)+⟨ω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='x⟩  = 1  2k  �  (w1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='wk)∈Fk  2  WH(w1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' wk)(−1)  Gβ(x)+⟨ω+  k  �  i=1  wiui,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='x⟩  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  which leads to  WFβ(ω) =  �  x∈V  (−1)Gβ(x)+H(x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='xk)+⟨ω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='x⟩  = 1  2k  �  x∈V  �  (w1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='wk)∈Fk  2  WH(w1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' wk)(−1)  Gβ(x)+⟨ω+  k  �  i=1  wiui,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='x⟩  = 1  2k  �  (w1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='wk)∈Fk  2  WH(w1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' wk)WGβ(ω +  k  �  i=1  wiui).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  By definition of the dual of a bent function, then  WFβ(ω) = 2m  2k  �  (w1,··· ,wk)∈Fk  2  WH(w1, · · · , wk)(−1)  G∗  β(ω+  k  �  i=1  wiui)  = 2m  2k (−1)G∗  β(ω)  �  (w1,··· ,wk)∈Fk  2  WH(w1, · · · , wk)(−1)  k  �  i=1  wiDui G∗  β(ω)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  This together with (2) yields  WFβ(ω) = 2m(−1)G∗  β(ω)+H(Du1 G∗  β(ω),Du2 G∗  β(ω),··· ,Duk G∗  β(ω)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  Hence Fβ is bent and  F ∗  β(x) = G∗  β(x) + H(Du1G∗  β(x), · · · , DukG∗  β(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  □  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' Construction via the Niho quadratic function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' Take V = F2n with the  inner product given by the trace map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' By Proposition 1 and Theorem 1, we have  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' Suppose G : F2n → F2n has 2n − 2m bent components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  Suppose  2 ≤ k ≤ n and {u1, · · · , uk} ⊆ F2n such that (Gβ(x);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' βu1, u2, · · · , uk) satisfies  6  XIANHONG XIE1,2, YI OUYANG3,4  Condition A for all β ∈ F2n−F2m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' Then for any reduced polynomial R(X2, · · · , Xk)  over F2, the vectorial function  F(x) := G(x) + u1xR(Tr2n/2(u2x), Tr2n/2(u3x), · · · , Tr2n/2(ukx))  has 2n −2m bent components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' Furthermore, the component Fβ(x) = Tr2n/2(βF(x))  is bent and its dual  F ∗  β(x) = G∗  β(x) + Dβu1G∗  β(x)R(Du2G∗  β(x), · · · , DukG∗  β(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' Comparing with the constructions in [21, 22], the vectorial function  constructed by Theorem 2 is new and its dual is explicitly given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  Moreover, the vectorial functions with maximal number of bent components by  our construction can have high algebraic degrees if we choose the reduced polynomial  R with high algebraic degree and u1, u2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' , uk linearly independent over F2, in this  case the algebraic degree of R(Tr2n/2(u1x), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' , Tr2n/2(ukx)) is equal to the algebraic  degree of R(X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' , Xk) (see [3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  From now on in this subsection, let G(x) = x2m+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' For β ∈ F2n − F2m, let  γ = β + β2m ∈ F∗  2m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  The component function of G at β is the monomial Niho quadratic function  Gβ : x ∈ F2n �→ Tr2n/2(βx2m+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  It is a bent function (see [20]) and its dual G∗  β is given by  G∗  β(x) = Tr2m/2(γ−1x2m+1) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  (3)  To apply Theorem 2, we first show that the function Gβ(x) satisfies Condition  A when u1, u2, · · · , uk are appropriately chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' Suppose β ∈ F2n − F2m, k ≤ m and {u1, u2, · · · , uk} ⊆ F2n such that  Tr2n/2(γ−1u2m  j ui) = 0 for all 1 ≤ i < j ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' Then (Gβ(x);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' u1, · · · , uk) satisfies  Condition A and for 1 ≤ j ≤ k,  DujG∗  β(x) = Tr2n/2(γ−1xu2m  j ) + Tr2m/2(γ−1u2m+1  j  ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' By Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' (3), the derivative of G∗  β(x) in the direction of uj ∈ F2n is  DujG∗  β(x) = Tr2m/2(γ−1x2m+1) + 1 + Tr2m/2(γ−1(x + uj)2m+1) + 1  = Tr2n/2(γ−1u2m  j x) + Tr2m/2(γ−1u2m+1  j  ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  Then the second order derivative in the direction of (ui, uj) is  DuiDujG∗  β(x) = DujG∗  β(x + ui) + DujG∗  β(x)  = Tr2n/2(γ−1u2m  j x) + Tr2n/2(γ−1u2m  j (x + ui))  = Tr2n/2(γ−1u2m  j ui) = 0,  with the last equality followed by our assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  □  By Lemma 3 and Theorem 1, then we have  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' Let β ∈ F2n − F2m and Gβ(x) = Tr2n/2(βx2m+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' If k ≤ m and  {u1, u2, · · · , uk} ⊆ F2n satisfying Tr2n/2(γ−1u2m  j ui) = 0 for any 1 ≤ i < j ≤ k,  then the function  Fβ(x) := Gβ(x) + H(Tr2n/2(u1x), Tr2n/2(u2x), · · · , Tr2n/2(ukx))  VECTORIAL FUNCTIONS WITH MAXIMAL NUMBER OF BENT COMPONENTS  7  where H(X1, X2, · · · , Xk) is any reduced polynomial over F2, is bent and its dual is  F ∗  β(x) = G∗  β(x) + H(Du1G∗  β(x), Du2G∗  β(x), · · · , DukG∗  β(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  Our first construction of vectorial functions with maximal number of bent com-  ponents is the following result:  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' Let 3 ≤ k ≤ m and {u1, u2, · · · , uk} ⊆ F2m satisfy Tr2m/2(u1uj) = 0  for j ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' Then for any reduced polynomial R(X2, · · · , Xk) over F2,  F(x) = x2m+1 + u1xR(Tr2n/2(u2x), Tr2n/2(u3x), · · · , Tr2n/2(ukx)),  has 2n − 2m bent components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' More precisely, for β ∈ F2n − F2m, Fβ is bent and  F ∗  β(x) = G∗  β(x) + Dβu1G∗  β(x)R(Du2G∗  β(x), · · · , DukG∗  β(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' Since ui ∈ F2m and Tr2m/2(u1uj) = 0, we have γ−1ujui ∈ F2m,  DuiDujG∗  β(x) = Tr2n/2(γ−1ujui) = 0, 2 ≤ i < j ≤ k ≤ m,  and  Dβu1DujG∗  β(x) = Tr2n/2(γ−1ujβu1) = Tr2m/2(uju1) = 0, 2 ≤ j ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  Therefore, (Gβ(x);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' βu1, u2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' , uk) satisfies Condition A for any β ∈ F2n −F2m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  □  In Theorem 4, if we take k = m and {ui : 1 ≤ i ≤ m} to be an orthogonal  basis of F2m over F2 under the inner product ⟨x, y⟩ = Tr2m/2(xy), then Condition  A holds, and what’s more, we have the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' Let R(X2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' , Xm) = X2 · · · Xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' Then the function  F(x) = x2m+1 + u1x  m  �  i=2  Tr2n/2(uix)  has maximal algebraic degree m and maximal number of bent components 2n − 2m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  In particular, for k = 2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=', R(X2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' , Xm) = X2, we can take u1, u2 ∈ F2n  such that (Gβ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' βu1, u2) satisfies Condition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' Thus we have  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' Suppose u1, u2 ∈ F2n such that u1u2m  2  ∈ F2m and Tr2m/2(u1u2m  2 ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  Then  F(x) = x2m+1 + u1xTr2n/2(u2x)  has 2n − 2m bent components: for β ∈ F2n − F2m, Fβ is bent and  F ∗  β(x) =Tr2m/2(λ−1x2m+1) + 1+  �  Tr2n/2(λ−1(βu1)2mx) + Tr2m/2(λ−1(βu1)2m+1)  �  ×  �  Tr2n/2(λ−1u2m  2 x) + Tr2m/2(λ−1u2m+1  2  )  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' The proof is similar to Theorem 4, so we omit it here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  □  Note that the function F(x) given by Theorem 5 is of the form xℓ(x), thus is a  solution of Problem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' We now show it is not equivalent to the functions in cases  (a), (b) and (e) in the Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' Recall for a vectorial function F and a, b ∈ V ,  δF (a, b) := |{x ∈ F2n : F(x + a) + F(x) = b}|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' The differential spectrum of F is  {δF(a, b) : a ∈ F∗  2n, b ∈ F2n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  It was shown in [26], [22] and [25] respectively that  δx2m+1(a, b) ∈ {0, 2m},  δx2i(x+x2m)(a, b) ∈ {0, 2gcd(i,m), 2m} (0 < i < m) and  8  XIANHONG XIE1,2, YI OUYANG3,4  δx2i(Tr2n/2m(Λ(x)))(a, b) ∈ {0, 2s, 2m},  where s is the dimension of the solution space (in F2m) of z2iΛ(Tr2n/2m(a)) +  (Tr2n/2m(a))2iΛ(z) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' Then the inequivalence of our function in Theorem 5 to  theirs follows from  Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' Suppose u1, u2 ∈ F2n − F2m, u1u2m  2  ∈ F2m and Tr2m/2(u1u2m  2 ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  Then the differential spectrum of F(x) = x2m+1 + u1xTr2n/2(u2x) is given by  δF (a, b) ∈  �  {0, 2},  if Tr2n/2(u2a) = 1,  {0, 2m−1, 2m},  if Tr2n/2(u2a) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' We have  F(x + a) + F(x)  = x2ma + xa2m + a2m+1 + u1xTr2n/2(u2a) + u1aTr2n/2(u2(x + a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  Notice that if x is a solution of F(x + a) + F(x) = b, so is x + a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  (A) Assume Tr2n/2(u2a) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' The equation F(x + a) + F(x) = b is reduced to  x2ma + xa2m + a2m+1 + u1x + u1aTr2n/2(u2x) + u1a = b,  and then to one of the following two systems of equations:  �  x2ma + xa2m + u1x = b + a2m+1 + u1a,  Tr2n/2(u2x) = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  �  x2ma + xa2m + u1x = b + a2m+1,  Tr2n/2(u2x) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  We claim that x2ma + xa2m + u1x is a permutation over F2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' Then δF (a, b) ∈  {0, 2} follows from the claim immediately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  For x, y ∈ F2n, let  x2ma + xa2m + u1x = y2ma + ya2m + u1y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  Set z = Tr2n/2m(xa2m) − Tr2n/2m(ya2m) ∈ F2m, then y = x + u−1  1 z and  z = Tr2n/2m(xa2m − ya2m) = −Tr2n/2m(a2mu−1  1 z) = −zTr2n/2m(a2mu−1  1 )  ⇒ z(1 + Tr2n/2m(a2mu−1  1 )) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  Suppose Tr2n/2m(a2mu−1  1 ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' Notice that u2m  1 u2 ∈ F∗  2m and au−2m  1  =  au2  u2m  1  u2 ,  thus  Tr2n/2m(a2mu−1  1 ) = Tr2n/2m(au−2m  1  ) = Tr2n/2m( au2  u2m  1 u2  ) = Tr2n/2m(au2)  u2m  1 u2  = 1  ⇒ Tr2n/2m(au2) = u2m  1 u2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  Since Tr2n/2(u2a) = 1, we have Tr2n/2(u2a) = Tr2m/2(u2m  1 u2) = 1, which is  a contradiction to the assumption Tr2m/2(u2m  1 u2) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' Thus z = 0 and x2ma +  xa2m + u1x is a linear permutation over F2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  (B) Assume Tr2n/2(u2a) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' The equation F(x + a) + F(x) = b is reduced to  x2ma + xa2m + a2m+1 + u1aTr2n/2(u2x) = b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  (4)  Assume that x, y are two solutions of (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' Then  x2ma + xa2m + a2m+1 + u1aTr2n/2(u2x) = b,  y2ma + ya2m + a2m+1 + u1aTr2n/2(u2y) = b,  VECTORIAL FUNCTIONS WITH MAXIMAL NUMBER OF BENT COMPONENTS  9  which means that z = x + y is a solution of  z2ma + za2m + u1aTr2n/2(u2z) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  or equivalently,  �  z2ma + za2m = 0,  Tr2n/2(u2z) = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  or  �  z2ma + za2m = u1a,  Tr2n/2(u2z) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  Thus δF (a, b) = 0 or the number of solutions of these two systems of equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  Let Xu = {x ∈ F2n : Tr2n/2(ux) = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  The zero set of the first system of  equations is the F2-vector space a−2mF2m ∩ Xu2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' Note that dimF2 a−2mF2m = m  and dimF2 Xu2 = n − 1, a−2mF2m ∩ Xu2 must be of dimension either m − 1 or m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  For the second system, note that z2ma + za2m ∈ F2m, we must have u1a ∈ F2m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  Hence u2a−2m =  u2m  1  u2  (u1a)2m ∈ F2m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' The solution of z2ma+za2m = u1a is z =  u1a  a2m(1+ξ)  with ξ2m+1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' Then  Tr2n/2(u2z) = Tr2n/2(  u2u1a  a2m(1 + ξ)) = Tr2n/2(u2a−2mu1a  1 + ξ  )  = Tr2m/2(u2a−2mu1a) = Tr2m/2(u2u2m  1 ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  Hence the second system has no zeros at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  □  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' Construction via the Maiorana-MacFarland class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' In this case, we let  V = F2m × F2m and the corresponding inner product be  ⟨(y1, z1), (y2, z2)⟩ = Tr2m/2(y1y2) + Tr2m/2(z1z2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  Let φ be a permutation of F2m, and G be the associated map defined by  G : F2m × F2m −→ F2m × F2m  (y, z) �−→ (yφ(z), z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  Then G has maximal number of bent components: for (a, b) ∈ F∗  2m × F2m, the  component function  Ga,b(y, z) = Tr2m/2(ayφ(z) + bz),  (5)  at (a, b) is a bent function in the Maiorana- MacFarland class, whose dual  G∗  a,b(y, z) = Tr2m/2((z + b)φ−1(a−1y)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  (6)  We assume that φ is an automorphism of F2m from now on in this subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' Let 2 ≤ k ≤ m, (a, b) ∈ F∗  2m × F2m, φ and G be given as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' If  the set {ui = (ui,1, ui,2) ∈ V : 1 ≤ i ≤ k} satisfies  Tr2m/2  �  uj,2φ−1(a−1ui,1) + ui,2φ−1(a−1uj,1)  �  = 0 for all 1 ≤ i < j ≤ k,  then (Ga,b(y, z);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' u1, · · · , uk) satisfies Condition A and for 1 ≤ i ≤ k,  DuiG∗  a,b(y, z) = Tr2m/2  �  (z + b)φ−1(a−1ui,1) + ui,2φ−1(a−1(y + ui,1))  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  As a consequence, for any reduced polynomial H(X1, · · · Xk) over F2, the function  Fa,b(y, z) = Ga,b(y, z) + H(Tr2m/2(u1,1y + u1,2z), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' , Tr2m/2(uk,1y + uk,2z))  is bent and its dual is  F ∗  a,b(y, z) = G∗  a,b(y, z) + H(Du1G∗  a,b(y, z), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' , DukG∗  a,b(y, z)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  10  XIANHONG XIE1,2, YI OUYANG3,4  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' It suffices to check that (Ga,b(y, z);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' u1, · · · , uk) satisfies Condition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' By  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' (6), the derivative of G∗  a,b(y, z) in the direction of ui is  DuiG∗  a,b(y, z) = G∗  a,b(y + ui,1, z + ui,2) + G∗  a,b(y, z)  = Tr2m/2  �  (z + b)φ−1(a−1ui,1) + ui,2φ−1(a−1(y + ui,1))  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  Then the second order derivative in the direction (ui, uj) is  DujDuiG∗  a,b(y, z)  = DujG∗  a,b(y + ui,1, z + ui,2) + DujG∗  a,b(y, z)  = Tr2m/2  �  φ−1(a−1)(uj,2φ−1(ui,1) + ui,2φ−1(uj,1))  �  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  □  Our second construction of vectorial functions with maximal number of bent  components is the following result:  Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' Let 2 ≤ k ≤ m, φ and G be given as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' Suppose u = (u1,1, 0)  and choose ui = (ui,1, ui,2) for 2 ≤ i ≤ k such that  Tr2m/2(φ−1(u1,1)ui,2) = 0 and ui,1 = φ(ui,2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  Then for any reduced polynomial R(X2, · · · Xk) over F2, the vectorial function  F(y, z) = (yφ(z), z) + (u1,1y, 0)R(Tr2m/2(u2,1y + u2,2z), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' , Tr2m/2(uk,1y + uk,2z))  has 2n − 2m bent components: for any (a, b) ∈ F∗  2m × F2m,  Fa,b(y, z) = ⟨(a, b), F(y, z)⟩ = Tr2m/2(ayφ(z) + bz)  + Tr2m/2(au1,1y)R(Tr2m/2(u2,1y + u2,2z), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' , Tr2m/2(uk,1y + uk,2z))  is bent and  F ∗  a,b(y, z) = G∗  a,b(y, z) + Du1,1a,0G∗  a,b(y, z)R(Du2G∗  a,b(y, z), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' , DukG∗  a,b(y, z)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' Since Tr2m/2(φ−1(u1,1)ui,2) = 0 and ui,1 = φ(ui,2), then for 2 ≤ i ≤ k,  DuiD(au1,1,0)G∗  a,b(y, z) = Tr2m/2(φ−1(a−1)ui,2φ−1(au1,1))  = Tr2m/2(ui,2φ−1(u1,1)) = 0,  and for 2 ≤ i < j ≤ k,  DujDuiG∗  a,b(y, z) = Tr2m/2  �  φ−1(a−1)(uj,2φ−1(ui,1) + ui,2φ−1(uj,1))  �  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  Therefore, (Ga,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' (au1,1, 0), u2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' , uk) satisfies Condition A for (a, b) ∈ F∗  2m × F2m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  □  Suppose m′ is a divisor of m, then (5) can be written as  Ga,b(y, z) = Tr2m/2(ayφ(z) + bz) = Tr2m′ /2(G′  a,b(y, z)),  where G′  a,b(y, z) = Tr2m/2m′ (ayφ(z) + bz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' Suppose m′ is a divisor of m, G′  a,b(y, z) = Tr2m/2m′ (ayφ(z) + bz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  Then G′  a,b(y, z) is a vectorial bent function for (a, b) ∈ F∗  2m × F2m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' Furthermore,  for c ∈ F∗  2m′, Gca,cb(y, z) = Tr2m′/2(cG′  a,b(y, z)) is a bent function and its dual is  G∗  ca,cb(y, z) = Tr2m/2  �  zφ−1((ac)−1y) + bcφ−1((ac)−1y)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' Note that for any c ∈ F∗  2m′, one has (ca, cb) ∈ F∗  2m × F2m, thus Gca,cb(y, z)  is bent function in the Maiorana-MacFarland class and the dual can be obtained  directly from (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  □  VECTORIAL FUNCTIONS WITH MAXIMAL NUMBER OF BENT COMPONENTS  11  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' Take a = b = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' Let c1, c2, c3 be three pairwise distinct elements in  F∗  2m′ such that c := c1 + c2 + c3 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' For s ∈ {c, c1, c2, c3}, let  Gs(y, z) := Gs,s(y, z) = Tr2m/2(syφ(z) + sz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  Then Gc and Gci are all bent functions and  Gc1(y, z) + Gc2(y, z) + Gc3(y, z) = Gc(y, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  To have the equality  G∗  c1(y, z) + G∗  c2(y, z) + G∗  c3(y, z) = G∗  c(y, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  (7)  It suffices to find an φ of F2m such that  (C1) φ−1(c−1y) = φ−1(c−1  1 y) + φ−1(c−1  2 y) + φ−1(c−1  3 y);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  (C2) cφ−1(c−1y) = c1φ−1(c−1  1 y) + c2φ−1(c−1  2 y) + c3φ−1(c−1  3 y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  Let φ be an involution and automorphism of F2m, then (C1) and (C2) are satisfied,  so is Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='(7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' This gives a solution of the open problem proposed by Mesnager ([20,  Open Problem 2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' Specially, the case of φ = φ−1 : z �→ z−1 was presented by [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' Binomial vectorial functions with maximal number of bent  components  Let n = 2m and v2(i) be the 2-adic valuation of i ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' The main result of this  section is  Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' The binomial vectorial function F(x) = x2m+1 + x2i+1 for 0 ≤ i ≤  m − 1 on F2n has 2n − 2m bent components if and only if i = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=', F(x) is affine  equivalent to x2m+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' The special case of odd m was proved by Zheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  From now on, fix i such that 0 ≤ i < m, and let  d = gcd(m + i, 2m) = gcd(m + i, 2i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  Let F(x) = x2m+1 + x2i+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  For a ∈ F2n, the component function Fa(x) =  Tr2n/2(ax2m+1 + ax2i+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' Let  La(y) := a2iy22i + (a + a2m)2iy2m+i + ay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  (8)  If a ∈ F2m, then Fa(x) = Tr2n/2(ax2i+1) and (8) is reduced to  La(y) := a2iy22i + ay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  For any y ∈ F∗  2n, the derivative of Fa(x) at direction y is  DyFa(x) = Tr2n/2(a((x + y)2m+1 + (x + y)2i+1)) + Tr2n/2(a(x2m+1 + x2i+1))  = Tr2n/2(x(ay2i + (a + a2m)y2m + a2n−iy2n−i)) = Tr2n/2(xLa(y)−2i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  The root set of La(y) in F2n forms an F2d-vector space, hence the number of the  roots of La(y) in F2n is either 1 or a power of 2d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' Assume v2(i) = v2(m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' For ξ ∈ F2d such that ξ2d/2+1 = 1, let a =  1  1+ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  Then a /∈ F2m and La(y) = 0 for any y ∈ F2d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  12  XIANHONG XIE1,2, YI OUYANG3,4  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' By v2(i) = v2(m), d is even, m = d  2 · m′ and i = d  2 · i′ with m′ and i′ both  odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' Then  ξ2m = ξ2i = ξ−1 =⇒ a2m = a2i = ξa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  This means that a /∈ F2m and  La(y) = a2iy22i + (a + a2m)2iy2m+i + ay = a(ξy22i + (1 + ξ)y2m+i + y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  For any y0 ∈ F2d = F22i ∩ F2m+i, one has y22i  0  = y2m+i  0  , hence La(y0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  □  We need the following two general results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' [29, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='30] Let χ′ be a multiplicative character of F∗  2m of order  2d − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' Then for any (a, b) ∈ F∗  2m × F2m,  �  x∈F2m  (−1)Tr2m/2(ax2d−1+b) = (−1)Tr2m/2(b)  2d−2  �  j=1  χ′j(a)G(χ′j),  where χ and G(χ) are the conjugate and the Gauss sum of χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' Suppose d < m is a factor of m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' Let gcd(2d − 1, m  d ) = t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' Then the set  N = {y ∈ F∗  2m : Tr2m/2d(y2d−1) = 0} has order  |N| =  \uf8f1  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f3  2m − 2d  2d  + (2d − 1)(−1)  m  d −1  2d  �  χ∈(�F∗  2d)  2d−1  t  \\{χ0}  G(χ)  m  d ,  if t ̸= 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  2m−d − 1,  if t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  where �F∗  2d is the set of the multiplicative characters of F∗  2d and χ0 is the trivial  character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' In particular, N is non-empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' We have  |N| = 1  2d  �  v∈F2d  �  y∈F∗  2m  (−1)Tr2d/2(vTr2m/2d(y2d−1))  = 2m − 1  2d  + 1  2d  �  v∈F∗  2d  �  y∈F∗  2m  (−1)Tr2m/2(vy2d−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  (9)  Suppose F∗  2m = ⟨β⟩, then F∗  2m = � 2m−1  2d−1 −1  i=0  βiF∗  2d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' Note that gcd( 2m−1  2d−1 , 2d − 1) =  gcd( m  d , 2d − 1) = t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' If t = 1, one has  |N| = 2m − 1  2d  + 2d − 1  2d  �  v∈F∗  2d  2m−1  2d−1 −1  �  i=0  (−1)Tr2m/2(vβi(2d−1))  = 2m − 1  2d  + 2d − 1  2d  �  v∈F∗  2d  2m−1  2d−1 −1  �  i=0  (−1)Tr2m/2(vβi)  = 2m − 1  2d  + 2d − 1  2d  �  v∈F∗  2m  (−1)Tr2m/2(v) = 2m − 2d  2d  ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  VECTORIAL FUNCTIONS WITH MAXIMAL NUMBER OF BENT COMPONENTS  13  If t ̸= 1, suppose χ′ is a multiplicative character of F∗  2m of order 2d − 1, then by  Lemma 5 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' (9),  |N| = 2m − 1  2d  + 1  2d  �  v∈F∗  2d  � �  y∈F2m  (−1)Tr2m/2(vy2d−1) − 1  �  = 2m − 1  2d  + 1  2d  �  v∈F∗  2d  �2d−2  �  j=1  χ′j(v)G(χ′j) − 1  �  = 2m − 1  2d  + 1  2d  �  v∈F∗  2d  2d−2  �  j=0  χ′j(v)G(χ′j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  (10)  Suppose N is the norm mapping from F2m to F2d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' For χ ∈ �F∗  2d, it can be lifted  from F2d to F2m by χ′ = χ ◦ N (see [29, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='28]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' Furthermore, χ is of order  2d − 1 if and only if χ′ is of order 2d − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' Then  2d−2  �  j=0  χ′j(v)G(χ′j) =  �  χ∈�F∗  2d  χ(N(v))G(χ ◦ N) = (−1)  m  d −1 �  χ∈�F∗  2d  χ(v  2m−1  2d−1 )G(χ)  m  d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  Suppose δ = β  2m−1  2d−1 ∈ F∗  2d, then F∗  2d =  2d−1  t  −1  �  j=0  δj⟨δ  2d−1  t  ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' By Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' (10), we get  |N| = 2m − 1  2d  + (−1)  m  d −1  2d  �  v∈F∗  2d  �  χ∈�F∗  2d  χ(v  2m−1  2d−1 )G(χ)  m  d  = 2m − 1  2d  + (−1)  m  d −1  2d  �  χ∈�F∗  2d  G(χ)  m  d  2d−1  t  −1  �  j=0  �  v∈δj⟨δ  2d−1  t  ⟩  χ(v  2m−1  2d−1 )  = 2m − 1  2d  + (−1)  m  d −1t  2d  �  χ∈�F∗  2d  G(χ)  m  d  2d−1  t  −1  �  j=0  χ(δ  j 2m−1  2d−1 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  Note that gcd( 2m−1  t(2d−1), 2d−1  t  ) = 1, then  2d−1  t  −1  �  i=0  χ(δi 2m−1  2d−1 ) =  2d−1  t  −1  �  i=0  χ(δit) =  �  x∈⟨δt⟩  χ(x) =  �  0,  if χ ̸= χ0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  2d−1  t  ,  if χ = χ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  Hence we have  |N| = 2m − 2d  2d  + (2d − 1)(−1)  m  d −1  2d  �  χ∈(�F∗  2d)  2d−1  t  \\{χ0}  G(χ)  m  d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  Note that |G(χ)| = 2  d  2 for χ ̸= χ0 (see [29, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='11]), then we have  |N| ≥ 2m − 2d − (2d − 1)(t − 1)2  m  2  2d  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  14  XIANHONG XIE1,2, YI OUYANG3,4  Note that t ̸= 1 and  2m − 2d − (2d − 1)(t − 1)2  m  2 = 2m − t2  m  2 +d + (t − 1)2  m  2 − 2d > 2m − t2  m  2 +d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  Since t = gcd( m  d , 2d − 1) ≤ 2d − 1, then 2m − t2  m  2 +d > 2m − 2  m  2 +2d ≥ 0 if m  d ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  If m = 3d, then t = gcd(3, 2d − 1) = 3 and d is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' One has  |N| = 23d − 2d − (2d − 1)2  3d  2 +1 > 23d − 2  5d  2 +1 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  If m = 2d, then t = gcd(2, 2d − 1) = 1, which contradicts to t ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' Thus we  complete the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  □  Back to our situation, we have the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' Suppose v2(m) < v2(i), then there exists a ∈ F2n −F2m such that La(y)  has roots in F∗  2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' For v2(m) < v2(i), note that d = gcd(m, i) = gcd(m + i, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' It suffices to  show that there exists a ∈ F2n − F2m such that La(y) has a root y0 ∈ F∗  2m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' Note  that for y ∈ F∗  2m,  La(y) = a2iy22i + (a + a2m)2iy2m+i + ay = a2iy22i + (a + a2m)2iy2i + ay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  (11)  Then we just need to find (a, y) ∈ F2n × F∗  2m such that  �  a + a2m = y−2i−1v,  (ay2i+1)2i + ay2i+1 = y2i−22iv,  (12)  for some v ∈ F∗  2d (here a /∈ F2m is automatic).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' Let z = av−1y2i+1, then we just  need to find (z, y) ∈ F2n × F∗  2m such that  �  z + z2m = 1,  (13)  z2i + z = y2i−22i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  (14)  We consider Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  Note that the F2d-linear maps ϕi : z �→ z2i + z and  ϕd : z �→ z2d + z from F2m to itself have the same kernel F2d and ϕi(z) = ϕd(z +  z2d + · · · + z2( i  d −1)d), then Im(ϕi) ⊆ Im(ϕd) and hence Im(ϕd) = Im(ϕi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' Note also  that the group homomorphisms y �→ y2i(1−2i) and y �→ y2d−1 from F∗  2m to itself  have the same kernel and image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' Then there is an one-to-one correspondents of  solutions (z, y) ∈ F2m × F∗  2m of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' (14) and of  z2d + z = y2d−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  (15)  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' (15) is soluble if and only if there exists y ∈ F∗  2m such that Tr2m/2d(y2d−1) = 0,  which is guaranteed by Lemma 6 as d < m in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' Thus there exists (z0, y0) ∈  F∗  2m × F∗  2m such that z2i  0 + z0 = y2i−22i  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  Let w ∈ F22d \\ F2d satisfy w2d + w = v0, then w2m−d = w2i = w and z =  z0 + w ∈ F2n \\ F2m is a solution of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' (13) and (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  Thus, y0 ∈ F∗  2m and  a = (z0 + w)y−2i−1  0  v satisfy the equation La(y) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  □  Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' For 0 ≤ i ≤ m − 1, if F(x) = x2m+1 + x2i+1 has 2n − 2m bent  components, then v2(m) ≤ v2(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  VECTORIAL FUNCTIONS WITH MAXIMAL NUMBER OF BENT COMPONENTS  15  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' Assume v2(m) > v2(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' In this case d = gcd(m, i) = gcd(n, i), and 2d =  gcd(2i, m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' This means 2d−1 = gcd(2m−1, 2i−1) and 22d−1 = gcd(2m−1, 22i−1),  which then implies that 2d + 1 is a factor of 2m − 1 and thus prime to 2m + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  Let α be a primitive element of F2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' Let a = αk(2m+1) ∈ F∗  2m such that a2i−1 =  α(2m+1)(2d−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' By Proposition 1, for this a, Fa(x) = Tr2n/2(ax2i+1) is not bent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' By  Lemma 1, DyFa(x) = Tr2n/2(x(ay2i + (ay)2n−i)) is not balanced for some y ∈ F2n,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=', a2i−1y22i−1 + 1 = 0 is soluble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' Let a2i−1 = α(2m+1)(2d−1) = y1−22i  0  and let  y1 ∈ F∗  2n such that y1−22i  0  = y22d−1  1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' Then the congruent equation  (22d − 1)x ≡ (2d − 1)(2m + 1) mod (2n − 1)  is soluble, equivalently, the equation  (2d + 1)x ≡ 2m + 1 mod (2d + 1) · 2n − 1  22d − 1  is soluble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' This is not possible since 2d + 1 is prime to 2m + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  □  Proof of Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' If i = 0, the result is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' We now assume i ̸= 0 and  v2(m) ≤ v2(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  If F(x) has maximal number of bent components, by Lemma 1, Fa(x) is bent  function for all a ∈ F2n −F2m and hence Dyfa(x) is balanced for any y ∈ F∗  2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' This  implies La(y) ̸= 0 for all y ∈ F∗  2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' Hence to show F(x) does not have maximal  number of bent components, it suffices to show there exists a ∈ F2n − F2m, such  that La(y) has a root in F∗  2n:  (i) If v2(m) = v2(i), this is implied by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  (ii) If v2(m) < v2(i), this is implied by Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  Thus for i ̸= 0, F(x) cannot have 2n − 2m bent components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  □  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' For a general binomial vectorial function F(x) = xd1 +xd2, our exper-  imental result indicates that F(x) is affine equivalent to x2m+1 or x2i(x + x2m) if  F(x) has maximal number of bent components, but so far we do not have a proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  We leave this as an open problem for future study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' Conclusion  We firstly give a generic construction of vectorial functions with maximal number  of bent components, and obtain two new classes of such vectorial functions based  on the Niho quadratic function and the Maiorana-MacFarland class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' Moreover,  we solve the open problem proposed by Mesnager, and partially answer the open  problem proposed by Pott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content=' We then show that the binomial function F(x) =  x2m+1 + x2i+1 : F22m → F22m has maximal number of bent components if and only  if i = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
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+page_content=' Press (1984).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='  1School of Information and Computer, Anhui Agricultural University, Hefei 230036,  China  2School of Cyber Science and Technology, University of Science and Technology  of China, Hefei 230027, China  Email address: xianhxie@ahau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='cn  3School of Mathematical Sciences, CAS Wu Wen-Tsun Key Laboratory of Mathe-  matics, University of Science and Technology of China, Hefei 230026, China  VECTORIAL FUNCTIONS WITH MAXIMAL NUMBER OF BENT COMPONENTS  17  4Hefei National Laboratory, Hefei 230088, China  Email address: yiouyang@ustc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
+page_content='cn' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE1T4oBgHgl3EQfAwJw/content/2301.02843v1.pdf'}
diff --git a/6NE3T4oBgHgl3EQfQwn7/content/tmp_files/2301.04417v1.pdf.txt b/6NE3T4oBgHgl3EQfQwn7/content/tmp_files/2301.04417v1.pdf.txt
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+Quantum Monte Carlo-based density functional for one-dimensional Bose-Bose
+mixtures
+Jakub Kopyci´nski,1, 2, ∗ Luca Parisi,2 Nick G. Parker,2 and Krzysztof Paw�lowski1
+1Center for Theoretical Physics, Polish Academy of Sciences, Al. Lotnik´ow 32/46, 02-668 Warsaw, Poland
+2Joint Quantum Centre (JQC) Durham–Newcastle,
+School of Mathematics, Statistics and Physics, Newcastle University,
+Newcastle upon Tyne, NE1 7RU, United Kingdom
+(Dated: January 12, 2023)
+We propose and benchmark a Gross-Pitaevskii-like equation for two-component Bose mixtures
+with competing interactions in 1D. Our approach follows the density-functional theory with the
+energy functional based on the exact Quantum Monte Carlo (QMC) simulations. Our model covers,
+but goes beyond, the popular approach with the Lee-Huang-Yang corrections. We first benchmark
+our approach against available QMC data in all interaction regimes and then study dynamical prop-
+erties, inaccessible by ab initio many-body simulations. Our analysis includes a study of monopole
+modes and reveals the presence of anomalous dark solitons.
+I.
+INTRODUCTION
+Recent studies of ultracold gases with competing in-
+teractions have led to a major change in the field. They
+undermined the validity of the mean-field approximation
+when attractive and repulsive interactions in the system
+almost cancel each other.
+In such a situation, it is necessary to include the effect
+of quantum fluctuations [1]. To account for it, one can
+include Lee-Huang-Yang (LHY) corrections [2, 3] to the
+mean-field equation using a local density approximation.
+One then derives a generalized Gross-Pitaevskii (GGP)
+equation.
+It has been widely used to theoretically in-
+vestigate the ground-state properties and excitations of
+Bose-Bose mixtures with particular attention given to
+the compressional mode (known also as the monopole or
+breathing mode) [4–7].
+The GGP theory predicts the existence of self-bound
+objects – ultradilute quantum droplets made of ultracold
+atoms [1]. The emergence of a liquid phase is marked by
+the presence of a local minimum in the energy density
+functional.
+Soon after having been proposed theoretically, quan-
+tum droplets were experimentally observed [8–12]. Some
+theoretical predictions indicate even the possibility of
+finding quantum droplets [13] in recently obtained het-
+eronuclear dipolar condensates [14, 15].
+Despite its remarkable usefulness, there are still fac-
+tors not included in the GGP theory. For instance, ab
+initio calculations show a liquid-gas transition in two-
+component mixtures [16], whereas the GGP does not.
+Moreover, the same work demonstrates a quantitative
+disagreement of the homogeneous state energy.
+Quite
+unexpectedly, the monopole mode frequencies happen to
+match the QMC calculations, though [17].
+Several attempts have been made to overcome the ex-
+isting imperfections of the GGP equation. One of the
+∗ jkopycinski@cft.edu.pl
+ideas, that follows the density functional theory, was
+to build an equation which would quantitatively repro-
+duce the spatially uniform state energy from a chosen
+ab initio method for any interaction strength.
+In this
+regard the 1D Bose contact gas is a special system as
+its ground-state energy has been already derived in the
+analytical ab initio calculations by E. Lieb and E. Lin-
+iger [18, 19]. Using this exact energy functional one gets
+the single-particle equation here referred to as the Lieb-
+Liniger Gross-Pitaevskii (LLGP) equation that was used
+in Refs. [20–27]. The equation proved to correctly de-
+scribe the ground state and low-lying excitations in all
+regimes – from the weakly-interacting one (which, con-
+trary to the 3D case happens at high gas densities) up
+to the Tonks-Girardeau regime (at low gas densities).
+The LLGP equation was recently used to study Bose gas
+with repulsive short-range and attractive dipolar inter-
+actions [28–32] to show the existence and properties of
+the dipolar quantum droplets. Concerning the droplets
+in quantum mixtures, a similar approach was employed
+to construct a quantum Monte Carlo (QMC)-based en-
+ergy density functional for bosonic mixtures in 3D [33],
+but so far the 1D Bose mixture was not investigated in
+such framework. For the latter system it was shown [16]
+that GGP fails to reproduce the phase diagram in certain
+regimes, in particular at low densities when atoms bind
+together into interacting dimers.
+In this article, we aim to formulate and benchmark
+the QMC-based single-orbital density functional theory
+that is applicable to two-component Bose mixtures with
+repulsive intra- and attractive intercomponent interac-
+tions. Our theoretical approach using a single orbital ψ
+and a QMC-based energy density functional E results in
+an equation of the following form:
+iℏ∂tψ(x, t) = − ℏ2
+2m∂2
+xψ(x, t) + δE
+δnψ(x, t),
+(1)
+where n is the particle density. We want it to be ap-
+plicable to two-component Bose mixtures with repulsive
+intra- and attractive intercomponent interactions. To do
+arXiv:2301.04417v1  [cond-mat.quant-gas]  11 Jan 2023
+
+2
+this, we analyse the phase diagram of the system and nu-
+merically study the static properties and monopole mode
+of quantum droplets.
+In lower-dimensional systems, we can name two sub-
+stantial beyond-LHY approaches. One of them is a pair-
+ing theory for bosons [34]. The other one is based on the
+inclusion of higher-order corrections to the GGP equa-
+tion [35]. Both of them generally give only a qualitative
+agreement with QMC calculations.
+Our approach shares similarities with a density func-
+tional theory [36] for Fermi systems at unitarity [37].
+The resulting density functional has been employed mul-
+tiple times to look into strongly interacting fermions [38–
+42], revealing a remarkable consistency with the experi-
+ments [43, 44].
+A great advantage of having a Gross-Pitaevskii-like
+equation, in comparison to the QMC methods, is the
+possibility of studying nonlinear and time-dependent ef-
+fects like the existence of dark solitons. This subject is
+particularly interesting as we may expect fundamentally
+different results than the solitons we know from single-
+species systems [45] or dark-dark solitons occurring in
+miscible bosonic mixtures [46]. Very recently there have
+been reports on wide soliton-like objects, both in mix-
+tures [47] and in dipolar Bose gases [27]. As such, last
+but not least, we show density and phase profiles of soli-
+tary waves evaluated with our theory.
+II.
+FRAMEWORK
+A.
+System
+We consider a one-dimensional Bose gas consisting of
+two components σ = {↑, ↓} in a box of size L.
+We
+assume that the components have equal atomic masses
+m↑ = m↓ = m. We also assume that the short-range
+interaction coupling constants are the same in the intra-
+component case g↑↑ = g↓↓ = g, whereas the intercom-
+ponent interactions can be independently tuned with a
+coupling constant g↑↓.
+Atoms of the same species re-
+pel each other while the intercomponent interactions are
+attractive. The binding energy of an atomic pair in vac-
+uum εb = −(mg2
+↑↓/4ℏ2) is a relevant energy scale in the
+system, while for the length scale we choose the intra-
+component scattering length a = 2ℏ2/mg.
+In experi-
+mental setups, such a system can be realised as a spin-
+balanced gas of a single bosonic isotope, where spins σ
+correspond to two different hyperfine levels and the in-
+teraction strengths can be tuned with magnetic field via
+Feshbach interactions.
+The single-component densities are locked according
+to the condition n↓/n↑ =
+�
+g↑↑/g↓↓, which holds even
+in inhomogeneous cases. In our system this implies that
+there are equal number of atoms in each component N↑ =
+N↓ = N/2 and that the single-component densities are
+half of the total density n↑ = n↓ = n/2. If the system is
+homogeneous, the overall density is equal to n = N/L.
+B.
+Generalized Gross-Pitaevskii and quantum
+Monte Carlo approaches
+In the weakly-interacting limit (corresponding to high
+densities na ≫ 1), one may expect the generalized GGP
+approach to be valid. The GGP energy density functional
+has the following part corresponding to interactions [16]:
+EGGP[n; g, g↑↓] = (g − g↑↓)n2
+4
+− mn3/2
+3
+√
+2πℏ
+�
+(g − g↑↓)3/2 + (g + g↑↓)3/2�
+.
+(2)
+The first term in Eq. (2) corresponds to the mean-field
+contribution to the interaction energy and the other – to
+the correction for quantum fluctuations, widely known as
+the LHY term.
+If we compare, however, the results from GGP equa-
+tion and ab initio calculations from diffusion Monte Carlo
+in a wide range of densities and interaction ratios, we ob-
+serve discrepancies at low ratios. It is due to one of the
+peculiarities of one-dimensional systems – the lower the
+density, the higher the interaction. Thus, the GGP model
+is correct in the high-density limit but cannot be trusted
+in the opposite case.
+First of all, the GGP predicts the existence of sta-
+ble quantum droplets for any ratio g↑↓/g < 1. In other
+words, there is always a local minimum present in the en-
+ergy density functional EGGP[n; g, g↑↓] as long as g↑↓/g <
+1. QMC predicts a certain critical value of the interac-
+tion ratio, below which the minimum disappears and we
+have a liquid-gas transition at (g↑↓/g)cr = 0.47(2) [16].
+Although there are other methods, like a general ex-
+tension to the LHY theory proposed in Ref. [35] or a
+10
+1
+100
+101
+Density na
+0
+1
+2
+3
+4
+Energy 2E/N b
+FIG. 1. Energy per particle as a function of density – com-
+parison of different models: QMC (markers), mLLGP (solid
+line), GGP (dashed line), pairing theory (dotted line) - and
+interaction ratios g↑↓/g = 0.45 (yellow [light grey]), 0.75 (ma-
+genta [grey]) and 0.9 (black).
+
+3
+pairing theory for bosons introduced in [34], which are
+able to predict such a transition, they do not enable us
+to quantitatively compute the homogeneous gas energy
+with their use. Neither does GGP, which results in an
+inaccurate estimate of a quantum droplet size and bulk
+density.
+Lastly, the GGP is not applicable to the strongly-
+interacting regime. When na ≪ 1, the gas energy quickly
+approaches half of the binding energy of a dimer, i.e.,
+−εb/2, indicating that the system could be understood
+as a weakly-interacting gas of dimers [34]. The energy per
+dimer approaches −εb in the limit of vanishing density,
+while according to the GGP theory it tends to zero.
+C.
+Lieb-Liniger Gross-Pitaevskii equation for
+two-component 1D bosonic mixtures (mLLGP
+equation)
+We aim to construct a novel approach to study bosonic
+mixtures in 1D, which gives (i) a quantitative agree-
+ment with QMC in terms of a homogeneous gas en-
+ergy E(n; g, g↑↓) in a wide range of interaction ra-
+tios [48], (ii) a proper limit of a uniform gas energy,
+i.e. limna→0 E(n; g, g↑↓) = −Nεb/2, and (iii) a correct
+value for the critical interaction ratio (g↑↓/g)cr, at which
+a liquid-gas transition occurs. It is more accurate than
+both the GGP and pairing theory, but, unlike QMC, en-
+ables us to study nonlinear and time-dependent effects,
+e.g. the properties of dark solitons.
+To do that, we fit QMC data from Ref. [16] to
+get a spline representation of the energy functional
+EmLLGP[n; g, g↑↓] and construct a single-orbital density
+functional theory for bosonic mixtures.
+To do this,
+we extrapolate the data in the low- and high-density
+regimes with two separate functions. This is necessary
+because the QMC data is covering only a part of densi-
+ties, omitting the low- and high-density regions. After-
+wards, we interpolate the data with a spline in densities
+and linearly in interaction ratios. In this way, we obtain
+EmLLGP[n; g, g↑↓] in a form which is convenient for nu-
+merical evaluation. This whole procedure is described in
+detail in Appendix A.
+Figure 1 shows us the energy per particle of a ho-
+mogeneous Bose-Bose mixture.
+For interaction ratios
+g↑↓/g ≃ 1 all three theories (GGP, pairing theory and
+mLLGP) are consistent with QMC calculations. In the
+case of the GGP and pairing theory, the smaller the ra-
+tio becomes, the higher the discrepancy is.
+For ratio
+g↑↓/g = 0.45, the energy per particle from the GGP
+model still possesses a pronounced minimum, whereas
+QMC, mLLGPE and the pairing theory predict a lack
+thereof.
+The latter deviates from the QMC data and
+matches it only qualitatively in this region. One can see
+the energy functional EmLLGP is constructed to fulfil all
+the conditions from the list above.
+The analysis of the energy functional in a state
+can
+provide
+us
+with
+important
+thermodynamic
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+Interaction ratio g
+/g
+100
+101
+Density na
+LIQUID
+UNSTABLE
+GAS
+FIG. 2.
+Phase diagram of a homogeneous two-component
+mixture: the unstable region is demarcated by spinodal den-
+sities, predicted from GGP (red [light grey] dashed line) and
+QMC (square markers).
+Equilibrium density given by the
+mLLGP (navy [dark grey] solid line), GGP (blue [grey] dashed
+line) and QMC (round markers).
+quantities.
+For
+instance,
+µmLLGP[n0; g, g↑↓]
+=
+δEmLLGP[n; g, g↑↓]/δn|n=n0
+is
+the
+chemical
+potential
+evaluated at density n0, and the speed of sound c
+is given by the following relation c =
+�
+n
+m
+dµ
+dn.
+The
+position of the energy per particle minimum plays a
+vital role in the context of quantum droplet studies: the
+equilibrium density neq where d(E/N)/dn = 0 is the
+value of the density in the droplet bulk, assuming the
+droplet is sufficiently large, i.e. N ≫ 1 and possesses a
+flat-top profile. In this limit, we may approximate the
+properties of the droplet bulk to be the same as those of
+a homogeneous system with density neq.
+With that knowledge we are able to explore the phase
+diagram and compare it to the one created with the QMC
+approach.
+We show it in Fig. 2.
+We are able to dis-
+tinguish 3 phases: gaseous, liquid and unstable.
+The
+gaseous one corresponds to the region where there is no
+minimum in the energy density functional. It happens
+when the interaction ratio g↑↓/g < 0.47.
+Above that
+value, the minimum exists and we enter the liquid phase.
+Nevertheless, in the region g↑↓/g > 0.47, there is a range
+of densities for which the speed of sound is complex. This
+signals a phonon instability.
+The unstable and stable
+liquid phases are demarcated by spinodal densities nins,
+where d2E/dn2 = 0. At this border, the compressibility
+is infinite. In Fig. 2 we also plot equilibrium densities
+neq (see solid navy line for mLLGP and a dashed blue
+one for GGP). The two comparisons QMC vs GGP and
+QMC vs mLLGP favour the latter approach. Wherever
+we have data from QMC simulations, the mLLGP pre-
+dicts the same equilibrium density as ab initio calcula-
+tions. On the other hand, the GGP extends both liquid
+
+4
+and unstable regions far beyond the critical interaction
+ratio (g↑↓/g)cr.
+For low interaction ratios g↑↓/g ≪ 1, the equilibrium
+densities are located in the low-density region. However,
+in this limit of densities, the gas cannot be treated any-
+more as weakly-interacting.
+The GGP approach, con-
+trary to QMC, gives us a rough estimate of neq only.
+Having established that the constructed energy func-
+tional reproduces the phase diagram according the QMC
+theory, we can now use this to construct an equation of
+the form of Eq. (1) which allows for modelling time de-
+pendence and inhomogeneity of the effective single par-
+ticle orbital. We now write this equation as:
+iℏ∂tψ(x, t) = − ℏ2
+2m∂2
+xψ(x, t)
++µmLLGP
+�
+|ψ(x, t)|2; g, g↑↓
+�
+ψ(x, t).
+(3)
+The square modulus of this orbital is interpreted as the
+particle density n(x).
+Next, in Sec. III A, we will nu-
+merically solve the mLLGP equation (3), with the use
+of imaginary time propagation to find broken-symmetry
+states in Bose-Bose mixtures. Following this, in Sec. III B
+we will additional solve the equation in real time to sim-
+ulate the breathing modes of a perturbed droplet. Our
+toolkit is provided under the link https://gitlab.com/
+jakkop/mudge/-/releases/v21Dec2022.
+0.5
+0.6
+0.7
+0.8
+0.9
+Interaction ratio g
+/g
+20
+40
+60
+80
+100
+Droplet width
+x2
+x 2 /a
+0
+25
+50
+Position xa
+1
+0.0
+0.5
+1.0
+1.5
+Density na
+FIG. 3.
+Droplet width
+�
+⟨x2⟩ − ⟨x⟩2 as a function of the
+interaction ratio g↑↓/g. Round cyan markers correspond to
+the GGP prediction, diamond black ones – to the mLLGP
+estimation. Number of particles forming the droplet N = 100.
+Inset: density profiles of quantum droplet evaluated at a ratio
+g↑↓/g = 0.6 using the mLLGPE (solid) and GGP (dashed)
+for different number of particles N = 20 (red [innermost]),
+60 (green [middle]) and 100 (blue [outermost]). Black dotted
+line corresponds to the equilibrium density given by the QMC
+calculations.
+III.
+QUANTUM DROPLETS
+A.
+Static properties
+The ground state (GS) of a two-component mixture
+in the liquid regime takes a form of a quantum droplet.
+Typical density profiles of one-dimensional droplets are
+shown in the inset of Fig. 3. The quantum droplets eval-
+uated with the mLLGP (see Appendix A for numerical
+details) exhibit a flat-top bulk when the number of par-
+ticles exceeds 20. For N = 60 and 100, we can observe
+a prominent plateau with the same density as the equi-
+librium value neq given by QMC calculations. We juxta-
+posed these density profiles with analogous ones given by
+the GGP equation. As we can see, their bulk densities
+do not match the QMC prediction. The discrepancy for
+g↑↓/g = 0.6 is equal to 14%, but grows up to 48% at the
+critical ratio (g↑↓/g)cr = 0.47(2) (cf. Fig. 2).
+As the number of particles in the droplet N, its bulk
+density neq and its width
+�
+⟨x2⟩ − ⟨x⟩2 are connected (in
+the first approximation neq ∝ N/
+�
+⟨x2⟩ − ⟨x⟩2), the dif-
+ference between the estimations of the equilibrium den-
+sity should be also visible when we plot the droplet width
+against the interaction ratio, but keeping a fixed number
+of atoms in the system. We show it in the main panel of
+Fig. 3.
+As we can see, the droplet width is a decreasing func-
+tion of g↑↓/g. When g↑↓ becomes larger, the interparticle
+attraction gets more pronounced and the droplet con-
+tracts. As expected, the GGP gives a qualitative agree-
+ment of the droplet width with the mLLGP. However,
+the lower the interaction ratio, the higher the discrep-
+ancy between the models.
+In classical physics the total energy of the droplet can
+be divided into the volume and surface terms Etot =
+EV + ES. In a one-dimensional system, the surface term
+should be N-independent and the volume term (for N ≫
+1) should be proportional to the number of particles in
+the droplet as we show it in the inset of Fig. 4. We are
+particularly interested in the value of the surface term. If
+a droplet gets split, the energy in the system increases by
+ES. Low values of the surface energy may be considered
+an issue in the experiment. Namely, thermal excitations
+might cause a fission of the droplet.
+Figure 4 depicts the surface energy of the droplet as
+a function of the interaction ratio g↑↓/g. In the case of
+mLLGP, the diminishing surface energy when approach-
+ing (g↑↓/g)cr is a signature of the liquid-gas transition
+proximity. The surface tension slowly decreases until it
+vanishes below the critical interaction ratio. The GGP
+does not predict such a transition, so the surface tension
+does not go to zero according to this theory.
+
+5
+0.5
+0.6
+0.7
+0.8
+0.9
+Interaction ratio g
+/g
+10
+2
+10
+1
+100
+101
+102
+103
+Surface energy Esma/
+0
+100
+Particle number N
+20
+0
+Energy Etotma/
+FIG. 4.
+Surface energies of quantum droplets for different
+interaction ratios g↑↓/g. Round cyan markers correspond to
+the GGP results, diamond black ones – to the mLLGP pre-
+diction.
+Inset: total energy of a quantum droplet obtained with Eq. 3
+for high particle numbers N ≫ 1. The dash-dotted line cor-
+responds to a linear fit Etot = eN + ES.
+100
+101
+102
+Particle number N
+10
+2
+10
+1
+Monopole mode frequency
+ma2/
+FIG. 5. Monopole mode frequency as a function of the num-
+ber of particles in the droplet. Round markers correspond to
+the linear response theory prediction based on QMC data, di-
+amonds – to the mLLGP, and the dashed lines – to the GGP
+predictions.
+Frequencies evaluated at a ratios g↑↓/g = 0.6
+(blue [dark grey]) and 0.8 (yellow [light grey]).
+B.
+Monopole mode excitation
+We now look into how the ground state reacts to a
+small perturbation. We choose to study the monopole
+mode. We evolve in real time a quantum droplet per-
+turbed by a factor exp(−iϵx2/a2), where ϵ is a small con-
+stant. It corresponds to a situation when the initial ve-
+locity field in a droplet has the form v(x) = −2ℏϵx/ma2
+(further details are provided in Appendix A). At the be-
+ginning, the droplet is squeezed and at some point it
+expands again. This process is periodic and has its char-
+acteristic frequency which we measure by looking at the
+standard deviation of the droplet width
+�
+⟨x2⟩ − ⟨x⟩2 in
+time.
+We show the results of this numerical analysis in Fig. 5
+altogether with the monopole mode frequencies evaluated
+with the GGP [4] and linear response theory predictions
+based on QMC data [17]. All three approaches give con-
+sistent results in the large particle number limit. The
+monopole mode frequency scales like ω ∝ N −1 there [4].
+Surprisingly, the QMC data also agree with the GGP-
+based results, even though the GGP equation is not ex-
+pected to be accurate for small N, due to the breakdown
+of the local density approximation (LDA), which requires
+fulfilling the condition N ≫ 1.
+The mLLGP simulations agree in most cases within
+the range of 2 uncertainties. The dominating source of
+uncertainty is the form of EmLLGP in the low-density re-
+gions n ≪ neq. As we lack Monte Carlo data there, we
+cannot control the quality of the fit below the equilibrium
+density. It is clearly visible when the number of particles
+in the droplet is low. The bulk density is lower than neq
+there (cf. the inset of Fig. 3), especially after a slight
+expansion happening due to the perturbation we apply.
+Thus, an accurate measurement of monopole mode fre-
+quencies seems to be the best choice to experimentally
+verify the validity of mLLGP-based study. It might be
+a daunting task, though. The difference is most strik-
+ing in the small-droplet limit, which might be difficult to
+achieve in an experimental setup.
+IV.
+DARK SOLITONS
+We supplement our study of Bose-Bose mixtures with a
+numerical analysis of dark solitons. They are an example
+of nonlinear effects which are beyond the range of QMC.
+We look for solitonic solutions of the mLLGP equa-
+tion (3) in the thermodynamic limit. By dark soliton we
+understand a density depletion travelling at a constant
+velocity vs without changing its shape. We may classify
+these solitons as grey solitons if vs > 0 and they have a
+non-zero density minimum, and as black solitons if they
+are motionless and their density minimum is equal to
+zero [45].
+We assume that the density and phase of the orbital
+ϕ = arg ψ far from the soliton are constant and equal to
+n∞ and ϕ∞. Our numerical methods (see Appendix B
+for details) enable us to find both motionless and moving
+dark solitons. We use a velocity relative to the speed of
+sound β = vs/c to characterize the soliton.
+If we take a look at the dark solitonic solutions in
+the weakly-interacting single-component Bose gas, we en-
+counter both moving and motionless solutions.
+Figure
+6
+presents
+the
+solitonic
+density
+minima
+
+6
+100
+101
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+(a)
+100
+101
+0
+1
+2
+3
+4
+5
+(b)
+6 × 10
+1
+8 × 10
+1
+100
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Density minimum nmin/n
+(c)
+6 × 10
+1
+8 × 10
+1
+100
+0
+5
+10
+15
+20
+Soliton width XFWHDa
+1
+(d)
+2 × 101
+3 × 101
+4 × 101
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+(e)
+2 × 101
+3 × 101
+4 × 101
+0
+5
+10
+15
+20
+(f)
+Background density n a
+= 0
+= 0.25
+= 0.5
+= 0.75
+FIG. 6. Minimum densities nmin (a, c, e) and full widths at half depth XFWHD (b, d, f) of dark solitons for different relative
+velocities of the soliton β and interaction ratios g↑↓/g = 0.4 (a, b), 0.55 (c, d), and 0.75 (e, f). Green shading corresponds to
+gaseous phase, blue – to the liquid one and red – to the unstable regime (cf. Fig 2). The vertical blue line marks the equilibrium
+density value.
+min n(x) and full widths at half depths XFWHD as func-
+tions of the density n∞ for three values of the interaction
+ratio g↑↓/g. The motionless solitons in the gaseous phase
+neq can be classified as standard ones – their density
+reaches zero (cf. black solid line in panel (a)). Moreover,
+the density minima of grey solitons increase with their
+velocity. Panel (b) shows us the soliton width, which di-
+verges as n∞ → 0. The chemical potential tends to zero
+and the healing length ξ ∝ µ−1/2 diverges.
+The situation changes when we cross the critical in-
+teraction ratio and enter the liquid phase. In the high-
+density limit the soliton minimum density is zero, but
+below neq we enter a region where the minimum density
+starts to increase (see panels (c) and (e)). One may say
+the motionless solitons greyen. These solitons have been
+first described in Ref. [27]. Due to their uncanny fea-
+tures, described at length later in this section, we call
+them anomalous.
+We have confirmed that these solu-
+tions maintain their form and phase profile during real
+time propagation in the presence of low-amplitude noise,
+confirming their stability.
+The solitonic solution (as it is shown in panels (d) and
+(f)) widens in two places. Once when n∞ → neq, both
+in the standard (n∞ → n+
+eq) and anomalous (n∞ → n−
+eq)
+
+7
+20
+10
+0
+10
+20
+0.0
+0.5
+1.0
+Density na
+(a)
+20
+10
+0
+10
+20
+/2
+0
+/2
+Phase
+(c)
+5
+0
+5
+0
+1
+2
+(b)
+5
+0
+5
+/2
+0
+/2
+(d)
+Position xa
+1
+FIG. 7. Motionless anomalous soliton density (a) and phase (c) profiles. Standard grey soliton density (b) and phase (d). The
+grey soliton is moving with relative velocity β = 0.5. Both solitons were evaluated at a ratio g↑↓/g = 0.6 using the mLLGPE.
+regimes and another time, while approaching the instabil-
+ity region. The most interesting regime to realise experi-
+mentally is in the vicinity of neq. The solitonic solutions
+there are both wide and deep, which may be easier to
+detect with in situ imaging procedure.
+Grey solitons also become shallower with decreasing
+density n∞. For β > 0, it is a gradual change though (cf.
+panels (c) and (e)). Another difference is that the grey
+soliton width does not diverge when n∞ → neq, it does
+so in the vicinity of the unstable regime only (cf. panels
+(d) and (f)).
+In Fig. 7(a) and (c) we show the density and phase
+profiles of motionless solitons evaluated at a ratio g↑↓/g =
+0.6 and density fulfilling the inequality nins < n∞ < neq.
+This soliton has a non-zero density minimum, normally
+characteristic to moving (grey) solitons. Moreover, there
+is no π-phase jump, as in a standard motionless (black)
+solitonic solution in the GPE [45].
+On the other hand, when n∞ > neq, no anomalous
+solutions are found. In this regime, solitons are similar
+to standard dark solitons. We show density and phase
+profiles of a grey soliton moving with velocity β = 0.5 in
+panels (b) and (d) of Fig. 7.
+To gain some insight into the large width of the soli-
+tons when n∞ ≈ neq, we shall consider again a homoge-
+neous gas. We can define the pressure as P = −dE/dL.
+Above the value of neq, the pressure is positive.
+But
+below the equilibrium density, the pressure becomes neg-
+ative. Thus, if we break the symmetry in the system by
+rarefying the density in one point, the pressure will make
+the gas on the sides of the defect contract and form a
+structure with wide density depletion.
+V.
+SUMMARY
+To conclude, we have presented a QMC-based single-
+orbital density functional theory for a two-component
+bosonic mixture in one dimension, which we call the
+mLLGP model. From construction, our approach pro-
+vides a quantitative agreement in terms of the energy
+and chemical potential of a homogeneous state with the
+ab initio QMC model from Ref. [17].
+We benchmark our equation by comparing the results
+with the original QMC data. This comparison shows the
+mLLGP can quantitatively predict the bulk density of a
+quantum droplet and the monopole mode frequency in
+the limit of a large number of particles in the droplet
+with a characteristic ω ∝ N −1 dependency. It also pre-
+dicts a correct phase diagram of Bose-Bose mixtures, in-
+cluding a transition from liquid to gas, not predicted by
+the mean-field model supplemented with the LHY cor-
+rection. Since our approach relies on fitting an energy
+functional to QMC data, it is limited by the range of
+underpinning QMC data, which is only currently avail-
+able in the literature for densities close to the equilibrium
+density and for specific interaction ratios. Should QMC
+data become available over a larger parameter space of
+density and interaction ratio, the model could be refined
+with an improved energy functional.
+Our work is limited to the specific case where the in-
+traspecies interactions are equal, g↓↓ = g↑↑, which leads
+
+8
+to the density profile of each component being equal
+to each other, n↓(x) = n↑(x).
+In principle the ap-
+proach could be extended to the more general case where
+g↓↓ ̸= g↑↑ and n↓(x) ̸= n↑(x), however this would re-
+quire QMC data over a wider parameter space. Given
+the computational intensity of QMC calculations, this is
+not tractable at the present time but may become possi-
+ble in the future.
+Lastly, we provide a brief study of solitonic solutions of
+the mLLGP equation, where we find ultrawide solitonic
+solutions. Moreover, anomalous motionless solitons were
+found as well. These solitons are characterized by the
+lack of a π-jump in the phase and a non-zero density
+minimum.
+The presence of such wide solitons can be an advantage
+for experimenters who would like to perform an in situ
+imaging of these objects.
+As far as we are concerned,
+the measurement of the monopole mode frequency for
+small droplets may be helpful to verify the validity of the
+mLLGP equation too. It would demand creating droplets
+consisting of very few particles, though, making such an
+experiment tougher to design and conduct. An avenue
+for further work would be to use the mLLGP model to
+study the dynamical properties of dark solitons in 1D
+Bose-Bose mixtures, particularly the anomalous solitons,
+including their collisions, stability and experimental gen-
+eration.
+Data availability — All the numerical data necessary to
+reproduce figures, including QMC data and the results of
+simulations with the MUDGE toolkit (https://gitlab.
+com/jakkop/mudge/-/releases/v21Dec2022) are avail-
+able in the Supplemental Material under the link [URL
+will be inserted by publisher].
+ACKNOWLEDGMENTS
+The authors acknowledge discussions with Dr Thomas
+Billam and Mr Thomas Flynn (Newcastle University).
+L.P. and N.P. acknowledge support from the UK Engi-
+neering and Physical Sciences Research Council (Grant
+No. EP/T015241/1). J.K. and K.P. acknowledge sup-
+port from the (Polish) National Science Center Grant
+No. 2019/34/E/ST2/00289.
+Center for Theoretical Physics of the Polish Academy
+of Sciences is a member of the National Laboratory of
+Atomic, Molecular and Optical Physics (KL FAMO).
+L.P. prepared the energy density functional, J.K. con-
+ducted the numerical simulations.
+K.P. and N.P. con-
+ceptualized and supervised the research. J.K. wrote the
+manuscript with input of all authors.
+Appendix A:
+Details of the numerical procedures
+Energy density functional
+In
+order
+to
+find
+the
+energy
+density
+func-
+tional
+EmLLGP[n; g, g↑↓],
+we
+use
+the
+QMC
+data
+from
+Ref.
+[16],
+namely
+the
+energy
+per
+particle
+E/N ≡ e(n; g, g↑↓) for the following interaction ratios
+g↑↓/g
+=
+{0.3, 0.4, 0.45, 0.5, 0.6, 0.65, 0.7, 0.75, 0.8, 0.9}.
+The data are extrapolated in the low density limit with
+a function fL(n) = −1 + c1n3/2 + c2n5/2 + c3n3 and
+fH(n) = c4n1/2 +c5n+c6n3/2 [in units of εb/2], where ci
+for i = {1, 2, . . . , 6} are constants to be fitted. Then, we
+perform a spline interpolation of the augmented QMC
+data and perform a linear interpolation between the
+ratios.
+The energy density functional is connected to
+the energy per particle function e(n; g, g↑↓) via a simple
+relation: EmLLGP[n; g, g↑↓] = ne(n; g, g↑↓).
+Imaginary and real time evolution details
+The mLLGP equation is a complex, nonlinear partial
+differential equation. The orbital ψ(x) is discretized on
+a spatial mesh with Nx fixed points and lattice spacing
+DX = L/Nx, where L is the box size. We assume peri-
+odic boundary conditions, i.e. ψ(−L/2) = ψ(L/2). The
+real-time evolution is done with the use of the split-step
+numerical method. The evolution with the kinetic term
+is done in the momentum domain, whereas the contact
+interaction term is calculated in the spatial domain. No
+external potential is used. The quantum droplet is ob-
+tained with the use of imaginary time evolution, where
+we use Wick rotation t → −iτ to find the ground state.
+The program written in C++ implementing the algo-
+rithm above is publicly available (see Data availability
+for link). The program uses the W-DATA format dedi-
+cated to store data in numerical experiments with ultra-
+cold Bose and Fermi gases. The W-DATA project is a
+part of the W-SLDA toolkit [49].
+When measuring the monopole mode frequency ω, we
+perturb the ground state by multiplying it by a fac-
+tor exp(−iϵx2/a2), where ϵ is of the order of 10−6 in
+our simulations.
+Afterwards, we fit the droplet width
+�
+⟨x2⟩ − ⟨x⟩2(t) to a function f(t) = A + B cos(ωt + C),
+where A, B, C, and ω are fitted constants.
+In order to estimate the uncertainty due to the quality
+of the energy density functional, we repeat the simula-
+tions with alternative spline representations of EmLLGP.
+Namely we reduce the number of points we use to extrap-
+olate the data with fL(n) and redo the whole procedure
+with a slightly different spline.
+Appendix B:
+Dark solitons in the mLLGP equation
+To find the dark solitonic solutions of the mLLGP
+equation, we go to the thermodynamic limit, i.e. L →
+∞, N → ∞ and N/L = const. We plug the following
+Ansatz for a wave travelling through the system at a con-
+stant velocity vs, i.e. ψ(x, t) = ˜ψ(ζ), where ζ = x − vst
+is a comoving coordinate, to Eq. (3) and obtain
+µs ˜ψ−imvs ˜ψ′ = − ℏ2
+2m
+˜ψ′′+µmLLGP
+�
+| ˜ψ|2; g, g↑↓
+�
+˜ψ. (B1)
+
+9
+If we assume that far away from the soliton, the den-
+sity and phase are constant limζ→∞ | ˜ψ(ζ)|2 = n∞ and
+limζ→∞ arg ˜ψ(ζ) = ϕ∞, we can find the value of the
+chemical potential µs = µmLLGP [n∞; g, g↑↓]. Then, we
+rewrite the equation above in a discretized form, assum-
+ing that we start from two points far away from the soli-
+ton ˜ψ0 = (1 − ϵ1)√n∞ and ˜ψ1 = (1 − ϵ2)√n∞ with
+ϵ1,2 ≪ 1 (typically ∼ 10−4) and ϵ1 > ϵ2.
+We have also checked that solitonic solutions are dy-
+namically stable.
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diff --git a/6NE3T4oBgHgl3EQfQwn7/content/tmp_files/load_file.txt b/6NE3T4oBgHgl3EQfQwn7/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..2d90cfe836aa3cd88e573b737120e517ce8886d8
--- /dev/null
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@@ -0,0 +1,813 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf,len=812
+page_content='Quantum Monte Carlo-based density functional for one-dimensional Bose-Bose  mixtures  Jakub Kopyci´nski,1, 2, ∗ Luca Parisi,2 Nick G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' Parker,2 and Krzysztof Paw�lowski1  1Center for Theoretical Physics, Polish Academy of Sciences, Al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' Lotnik´ow 32/46, 02-668 Warsaw, Poland  2Joint Quantum Centre (JQC) Durham–Newcastle,  School of Mathematics, Statistics and Physics, Newcastle University,  Newcastle upon Tyne, NE1 7RU, United Kingdom  (Dated: January 12, 2023)  We propose and benchmark a Gross-Pitaevskii-like equation for two-component Bose mixtures  with competing interactions in 1D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' Our approach follows the density-functional theory with the  energy functional based on the exact Quantum Monte Carlo (QMC) simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' Our model covers,  but goes beyond, the popular approach with the Lee-Huang-Yang corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' We first benchmark  our approach against available QMC data in all interaction regimes and then study dynamical prop-  erties, inaccessible by ab initio many-body simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' Our analysis includes a study of monopole  modes and reveals the presence of anomalous dark solitons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  INTRODUCTION  Recent studies of ultracold gases with competing in-  teractions have led to a major change in the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' They  undermined the validity of the mean-field approximation  when attractive and repulsive interactions in the system  almost cancel each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  In such a situation, it is necessary to include the effect  of quantum fluctuations [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' To account for it, one can  include Lee-Huang-Yang (LHY) corrections [2, 3] to the  mean-field equation using a local density approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  One then derives a generalized Gross-Pitaevskii (GGP)  equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  It has been widely used to theoretically in-  vestigate the ground-state properties and excitations of  Bose-Bose mixtures with particular attention given to  the compressional mode (known also as the monopole or  breathing mode) [4–7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  The GGP theory predicts the existence of self-bound  objects – ultradilute quantum droplets made of ultracold  atoms [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' The emergence of a liquid phase is marked by  the presence of a local minimum in the energy density  functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  Soon after having been proposed theoretically, quan-  tum droplets were experimentally observed [8–12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' Some  theoretical predictions indicate even the possibility of  finding quantum droplets [13] in recently obtained het-  eronuclear dipolar condensates [14, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  Despite its remarkable usefulness, there are still fac-  tors not included in the GGP theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' For instance, ab  initio calculations show a liquid-gas transition in two-  component mixtures [16], whereas the GGP does not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  Moreover, the same work demonstrates a quantitative  disagreement of the homogeneous state energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  Quite  unexpectedly, the monopole mode frequencies happen to  match the QMC calculations, though [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  Several attempts have been made to overcome the ex-  isting imperfections of the GGP equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' One of the  ∗ jkopycinski@cft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='pl  ideas, that follows the density functional theory, was  to build an equation which would quantitatively repro-  duce the spatially uniform state energy from a chosen  ab initio method for any interaction strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  In this  regard the 1D Bose contact gas is a special system as  its ground-state energy has been already derived in the  analytical ab initio calculations by E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' Lieb and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' Lin-  iger [18, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' Using this exact energy functional one gets  the single-particle equation here referred to as the Lieb-  Liniger Gross-Pitaevskii (LLGP) equation that was used  in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' [20–27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' The equation proved to correctly de-  scribe the ground state and low-lying excitations in all  regimes – from the weakly-interacting one (which, con-  trary to the 3D case happens at high gas densities) up  to the Tonks-Girardeau regime (at low gas densities).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  The LLGP equation was recently used to study Bose gas  with repulsive short-range and attractive dipolar inter-  actions [28–32] to show the existence and properties of  the dipolar quantum droplets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' Concerning the droplets  in quantum mixtures, a similar approach was employed  to construct a quantum Monte Carlo (QMC)-based en-  ergy density functional for bosonic mixtures in 3D [33],  but so far the 1D Bose mixture was not investigated in  such framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' For the latter system it was shown [16]  that GGP fails to reproduce the phase diagram in certain  regimes, in particular at low densities when atoms bind  together into interacting dimers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  In this article, we aim to formulate and benchmark  the QMC-based single-orbital density functional theory  that is applicable to two-component Bose mixtures with  repulsive intra- and attractive intercomponent interac-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' Our theoretical approach using a single orbital ψ  and a QMC-based energy density functional E results in  an equation of the following form:  iℏ∂tψ(x, t) = − ℏ2  2m∂2  xψ(x, t) + δE  δnψ(x, t),  (1)  where n is the particle density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' We want it to be ap-  plicable to two-component Bose mixtures with repulsive  intra- and attractive intercomponent interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' To do  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='04417v1  [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='quant-gas]  11 Jan 2023  2  this, we analyse the phase diagram of the system and nu-  merically study the static properties and monopole mode  of quantum droplets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  In lower-dimensional systems, we can name two sub-  stantial beyond-LHY approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' One of them is a pair-  ing theory for bosons [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' The other one is based on the  inclusion of higher-order corrections to the GGP equa-  tion [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' Both of them generally give only a qualitative  agreement with QMC calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  Our approach shares similarities with a density func-  tional theory [36] for Fermi systems at unitarity [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  The resulting density functional has been employed mul-  tiple times to look into strongly interacting fermions [38–  42], revealing a remarkable consistency with the experi-  ments [43, 44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  A great advantage of having a Gross-Pitaevskii-like  equation, in comparison to the QMC methods, is the  possibility of studying nonlinear and time-dependent ef-  fects like the existence of dark solitons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' This subject is  particularly interesting as we may expect fundamentally  different results than the solitons we know from single-  species systems [45] or dark-dark solitons occurring in  miscible bosonic mixtures [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' Very recently there have  been reports on wide soliton-like objects, both in mix-  tures [47] and in dipolar Bose gases [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' As such, last  but not least, we show density and phase profiles of soli-  tary waves evaluated with our theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  FRAMEWORK  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  System  We consider a one-dimensional Bose gas consisting of  two components σ = {↑, ↓} in a box of size L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  We  assume that the components have equal atomic masses  m↑ = m↓ = m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' We also assume that the short-range  interaction coupling constants are the same in the intra-  component case g↑↑ = g↓↓ = g, whereas the intercom-  ponent interactions can be independently tuned with a  coupling constant g↑↓.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  Atoms of the same species re-  pel each other while the intercomponent interactions are  attractive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' The binding energy of an atomic pair in vac-  uum εb = −(mg2  ↑↓/4ℏ2) is a relevant energy scale in the  system, while for the length scale we choose the intra-  component scattering length a = 2ℏ2/mg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  In experi-  mental setups, such a system can be realised as a spin-  balanced gas of a single bosonic isotope, where spins σ  correspond to two different hyperfine levels and the in-  teraction strengths can be tuned with magnetic field via  Feshbach interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  The single-component densities are locked according  to the condition n↓/n↑ =  �  g↑↑/g↓↓, which holds even  in inhomogeneous cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' In our system this implies that  there are equal number of atoms in each component N↑ =  N↓ = N/2 and that the single-component densities are  half of the total density n↑ = n↓ = n/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' If the system is  homogeneous, the overall density is equal to n = N/L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  Generalized Gross-Pitaevskii and quantum  Monte Carlo approaches  In the weakly-interacting limit (corresponding to high  densities na ≫ 1), one may expect the generalized GGP  approach to be valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' The GGP energy density functional  has the following part corresponding to interactions [16]:  EGGP[n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' g, g↑↓] = (g − g↑↓)n2  4  − mn3/2  3  √  2πℏ  �  (g − g↑↓)3/2 + (g + g↑↓)3/2�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  (2)  The first term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' (2) corresponds to the mean-field  contribution to the interaction energy and the other – to  the correction for quantum fluctuations, widely known as  the LHY term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  If we compare, however, the results from GGP equa-  tion and ab initio calculations from diffusion Monte Carlo  in a wide range of densities and interaction ratios, we ob-  serve discrepancies at low ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' It is due to one of the  peculiarities of one-dimensional systems – the lower the  density, the higher the interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' Thus, the GGP model  is correct in the high-density limit but cannot be trusted  in the opposite case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  First of all, the GGP predicts the existence of sta-  ble quantum droplets for any ratio g↑↓/g < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' In other  words, there is always a local minimum present in the en-  ergy density functional EGGP[n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' g, g↑↓] as long as g↑↓/g <  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' QMC predicts a certain critical value of the interac-  tion ratio, below which the minimum disappears and we  have a liquid-gas transition at (g↑↓/g)cr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='47(2) [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  Although there are other methods, like a general ex-  tension to the LHY theory proposed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' [35] or a  10  1  100  101  Density na  0  1  2  3  4  Energy 2E/N b  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' Energy per particle as a function of density – com-  parison of different models: QMC (markers), mLLGP (solid  line), GGP (dashed line), pairing theory (dotted line) - and  interaction ratios g↑↓/g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='45 (yellow [light grey]), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='75 (ma-  genta [grey]) and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='9 (black).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  3  pairing theory for bosons introduced in [34], which are  able to predict such a transition, they do not enable us  to quantitatively compute the homogeneous gas energy  with their use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' Neither does GGP, which results in an  inaccurate estimate of a quantum droplet size and bulk  density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  Lastly, the GGP is not applicable to the strongly-  interacting regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' When na ≪ 1, the gas energy quickly  approaches half of the binding energy of a dimer, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=',  −εb/2, indicating that the system could be understood  as a weakly-interacting gas of dimers [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' The energy per  dimer approaches −εb in the limit of vanishing density,  while according to the GGP theory it tends to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  Lieb-Liniger Gross-Pitaevskii equation for  two-component 1D bosonic mixtures (mLLGP  equation)  We aim to construct a novel approach to study bosonic  mixtures in 1D, which gives (i) a quantitative agree-  ment with QMC in terms of a homogeneous gas en-  ergy E(n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' g, g↑↓) in a wide range of interaction ra-  tios [48], (ii) a proper limit of a uniform gas energy,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' limna→0 E(n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' g, g↑↓) = −Nεb/2, and (iii) a correct  value for the critical interaction ratio (g↑↓/g)cr, at which  a liquid-gas transition occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' It is more accurate than  both the GGP and pairing theory, but, unlike QMC, en-  ables us to study nonlinear and time-dependent effects,  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' the properties of dark solitons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  To do that, we fit QMC data from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' [16] to  get a spline representation of the energy functional  EmLLGP[n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' g, g↑↓] and construct a single-orbital density  functional theory for bosonic mixtures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  To do this,  we extrapolate the data in the low- and high-density  regimes with two separate functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' This is necessary  because the QMC data is covering only a part of densi-  ties, omitting the low- and high-density regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' After-  wards, we interpolate the data with a spline in densities  and linearly in interaction ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' In this way, we obtain  EmLLGP[n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' g, g↑↓] in a form which is convenient for nu-  merical evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' This whole procedure is described in  detail in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  Figure 1 shows us the energy per particle of a ho-  mogeneous Bose-Bose mixture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  For interaction ratios  g↑↓/g ≃ 1 all three theories (GGP, pairing theory and  mLLGP) are consistent with QMC calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' In the  case of the GGP and pairing theory, the smaller the ra-  tio becomes, the higher the discrepancy is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  For ratio  g↑↓/g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='45, the energy per particle from the GGP  model still possesses a pronounced minimum, whereas  QMC, mLLGPE and the pairing theory predict a lack  thereof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  The latter deviates from the QMC data and  matches it only qualitatively in this region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' One can see  the energy functional EmLLGP is constructed to fulfil all  the conditions from the list above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  The analysis of the energy functional in a state  can  provide  us  with  important  thermodynamic  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='9  Interaction ratio g  /g  100  101  Density na  LIQUID  UNSTABLE  GAS  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  Phase diagram of a homogeneous two-component  mixture: the unstable region is demarcated by spinodal den-  sities, predicted from GGP (red [light grey] dashed line) and  QMC (square markers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  Equilibrium density given by the  mLLGP (navy [dark grey] solid line), GGP (blue [grey] dashed  line) and QMC (round markers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  For  instance,  µmLLGP[n0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' g, g↑↓]  =  δEmLLGP[n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' g, g↑↓]/δn|n=n0  is  the  chemical  potential  evaluated at density n0, and the speed of sound c  is given by the following relation c =  �  n  m  dµ  dn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  The  position of the energy per particle minimum plays a  vital role in the context of quantum droplet studies: the  equilibrium density neq where d(E/N)/dn = 0 is the  value of the density in the droplet bulk, assuming the  droplet is sufficiently large, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' N ≫ 1 and possesses a  flat-top profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' In this limit, we may approximate the  properties of the droplet bulk to be the same as those of  a homogeneous system with density neq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  With that knowledge we are able to explore the phase  diagram and compare it to the one created with the QMC  approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  We show it in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  We are able to dis-  tinguish 3 phases: gaseous, liquid and unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  The  gaseous one corresponds to the region where there is no  minimum in the energy density functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' It happens  when the interaction ratio g↑↓/g < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  Above that  value, the minimum exists and we enter the liquid phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  Nevertheless, in the region g↑↓/g > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='47, there is a range  of densities for which the speed of sound is complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' This  signals a phonon instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  The unstable and stable  liquid phases are demarcated by spinodal densities nins,  where d2E/dn2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' At this border, the compressibility  is infinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' 2 we also plot equilibrium densities  neq (see solid navy line for mLLGP and a dashed blue  one for GGP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' The two comparisons QMC vs GGP and  QMC vs mLLGP favour the latter approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' Wherever  we have data from QMC simulations, the mLLGP pre-  dicts the same equilibrium density as ab initio calcula-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' On the other hand, the GGP extends both liquid  4  and unstable regions far beyond the critical interaction  ratio (g↑↓/g)cr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  For low interaction ratios g↑↓/g ≪ 1, the equilibrium  densities are located in the low-density region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' However,  in this limit of densities, the gas cannot be treated any-  more as weakly-interacting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  The GGP approach, con-  trary to QMC, gives us a rough estimate of neq only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  Having established that the constructed energy func-  tional reproduces the phase diagram according the QMC  theory, we can now use this to construct an equation of  the form of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' (1) which allows for modelling time de-  pendence and inhomogeneity of the effective single par-  ticle orbital.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' We now write this equation as:  iℏ∂tψ(x, t) = − ℏ2  2m∂2  xψ(x, t)  +µmLLGP  �  |ψ(x, t)|2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' g, g↑↓  �  ψ(x, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  (3)  The square modulus of this orbital is interpreted as the  particle density n(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  Next, in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' III A, we will nu-  merically solve the mLLGP equation (3), with the use  of imaginary time propagation to find broken-symmetry  states in Bose-Bose mixtures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' Following this, in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' III B  we will additional solve the equation in real time to sim-  ulate the breathing modes of a perturbed droplet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' Our  toolkit is provided under the link https://gitlab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='com/  jakkop/mudge/-/releases/v21Dec2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='9  Interaction ratio g  /g  20  40  60  80  100  Droplet width  x2  x 2 /a  0  25  50  Position xa  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='5  Density na  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  Droplet width  �  ⟨x2⟩ − ⟨x⟩2 as a function of the  interaction ratio g↑↓/g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' Round cyan markers correspond to  the GGP prediction, diamond black ones – to the mLLGP  estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' Number of particles forming the droplet N = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  Inset: density profiles of quantum droplet evaluated at a ratio  g↑↓/g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='6 using the mLLGPE (solid) and GGP (dashed)  for different number of particles N = 20 (red [innermost]),  60 (green [middle]) and 100 (blue [outermost]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' Black dotted  line corresponds to the equilibrium density given by the QMC  calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  QUANTUM DROPLETS  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  Static properties  The ground state (GS) of a two-component mixture  in the liquid regime takes a form of a quantum droplet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  Typical density profiles of one-dimensional droplets are  shown in the inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' The quantum droplets eval-  uated with the mLLGP (see Appendix A for numerical  details) exhibit a flat-top bulk when the number of par-  ticles exceeds 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' For N = 60 and 100, we can observe  a prominent plateau with the same density as the equi-  librium value neq given by QMC calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' We juxta-  posed these density profiles with analogous ones given by  the GGP equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' As we can see, their bulk densities  do not match the QMC prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' The discrepancy for  g↑↓/g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='6 is equal to 14%, but grows up to 48% at the  critical ratio (g↑↓/g)cr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='47(2) (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  As the number of particles in the droplet N, its bulk  density neq and its width  �  ⟨x2⟩ − ⟨x⟩2 are connected (in  the first approximation neq ∝ N/  �  ⟨x2⟩ − ⟨x⟩2), the dif-  ference between the estimations of the equilibrium den-  sity should be also visible when we plot the droplet width  against the interaction ratio, but keeping a fixed number  of atoms in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' We show it in the main panel of  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  As we can see, the droplet width is a decreasing func-  tion of g↑↓/g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' When g↑↓ becomes larger, the interparticle  attraction gets more pronounced and the droplet con-  tracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' As expected, the GGP gives a qualitative agree-  ment of the droplet width with the mLLGP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' However,  the lower the interaction ratio, the higher the discrep-  ancy between the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  In classical physics the total energy of the droplet can  be divided into the volume and surface terms Etot =  EV + ES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' In a one-dimensional system, the surface term  should be N-independent and the volume term (for N ≫  1) should be proportional to the number of particles in  the droplet as we show it in the inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' We are  particularly interested in the value of the surface term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' If  a droplet gets split, the energy in the system increases by  ES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' Low values of the surface energy may be considered  an issue in the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' Namely, thermal excitations  might cause a fission of the droplet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  Figure 4 depicts the surface energy of the droplet as  a function of the interaction ratio g↑↓/g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' In the case of  mLLGP, the diminishing surface energy when approach-  ing (g↑↓/g)cr is a signature of the liquid-gas transition  proximity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' The surface tension slowly decreases until it  vanishes below the critical interaction ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' The GGP  does not predict such a transition, so the surface tension  does not go to zero according to this theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='9  Interaction ratio g  /g  10  2  10  1  100  101  102  103  Surface energy Esma/  0  100  Particle number N  20  0  Energy Etotma/  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  Surface energies of quantum droplets for different  interaction ratios g↑↓/g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' Round cyan markers correspond to  the GGP results, diamond black ones – to the mLLGP pre-  diction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  Inset: total energy of a quantum droplet obtained with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' 3  for high particle numbers N ≫ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' The dash-dotted line cor-  responds to a linear fit Etot = eN + ES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  100  101  102  Particle number N  10  2  10  1  Monopole mode frequency  ma2/  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' Monopole mode frequency as a function of the num-  ber of particles in the droplet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' Round markers correspond to  the linear response theory prediction based on QMC data, di-  amonds – to the mLLGP, and the dashed lines – to the GGP  predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  Frequencies evaluated at a ratios g↑↓/g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='6  (blue [dark grey]) and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='8 (yellow [light grey]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  Monopole mode excitation  We now look into how the ground state reacts to a  small perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' We choose to study the monopole  mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' We evolve in real time a quantum droplet per-  turbed by a factor exp(−iϵx2/a2), where ϵ is a small con-  stant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' It corresponds to a situation when the initial ve-  locity field in a droplet has the form v(x) = −2ℏϵx/ma2  (further details are provided in Appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' At the be-  ginning, the droplet is squeezed and at some point it  expands again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' This process is periodic and has its char-  acteristic frequency which we measure by looking at the  standard deviation of the droplet width  �  ⟨x2⟩ − ⟨x⟩2 in  time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  We show the results of this numerical analysis in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' 5  altogether with the monopole mode frequencies evaluated  with the GGP [4] and linear response theory predictions  based on QMC data [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' All three approaches give con-  sistent results in the large particle number limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' The  monopole mode frequency scales like ω ∝ N −1 there [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  Surprisingly, the QMC data also agree with the GGP-  based results, even though the GGP equation is not ex-  pected to be accurate for small N, due to the breakdown  of the local density approximation (LDA), which requires  fulfilling the condition N ≫ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  The mLLGP simulations agree in most cases within  the range of 2 uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' The dominating source of  uncertainty is the form of EmLLGP in the low-density re-  gions n ≪ neq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' As we lack Monte Carlo data there, we  cannot control the quality of the fit below the equilibrium  density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' It is clearly visible when the number of particles  in the droplet is low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' The bulk density is lower than neq  there (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' the inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' 3), especially after a slight  expansion happening due to the perturbation we apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  Thus, an accurate measurement of monopole mode fre-  quencies seems to be the best choice to experimentally  verify the validity of mLLGP-based study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' It might be  a daunting task, though.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' The difference is most strik-  ing in the small-droplet limit, which might be difficult to  achieve in an experimental setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  DARK SOLITONS  We supplement our study of Bose-Bose mixtures with a  numerical analysis of dark solitons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' They are an example  of nonlinear effects which are beyond the range of QMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  We look for solitonic solutions of the mLLGP equa-  tion (3) in the thermodynamic limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' By dark soliton we  understand a density depletion travelling at a constant  velocity vs without changing its shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' We may classify  these solitons as grey solitons if vs > 0 and they have a  non-zero density minimum, and as black solitons if they  are motionless and their density minimum is equal to  zero [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  We assume that the density and phase of the orbital  ϕ = arg ψ far from the soliton are constant and equal to  n∞ and ϕ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' Our numerical methods (see Appendix B  for details) enable us to find both motionless and moving  dark solitons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' We use a velocity relative to the speed of  sound β = vs/c to characterize the soliton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  If we take a look at the dark solitonic solutions in  the weakly-interacting single-component Bose gas, we en-  counter both moving and motionless solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  Figure  6  presents  the  solitonic  density  minima  6  100  101  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='0  (a)  100  101  0  1  2  3  4  5  (b)  6 × 10  1  8 × 10  1  100  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='0  Density minimum nmin/n  (c)  6 × 10  1  8 × 10  1  100  0  5  10  15  20  Soliton width XFWHDa  1  (d)  2 × 101  3 × 101  4 × 101  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='0  (e)  2 × 101  3 × 101  4 × 101  0  5  10  15  20  (f)  Background density n a  = 0  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='25  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='5  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='75  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' Minimum densities nmin (a, c, e) and full widths at half depth XFWHD (b, d, f) of dark solitons for different relative  velocities of the soliton β and interaction ratios g↑↓/g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='4 (a, b), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='55 (c, d), and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='75 (e, f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' Green shading corresponds to  gaseous phase, blue – to the liquid one and red – to the unstable regime (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' Fig 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' The vertical blue line marks the equilibrium  density value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  min n(x) and full widths at half depths XFWHD as func-  tions of the density n∞ for three values of the interaction  ratio g↑↓/g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' The motionless solitons in the gaseous phase  neq can be classified as standard ones – their density  reaches zero (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' black solid line in panel (a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' Moreover,  the density minima of grey solitons increase with their  velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' Panel (b) shows us the soliton width, which di-  verges as n∞ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' The chemical potential tends to zero  and the healing length ξ ∝ µ−1/2 diverges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  The situation changes when we cross the critical in-  teraction ratio and enter the liquid phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' In the high-  density limit the soliton minimum density is zero, but  below neq we enter a region where the minimum density  starts to increase (see panels (c) and (e)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' One may say  the motionless solitons greyen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' These solitons have been  first described in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' Due to their uncanny fea-  tures, described at length later in this section, we call  them anomalous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  We have confirmed that these solu-  tions maintain their form and phase profile during real  time propagation in the presence of low-amplitude noise,  confirming their stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  The solitonic solution (as it is shown in panels (d) and  (f)) widens in two places.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' Once when n∞ → neq, both  in the standard (n∞ → n+  eq) and anomalous (n∞ → n−  eq)  7  20  10  0  10  20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='0  Density na  (a)  20  10  0  10  20  /2  0  /2  Phase  (c)  5  0  5  0  1  2  (b)  5  0  5  /2  0  /2  (d)  Position xa  1  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' Motionless anomalous soliton density (a) and phase (c) profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' Standard grey soliton density (b) and phase (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' The  grey soliton is moving with relative velocity β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' Both solitons were evaluated at a ratio g↑↓/g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='6 using the mLLGPE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  regimes and another time, while approaching the instabil-  ity region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' The most interesting regime to realise experi-  mentally is in the vicinity of neq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' The solitonic solutions  there are both wide and deep, which may be easier to  detect with in situ imaging procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  Grey solitons also become shallower with decreasing  density n∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' For β > 0, it is a gradual change though (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  panels (c) and (e)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' Another difference is that the grey  soliton width does not diverge when n∞ → neq, it does  so in the vicinity of the unstable regime only (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' panels  (d) and (f)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' 7(a) and (c) we show the density and phase  profiles of motionless solitons evaluated at a ratio g↑↓/g =  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='6 and density fulfilling the inequality nins < n∞ < neq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  This soliton has a non-zero density minimum, normally  characteristic to moving (grey) solitons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' Moreover, there  is no π-phase jump, as in a standard motionless (black)  solitonic solution in the GPE [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  On the other hand, when n∞ > neq, no anomalous  solutions are found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' In this regime, solitons are similar  to standard dark solitons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' We show density and phase  profiles of a grey soliton moving with velocity β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='5 in  panels (b) and (d) of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  To gain some insight into the large width of the soli-  tons when n∞ ≈ neq, we shall consider again a homoge-  neous gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' We can define the pressure as P = −dE/dL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  Above the value of neq, the pressure is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  But  below the equilibrium density, the pressure becomes neg-  ative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' Thus, if we break the symmetry in the system by  rarefying the density in one point, the pressure will make  the gas on the sides of the defect contract and form a  structure with wide density depletion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  SUMMARY  To conclude, we have presented a QMC-based single-  orbital density functional theory for a two-component  bosonic mixture in one dimension, which we call the  mLLGP model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' From construction, our approach pro-  vides a quantitative agreement in terms of the energy  and chemical potential of a homogeneous state with the  ab initio QMC model from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  We benchmark our equation by comparing the results  with the original QMC data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' This comparison shows the  mLLGP can quantitatively predict the bulk density of a  quantum droplet and the monopole mode frequency in  the limit of a large number of particles in the droplet  with a characteristic ω ∝ N −1 dependency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' It also pre-  dicts a correct phase diagram of Bose-Bose mixtures, in-  cluding a transition from liquid to gas, not predicted by  the mean-field model supplemented with the LHY cor-  rection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' Since our approach relies on fitting an energy  functional to QMC data, it is limited by the range of  underpinning QMC data, which is only currently avail-  able in the literature for densities close to the equilibrium  density and for specific interaction ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' Should QMC  data become available over a larger parameter space of  density and interaction ratio, the model could be refined  with an improved energy functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  Our work is limited to the specific case where the in-  traspecies interactions are equal, g↓↓ = g↑↑, which leads  8  to the density profile of each component being equal  to each other, n↓(x) = n↑(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  In principle the ap-  proach could be extended to the more general case where  g↓↓ ̸= g↑↑ and n↓(x) ̸= n↑(x), however this would re-  quire QMC data over a wider parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' Given  the computational intensity of QMC calculations, this is  not tractable at the present time but may become possi-  ble in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  Lastly, we provide a brief study of solitonic solutions of  the mLLGP equation, where we find ultrawide solitonic  solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' Moreover, anomalous motionless solitons were  found as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' These solitons are characterized by the  lack of a π-jump in the phase and a non-zero density  minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  The presence of such wide solitons can be an advantage  for experimenters who would like to perform an in situ  imaging of these objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  As far as we are concerned,  the measurement of the monopole mode frequency for  small droplets may be helpful to verify the validity of the  mLLGP equation too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' It would demand creating droplets  consisting of very few particles, though, making such an  experiment tougher to design and conduct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' An avenue  for further work would be to use the mLLGP model to  study the dynamical properties of dark solitons in 1D  Bose-Bose mixtures, particularly the anomalous solitons,  including their collisions, stability and experimental gen-  eration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  Data availability — All the numerical data necessary to  reproduce figures, including QMC data and the results of  simulations with the MUDGE toolkit (https://gitlab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  com/jakkop/mudge/-/releases/v21Dec2022) are avail-  able in the Supplemental Material under the link [URL  will be inserted by publisher].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  ACKNOWLEDGMENTS  The authors acknowledge discussions with Dr Thomas  Billam and Mr Thomas Flynn (Newcastle University).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' acknowledge support from the UK Engi-  neering and Physical Sciences Research Council (Grant  No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' EP/T015241/1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' acknowledge sup-  port from the (Polish) National Science Center Grant  No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' 2019/34/E/ST2/00289.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  Center for Theoretical Physics of the Polish Academy  of Sciences is a member of the National Laboratory of  Atomic, Molecular and Optical Physics (KL FAMO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' prepared the energy density functional, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' con-  ducted the numerical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' con-  ceptualized and supervised the research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' wrote the  manuscript with input of all authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  Appendix A:  Details of the numerical procedures  Energy density functional  In  order  to  find  the  energy  density  func-  tional  EmLLGP[n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' g, g↑↓],  we  use  the  QMC  data  from  Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  [16],  namely  the  energy  per  particle  E/N ≡ e(n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' g, g↑↓) for the following interaction ratios  g↑↓/g  =  {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='45, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='6, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='65, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='7, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='75, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='8, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='9}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  The data are extrapolated in the low density limit with  a function fL(n) = −1 + c1n3/2 + c2n5/2 + c3n3 and  fH(n) = c4n1/2 +c5n+c6n3/2 [in units of εb/2], where ci  for i = {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' , 6} are constants to be fitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' Then, we  perform a spline interpolation of the augmented QMC  data and perform a linear interpolation between the  ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  The energy density functional is connected to  the energy per particle function e(n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' g, g↑↓) via a simple  relation: EmLLGP[n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' g, g↑↓] = ne(n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' g, g↑↓).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  Imaginary and real time evolution details  The mLLGP equation is a complex, nonlinear partial  differential equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' The orbital ψ(x) is discretized on  a spatial mesh with Nx fixed points and lattice spacing  DX = L/Nx, where L is the box size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' We assume peri-  odic boundary conditions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' ψ(−L/2) = ψ(L/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' The  real-time evolution is done with the use of the split-step  numerical method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' The evolution with the kinetic term  is done in the momentum domain, whereas the contact  interaction term is calculated in the spatial domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' No  external potential is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' The quantum droplet is ob-  tained with the use of imaginary time evolution, where  we use Wick rotation t → −iτ to find the ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  The program written in C++ implementing the algo-  rithm above is publicly available (see Data availability  for link).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' The program uses the W-DATA format dedi-  cated to store data in numerical experiments with ultra-  cold Bose and Fermi gases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' The W-DATA project is a  part of the W-SLDA toolkit [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  When measuring the monopole mode frequency ω, we  perturb the ground state by multiplying it by a fac-  tor exp(−iϵx2/a2), where ϵ is of the order of 10−6 in  our simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  Afterwards, we fit the droplet width  �  ⟨x2⟩ − ⟨x⟩2(t) to a function f(t) = A + B cos(ωt + C),  where A, B, C, and ω are fitted constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  In order to estimate the uncertainty due to the quality  of the energy density functional, we repeat the simula-  tions with alternative spline representations of EmLLGP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  Namely we reduce the number of points we use to extrap-  olate the data with fL(n) and redo the whole procedure  with a slightly different spline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  Appendix B:  Dark solitons in the mLLGP equation  To find the dark solitonic solutions of the mLLGP  equation, we go to the thermodynamic limit, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' L →  ∞, N → ∞ and N/L = const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' We plug the following  Ansatz for a wave travelling through the system at a con-  stant velocity vs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' ψ(x, t) = ˜ψ(ζ), where ζ = x − vst  is a comoving coordinate, to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' (3) and obtain  µs ˜ψ−imvs ˜ψ′ = − ℏ2  2m  ˜ψ′′+µmLLGP  �  | ˜ψ|2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' g, g↑↓  �  ˜ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' (B1)  9  If we assume that far away from the soliton, the den-  sity and phase are constant limζ→∞ | ˜ψ(ζ)|2 = n∞ and  limζ→∞ arg ˜ψ(ζ) = ϕ∞, we can find the value of the  chemical potential µs = µmLLGP [n∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' g, g↑↓].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content=' Then, we  rewrite the equation above in a discretized form, assum-  ing that we start from two points far away from the soli-  ton ˜ψ0 = (1 − ϵ1)√n∞ and ˜ψ1 = (1 − ϵ2)√n∞ with  ϵ1,2 ≪ 1 (typically ∼ 10−4) and ϵ1 > ϵ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
+page_content='  We have also checked that solitonic solutions are dy-  namically stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfQwn7/content/2301.04417v1.pdf'}
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+ 
+Abstract—Network is composed of logical nodes and edges for 
+communications. Atomistic component of things connected to the 
+network is a memory chip. Accordingly, the unique linkage of a 
+memory chip and a logical node can be a promising to resolve the 
+root-of-trust problem on the Internet-of-Things. For this aim, we 
+propose a protocol of challenge-response using a memory chip.  
+For the central management, a central node controls the entry 
+of electronic appliances with a memory chip into the network, and 
+excludes a fake node (e.g., the spoofing entity) from the network 
+that 
+the 
+central 
+node 
+manages. 
+For 
+the 
+decentralized 
+communications, Merkle’s tree turns out being composed of 
+memory chips to which the logical nodes are uniquely linked, 
+respectively. The root of Merkle turns out being the memory chip 
+that stores the latest record of data transaction. We can register 
+this memory chip as a new block by satisfying the requirement of 
+the proof-of-consensus. After blocks are chained, it gets harder for 
+even the central node to manipulate transaction record among 
+memory chips. By this way, the decentralized system (e.g., 
+blockchain) and the central management can coexist.  A new idea 
+of security state is also discussed briefly.  
+Index Terms—Blockchain, Merkle’s tree, central management, 
+decentralized system, memory chip, IC chip.  
+I. INTRODUCTION 
+Decentralized systems (e.g., blockchain) can maximize the 
+value of the network applications. By uploading hardware to the 
+network, the Internet-of-Things (IoT) can heighten the value of 
+the network applications. However, the entry of hardware to the 
+network should be under the central management for the 
+authentication reason. The decentralized system and the central 
+management should therefore coexist in the IoT network, 
+though it has not been discussed enough.  
+What is the upload of hardware to the network? --- The 
+network comprises logical nodes and communication edges 
+(lines) connecting logical nodes. In general terms, the upload of 
+hardware to the network is to uniquely link a physical entity to 
+a logical node. The uploaded physical entity is a physical node.  
+It is an electronic appliance whose atomistic component is an 
+integrated circuit (IC) chip, in particular, a memory chip in the 
+Neumann-type computers. The falsification of the linkage of a 
+physical node and a logical node is the spoofing. The proof of 
+no spoofing is the root-of-trust. The upload of hardware (i.e., 
+the linkage of a physical node and a logical node) without the 
+root-of-trust causes a kind of oracle problem. Because one may 
+be forced to trust it with no proof and any existing Blockchain 
+cannot resolve it. A practical reinforcement of Blockchain is 
+therefore necessary to use Blockchain on the IoT network.  
+A logical node is allocated with an address on the network. 
+This address is public on the network, such that an arbitral entry 
+in the network can know it to reach the logical node. Following 
+the concept of Diffie-Hellman [1], we can think that a public 
+key can play a role of (public) address on the network. Allice 
+has distributed her public key on the network. Anyone who 
+entries to the network can receive her public key. Both Bob and 
+Mike can encrypt their messages by using Allice’s public key. 
+They can distribute (or expose) their messages to her on the 
+network. Any entry can receive those messages but only Allice 
+can read the messages by decrypting them using her secret key 
+that is not distributed on the network. While only Allice can 
+read the messages, her public key can play a role of Allice’s 
+address on the network, which is public on the network. Like 
+this, the atomistic component of a physical node linked to a 
+logical address is a memory chip and a public key can play a 
+role of address of the logical node (named, logical address). The 
+root-of-trust is therefore the proof of the unique linkage of a 
+memory chip and a public key.  
+Data can be transferred from a logical node to another, which 
+data transfer can be denoted by an arrow. Suppose that a logical 
+node can receive data transferred from plural logical nodes. 
+Plural data transfers result in a tree diagram, whose root is the 
+destination having all transfer records as well as the latest one. 
+This is called Merkle’s tree and its root is called the root of 
+Merkle  [2]. If we can prove the unique linkage of a memory 
+chip and a logical node (represented by its public key) in some 
+way, then it turns out being Merkle’s tree of memory chips. The 
+Merkle root is thus a memory chip which stores the latest 
+transaction record of data transfer. In the network, miners look 
+for a root of Merkle (represented by its public key) and then 
+register it as a new block by satisfying the requirement for the 
+proof-of-consensus. Accordingly, the registered new block is a 
+memory chip having the latest transaction record and uniquely 
+linked to the public key of the registered root of Merkle. By 
+repeating this procedure, plural blocks are serially registered to 
+construct the blockchain of memory chips with the root-of-trust 
+(i.e., the unique linkage of memory chips and public keys).  
+A central node can control the upload of an electronic 
+appliance (hardware) with a memory chip to the network. A 
+logical node uniquely linked to this memory chip is under the 
+central management by the central node. If the central node 
+permits the upload of this electronic appliance, then the logical 
+node uniquely linked to the memory chip mounted in this 
+electronic appliance is permitted to entry to the network. If the 
+central node denies the upload of this electronic appliance, then 
+the logical node uniquely linked to the memory chip mounted 
+in this electronic appliance is denied to entry to the network. 
+But, after the blocks of memory chips are chained, the central 
+node can hardly manipulate transaction record in the blockchain 
+of memory chips that the central node has permitted to entry. 
+Because it must be necessary to redo the proof-of-consensus for 
+the chained blocks [3]. As the blockchain length increases, it 
+On coexistence of decentralized system (blockchain) and 
+central management in Internet-of-Things 
+Hiroshi Watanabe, National Yang Ming Chiao Tung University, hwhpnabe@gmail.com  
+
+ 
+becomes harder. That is, the central node can control the entry 
+of an electronic appliance (represented by its memory chip) to 
+the network but can hardly manipulate the transaction record 
+among memory chips that the central node has permitted to 
+entry. By this way, the decentralized system and the central 
+management can coexist with the root-of-trust (i.e., the unique 
+linkage of memory chips and public keys).  
+In this work, we illustrate a conceptual solution for this in as 
+a clear manner as possible. In II, we describe the method to 
+realize the unique linkage of memory chips and public keys. In 
+III, we illustrate the blockchain of memory chips. IV and V are 
+devoted to discussion and summary, respectively.  
+II. UNIQUE LINKAGE OF MEMORY CHIPS AND PUBLIC KEYS 
+This can be realized by the challenge-response protocol 
+(CRP) using a memory chip and a physical random number 
+(PRN) which is specific to the memory chip [4]. Fig. 1 
+illustrates the challenge-response (a) without and (b) with the 
+spoofing. The connected device A inspects the connected 
+device B by asking “Hey B, who are you?" (Challenge C). In 
+(a), without the spoofing, the connected device B replies to the 
+connected device A, “I am logical address B” (Response R). 
+However, without the root-of-trust, we are unsure if the 
+connected device B and the logical address B are really linked. 
+In (b), a hacker spoofs the logical address B, to "Hey B, who 
+are you?” (Challenge C), the response is “I am logical address 
+B” (Response R). That is, the CRP without the root-of-trust 
+makes nonsense in the IoT network.  
+A. Root-of-Trust using a memory chip 
+First, suppose that a public key can play a role of a logical 
+address B. Next, suppose that a secret key is irreversibly 
+generated from a code specific to a memory chip which is an 
+atomistic component of a physical node (e.g., connected device 
+B). Using some algorithm for the public key infrastructure 
+(PKI), we can uniquely link the secret and public keys. Using 
+this specific code and the PKI, we can uniquely link the memory 
+chip and the logical address B.  
+In general, we can obtain the response (�����) from the 
+physical random number of chip ( � ), ������ , and the 
+challenge (C) using a function, �, as follows.    
+����� = ���, ������� 
+ 
+(1) 
+In Fig. 2, the connected device A sends the challenge (C) 
+“Hey B, who are you?” to the memory chip (�) having ��� ��� 
+of the connected device B. The response (R) generated using 
+(1) is “I am chip B.” If the connected device B was spoofed by 
+hacker’s laptop, the response would be “I am a chip in hacker’s 
+laptop”. The CRP using a memory chip makes sense for the 
+cyber-attacks. Because no cyber-attack can tear off memory 
+chip from the motherboard of the connected device B.  
+B. PRN and PKI 
+We can generate one or two prime number(s) from this 
+����� in some way. Thus, we can generate a “uniquely linked” 
+pair of secret and public keys using an algorithm of PKI --- RSA 
+[5], ElGamal [6], etc. Therefore, the secret key (SK) and public 
+key (PK) of chip (�) are respectively written as follows.    
+������ = �������� 
+(2) 
+������ = �̅������� 
+(3) 
+The ��, �̅� are the key generation functions. It is preferable that 
+prime numbers are great enough. The (3) connects chip (�) and 
+public key ������ through (1). Below we evaluate randomness 
+to validate if ��� ��� is specific to chip (�) in practice.  
+C. A concrete example of retrieving PRN from memory chips 
+Fig. 3 illustrates the redundancy mechanism of a memory 
+chip [7]. There are several blocks on chip. Each block is 
+composed of many integrated memory cells (corresponding to 
+bits), which are arrayed in the X (Row)-Y (Column) plane. It is 
+further divided into two cell arrays (redundancy and regular 
+arrays). The Y-decoders A and B control the row access in the 
+redundancy and regular arrays, respectively. An access code is 
+input to a peripheral circuit to choose either Y-decoder A or B 
+or both. Since memory chip is a mass-product, it is impossible 
+ 
+ 
+Fig. 2. CRP with physical randomness specific to a memory 
+chip.  
+Logical address B
+Logical address A
+Logical address C
+Connected 
+device B
+Connected 
+device C
+Connected 
+device A
+Hey, B
+Who are you?
+Secret
+Key
+1-to-1
+Public 
+Key
+Memory 
+Chip
+PRN (n)
+Challenge (C)
+I am Chip B.
+ 
+(a) 
+ 
+ 
+(b) 
+Fig. 1. Challenge-Response Protocol. (a) without spoofing. 
+(b) with spoofing. 
+Logical address B
+Logical address A
+Logical address C
+Cybernetwork 
+(logical address to 
+logical address)
+Connected 
+device B
+Connected 
+device C
+Connected 
+device A
+IoT network 
+(device to device)
+Hey B,
+Who are you?
+I am logical address B.
+Logical address B
+Logical address A
+Logical address C
+Cybernetwork 
+(logical address to 
+logical address)
+Connected 
+device B
+Connected 
+device C
+Connected 
+device A
+IoT network 
+(device to device)
+Hey B, 
+Who are you?
+spoofing
+Laptop
+I am logical address B.
+
+ 
+to exclude all failures from the cell arrays perfectly. Suppose 
+there are plural failure bits in the regular array, wherein the row 
+number with a failure bit is denoted by �� for � = 1, 2, … � with 
+� being the number of rows with a failure bit in the regular 
+array. While this � is small enough, that is, the failure is 
+controllable under the manufacturing specification, we can 
+apparently exclude the failure bits from the memory access 
+operations (i.e., read, write, and erase in the regular array). 
+Inputting an access code into the peripheral circuit, we can 
+choose a normal access mode to use only Y-decoder B. We, 
+thus, access the addresses from the top of the regular array along 
+a column (�) chosen by X-decoder. When arriving at ��, we 
+swap the bit with another bit on the row number ��  in the 
+redundancy array. If the bit at the address (��, �) is not a failure 
+bit, we can apparently exclude the failure bit at (��, �) from the 
+memory access operations in the regular array. In usual, the 
+users of DRAM can access only the regular array. As � 
+decreases, we can reduce the necessary number of rows in the 
+redundancy array. It can suppress the probability that a failure 
+bit is found in the redundancy array as well. As improving the 
+manufacturing quality, the average of � gets smaller. We can 
+expect that � is small enough in the mass-produced memory 
+chips, but it can hardly be zero. In general, the distribution of 
+rows with a failure bit in the regular array of chip (�), denoted 
+by �����, is out of the manufacturing control. It can cause a 
+physical randomness specific to chip (�). The inner memory 
+stores this ����� to exclude failure bits from the memory access 
+operation. Below we illustrate the method to retrieve ��� ���.  
+Preprocess) We input a normal access code to the peripheral 
+circuit to use Y-decoder B. This is the normal access mode. We 
+write state-0 along the �-th column (chosen in advance) in the 
+regular array. See Fig. 4 (a). Next, inputting a special access 
+code to the peripheral circuit, we change the access mode to the 
+special access mode to use Y-decoder A. Then, we write state-
+1 along the �-th column in the redundancy array. See Fig. 4 (a).  
+Read) In a normal access mode (using Y-decoder B), we read 
+bits along the �-th column in the regular array. By the swapping, 
+readout turns out being state-1 on rows with a failure bit (��) 
+and state-0 on the other rows. See Fig. 4 (b). Thus ����� specific 
+to chip (�) turns out being a physical random number, ������.  
+PRN Stability) Reliability of the inner memory storing ����� 
+determines that of  ������. In general, the inner memory (e.g., 
+a fuse memory) is more stable to the environmental change than 
+cell arrays. In experiment using the dynamical random access 
+memory (DRAM) chips, that is the most popular IC product for 
+the main memory, the excellent characteristics of retention and 
+stability to temperature change were clearly shown [4], [7].  
+Randomness) As an example, we choose 4 Mbits - 16 Gbits 
+(i.e., the old and new generations) DRAM for evaluating the 
+entropy of chip-specific randomness. For the ease of discussion, 
+we assume that � is much smaller than the total row number 
+(� ), and the row and column numbers are the same (i.e., 
+isotropic layout) in each block. The number of blocks on a chip 
+is thus ��. For � = 1, the entire cell array on chip is one block. 
+The � is in the order of 100,000 for 16 Gbits, which is most 
+advanced recently. The randomness can be estimated by 
+calculating the combination product of � and �, i.e., � 
+!
+. For 
+an arbitral � , the randomness can be estimated by 
+� 
+"×!
+. 
+Following Boltzmann, the entropy divided by $% is ln
+� 
+"×!
+, 
+where $% is the Boltzmann constant. For � = 1, although it is 
+an underestimation a little bit, we plot information entropies of 
+PRN from DRAM chips (4 Mbits to 16 Gbits) in Fig. 5. For 
+� = 10, ln
+� 
+"×!
+> 10�* even for 4 Mbits with � = 1 (the 
+worst case in this estimation). If one hundred trillion (10+,) IoT 
+devices will be deployed all over the world, the probability that 
+two chips have a same PRN is less than 10-++ (i.e., 0.00001 
+ppm). Thus, the randomness is good enough in the 4 Mbits – 16 
+Gbits products of DRAM. In addition, a couple of ten rows (≫
+�) in the redundancy array is good enough for swapping to hide 
+most of failure bits from the memory access operations. The 
+redundancy array comprising twenty rows is much narrower 
+than the regular array (20 ≪ 2,000 and 100,000 in 4 Mbits and 
+16 Gbits products, respectively). It does not matter in the rough 
+ 
+Fig. 3. Redundancy mechanism. 
+Redundancy array
+Regular 
+array
+X-direction (Row)
+Y-direction (Column)
+Y-decoder B
+Y-decoder A
+Inner 
+memory
+Peripheral
+circuit
+R1
+R2
+F1
+F2
+Swap-1
+Swap-2
+Failure bit
+Choose column ( )
+Access code
+X-decoder
+ 
+Fig. 4. Retrieve PRN from memory chip. (a) preprocess. 
+(b) read.  
+0
+00
+000
+1
+11
+1
+1
+0
+01
+0
+Chosen column ( )
+Swap-1
+Swap-2
+Redundancy 
+array
+Regular 
+array
+R1
+R2
+F1
+F2
+(a)
+(b)
+Chosen column ( )
+0
+0
+00
+0
+1
+0
+ 
+Fig. 5. Entropy of randomness with memory chips 
+10
+20
+40
+80
+160
+320
+10
+60
+110
+160
+ENTROPY / KB
+ROW NUMBER WITH A FAILURE BIT
+INFORMATION QUANTITY
+16 Gbits
+4 Gbits
+1 Gbits
+256 Mbits
+64 Mbits
+16 Mbits
+4 Mbits
+100 Trillion
+11.36(4)
+
+ 
+estimation of the entropy even though increasing �. Therefore, 
+we can obtain a sufficient randomness to retrieve ��� with no 
+additional area for cell array. It tells us there is no chip area 
+penalty. The present method is feasible to the wide generations 
+of the existing DRAM products (4 Mbits – 16 Gbits). It is self-
+evident that the present method is also feasible to the coming 
+generations beyond 16 Gbits.  
+D. Independencies of chips 
+While the entropy of randomness specific to a memory chip 
+is good enough, we can hardly retrieve a same random number 
+which is specific to two different chips, as discussed above.  
+�����1� 0 �����2� 1ℎ�34 �1 0 �2 
+(4) 
+While satisfying (4), we respectively convert (1) – (3) to:   
+��+��� 0 ������ 1ℎ�34 �1 0 �2 
+(5) 
+���+��� 0 ������� 1ℎ�34 �1 0 �2 
+(6) 
+���+��� 0 ������� 1ℎ�34 �1 0 �2 
+(7) 
+Furthermore, we can write down as follows.  
+����+� 0 ������ 1ℎ�34 �+ 0 �� 
+(8) 
+�����+� 0 ������� 1ℎ�34 �+ 0 �� 
+(9) 
+�����+� 0 ������� 1ℎ�34 �+  0 �� 
+(10) 
+The (5) – (10) characterize functions that are necessary for 
+memory chips to satisfy the root-of-trust (i.e., the unique 
+linkage of memory chips and public keys). In Fig. 6 (a), 
+physical nodes satisfying these configures firewall of things [8].  
+E. Auto-detection and auto-remove (Entry control) 
+Auto-detection and auto-remove of the fake and vulnerable 
+nodes can be performed using (5) – (10) and the smart contract. 
+Details will be discussed elsewhere.   
+F. Security State 
+For each �, we generate response, secret and public keys 
+using a challenge �5, where 3 is a natural number. Regard the 
+generation of them as appearing similar to an observation in 
+quantum mechanics (denoted by �6,  �7 and �6, respectively). To 
+deduce (8) – (10), we can begin with the following equations, 
+respectively.  
+�6�|3⟩ = ����5�|3⟩ 
+(11) 
+�7�|3⟩ = �����5�|3⟩  
+(12) 
+�6�|3⟩ = �����5�|3⟩  
+(13) 
+The ����5�, �����5� and �����5� of chip (�) can describe a 
+security state denoted by |3⟩ (like an eigenstate in quantum 
+mechanics). However, the security state of a physical node (�) 
+can be determined by choosing a challenge �5. Since the state 
+|3⟩ has been chosen before the observation (i.e., the generation 
+of response and secret and public keys), (11) – (13) are closer 
+to Einstein’s hidden variables [9] than Schrödinger’s cat. 
+Anyway, this can confuse adversaries. Fig. 6 (b) illustrates the 
+merit of security state. In the network, another central node 
+(security node) can change response and secret and public keys 
+of any physical nodes by replacing �5  periodically or at his 
+convenience. It is effective when some kind of security 
+problems is found. Details will be discussed elsewhere.  
+The �5 used by a security node should differ from challenge 
+used by a management node. The entry and the security state 
+are controlled by two independent central nodes (management 
+and security nodes, respectively). This reminds us of the 
+separation of three powers in the democracy, even though this 
+is a central control indeed.  
+III. BLOCKCHAIN OF MEMORY CHIPS 
+Most of physical nodes is a Neumann computer which is 
+composed of an input-output (I/O), an arithmetic unit and a 
+memory chip. Data, stored in a memory chip of a physical node, 
+is retrieved and processed by the arithmetic unit and then output 
+ 
+Fig. 7. Merkle tree of memory chips. 
+Root of Merkle (
+)
+Memory 
+Chip
+PRN (
+)
+Memory
+chip
+PRN (
+)
+Memory
+chip
+PRN (
+)
+Memory
+chip
+PRN (
+)
+Memory
+chip
+PRN (
+)
+Memory
+chip
+PRN (
+)
+Memory
+chip
+PRN (
+)
+Memory
+chip
+PRN 
+)
+Memory
+chip
+PRN (
+)
+        
+ 
+(a)                                                                                              (b) 
+ 
+Fig. 6. Central management. (a) by management node. (b) by security node. 
+
+R/W
+Security
+Security
+State
+node
+SmartmeterFirewall of things
+R/W
+Entry
+Management
+certificate
+node
+Smartmeter 
+from I/O to the external. This data is received by I/O of another 
+physical node and then processed by the arithmetic unit and 
+then stored in the memory chip of the receiving physical node. 
+That is, data is really transferred between memory chips.  
+A. Merkle tree of memory chips  
+In Fig. 7, we suppose that data stored in chip (�0) has been 
+transferred from chips (�1, �4, and �6) with recording the 
+updated history of data transfer to chip (�0). Each data transfer 
+is modeled in Fig. 8, which illustrates that from node (� < 1) to 
+node (�). In chip (�), the ����5� having been obtained from �5 
+and ������ at security state |3⟩ according to (1) is used to 
+generate �����5� and �����5� using (2) and (3), respectively. 
+While (4) is satisfied, we regard (5) – (13) as all satisfied. The 
+chip-specific ����5� is related to �����5�. The unique linkage 
+of memory chip and public key is thus assured by the unique 
+linkage of secret and public keys that are generated using a 
+same chip-specific ����5�. We can regard logical node (�) as 
+equal to physical node (�) on the network. We can combine 
+them to call it as node (�), if not specially noted.   
+In the node (� < 1), the ���-+��5�,  hash value (� < 2), and 
+electronic signature (� < 2) are converted to a new hash value 
+(� < 1) using a hash function (e.g., SHA256). Subsequently, 
+the node (� < 1) gets �����5� on the network. (Public key is 
+public on the network.) The got public key and the generated 
+hash value ( � < 1 ) are encrypted to be a new electronic 
+signature (� < 1) using ���-+��5� . The node (� < 1) sends the 
+hash value (� < 1) and the electronic signature (� < 1) to the 
+node (�). This transfer is as same as used in the existing 
+blockchain if we exclude memory chips from the illustration 
+(drawn in red). That is, our solution for the unique linkage of 
+memory chips and public keys is fully compatible to the 
+existing blockchain.  
+Move back to Fig. 7. Data stored in chip (�1) has been 
+transferred from chips (�2 and �3) with recording the updated 
+transferring history in chip (�1). Data stored in chip (�4) has 
+been transferred from chip (�5) with recording the updated 
+transferring history therein. Data stored in chip (�6) has been 
+transferred from chips (�7 and �8) with recording the updated 
+transferring history therein. It turns out being a tree of Merkle. 
+There is chip (�0) at the bottom, named, the Merkle root (�0), 
+with recording the entire history of data transfers from plural 
+chips to the root chip (�0).  
+B. Blockchain of memory chips 
+In Fig. 9, we follow the conventional mining process to chain 
+blocks. A miner can register the newest block hash (A) by 
+hashing an added nonce (A), a stamp of the root chip (�0) (e.g., 
+���B��5�, hash value (�0 < 1), etc.), and the previous block 
+hash (A < 1), such that the newly generated block hash (A) 
+satisfies the requirement for the poof-of-consensus by tuning 
+nonce (A). The block hash (A < 1) was registered by hashing an 
+added nonce (A < 1), a stamp of the root node (�00) (e.g., 
+���BB��5�, hash value (�00 < 1), etc.), and the previous block 
+hash (A < 2), so that the block hash (A < 1) satisfies the 
+requirement for the poof-of-consensus by tuning nonce (A < 1). 
+Repeating this procedure, the miners have constructed the 
+blockchain of memory chips.  
+There are plenty of diverse types of electronic appliances 
+(physical nodes) --- routers, PCs, smartphones, card readers, 
+RFID reader-writers (R/Ws), surveillance cameras, industrial 
+robots, smart meters, printers etc. in Fig. 9. However, DRAM 
+is a general-purpose memory and installed in most of electronic 
+appliances. In Fig. 10, any nodes are approved to communicate 
+each other without being monitored by any central nodes inside 
+the firewall of things [8]. One of central nodes (management 
+node) controls the entry of physical nodes to the network and 
+prohibits the spoofing by a hacker’s laptop with non-registered 
+chip (see Fig. 1 (b) as well). Another central node (security node) 
+controls security state. The unique linkage of memory chips and 
+public keys is fully compatible to the blockchain construction. 
+ 
+Fig. 9. Blockchain of memory chips. 
+ 
+Fig. 10. The coexistence of central management and 
+decentralized system (blockchain).  
+ 
+Fig. 8. Data transaction. See it by removing red ones.  
+
+Firewall of things (Hardware
+Laptop
+Blockchain
+Non-registered
+chip
+Smartmeter
+spoxfing
+R/W
+Entry
+Management
+node
+Security
+state
+Security
+nodeR/W
+Smartmeter
+PRN (n000)
+PRN (n00)
+PRN (n0)
+Memory
+Memory
+Memory
+Chip
+Chip
+Chip
+Nonce (r - 2)
+Nonce (r - 1)
+Nonce (r)
+Block hash (7 - 3)
+Blockhash (n-2)
+Block hash (r.- 1)
+Block hash (r)
+Block (r- 2)
+Block (r - 1)
+Block (r)Chip (n - 1)
+Chip (n)
+Physical node
+SKn-1(C)
+SKn(C)
+Key
+Key
+Rn-1(C)
+R,(C)
+Generator
+Generator
+PKn-1(C)
+PKn(C)
+Logical node
+Hash value
+Hash value
+(n - 2)
+(n - 1)
+Electronic Signature
+Electronic Signature
+(n 2)
+(n - 1)
+Node(n - 1)
+Node(n) 
+Blockchain protects communications in the firewall of things. 
+This is the decentralized communications under central control.  
+C. Resilient Blockchain of things 
+There is no risk of mechanical failures in logical accounts. 
+But, in the IoT network, there is the risk of mechanical troubles. 
+If we replace a defected device having a memory chip (�C) by 
+a new device having a memory chip (�D) in the blockchain of 
+memory chips, the response and public and secret keys will 
+change since �����C� and �����D� differ. It requires the 
+reproduction of the Merkle root using the entire tree. Another 
+case to require the reproduction of the Merkle root is the change 
+of security state for some security reason.  
+However, we can follow the idea of Beyond Blockchain one 
+(BBc-1) [10], which can be regarded as resilient blockchain. 
+Only limited subtrees are good enough to reproduce the Merkle 
+root. It can reduce the reproduction cost. The reproduced tree 
+can be registered as a new block after the proof-of-consensus.  
+IV. DISCUSSION 
+Since the version of security of IoT devices vary greatly from 
+old to new, a hacker may get an authenticate information of a 
+vulnerable IoT device. Any advanced security parts (PUC, 
+protected memory [11] etc.) cannot protect old IoT devices 
+having been shipped with the security parts in the former 
+generations or with no security parts.  
+The present method can be an application software, which 
+retrieves PRN from existing IC chips in the distributed IoT 
+devices. It was demonstrated that (4) can be satisfied within the 
+quality control of mass-produced DRAM chips (4 Mbits – 16 
+Gbits), with no chip area penalty. Most of existing IoT devices 
+(old and new) mounts a DRAM chip with the bit capacity being 
+4 Mbits – 16 Gbits. To adopt the present method, we may install 
+the application software and not replace old IoT devices by new 
+ones with the advanced security parts (PUC etc.).   
+Smartphone suppliers can construct the unique linkage of 
+smartphones and public keys using the present method. This is 
+the firewall of smartphones. End users are required to download 
+an application to retrieve PRN from DRAM chip existing in 
+their smartphones but not to replace their old smartphones with 
+new ones so that their smartphones can enter into the firewall.  
+In the industrial manufacturing, the supervisory control and 
+data acquisition (SCADA) is collecting the attention. In the 
+semiconductor manufacturing, for example, a medium class 
+factory with ~40,000 wafers per month has about 3,000 units 
+of equipment. If each unit has 100 parameters in average, then 
+about 300,000 parameters are to be optimized even in a medium 
+class factory. Twice or more in a huge class factory. In the 
+SCADA, those parameters are collected from plural smart 
+factories by IoT sensors. A remote facility of AI dynamically 
+and consistently optimizes data (parameters) collected from the 
+plural smart factories. Therefore, there is an increasing risk that 
+the cyber-attack will upset the global supply chain and theft 
+technical know-how. Since each IoT sensor has a DRAM chip, 
+we can construct the unique linkage of IoT sensors and public 
+keys over the remote-controlled smart factories distributed 
+globally. This is the firewall of IoT sensors and industrial robots.  
+In the logistics control, the security of RFID tags has been 
+discussed. However, a spoofed R/W can deliver parcels to an 
+illegal destination. Thus, we propose the unique linkage of 
+R/Ws and public keys [12]. This is the firewall of R/Ws.  
+For the traceability in the supply chain of IC chips, we can 
+construct the firewall of IC chips using the present method.  
+V. SUMMARY 
+Taking into account the quality control of mass-produced 
+memory chips, i.e., atomistic component of IoT devices, we can 
+construct the unique linkage of memory chips and public keys 
+with a full compatibility to the existing blockchains. The central 
+management of IoT devices and decentralized communications 
+among IoT devices can therefore coexist. The present method 
+is valid to various IC products as well as DRAM. Application 
+range is pretty-wide and not limited to discussed here. 
+ACKNOWLEDGMENT 
+The author thanks to T. Hamamoto, Y. Nagai, J. Liang, M. 
+Chang, E. Tseng, J. Moon, K. Saito, K. Taniguchi, S. Torisawa, 
+T. Kato, KY Tsai, J. Chen, L. Chang, Y. Hirota, A. Kinoshita 
+for valuable and stimulating discussions, and T. Okada, S. 
+Miyazaki, and H. Fukuyama for support on experiment.  
+REFERENCES 
+[1] 
+W. Diffie and M. E. Hellman, "New Directions in Cryptography", IEEE 
+Tran. Information Theory, vol.IT-22, No.6, pp.644-654, 1976. 
+[2] 
+R.C. Merkle, "Protocols for public key cryptosystems," In Proc. 1980 
+Symposium on Security and Privacy, IEEE Computer Society, pages 
+122-133, April 1980. 
+[3] 
+Anonymous under the name of Satoshi Nakamoto, “Bitcoin: A Peer-To-
+Peer Electronic Cash System”, https://bitcoin.org/bitcoin.pdf. 
+[4] 
+H. Watanabe, “Can Blockchain protect Internet-of-Things?”, Future 
+Technologies Conference (FTC) 2017, Vancouver, 29-30, Nov. 2017.  
+[5] 
+Rivest, Ronald L.; Shamir, Adi; Adelman, Len M. (1977-07-04), “A 
+Method for Obtaining Digital Signature and Public-key Cryptsystems”, 
+MIT-LCS-TM-082 (MIT Laboratory for Computer Science).  
+[6] 
+T. Elgamal, “A Public-Key Cryptosystem and a Signature Scheme 
+Based on Discrete Logarithms”, IEEE Transactions on Information 
+Theory, v. IT-31, n. 4, 1985, pp. 469-472.  
+[7] 
+H. Watanabe, T. Hamamoto, “The cheapest physical chip-ID fully 
+compatible to mass-product DRAM process”, Proc. AWAD, 399, 
+Gyeongju, Korea, July 3-5, 2017.  
+[8] 
+H. Watanabe, “Firewall of Things, Firewall of Memory Chips”, Flash 
+Memory Summit 2022, Aug. 2-4, Santa Clara, 2022.  
+[9] 
+A. Einstein, B. Podolsky and N. Rosen, “Can quantum-mechanical 
+description of physical reality be considered complete?”, Phys. Rev. 47, 
+pp. 777 – 780, 1935.  
+[10] 
+K. Saito and T. Kubo, “BBc-1: Beyond Blockchain One – An 
+architecture for promise-fixation device in the air --”, white paper 
+(2017) on https://beyond-blockchain.org; 
+https://github.com/beyond-blockchain/bbc1/blob/develop/docs/BBc-
+1%5C_design%5C_paper.pdf  
+[11] 
+C. P. Gorog, “cryptographic trust enabled devices of cybersecurity 
+systems”, WO/2022/072609.  
+[12] 
+H. Watanabe, K. Saito, S. Miyazaki, T. Okada, H. Fukuyama, T. Kato, 
+and K. Taniguchi, “Proof of authenticity of logistics information with 
+passive RFID tags and blockchain”, the 2021 IEEE International 
+Conference on Electronic Communications, Internet of Things and Big 
+Data (ICEIB 2021), Yilan, Taiwan, Dec. 10-12, 2021. 
+
diff --git a/BNAzT4oBgHgl3EQfTfw1/content/tmp_files/load_file.txt b/BNAzT4oBgHgl3EQfTfw1/content/tmp_files/load_file.txt
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@@ -0,0 +1,437 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf,len=436
+page_content='Abstract—Network is composed of logical nodes and edges for  communications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' Atomistic component of things connected to the  network is a memory chip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' Accordingly, the unique linkage of a  memory chip and a logical node can be a promising to resolve the  root-of-trust problem on the Internet-of-Things.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' For this aim, we  propose a protocol of challenge-response using a memory chip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='  For the central management, a central node controls the entry  of electronic appliances with a memory chip into the network, and  excludes a fake node (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=', the spoofing entity) from the network  that  the  central  node  manages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='  For  the  decentralized  communications, Merkle’s tree turns out being composed of  memory chips to which the logical nodes are uniquely linked,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' The root of Merkle turns out being the memory chip  that stores the latest record of data transaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' We can register  this memory chip as a new block by satisfying the requirement of  the proof-of-consensus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' After blocks are chained, it gets harder for  even the central node to manipulate transaction record among  memory chips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' By this way, the decentralized system (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=',  blockchain) and the central management can coexist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' A new idea  of security state is also discussed briefly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='  Index Terms—Blockchain, Merkle’s tree, central management,  decentralized system, memory chip, IC chip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' INTRODUCTION  Decentralized systems (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=', blockchain) can maximize the  value of the network applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' By uploading hardware to the  network, the Internet-of-Things (IoT) can heighten the value of  the network applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' However, the entry of hardware to the  network should be under the central management for the  authentication reason.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' The decentralized system and the central  management should therefore coexist in the IoT network,  though it has not been discussed enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='  What is the upload of hardware to the network?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' --- The  network comprises logical nodes and communication edges  (lines) connecting logical nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' In general terms, the upload of  hardware to the network is to uniquely link a physical entity to  a logical node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' The uploaded physical entity is a physical node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='  It is an electronic appliance whose atomistic component is an  integrated circuit (IC) chip, in particular, a memory chip in the  Neumann-type computers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' The falsification of the linkage of a  physical node and a logical node is the spoofing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' The proof of  no spoofing is the root-of-trust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' The upload of hardware (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=',  the linkage of a physical node and a logical node) without the  root-of-trust causes a kind of oracle problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' Because one may  be forced to trust it with no proof and any existing Blockchain  cannot resolve it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' A practical reinforcement of Blockchain is  therefore necessary to use Blockchain on the IoT network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='  A logical node is allocated with an address on the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='  This address is public on the network, such that an arbitral entry  in the network can know it to reach the logical node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' Following  the concept of Diffie-Hellman [1], we can think that a public  key can play a role of (public) address on the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' Allice  has distributed her public key on the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' Anyone who  entries to the network can receive her public key.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' Both Bob and  Mike can encrypt their messages by using Allice’s public key.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='  They can distribute (or expose) their messages to her on the  network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' Any entry can receive those messages but only Allice  can read the messages by decrypting them using her secret key  that is not distributed on the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' While only Allice can  read the messages, her public key can play a role of Allice’s  address on the network, which is public on the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' Like  this, the atomistic component of a physical node linked to a  logical address is a memory chip and a public key can play a  role of address of the logical node (named, logical address).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' The  root-of-trust is therefore the proof of the unique linkage of a  memory chip and a public key.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='  Data can be transferred from a logical node to another, which  data transfer can be denoted by an arrow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' Suppose that a logical  node can receive data transferred from plural logical nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='  Plural data transfers result in a tree diagram, whose root is the  destination having all transfer records as well as the latest one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='  This is called Merkle’s tree and its root is called the root of  Merkle  [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' If we can prove the unique linkage of a memory  chip and a logical node (represented by its public key) in some  way, then it turns out being Merkle’s tree of memory chips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' The  Merkle root is thus a memory chip which stores the latest  transaction record of data transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' In the network, miners look  for a root of Merkle (represented by its public key) and then  register it as a new block by satisfying the requirement for the  proof-of-consensus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' Accordingly, the registered new block is a  memory chip having the latest transaction record and uniquely  linked to the public key of the registered root of Merkle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' By  repeating this procedure, plural blocks are serially registered to  construct the blockchain of memory chips with the root-of-trust  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=', the unique linkage of memory chips and public keys).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='  A central node can control the upload of an electronic  appliance (hardware) with a memory chip to the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' A  logical node uniquely linked to this memory chip is under the  central management by the central node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' If the central node  permits the upload of this electronic appliance, then the logical  node uniquely linked to the memory chip mounted in this  electronic appliance is permitted to entry to the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' If the  central node denies the upload of this electronic appliance, then  the logical node uniquely linked to the memory chip mounted  in this electronic appliance is denied to entry to the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='  But, after the blocks of memory chips are chained, the central  node can hardly manipulate transaction record in the blockchain  of memory chips that the central node has permitted to entry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='  Because it must be necessary to redo the proof-of-consensus for  the chained blocks [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' As the blockchain length increases, it  On coexistence of decentralized system (blockchain) and  central management in Internet-of-Things  Hiroshi Watanabe, National Yang Ming Chiao Tung University, hwhpnabe@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='com  becomes harder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' That is, the central node can control the entry  of an electronic appliance (represented by its memory chip) to  the network but can hardly manipulate the transaction record  among memory chips that the central node has permitted to  entry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' By this way, the decentralized system and the central  management can coexist with the root-of-trust (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=', the unique  linkage of memory chips and public keys).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='  In this work, we illustrate a conceptual solution for this in as  a clear manner as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' In II, we describe the method to  realize the unique linkage of memory chips and public keys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' In  III, we illustrate the blockchain of memory chips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' IV and V are  devoted to discussion and summary, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' UNIQUE LINKAGE OF MEMORY CHIPS AND PUBLIC KEYS  This can be realized by the challenge-response protocol  (CRP) using a memory chip and a physical random number  (PRN) which is specific to the memory chip [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' 1  illustrates the challenge-response (a) without and (b) with the  spoofing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' The connected device A inspects the connected  device B by asking “Hey B, who are you?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='" (Challenge C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' In  (a), without the spoofing, the connected device B replies to the  connected device A, “I am logical address B” (Response R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='  However, without the root-of-trust, we are unsure if the  connected device B and the logical address B are really linked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='  In (b), a hacker spoofs the logical address B, to "Hey B, who  are you?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' (Challenge C), the response is “I am logical address  B” (Response R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' That is, the CRP without the root-of-trust  makes nonsense in the IoT network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' Root-of-Trust using a memory chip  First, suppose that a public key can play a role of a logical  address B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' Next, suppose that a secret key is irreversibly  generated from a code specific to a memory chip which is an  atomistic component of a physical node (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=', connected device  B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' Using some algorithm for the public key infrastructure  (PKI), we can uniquely link the secret and public keys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' Using  this specific code and the PKI, we can uniquely link the memory  chip and the logical address B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='  In general, we can obtain the response (�����) from the  physical random number of chip ( � ), ������ , and the  challenge (C) using a function, �, as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='  ����� = ���, �������  (1)  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' 2, the connected device A sends the challenge (C)  “Hey B, who are you?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' to the memory chip (�) having ��� ���  of the connected device B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' The response (R) generated using  (1) is “I am chip B.” If the connected device B was spoofed by  hacker’s laptop, the response would be “I am a chip in hacker’s  laptop”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' The CRP using a memory chip makes sense for the  cyber-attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' Because no cyber-attack can tear off memory  chip from the motherboard of the connected device B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' PRN and PKI  We can generate one or two prime number(s) from this  ����� in some way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' Thus, we can generate a “uniquely linked”  pair of secret and public keys using an algorithm of PKI --- RSA  [5], ElGamal [6], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' Therefore, the secret key (SK) and public  key (PK) of chip (�) are respectively written as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='  ������ = ��������  (2)  ������ = �̅�������  (3)  The ��, �̅� are the key generation functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' It is preferable that  prime numbers are great enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' The (3) connects chip (�) and  public key ������ through (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' Below we evaluate randomness  to validate if ��� ��� is specific to chip (�) in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' A concrete example of retrieving PRN from memory chips  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' 3 illustrates the redundancy mechanism of a memory  chip [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' There are several blocks on chip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' Each block is  composed of many integrated memory cells (corresponding to  bits), which are arrayed in the X (Row)-Y (Column) plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' It is  further divided into two cell arrays (redundancy and regular  arrays).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' The Y-decoders A and B control the row access in the  redundancy and regular arrays, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' An access code is  input to a peripheral circuit to choose either Y-decoder A or B  or both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' Since memory chip is a mass-product, it is impossible  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' CRP with physical randomness specific to a memory  chip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='  Logical address B  Logical address A  Logical address C  Connected  device B  Connected  device C  Connected  device A  Hey, B  Who are you?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='  Secret  Key  1-to-1  Public  Key  Memory  Chip  PRN (n)  Challenge (C)  I am Chip B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='  (a)  (b)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' Challenge-Response Protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' (a) without spoofing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='  (b) with spoofing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='  Logical address B  Logical address A  Logical address C  Cybernetwork  (logical address to  logical address)  Connected  device B  Connected  device C  Connected  device A  IoT network  (device to device)  Hey B,  Who are you?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='  I am logical address B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='  Logical address B  Logical address A  Logical address C  Cybernetwork  (logical address to  logical address)  Connected  device B  Connected  device C  Connected  device A  IoT network  (device to device)  Hey B,  Who are you?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='  spoofing  Laptop  I am logical address B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='  to exclude all failures from the cell arrays perfectly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' Suppose  there are plural failure bits in the regular array, wherein the row  number with a failure bit is denoted by �� for � = 1, 2, … � with  � being the number of rows with a failure bit in the regular  array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' While this � is small enough, that is, the failure is  controllable under the manufacturing specification, we can  apparently exclude the failure bits from the memory access  operations (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=', read, write, and erase in the regular array).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='  Inputting an access code into the peripheral circuit, we can  choose a normal access mode to use only Y-decoder B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' We,  thus, access the addresses from the top of the regular array along  a column (�) chosen by X-decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' When arriving at ��, we  swap the bit with another bit on the row number ��  in the  redundancy array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' If the bit at the address (��, �) is not a failure  bit, we can apparently exclude the failure bit at (��, �) from the  memory access operations in the regular array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' In usual, the  users of DRAM can access only the regular array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' As �  decreases, we can reduce the necessary number of rows in the  redundancy array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' It can suppress the probability that a failure  bit is found in the redundancy array as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' As improving the  manufacturing quality, the average of � gets smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' We can  expect that � is small enough in the mass-produced memory  chips, but it can hardly be zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' In general, the distribution of  rows with a failure bit in the regular array of chip (�), denoted  by �����, is out of the manufacturing control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' It can cause a  physical randomness specific to chip (�).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' The inner memory  stores this ����� to exclude failure bits from the memory access  operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' Below we illustrate the method to retrieve ��� ���.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='  Preprocess) We input a normal access code to the peripheral  circuit to use Y-decoder B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' This is the normal access mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' We  write state-0 along the �-th column (chosen in advance) in the  regular array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' 4 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' Next, inputting a special access  code to the peripheral circuit, we change the access mode to the  special access mode to use Y-decoder A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' Then, we write state-  1 along the �-th column in the redundancy array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' 4 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='  Read) In a normal access mode (using Y-decoder B), we read  bits along the �-th column in the regular array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' By the swapping,  readout turns out being state-1 on rows with a failure bit (��)  and state-0 on the other rows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' 4 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' Thus ����� specific  to chip (�) turns out being a physical random number, ������.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='  PRN Stability) Reliability of the inner memory storing �����  determines that of  ������.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' In general, the inner memory (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=',  a fuse memory) is more stable to the environmental change than  cell arrays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' In experiment using the dynamical random access  memory (DRAM) chips, that is the most popular IC product for  the main memory, the excellent characteristics of retention and  stability to temperature change were clearly shown [4], [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='  Randomness) As an example, we choose 4 Mbits - 16 Gbits  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=', the old and new generations) DRAM for evaluating the  entropy of chip-specific randomness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' For the ease of discussion,  we assume that � is much smaller than the total row number  (� ), and the row and column numbers are the same (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=',  isotropic layout) in each block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' The number of blocks on a chip  is thus ��.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' For � = 1, the entire cell array on chip is one block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='  The � is in the order of 100,000 for 16 Gbits, which is most  advanced recently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' The randomness can be estimated by  calculating the combination product of � and �, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=', �  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' For  an arbitral � , the randomness can be estimated by  �  "×!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='  Following Boltzmann, the entropy divided by $% is ln  �  "×!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='  ,  where $% is the Boltzmann constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' For � = 1, although it is  an underestimation a little bit, we plot information entropies of  PRN from DRAM chips (4 Mbits to 16 Gbits) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' For  � = 10, ln  �  "×!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='  > 10�* even for 4 Mbits with � = 1 (the  worst case in this estimation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' If one hundred trillion (10+,) IoT  devices will be deployed all over the world, the probability that  two chips have a same PRN is less than 10-++ (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=', 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='00001  ppm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' Thus, the randomness is good enough in the 4 Mbits – 16  Gbits products of DRAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' In addition, a couple of ten rows (≫  �) in the redundancy array is good enough for swapping to hide  most of failure bits from the memory access operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' The  redundancy array comprising twenty rows is much narrower  than the regular array (20 ≪ 2,000 and 100,000 in 4 Mbits and  16 Gbits products, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' It does not matter in the rough  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' Redundancy mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='  Redundancy array  Regular  array  X-direction (Row)  Y-direction (Column)  Y-decoder B  Y-decoder A  Inner  memory  Peripheral  circuit  R1  R2  F1  F2  Swap-1  Swap-2  Failure bit  Choose column ( )  Access code  X-decoder  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' Retrieve PRN from memory chip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' (a) preprocess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='  (b) read.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='  0  00  000  1  11  1  1  0  01  0  Chosen column ( )  Swap-1  Swap-2  Redundancy  array  Regular  array  R1  R2  F1  F2  (a)  (b)  Chosen column ( )  0  0  00  0  1  0  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' Entropy of randomness with memory chips  10  20  40  80  160  320  10  60  110  160  ENTROPY / KB  ROW NUMBER WITH A FAILURE BIT  INFORMATION QUANTITY  16 Gbits  4 Gbits  1 Gbits  256 Mbits  64 Mbits  16 Mbits  4 Mbits  100 Trillion  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='36(4)  estimation of the entropy even though increasing �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' Therefore,  we can obtain a sufficient randomness to retrieve ��� with no  additional area for cell array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' It tells us there is no chip area  penalty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' The present method is feasible to the wide generations  of the existing DRAM products (4 Mbits – 16 Gbits).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' It is self-  evident that the present method is also feasible to the coming  generations beyond 16 Gbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' Independencies of chips  While the entropy of randomness specific to a memory chip  is good enough, we can hardly retrieve a same random number  which is specific to two different chips, as discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='  �����1� 0 �����2� 1ℎ�34 �1 0 �2  (4)  While satisfying (4), we respectively convert (1) – (3) to:  ��+��� 0 ������ 1ℎ�34 �1 0 �2  (5)  ���+��� 0 ������� 1ℎ�34 �1 0 �2  (6)  ���+��� 0 ������� 1ℎ�34 �1 0 �2  (7)  Furthermore, we can write down as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='  ����+� 0 ������ 1ℎ�34 �+ 0 ��  (8)  �����+� 0 ������� 1ℎ�34 �+ 0 ��  (9)  �����+� 0 ������� 1ℎ�34 �+  0 ��  (10)  The (5) – (10) characterize functions that are necessary for  memory chips to satisfy the root-of-trust (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=', the unique  linkage of memory chips and public keys).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' 6 (a),  physical nodes satisfying these configures firewall of things [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' Auto-detection and auto-remove (Entry control)  Auto-detection and auto-remove of the fake and vulnerable  nodes can be performed using (5) – (10) and the smart contract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='  Details will be discussed elsewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' Security State  For each �, we generate response, secret and public keys  using a challenge �5, where 3 is a natural number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' Regard the  generation of them as appearing similar to an observation in  quantum mechanics (denoted by �6,  �7 and �6, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' To  deduce (8) – (10), we can begin with the following equations,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='  �6�|3⟩ = ����5�|3⟩  (11)  �7�|3⟩ = �����5�|3⟩  (12)  �6�|3⟩ = �����5�|3⟩  (13)  The ����5�, �����5� and �����5� of chip (�) can describe a  security state denoted by |3⟩ (like an eigenstate in quantum  mechanics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' However, the security state of a physical node (�)  can be determined by choosing a challenge �5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' Since the state  |3⟩ has been chosen before the observation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=', the generation  of response and secret and public keys), (11) – (13) are closer  to Einstein’s hidden variables [9] than Schrödinger’s cat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='  Anyway, this can confuse adversaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' 6 (b) illustrates the  merit of security state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' In the network, another central node  (security node) can change response and secret and public keys  of any physical nodes by replacing �5  periodically or at his  convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' It is effective when some kind of security  problems is found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' Details will be discussed elsewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='  The �5 used by a security node should differ from challenge  used by a management node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' The entry and the security state  are controlled by two independent central nodes (management  and security nodes, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' This reminds us of the  separation of three powers in the democracy, even though this  is a central control indeed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' BLOCKCHAIN OF MEMORY CHIPS  Most of physical nodes is a Neumann computer which is  composed of an input-output (I/O), an arithmetic unit and a  memory chip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' Data, stored in a memory chip of a physical node,  is retrieved and processed by the arithmetic unit and then output  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' Merkle tree of memory chips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='  Root of Merkle (  )  Memory  Chip  PRN (  )  Memory  chip  PRN (  )  Memory  chip  PRN (  )  Memory  chip  PRN (  )  Memory  chip  PRN (  )  Memory  chip  PRN (  )  Memory  chip  PRN (  )  Memory  chip  PRN  )  Memory  chip  PRN (  )  (a)                                                                                              (b)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' Central management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' (a) by management node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' (b) by security node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='  R/W  Security  Security  State  node  SmartmeterFirewall of things  R/W  Entry  Management  certificate  node  Smartmeter  from I/O to the external.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' This data is received by I/O of another  physical node and then processed by the arithmetic unit and  then stored in the memory chip of the receiving physical node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='  That is, data is really transferred between memory chips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' Merkle tree of memory chips  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' 7, we suppose that data stored in chip (�0) has been  transferred from chips (�1, �4, and �6) with recording the  updated history of data transfer to chip (�0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' Each data transfer  is modeled in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' 8, which illustrates that from node (� < 1) to  node (�).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' In chip (�), the ����5� having been obtained from �5  and ������ at security state |3⟩ according to (1) is used to  generate �����5� and �����5� using (2) and (3), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='  While (4) is satisfied, we regard (5) – (13) as all satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' The  chip-specific ����5� is related to �����5�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' The unique linkage  of memory chip and public key is thus assured by the unique  linkage of secret and public keys that are generated using a  same chip-specific ����5�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' We can regard logical node (�) as  equal to physical node (�) on the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' We can combine  them to call it as node (�), if not specially noted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='  In the node (� < 1), the ���-+��5�,  hash value (� < 2), and  electronic signature (� < 2) are converted to a new hash value  (� < 1) using a hash function (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=', SHA256).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' Subsequently,  the node (� < 1) gets �����5� on the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' (Public key is  public on the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=') The got public key and the generated  hash value ( � < 1 ) are encrypted to be a new electronic  signature (� < 1) using ���-+��5� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' The node (� < 1) sends the  hash value (� < 1) and the electronic signature (� < 1) to the  node (�).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' This transfer is as same as used in the existing  blockchain if we exclude memory chips from the illustration  (drawn in red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' That is, our solution for the unique linkage of  memory chips and public keys is fully compatible to the  existing blockchain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='  Move back to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' Data stored in chip (�1) has been  transferred from chips (�2 and �3) with recording the updated  transferring history in chip (�1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' Data stored in chip (�4) has  been transferred from chip (�5) with recording the updated  transferring history therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' Data stored in chip (�6) has been  transferred from chips (�7 and �8) with recording the updated  transferring history therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' It turns out being a tree of Merkle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='  There is chip (�0) at the bottom, named, the Merkle root (�0),  with recording the entire history of data transfers from plural  chips to the root chip (�0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' Blockchain of memory chips  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' 9, we follow the conventional mining process to chain  blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' A miner can register the newest block hash (A) by  hashing an added nonce (A), a stamp of the root chip (�0) (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=',  ���B��5�, hash value (�0 < 1), etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' ), and the previous block  hash (A < 1), such that the newly generated block hash (A)  satisfies the requirement for the poof-of-consensus by tuning  nonce (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' The block hash (A < 1) was registered by hashing an  added nonce (A < 1), a stamp of the root node (�00) (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=',  ���BB��5�, hash value (�00 < 1), etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' ), and the previous block  hash (A < 2), so that the block hash (A < 1) satisfies the  requirement for the poof-of-consensus by tuning nonce (A < 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='  Repeating this procedure, the miners have constructed the  blockchain of memory chips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='  There are plenty of diverse types of electronic appliances  (physical nodes) --- routers, PCs, smartphones, card readers,  RFID reader-writers (R/Ws), surveillance cameras, industrial  robots, smart meters, printers etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' However, DRAM  is a general-purpose memory and installed in most of electronic  appliances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' 10, any nodes are approved to communicate  each other without being monitored by any central nodes inside  the firewall of things [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' One of central nodes (management  node) controls the entry of physical nodes to the network and  prohibits the spoofing by a hacker’s laptop with non-registered  chip (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' 1 (b) as well).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' Another central node (security node)  controls security state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' The unique linkage of memory chips and  public keys is fully compatible to the blockchain construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' Blockchain of memory chips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' The coexistence of central management and  decentralized system (blockchain).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' Data transaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' See it by removing red ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='  Firewall of things (Hardware  Laptop  Blockchain  Non-registered  chip  Smartmeter  spoxfing  R/W  Entry  Management  node  Security  state  Security  nodeR/W  Smartmeter  PRN (n000)  PRN (n00)  PRN (n0)  Memory  Memory  Memory  Chip  Chip  Chip  Nonce (r - 2)  Nonce (r - 1)  Nonce (r)  Block hash (7 - 3)  Blockhash (n-2)  Block hash (r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='- 1)  Block hash (r)  Block (r- 2)  Block (r - 1)  Block (r)Chip (n - 1)  Chip (n)  Physical node  SKn-1(C)  SKn(C)  Key  Key  Rn-1(C)  R,(C)  Generator  Generator  PKn-1(C)  PKn(C)  Logical node  Hash value  Hash value  (n - 2)  (n - 1)  Electronic Signature  Electronic Signature  (n 2)  (n - 1)  Node(n - 1)  Node(n)  Blockchain protects communications in the firewall of things.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='  This is the decentralized communications under central control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' Resilient Blockchain of things  There is no risk of mechanical failures in logical accounts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='  But, in the IoT network, there is the risk of mechanical troubles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='  If we replace a defected device having a memory chip (�C) by  a new device having a memory chip (�D) in the blockchain of  memory chips, the response and public and secret keys will  change since �����C� and �����D� differ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' It requires the  reproduction of the Merkle root using the entire tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' Another  case to require the reproduction of the Merkle root is the change  of security state for some security reason.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='  However, we can follow the idea of Beyond Blockchain one  (BBc-1) [10], which can be regarded as resilient blockchain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='  Only limited subtrees are good enough to reproduce the Merkle  root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' It can reduce the reproduction cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' The reproduced tree  can be registered as a new block after the proof-of-consensus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' DISCUSSION  Since the version of security of IoT devices vary greatly from  old to new, a hacker may get an authenticate information of a  vulnerable IoT device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' Any advanced security parts (PUC,  protected memory [11] etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=') cannot protect old IoT devices  having been shipped with the security parts in the former  generations or with no security parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='  The present method can be an application software, which  retrieves PRN from existing IC chips in the distributed IoT  devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' It was demonstrated that (4) can be satisfied within the  quality control of mass-produced DRAM chips (4 Mbits – 16  Gbits), with no chip area penalty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' Most of existing IoT devices  (old and new) mounts a DRAM chip with the bit capacity being  4 Mbits – 16 Gbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' To adopt the present method, we may install  the application software and not replace old IoT devices by new  ones with the advanced security parts (PUC etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='  Smartphone suppliers can construct the unique linkage of  smartphones and public keys using the present method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' This is  the firewall of smartphones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' End users are required to download  an application to retrieve PRN from DRAM chip existing in  their smartphones but not to replace their old smartphones with  new ones so that their smartphones can enter into the firewall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='  In the industrial manufacturing, the supervisory control and  data acquisition (SCADA) is collecting the attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' In the  semiconductor manufacturing, for example, a medium class  factory with ~40,000 wafers per month has about 3,000 units  of equipment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' If each unit has 100 parameters in average, then  about 300,000 parameters are to be optimized even in a medium  class factory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' Twice or more in a huge class factory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' In the  SCADA, those parameters are collected from plural smart  factories by IoT sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' A remote facility of AI dynamically  and consistently optimizes data (parameters) collected from the  plural smart factories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' Therefore, there is an increasing risk that  the cyber-attack will upset the global supply chain and theft  technical know-how.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' Since each IoT sensor has a DRAM chip,  we can construct the unique linkage of IoT sensors and public  keys over the remote-controlled smart factories distributed  globally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' This is the firewall of IoT sensors and industrial robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='  In the logistics control, the security of RFID tags has been  discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' However, a spoofed R/W can deliver parcels to an  illegal destination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' Thus, we propose the unique linkage of  R/Ws and public keys [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' This is the firewall of R/Ws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='  For the traceability in the supply chain of IC chips, we can  construct the firewall of IC chips using the present method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' SUMMARY  Taking into account the quality control of mass-produced  memory chips, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=', atomistic component of IoT devices, we can  construct the unique linkage of memory chips and public keys  with a full compatibility to the existing blockchains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' The central  management of IoT devices and decentralized communications  among IoT devices can therefore coexist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' The present method  is valid to various IC products as well as DRAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' Application  range is pretty-wide and not limited to discussed here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='  ACKNOWLEDGMENT  The author thanks to T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' Hamamoto, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' Nagai, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' Liang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='  Chang, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' Tseng, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' Moon, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' Saito, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' Taniguchi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' Torisawa,  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' Kato, KY Tsai, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' Chen, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' Chang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' Hirota, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' Kinoshita  for valuable and stimulating discussions, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' Okada, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='  Miyazaki, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' Fukuyama for support on experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
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+page_content=' Hamamoto, “The cheapest physical chip-ID fully  compatible to mass-product DRAM process”, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' AWAD, 399,  Gyeongju, Korea, July 3-5, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='  [8]  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' Watanabe, “Firewall of Things, Firewall of Memory Chips”, Flash  Memory Summit 2022, Aug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' 2-4, Santa Clara, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='  [9]  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' Einstein, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' Podolsky and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' Rosen, “Can quantum-mechanical  description of physical reality be considered complete?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=', Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' 47,  pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' 777 – 780, 1935.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='  [10]  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' Saito and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' Kubo, “BBc-1: Beyond Blockchain One – An  architecture for promise-fixation device in the air --”, white paper  (2017) on https://beyond-blockchain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='org;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='  https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='com/beyond-blockchain/bbc1/blob/develop/docs/BBc-  1%5C_design%5C_paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='pdf  [11]  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' Gorog, “cryptographic trust enabled devices of cybersecurity  systems”, WO/2022/072609.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content='  [12]  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' Watanabe, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' Saito, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' Miyazaki, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' Okada, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' Fukuyama, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' Kato,  and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' Taniguchi, “Proof of authenticity of logistics information with  passive RFID tags and blockchain”, the 2021 IEEE International  Conference on Electronic Communications, Internet of Things and Big  Data (ICEIB 2021), Yilan, Taiwan, Dec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
+page_content=' 10-12, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfTfw1/content/2301.01249v1.pdf'}
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+arXiv:2301.13457v1  [math.NT]  31 Jan 2023
+THE PRIME NUMBER THEOREM FOR PRIMES IN ARITHMETIC
+PROGRESSIONS UNDER THE GENERALISED RIEMANN
+HYPOTHESIS
+ETHAN SIMPSON LEE
+Abstract. We assume the generalised Riemann hypothesis for Dirichlet L-functions L(s, χ)
+and establish explicit formulae for ψ(x, χ), θ(x, χ), and an explicit version of the prime num-
+ber theorem for primes in arithmetic progressions.
+1. Introduction
+Suppose that x ≥ 2, p are prime numbers, χ is a Dirichlet character modulo q ≥ 3,
+ψ(x, χ) =
+�
+n≤x
+χ(n)Λ(n),
+and
+ψ1(x, χ) =
+� x
+0
+ψ(t, χ) dt =
+�
+n≤x
+χ(n)Λ(n)(x − n).
+The purpose of this note is to update the latest explicit and conditional version of the prime
+number theorem for primes in arithmetic progressions, which are asymptotic bounds for
+π(x; q, a) =
+�
+p≤x
+p≡a (mod q)
+1,
+θ(x; q, a) =
+�
+p≤x
+p≡a (mod q)
+log p,
+and
+ψ(x; q, a) =
+�
+n≤x
+n≡a (mod q)
+Λ(n),
+in which 0 ≤ a < q is an integer such that (a, q) = 1. Explicit bounds for each of these
+counting functions is a natural consequence of an explicit bound for ψ(x, χ), since
+ψ(x; q, a) =
+1
+ϕ(q)
+�
+χ
+χ(a)ψ(x, χ),
+(1)
+where ϕ(q) is the Euler-totient function evaluated at q; we will see how a result for ψ(x; q, a)
+naturally leads to results for θ(x; q, a) and π(x; q, a) later.
+Assuming the Generalised Riemann Hypothesis (GRH), the latest explicit version of the
+prime number theorem for primes in arithmetic progressions was established by Ernvall-
+Hyt¨onen and Paloj¨arvi in [9]. We refine their results using Theorem 1.1.
+Theorem 1.1. Suppose that the GRH is true, δχ = 1 if χ = χ0, and δχ = 0 if χ ̸= χ0, where
+χ0 is the principal character. If x ≥ max{e10, q} and q ≥ 3, then
+|ψ(x, χ) − δχx| ≤
+� √x(log x)2
+8π
++ 1.12(log x)2
+if δχ = 1,
+0.223√x(log x)2 + 3.446√x log x + 317.501
+if δχ = 0.
+The first case of Theorem 1.1 is a simple application of [16, (6.2)]. To prove the second
+case of Theorem 1.1, we prove the result for primitive non-principal characters and extend
+ESL thanks the Heilbronn Institute for Mathematical Research for their support.
+1
+
+that observation to any non-principal character in the final steps. Now, to prove the result
+for these primitive characters, we note that ψ(x, χ) differs from
+∆(x, χ) = ψ1(x + √x log x, χ) − ψ1(x, χ)
+√x log x
+by a small error of order √x log x; this is a consequence of the prime number theorem for
+short intervals in [5] which requires log x ≥ 10. All that remains is to bound ∆(x, χ) using
+an explicit formula for ψ1(x, χ); this will establish the relationship
+∆(x, χ)√x log x ≈ −
+�
+̺χ
+(x + h)̺χ+1 − x̺χ+1
+̺χ(̺χ + 1)
+,
+(2)
+in which ̺χ = 1/2 + iγχ are the non-trivial zeros of L(s, χ) and the error is well-understood.
+In the end, the main challenge will be to obtain good bounds for the absolute value of this
+sum over zeros.
+Using the approach described above, we actually prove a more precise statement than
+Theorem 1.1, which involves terms that depend on q (see (10)), then we assert the bound
+x ≥ q and collect like terms so that the upper bound depends on x only. We purposely do
+these extra steps to make the result user-friendly for applications. To this end, we present
+two corollaries of Theorem 1.1 below; these will be proved in Sections 2.3 and 2.4 respectively.
+The latter of these corollaries is an explicit version of the prime number theorem for primes
+in arithmetic progressions.
+Corollary 1.2. If the GRH is true, x ≥ max{e10, q}, and q ≥ 3, then
+|θ(x, χ) − δχx| ≤
+�√x(log x)2
+8π
++ 1.443√x log x + 1.12(log x)2
+if δχ = 1,
+0.223√x(log x)2 + 4.889√x log x + 317.501
+if δχ = 0.
+Corollary 1.3. If the GRH is true, (a, q) = 1, x ≥ max{e10, q}, q ≥ 3, and Li(x) =
+� x
+2
+dt
+log t,
+then
+����π(x; q, a) − Li(x)
+ϕ(q)
+���� <
+�
+0.223 + 0.0474
+ϕ(q)
+� √x log x + 6.710√x + 460.313,
+(3)
+����θ(x; q, a) −
+x
+ϕ(q)
+���� <
+�
+0.223 + 0.0474
+ϕ(q)
+� √x(log x)2 + 4.889√x log x + 317.501,
+����ψ(x; q, a) −
+x
+ϕ(q)
+���� <
+�
+0.223 + 0.0474
+ϕ(q)
+� √x(log x)2 + 3.446√x log x + 317.501.
+Unconditional versions of the results we prove also exist. For example, Bennett, Martin,
+O’Bryant, and Rechnitzer [1, Thm. 1.3] give constants cπ(q) and xπ(q) such that
+����π(x; q, a) − Li(x)
+ϕ(q)
+���� < cπ(q)x
+(log x)2
+for all
+x ≥ xπ(q).
+Other authors including Bordignon [2], Chen–Wang [4], Dusart [8], Liu–Wang [13], Mc-
+Curley [14], and Ramar´e–Rumely [15] have done unconditional work in this area too. In
+the remainder of this note, we prove all the results above in Section 2 and an important
+(technical) ingredient in Section 3.
+2
+
+Remark. To justify that our results refine the results in [9], we will compare (3) against
+their [9, Cor. 2], because this is the clearest like-for-like comparison that we could draw
+between our respective expositions. In particular, both results were proved using (1) and
+a version of Theorem 1.1. Upon asserting x ≥ q, recall that they have proved (3) with
+respective coefficients
+0.184 +
+1
+8πϕ(q) + 1
+6π ≈ 0.257 + 0.0398
+ϕ(q) ,
+12 969.946,
+and
+− 237.934
+in the upper bound. The first and second of these coefficients are clearly worse than their
+counterparts in (3) and the final constant does not have much effect in general. The main
+benefit of their result over ours is that it holds for x ≥ q, whereas ours holds for x ≥
+max{e10, q} ≈ max{22 026.47, q}; this is not too restrictive, but it is an important distinction.
+Remark. To refine each of our results, an effective approach would be to refine the bound
+on the sum over zeros in (2) that we obtain in Theorem 3.6. In particular, we will bound
+the sum by considering the ranges
+|γχ| ≤ 5
+7,
+5
+7 < |γχ| < T,
+and
+|γχ| ≥ T
+separately; the definition of T and reasons for choosing these split points will become ap-
+parent in Section 3. However, for fixed q and η > 0, if one can use computations to evaluate
+the sum exactly in the range |γχ| ≤ η for each of the primitive and non-principal characters
+modulo q, then we can reduce the effect of wasteful bounds that inevitably occur whenever
+small zeros are considered. This method of refinement has been used to great effect when
+considering sums over the non-trivial zeros of the Riemann and Dedekind zeta-functions; for
+examples see [6,10,11].
+2. Main results
+In this section, we will prove each of the main results that were presented in the intro-
+duction. First, we prove Theorem 1.1 by considering the cases δχ = 1 and δχ = 0 separately
+in Sections 2.1 and 2.2 respectively. Next, we will describe how one can prove Corollary 1.2
+using Theorem 1.1 in Section 2.3. Finally, we will prove Corollary 1.3, which is an explicit
+version of the prime number theorem for primes in arithmetic progressions, in Section 2.4.
+2.1. Theorem 1.1: Case I. If x ≥ 2 and χ = χ0 modulo q ≥ 3, then
+|ψ(x, χ0) − ψ(x)| ≤
+�
+n≤x
+(n,q)>1
+Λ(n),
+where
+ψ(x) =
+�
+n≤x
+Λ(n).
+(4)
+Insert Lemmas 2.1-2.2 (below) into (4) to see
+|ψ(x, χ0) − x| ≤
+√x(log x)2
+8π
++ 1.12 log q log x,
+for all
+x ≥ 73.2.
+Lemma 2.1. If the Riemann hypothesis is true and x ≥ 73.2, then
+|ψ(x) − x| <
+√x(log x)2
+8π
+.
+Proof. This is [16, (6.2)].
+□
+3
+
+Lemma 2.2. If x ≥ 2 and q ≥ 3, then
+1
+log x
+�
+n≤x
+(n,q)>1
+Λ(n) ≤ τ(q) ≤ 1.12 log q,
+where
+τ(q) =
+�
+2
+if q = 6,
+log q
+if q ̸= 6.
+Proof. This is [9, Lem. 12].
+□
+2.2. Theorem 1.1: Case II. Suppose that x ≥ 2 and χ ̸= χ0 modulo q ≥ 3 is primitive.
+To begin, we observe that
+����ψ(x, χ) − ψ1(x + h, χ) − ψ1(x, χ)
+h
+���� ≤
+�
+x<n≤x+h
+Λ(n),
+(5)
+in which h > 0 is a parameter to be chosen. Note that (5) follows from the relationship
+ψ1(x + h, χ) − ψ1(x, χ)
+h
+= ψ(x, χ) +
+�
+x<n≤x+h
+χ(n)Λ(n)
+�
+1 + x − n
+h
+�
+.
+It follows from (5) and Lemma 2.3 (below) that we have the relationship
+����ψ(x, χ) − ψ1(x + √x log x, χ) − ψ1(x, χ)
+√x log x
+���� ≤ 3√x log x
+for
+log x ≥ 10.
+(6)
+Using (6) and the upcoming explicit formula for ψ1(x, χ), we can estimate ψ(x, χ).
+Lemma 2.3. If log x ≥ 10, h = √x log x, and the Riemann hypothesis is true, then
+�
+x<n≤x+h
+Λ(n) ≤ 3√x log x.
+Proof. Cully-Hugill and Dudek [5, Thm. 1] tell us that for all log x ≥ 10 and any choice
+√x log x ≤ h ≤ x3/4, we have
+|ψ(x + h) − ψ(x) − h| ≤ 1
+π
+√x log x log
+�
+h
+√x log x
+�
++ 2√x log x.
+Insert the choice h = √x log x to deduce the result.
+□
+Lemma 2.4. Suppose that ̺χ = 1/2 + iγχ are the non-trivial zeros of L(s, χ). If x ≥ 2 and
+χ is principal or primitive, then
+ψ1(x, χ) = δχ
+x2
+2 −
+�
+̺χ
+x̺χ+1
+̺χ(̺χ + 1) + a(χ)x + b(χ) + c(x, χ),
+in which aχ = 1−χ(−1)
+2
+, −L′(s,χ)
+L′(s,χ) = −1
+s + bχ + O(s), −L′(s−1,χ)
+L′(s−1,χ) = −1
+s + cχ + O(s),
+a(χ) = (1 + bχ)(1 − aχ)(1 − δχ) − L′(0, χ)
+L(0, χ) (aχ(1 − δχ) + δχ),
+b(χ) = (1 − cχ)aχ + L′(−1, χ)
+L(−1, χ) (1 − aχ),
+c(x, χ) = aχ log x − (1 − aχ)(1 − δχ)x log x −
+∞
+�
+m=1
+x1−2m−aχ
+(2m + aχ)(2m − 1 + aχ).
+4
+
+Proof. This is a consequence of Perron’s formula; see [7, Lem. 2.1].
+□
+It follows from Lemma 2.4 that if 0 < h < x, then
+ψ1(x + h, χ) − ψ1(x, χ)
+h
+= 1
+h
+
+δχ
+(x + h)2 − x2
+2
+−
+�
+̺χ
+(x + h)̺χ+1 − x̺χ+1
+̺χ(̺χ + 1)
++ a(χ)h + c(x + h, χ) − c(x, χ)
+
+
+= δχ
+�
+x + h
+2
+�
++ a(χ) + 1
+h
+
+c(x + h, χ) − c(x, χ) −
+�
+̺χ
+(x + h)̺χ+1 − x̺χ+1
+̺χ(̺χ + 1)
+
+ .
+(7)
+Using this observation, we will prove the following result using (7) in the next section.
+Theorem 2.5. If χ ̸= χ0 modulo q ≥ 3 is primitive, h = √x log x, and x ≥ max{e10, q},
+then
+����
+ψ1(x + h, χ) − ψ1(x, χ)
+h
+���� < 0.215√x(log x)2 + 3.442√x log x + 317.501.
+Further, Ernvall-Hyt¨onen and Paloj¨arvi proved in [9, Lem. 13] that if χ is a non-primitive
+Dirichlet character modulo q ≥ 3 that is induced by a primitive character χ∗, then
+|ψ0(x, χ) − ψ0(x, χ∗)|
+log x
+≤ τ(q) ≤ 1.12 log q,
+where
+ψ0(x, χ) = ψ(x+, χ) + ψ(x−, χ)
+2
+;
+(8)
+this modification to the definition of ψ(x, χ) accounts for discontinuities that may present
+at prime powers. A simple consequence of this definition is that
+|ψ(x, χ) − ψ0(x, χ)| ≤ log x
+2
+.
+(9)
+Finally, insert Theorem 2.5 into (6) and apply (8), (9) to extend the result to any non-
+principal χ; the end result follows.
+Theorem 2.6. If x ≥ max{e10, q} and χ ̸= χ0 modulo q ≥ 3, then
+|ψ(x, χ)| <
+�
+0.215 + 1.12
+√x
+� √x(log x)2 +
+�
+3.442 + 0.5
+√x
+� √x log x + 317.501
+< 0.223√x(log x)2 + 3.446√x log x + 317.501.
+Remark. A more precise version of Theorem 2.5 that holds for all log x ≥ 10 will be given
+in (19) in the following section. If we were to use (19) instead of Theorem 2.5 to deduce
+Theorem 2.6, then we would see the more precise
+|ψ(x, χ)| <
+�
+g1(x)
+�� 1
+8π + 1.12
+√x
+�
+log x + log q
+2π
+�
++ g2(x) + 0.5
+√x
+� √x log x
++ 0.74√x log q + g3(q)
+(10)
+holds for all log x ≥ 10, χ ̸= χ0, and q ≥ 3; the functions gi(x) will be defined below (19). To
+keep our exposition clean in the end, we asserted the bound x ≥ q onto each term containing
+q and collected like terms together.
+5
+
+2.3. Corollary 1.2. To use Theorem 1.1 to obtain explicit knowledge about θ(x, χ), we
+require the following simple relationship:
+|ψ(x, χ) − θ(x, χ)| ≤
+�
+2≤k≤
+�log x
+log 2
+� θ(x
+1
+k ) < log x
+log 2θ(√x),
+where
+θ(x) =
+�
+p≤x
+log p.
+Clearly, we will require [3, Cor. 2.1], which states that for all x ≥ 0, there is a constant
+cθ = 1 + 1.93378 · 10−8 such that θ(x) ≤ cθx. It follows that
+|ψ(x, χ) − θ(x, χ)|
+√x log x
+<
+cθ
+log 2 < 1.44270.
+(11)
+Therefore, Corollary 1.2 is a straightforward consequence of Theorem 1.1 and (11).
+2.4. Corollary 1.3. Insert Theorem 1.1 into (1) to see for all x ≥ max{e10, q}, we have
+����ψ(x; q, a) −
+x
+ϕ(q)
+���� ≤
+1
+ϕ(q)
+�√x(log x)2
+8π
++ 1.12(log x)2 +
+�
+χ̸=χ0
+|ψ(x, χ)|
+�
+≤
+�
+0.223 +
+1
+ϕ(q)
+� 1
+8π + 1.12
+√x
+�� √x(log x)2 + 3.446√x log x + 317.501
+≤
+�
+0.223 + 0.0474
+ϕ(q)
+� √x(log x)2 + 3.446√x log x + 317.501,
+(12)
+since there are ϕ(q) Dirichlet characters modulo q. Use [3, Cor. 2.1] in a similar way to how
+we used it in Section 2.3 to see
+|ψ(x; q, a) − θ(x; q, a)| ≤ 1.44270√x log x.
+Now, combine the preceding observations to see that for all x ≥ max{e10, q}, we have
+����θ(x; q, a) −
+x
+ϕ(q)
+���� ≤
+�
+0.223 + 0.0474
+ϕ(q)
+� √x(log x)2 + 4.889√x log x + 317.501.
+(13)
+Finally, partial summation yields the relationship
+π(x; q, a) = θ(x; q, a)
+log x
++
+� x
+2
+θ(t; q, a)
+t(log t)2 dt.
+Therefore, the contribution from the main term in (13) is
+1
+ϕ(q)
+�
+x
+log x +
+� x
+2
+dt
+(log t)2
+�
+=
+1
+ϕ(q)
+�
+Li(x) +
+2
+log 2
+�
+.
+We also know that for all log x ≥ 10, we have
+4.889
+� x
+2
+dt
+√
+t log t = 4.889
+�
+Ei
+�log x
+2
+�
+− Ei
+�log 2
+2
+��
+≤ 1.32718√x.
+6
+
+It follows that for all x ≥ max{e10, q}, we have
+����π(x; q, a) − Li(x)
+ϕ(q)
+���� ≤
+�
+0.223 + 0.0474
+ϕ(q)
+� �√x log x +
+� x
+2
+dt
+√
+t
+�
++ 4.889
+�√x +
+� x
+2
+dt
+√
+t log t
+�
++ 317.501
+�
+1
+log x +
+� x
+2
+dt
+t(log t)2
+�
++
+2
+ϕ(q) log 2
+≤
+�
+0.223 + 0.0474
+ϕ(q)
+� √x log x +
+�
+6.21618 + 2
+�
+0.223 + 0.0474
+ϕ(q)
+�� √x
++
+1
+log 2 − 2
+√
+2
+�
+0.223 + 0.0474
+ϕ(q)
+�
++ 317.501
+log 2
+<
+�
+0.223 + 0.0474
+ϕ(q)
+� √x log x + 6.710√x + 460.313,
+(14)
+since ϕ(q) ≥ 2. Combine the observations (12), (13), and (14) to retrieve Corollary 1.3.
+3. Smoothed difference
+The purpose of this section is to prove Theorem 2.5, which was an important ingredient
+in the derivation of Theorem 2.6.
+3.1. Preliminary bounds. We will need the following lemmas in what follows.
+Lemma 3.1. If χ is a primitive, non-principal character modulo q with χ(−1) = 1, then
+|bχ| < 2.751 log q + 23.878.
+Proof. This is [9, Lem. 17].
+□
+Lemma 3.2. If χ is primitive modulo q ≥ 3 such that χ(−1) = −1, then
+����
+L′(0, χ)
+L(0, χ)
+���� < f(q) < 316.5 + 0.593 log log q(log q)2 + 0.0758√q log q + log q,
+where
+f(q) := γ + log 2 +
+���log q
+π
+��� +
+
+
+
+
+
+0.027√q log q + 0.067√q + 316.229
+if 3 ≤ q < 4 · 105,
+3.715(log q)2
+if 4 · 105 ≤ q < 1010,
+(0.593 log log q + 1.205)(log q)2
+if q ≥ 1010.
+Proof. Apply [9, Cor. 20] to obtain the first inequality. For all q ≥ 1010, we have
+f(q) < γ + log 2
+π + 0.593 log log q(log q)2 + 0.000278√q log q + log q.
+For all 4 · 105 ≤ q < 1010, we have
+f(q) < γ + log 2
+π + 0.0758√q log q + log q.
+For all 3 < q < 4 · 105, we have
+f(q) < γ + log 2
+π + 316.229 + 0.0754√q log q + log q.
+7
+
+It follows that for all q > 3, we have
+f(q) < 316.5 + 0.593 log log q(log q)2 + 0.0758√q log q + log q
+Moreover, if q = 3, then f(q) ≈ 317.72, which is also majorised by the preceding bound.
+□
+Lemma 3.3. If x ≥ 2, then
+∞
+�
+m=1
+x1−2m−aχ
+(2m + aχ)(2m − 1 + aχ) = aχ + (−x)aχ tanh−1(x−1) − x1−aχ log
+√
+1 − x−2.
+Proof. If χ(−1) = −1, then aχ = 1 and
+∞
+�
+m=1
+t2m
+2m(2m + 1)
+����
+t= 1
+x
+=
+� 1/x
+0
+t−2
+∞
+�
+m=1
+t2m+1
+2m + 1 dt =
+� 1/x
+0
+tanh−1 t − t
+t2
+dt
+= 1 − x tanh−1(x−1) − log
+√
+1 − x−2.
+Similarly, if χ(−1) = 1, then aχ = 0 and
+∞
+�
+m=1
+t2m−1
+2m(2m − 1)
+����
+t= 1
+x
+=
+� 1/x
+0
+t−2
+∞
+�
+m=1
+t2m
+2m dt = −
+� 1/x
+0
+log
+√
+1 − t2
+t2
+dt
+= tanh−1(x−1) − x log
+√
+1 − x−2.
+□
+3.2. The sum over zeros. We will need explicit estimates for N(T, χ), which is the number
+of non-trivial zeros of L(s, χ) such that |γχ| ≤ T. To this end, we import [1, Cor. 1.2] below.
+Theorem 3.4. If χ is a character with conductor q ≥ 2 and T ≥ 5/7, then
+����N(T, χ) − T
+π log qT
+2πe
+���� ≤ 0.247 log qT
+2π + 6.894.
+Using Theorem 3.4, we obtain the following generalisation of [12, Lem. 1].
+Lemma 3.5. Let γ denote the ordinates of the non-trivial (and non-exceptional) zeros of
+L(s, χ) and φ : [T0, ∞) → [0, ∞) be a monotone, non-increasing function on [T0, ∞) for
+some T0 ≥ 5/7 that is continuously differentiable such that φ′(t) ≤ 0. If 5/7 ≤ U ≤ V , then
+�
+U≤|γχ|≤V
+φ(γ) = log q
+π
+� V
+U
+φ(t) dt + 1
+π
+� V
+U
+φ(t) log
+� t
+2π
+�
+dt + E(U, V ),
+(15)
+in which
+|E(U, V )| ≤ 2φ(U)
+�
+0.247 log qU
+2π + 6.894
+�
++ 0.247
+� V
+U
+φ(t)
+t
+dt.
+Proof. Theorem 3.4 implies
+�
+U≤|γχ|≤V
+φ(γ) =
+� V
+U
+φ(t) dN(t, χ)
+= log q
+π
+� V
+U
+φ(t) dt + 1
+π
+� V
+U
+φ(t) log
+� t
+2π
+�
+dt +
+� V
+U
+φ(t) dQ(t, χ),
+8
+
+in which |Q(t, χ)| ≤ 0.247 log qT
+2π + 6.894. Therefore, two applications of partial summation
+imply that the result holds with
+|E(U, V )| =
+����φ(V )Q(V, χ) − φ(U)Q(U, χ) −
+� V
+U
+φ′(t)Q(t, χ) dt
+����
+≤
+�
+m∈{U,V }
+φ(m)
+�
+0.247 log qm
+2π + 6.894
+�
+−
+� V
+U
+φ′(t)
+�
+0.247 log qt
+2π + 6.894
+�
+dt
+= 2φ(U)
+�
+0.247 log qU
+2π + 6.894
+�
++ 0.247
+� V
+U
+φ(t)
+t
+dt.
+□
+Using Lemma 3.5, we can prove the following result, which is important later.
+Theorem 3.6. If log x ≥ 10 and h = √x log x, then
+������
+�
+̺χ
+(x + h)̺χ+1 − x̺χ+1
+h̺χ(̺χ + 1)
+������
+≤
+�
+1 + log x
+√x
+� 1
+2 �√x(log x)2
+8π
++
+√x log q log x
+2π
+�
++ 0.74√x log q + 2.60√x log x.
+Proof. For T > 5/7, apply Lemma 3.5 with φ(t) = t−1 and φ(t) = t−2 respectively to obtain
+�
+5
+7≤|γχ|<T
+1
+γχ
+≤ log q log 7T
+5
+π
++ log 7T
+5 log
+5T
+28π2
+2π
++ 14
+5
+�
+0.247 log 5q
+14π + 6.894
+�
++ 247(7T − 5)
+5 000T
+=
+�
+0.7988 + log T
+π
+�
+log q + (log T)2
+2π
+− 0.5850 log T + 17.9303 − 0.247
+T
+,
+(16)
+�
+|γχ|≥T
+1
+γ2
+χ
+≤ log q
+πT + 1 + log T
+2π
+πT
++ 2
+T 2
+�
+0.247 log qT
+2π + 6.894
+�
++ 0.1235
+T 2
+=
+� 1
+π + 0.494
+T
+� log q
+T
++
+� 1
+π + 0.494
+T
+� log T
+T
+− 0.2667
+T
++ 13.004
+T 2
+.
+(17)
+Therefore, to estimate the objective sum over ̺χ, observe that
+������
+�
+|γχ|≥T
+(x + h)̺χ+1 − x̺χ+1
+̺χ(̺χ + 1)h
+������
+≤
+�
+1 +
+�
+1 + h
+x
+� 3
+2 �x
+3
+2
+h
+�
+|γχ|≥T
+1
+γ2χ
+and
+������
+�
+|γχ|<T
+(x + h)̺χ+1 − x̺χ+1
+h̺χ(̺χ + 1)
+������
+=
+������
+�
+|γχ|<T
+1
+h̺χ
+� x+h
+x
+t̺χ dt
+������
+≤
+�
+|γχ|<T
+(x + h)
+1
+2
+γχ
+≤ √x
+�
+1 + h
+x
+� 1
+2 �
+|γχ|<T
+1
+γχ
+.
+9
+
+Combine these observations to see
+������
+�
+̺χ
+(x + h)̺χ+1 − x̺χ+1
+h̺χ(̺χ + 1)
+������
+≤ √x
+�
+1 + h
+x
+� 1
+2 �
+|γχ|<T
+1
+γχ
++
+�
+1 +
+�
+1 + h
+x
+� 3
+2 �x
+3
+2
+h
+�
+|γχ|≥T
+1
+γ2χ
+. (18)
+Finally, insert the choices h = √x log x, T = √x/ log x (so that h = x/T), and apply
+Theorem 3.4, (16), (17) to yield
+������
+�
+̺χ
+(x + h)̺χ+1 − x̺χ+1
+h̺χ(̺χ + 1)
+������
+≤ √x
+� �
+1 + h
+x
+� 1
+2 �
+N(5/7, χ) +
+�
+5
+7 ≤|γχ|<T
+1
+γχ
+�
++
+�
+1 +
+�
+1 + h
+x
+� 3
+2 � �
+|γχ|≥T
+T
+γ2
+χ
+�
+≤ f1(x)√x(log x)2 + f2(x)√x log q log x + f3(x)√x log x,
+where
+f1(x) =
+�
+1 + log x
+√x
+� 1
+2 �
+1
+8π − log log x
+2π log x + 1
+2π
+�log log x
+log x
+�2�
+,
+f2(x) =
+�
+1 + log x
+√x
+� 1
+2 � 1
+2π − log log x
+π log x + 1.2732
+log x
+�
++
+1 +
+�
+1 + log x
+√x
+� 3
+2
+log x
+� 1
+π + 0.494 log x
+√x
+�
+,
+f3(x) =
+�
+1 +
+�
+1 + log x
+√x
+� 3
+2 � ��1
+π + 0.494 log x
+√x
+� �1
+2 − log log x
+log x
+�
+− 0.266
+log x + 13.004
+√x
+�
+−
+�
+1 + log x
+√x
+� 1
+2 �
+0.2925 − 0.5850 log log x + 23.5655
+log x
++ 0.247
+√x
+�
+.
+Finally, if log x ≥ 10, then f3(x) < 2.5977,
+f1(x) ≤ 1
+8π
+�
+1 + log x
+√x
+� 1
+2
+,
+and
+f2(x) ≤ 1
+2π
+�
+1 + log x
+√x
+� 1
+2
++ 0.73919
+log x .
+□
+3.3. Proof of Theorem 2.5. We begin by using the earlier preparatory lemmas to establish
+the following ingredients.
+Lemma 3.7. If χ is non-principal and primitive modulo q ≥ 3, then
+|a(χ)| ≤
+�
+2.751 log q + 24.878
+if aχ = 0
+0.593 log log q(log q)2 + 0.0758√q log q + log q + 316.5,
+if aχ = 1
+< 0.593 log log q(log q)2 + 0.0758√q log q + 2.751 log q + 316.5.
+Proof. If χ is non-principal, then
+a(χ) =
+�
+1 + bχ
+if χ(−1) = 1,
+−L′(0,χ)
+L(0,χ)
+if χ(−1) = −1.
+Apply Lemma 3.1 when χ(−1) = 1 and Lemma 3.2 when χ(−1) = −1.
+□
+10
+
+Lemma 3.8. If log x ≥ 10, h = √x log x, and χ ̸= χ0 modulo q ≥ 3 is primitive, then
+����
+c(x + h, χ) − c(x, χ)
+h
+���� ≤ aχ
+x + (1 − aχ)
+�
+log x + 1 + log x
+√x
+�
++ 7 · 10−5
+h
+.
+Proof. For any 0 < h < x, we have
+c(x + h, χ) − c(x, χ)
+h
+= aχ
+h log
+�
+1 + h
+x
+�
+− (1 − aχ)
+�x + h
+h
+log
+�
+1 + h
+x
+�
++ log x
+�
++
+∞
+�
+m=1
+�
+1 −
+�
+1 + h
+x
+�1−2m−aχ�
+x1−2m−aχ
+(2m + aχ)(2m − 1 + aχ)h.
+Apply Lemma 3.3, log(1 + x) < x for all x > 0, and h = √x log x to see
+����
+c(x + h, χ) − c(x, χ)
+h
+���� ≤ aχ
+x + (1 − aχ)
+�
+log x + 1 + log x
+√x
+�
++ aχ + (−x)aχ tanh−1(x−1) − x1−aχ log
+√
+1 − x−2
+h
+.
+To complete the proof, observe that the numerator in the final fraction decreases in x whether
+aχ = 0 or aχ = 1, so it is is majorised for all log x ≥ 10 by
+max{7 · 10−5, 4 · 10−10} = 7 · 10−5.
+□
+Insert Theorem 3.6, Lemma 3.7, and Lemma 3.8 into (7) to bound the difference
+ψ1(x + h, χ) − ψ1(x, χ)
+h
+for non-principal and primitive χ modulo q ≥ 3 and log x ≥ 10 under the choice h = √x log x;
+the result is
+����
+ψ1(x + h, χ) − ψ1(x, χ)
+h
+����
+≤ |a(χ)| +
+����
+c(x + h, χ) − c(x, χ)
+h
+���� +
+����
+�
+̺χ
+(x + h)̺χ+1 − x̺χ+1
+h̺χ(̺χ + 1)
+����
+≤ 0.593 log log q(log q)2 + 0.0758√q log q + 2.751 log q + 317.5
++ 1
+x +
+�
+1 + 1
+√x
+�
+log x + 7 · 10−5
+h
++
+�
+1 + log x
+√x
+� 1
+2 �log x
+8π
++ log q
+2π
+� √x log x + 2.60√x log x + 0.74√x log q
+<
+�
+g1(x)
+�log x
+8π
++ log q
+2π
+�
++ g2(x)
+� √x log x + 0.74√x log q + g3(q),
+(19)
+in which
+g1(x) =
+�
+1 + log x
+√x
+� 1
+2
+,
+g2(x) = 2.60 + 1
+√x + 1
+x,
+g3(q) = 317.50005 + 0.593 log log q(log q)2 + 0.0758√q log q + 2.751 log q.
+11
+
+Since g1(x) and g2(x) decrease in x, note that g1(x) < 1.034 and g2(x) < 2.607. Furthermore,
+if x ≥ max{e10, q}, then
+����
+ψ1(x + h, χ) − ψ1(x, χ)
+h
+���� < g4(x)√x(log x)2 + g5(x)√x log x + 317.50005,
+(20)
+in which
+g4(x) = 5g1(x)
+8π
++ 0.593 log log x
+√x
+< 0.215, g5(x) = 1.46g1(x)+g2(x)+0.0758+ 2.751
+√x < 3.442.
+Remark. To see the limiting behaviour of our bounds, suppose x ≥ max{e100, q} and note
+that we would have (20) with g4(x) < 0.199 and g5(x) < 3.416. Taking x even larger will
+not yield any extra benefits in (20).
+References
+1. M. A. Bennett, G. Martin, K. O’Bryant, and A. Rechnitzer, Explicit bounds for primes in arithmetic
+progressions, Illinois J. Math. 62 (2018), no. 1-4, 427–532. MR 3922423
+2. M. Bordignon, Medium-sized values for the prime number theorem for primes in arithmetic progressions,
+New York J. Math. 27 (2021), 1415–1438. MR 4319995
+3. S. Broadbent, H. Kadiri, A. Lumley, N. Ng, and K. Wilk, Sharper bounds for the Chebyshev function
+θ(x), Math. Comp. 90 (2021), no. 331, 2281–2315. MR 4280302
+4. J. R. Chen and T. Z. Wang, On distribution of primes in an arithmetical progression, Sci. China Ser. A
+33 (1990), no. 4, 397–408. MR 1070685
+5. M. Cully-Hugill and A. W. Dudek, A conditional explicit result for the prime number theorem in short
+intervals, Res. Number Theory 8 (2022), no. 3, Paper No. 61, 8. MR 4468112
+6. M. Cully-Hugill and E. S. Lee, Explicit interval estimates for prime numbers, Math. Comp. 91 (2022),
+no. 336, 1955–1970. MR 4435953
+7. A. W. Dudek, L. Greni´e, and G. Molteni, Explicit short intervals for primes in arithmetic progressions
+on GRH, Int. J. Number Theory 15 (2019), no. 4, 825–862. MR 3943895
+8. P. Dusart, Estimates of θ(x; k, l) for large values of x, Math. Comp. 71 (2002), no. 239, 1137–1168.
+MR 1898748
+9. A.-M. Ernvall-Hyt¨onen and N. Paloj¨arvi, Explicit bound for the number of primes in arithmetic pro-
+gressions assuming the generalized Riemann hypothesis, Math. Comp. 91 (2022), no. 335, 1317–1365.
+MR 4405497
+10. L. Greni´e and G. Molteni, Explicit versions of the prime ideal theorem for Dedekind zeta functions under
+GRH, Math. Comp. 85 (2016), no. 298, 889–906. MR 3434887
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+, An explicit Chebotarev density theorem under GRH, J. Number Theory 200 (2019), 441–485.
+MR 3944447
+12. R. S. Lehman, On the difference π(x) − li(x), Acta Arith. 11 (1966), 397–410. MR 202686
+13. M.-C. Liu and T. Wang, Distribution of zeros of Dirichlet L-functions and an explicit formula for ψ(t, χ),
+Acta Arith. 102 (2002), no. 3, 261–293. MR 1884719
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+gressions, Math. Comp. 42 (1984), no. 165, 265–285. MR 726004
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+University of Bristol, School of Mathematics, Fry Building, Woodland Road, Bristol,
+BS8 1UG
+Email address: ethan.lee@bristol.ac.uk
+URL: https://sites.google.com/view/ethansleemath/home
+12
+
diff --git a/CtFQT4oBgHgl3EQf_Tct/content/tmp_files/load_file.txt b/CtFQT4oBgHgl3EQf_Tct/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..554d0f102e8a0c7bc1e3df54adf11cf9ccf0d599
--- /dev/null
+++ b/CtFQT4oBgHgl3EQf_Tct/content/tmp_files/load_file.txt
@@ -0,0 +1,589 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf,len=588
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='13457v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='NT]  31 Jan 2023  THE PRIME NUMBER THEOREM FOR PRIMES IN ARITHMETIC  PROGRESSIONS UNDER THE GENERALISED RIEMANN  HYPOTHESIS  ETHAN SIMPSON LEE  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' We assume the generalised Riemann hypothesis for Dirichlet L-functions L(s, χ)  and establish explicit formulae for ψ(x, χ), θ(x, χ), and an explicit version of the prime num-  ber theorem for primes in arithmetic progressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' Introduction  Suppose that x ≥ 2, p are prime numbers, χ is a Dirichlet character modulo q ≥ 3,  ψ(x, χ) =  �  n≤x  χ(n)Λ(n),  and  ψ1(x, χ) =  � x  0  ψ(t, χ) dt =  �  n≤x  χ(n)Λ(n)(x − n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='  The purpose of this note is to update the latest explicit and conditional version of the prime  number theorem for primes in arithmetic progressions, which are asymptotic bounds for  π(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' q, a) =  �  p≤x  p≡a (mod q)  1,  θ(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' q, a) =  �  p≤x  p≡a (mod q)  log p,  and  ψ(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' q, a) =  �  n≤x  n≡a (mod q)  Λ(n),  in which 0 ≤ a < q is an integer such that (a, q) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' Explicit bounds for each of these  counting functions is a natural consequence of an explicit bound for ψ(x, χ), since  ψ(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' q, a) =  1  ϕ(q)  �  χ  χ(a)ψ(x, χ),  (1)  where ϕ(q) is the Euler-totient function evaluated at q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' we will see how a result for ψ(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' q, a)  naturally leads to results for θ(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' q, a) and π(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' q, a) later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='  Assuming the Generalised Riemann Hypothesis (GRH), the latest explicit version of the  prime number theorem for primes in arithmetic progressions was established by Ernvall-  Hyt¨onen and Paloj¨arvi in [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' We refine their results using Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' Suppose that the GRH is true, δχ = 1 if χ = χ0, and δχ = 0 if χ ̸= χ0, where  χ0 is the principal character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' If x ≥ max{e10, q} and q ≥ 3, then  |ψ(x, χ) − δχx| ≤  � √x(log x)2  8π  + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='12(log x)2  if δχ = 1,  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='223√x(log x)2 + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='446√x log x + 317.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='501  if δχ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='  The first case of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='1 is a simple application of [16, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='2)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' To prove the second  case of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='1, we prove the result for primitive non-principal characters and extend  ESL thanks the Heilbronn Institute for Mathematical Research for their support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='  1  that observation to any non-principal character in the final steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' Now, to prove the result  for these primitive characters, we note that ψ(x, χ) differs from  ∆(x, χ) = ψ1(x + √x log x, χ) − ψ1(x, χ)  √x log x  by a small error of order √x log x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' this is a consequence of the prime number theorem for  short intervals in [5] which requires log x ≥ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' All that remains is to bound ∆(x, χ) using  an explicit formula for ψ1(x, χ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' this will establish the relationship  ∆(x, χ)√x log x ≈ −  �  ̺χ  (x + h)̺χ+1 − x̺χ+1  ̺χ(̺χ + 1)  ,  (2)  in which ̺χ = 1/2 + iγχ are the non-trivial zeros of L(s, χ) and the error is well-understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='  In the end, the main challenge will be to obtain good bounds for the absolute value of this  sum over zeros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='  Using the approach described above, we actually prove a more precise statement than  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='1, which involves terms that depend on q (see (10)), then we assert the bound  x ≥ q and collect like terms so that the upper bound depends on x only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' We purposely do  these extra steps to make the result user-friendly for applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' To this end, we present  two corollaries of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='1 below;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' these will be proved in Sections 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='3 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='4 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='  The latter of these corollaries is an explicit version of the prime number theorem for primes  in arithmetic progressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='  Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' If the GRH is true, x ≥ max{e10, q}, and q ≥ 3, then  |θ(x, χ) − δχx| ≤  �√x(log x)2  8π  + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='443√x log x + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='12(log x)2  if δχ = 1,  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='223√x(log x)2 + 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='889√x log x + 317.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='501  if δχ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='  Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' If the GRH is true, (a, q) = 1, x ≥ max{e10, q}, q ≥ 3, and Li(x) =  � x  2  dt  log t,  then  ����π(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' q, a) − Li(x)  ϕ(q)  ���� <  �  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='223 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='0474  ϕ(q)  � √x log x + 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='710√x + 460.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='313,  (3)  ����θ(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' q, a) −  x  ϕ(q)  ���� <  �  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='223 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='0474  ϕ(q)  � √x(log x)2 + 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='889√x log x + 317.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='501,  ����ψ(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' q, a) −  x  ϕ(q)  ���� <  �  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='223 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='0474  ϕ(q)  � √x(log x)2 + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='446√x log x + 317.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='501.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='  Unconditional versions of the results we prove also exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' For example, Bennett, Martin,  O’Bryant, and Rechnitzer [1, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='3] give constants cπ(q) and xπ(q) such that  ����π(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' q, a) − Li(x)  ϕ(q)  ���� < cπ(q)x  (log x)2  for all  x ≥ xπ(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='  Other authors including Bordignon [2], Chen–Wang [4], Dusart [8], Liu–Wang [13], Mc-  Curley [14], and Ramar´e–Rumely [15] have done unconditional work in this area too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' In  the remainder of this note, we prove all the results above in Section 2 and an important  (technical) ingredient in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='  2  Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' To justify that our results refine the results in [9], we will compare (3) against  their [9, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' 2], because this is the clearest like-for-like comparison that we could draw  between our respective expositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' In particular, both results were proved using (1) and  a version of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' Upon asserting x ≥ q, recall that they have proved (3) with  respective coefficients  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='184 +  1  8πϕ(q) + 1  6π ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='257 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='0398  ϕ(q) ,  12 969.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='946,  and  − 237.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='934  in the upper bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' The first and second of these coefficients are clearly worse than their  counterparts in (3) and the final constant does not have much effect in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' The main  benefit of their result over ours is that it holds for x ≥ q, whereas ours holds for x ≥  max{e10, q} ≈ max{22 026.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='47, q};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' this is not too restrictive, but it is an important distinction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='  Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' To refine each of our results, an effective approach would be to refine the bound  on the sum over zeros in (2) that we obtain in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' In particular, we will bound  the sum by considering the ranges  |γχ| ≤ 5  7,  5  7 < |γχ| < T,  and  |γχ| ≥ T  separately;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' the definition of T and reasons for choosing these split points will become ap-  parent in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' However, for fixed q and η > 0, if one can use computations to evaluate  the sum exactly in the range |γχ| ≤ η for each of the primitive and non-principal characters  modulo q, then we can reduce the effect of wasteful bounds that inevitably occur whenever  small zeros are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' This method of refinement has been used to great effect when  considering sums over the non-trivial zeros of the Riemann and Dedekind zeta-functions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' for  examples see [6,10,11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' Main results  In this section, we will prove each of the main results that were presented in the intro-  duction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' First, we prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='1 by considering the cases δχ = 1 and δχ = 0 separately  in Sections 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='2 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' Next, we will describe how one can prove Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='2  using Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='1 in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' Finally, we will prove Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='3, which is an explicit  version of the prime number theorem for primes in arithmetic progressions, in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='1: Case I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' If x ≥ 2 and χ = χ0 modulo q ≥ 3, then  |ψ(x, χ0) − ψ(x)| ≤  �  n≤x  (n,q)>1  Λ(n),  where  ψ(x) =  �  n≤x  Λ(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='  (4)  Insert Lemmas 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='1-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='2 (below) into (4) to see  |ψ(x, χ0) − x| ≤  √x(log x)2  8π  + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='12 log q log x,  for all  x ≥ 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' If the Riemann hypothesis is true and x ≥ 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='2, then  |ψ(x) − x| <  √x(log x)2  8π  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' This is [16, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='2)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='  □  3  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' If x ≥ 2 and q ≥ 3, then  1  log x  �  n≤x  (n,q)>1  Λ(n) ≤ τ(q) ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='12 log q,  where  τ(q) =  �  2  if q = 6,  log q  if q ̸= 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' This is [9, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='  □  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='1: Case II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' Suppose that x ≥ 2 and χ ̸= χ0 modulo q ≥ 3 is primitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='  To begin, we observe that  ����ψ(x, χ) − ψ1(x + h, χ) − ψ1(x, χ)  h  ���� ≤  �  x<n≤x+h  Λ(n),  (5)  in which h > 0 is a parameter to be chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' Note that (5) follows from the relationship  ψ1(x + h, χ) − ψ1(x, χ)  h  = ψ(x, χ) +  �  x<n≤x+h  χ(n)Λ(n)  �  1 + x − n  h  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='  It follows from (5) and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='3 (below) that we have the relationship  ����ψ(x, χ) − ψ1(x + √x log x, χ) − ψ1(x, χ)  √x log x  ���� ≤ 3√x log x  for  log x ≥ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='  (6)  Using (6) and the upcoming explicit formula for ψ1(x, χ), we can estimate ψ(x, χ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' If log x ≥ 10, h = √x log x, and the Riemann hypothesis is true, then  �  x<n≤x+h  Λ(n) ≤ 3√x log x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' Cully-Hugill and Dudek [5, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' 1] tell us that for all log x ≥ 10 and any choice  √x log x ≤ h ≤ x3/4, we have  |ψ(x + h) − ψ(x) − h| ≤ 1  π  √x log x log  �  h  √x log x  �  + 2√x log x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='  Insert the choice h = √x log x to deduce the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='  □  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' Suppose that ̺χ = 1/2 + iγχ are the non-trivial zeros of L(s, χ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' If x ≥ 2 and  χ is principal or primitive, then  ψ1(x, χ) = δχ  x2  2 −  �  ̺χ  x̺χ+1  ̺χ(̺χ + 1) + a(χ)x + b(χ) + c(x, χ),  in which aχ = 1−χ(−1)  2  , −L′(s,χ)  L′(s,χ) = −1  s + bχ + O(s), −L′(s−1,χ)  L′(s−1,χ) = −1  s + cχ + O(s),  a(χ) = (1 + bχ)(1 − aχ)(1 − δχ) − L′(0, χ)  L(0, χ) (aχ(1 − δχ) + δχ),  b(χ) = (1 − cχ)aχ + L′(−1, χ)  L(−1, χ) (1 − aχ),  c(x, χ) = aχ log x − (1 − aχ)(1 − δχ)x log x −  ∞  �  m=1  x1−2m−aχ  (2m + aχ)(2m − 1 + aχ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='  4  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' This is a consequence of Perron’s formula;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' see [7, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='  □  It follows from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='4 that if 0 < h < x, then  ψ1(x + h, χ) − ψ1(x, χ)  h  = 1  h  \uf8eb  \uf8edδχ  (x + h)2 − x2  2  −  �  ̺χ  (x + h)̺χ+1 − x̺χ+1  ̺χ(̺χ + 1)  + a(χ)h + c(x + h, χ) − c(x, χ)  \uf8f6  \uf8f8  = δχ  �  x + h  2  �  + a(χ) + 1  h  \uf8eb  \uf8edc(x + h, χ) − c(x, χ) −  �  ̺χ  (x + h)̺χ+1 − x̺χ+1  ̺χ(̺χ + 1)  \uf8f6  \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='  (7)  Using this observation, we will prove the following result using (7) in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' If χ ̸= χ0 modulo q ≥ 3 is primitive, h = √x log x, and x ≥ max{e10, q},  then  ����  ψ1(x + h, χ) − ψ1(x, χ)  h  ���� < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='215√x(log x)2 + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='442√x log x + 317.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='501.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='  Further, Ernvall-Hyt¨onen and Paloj¨arvi proved in [9, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' 13] that if χ is a non-primitive  Dirichlet character modulo q ≥ 3 that is induced by a primitive character χ∗, then  |ψ0(x, χ) − ψ0(x, χ∗)|  log x  ≤ τ(q) ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='12 log q,  where  ψ0(x, χ) = ψ(x+, χ) + ψ(x−, χ)  2  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='  (8)  this modification to the definition of ψ(x, χ) accounts for discontinuities that may present  at prime powers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' A simple consequence of this definition is that  |ψ(x, χ) − ψ0(x, χ)| ≤ log x  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='  (9)  Finally, insert Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='5 into (6) and apply (8), (9) to extend the result to any non-  principal χ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' the end result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' If x ≥ max{e10, q} and χ ̸= χ0 modulo q ≥ 3, then  |ψ(x, χ)| <  �  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='215 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='12  √x  � √x(log x)2 +  �  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='442 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='5  √x  � √x log x + 317.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='501  < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='223√x(log x)2 + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='446√x log x + 317.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='501.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='  Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' A more precise version of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='5 that holds for all log x ≥ 10 will be given  in (19) in the following section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' If we were to use (19) instead of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='5 to deduce  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='6, then we would see the more precise  |ψ(x, χ)| <  �  g1(x)  �� 1  8π + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='12  √x  �  log x + log q  2π  �  + g2(x) + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='5  √x  � √x log x  + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='74√x log q + g3(q)  (10)  holds for all log x ≥ 10, χ ̸= χ0, and q ≥ 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' the functions gi(x) will be defined below (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' To  keep our exposition clean in the end, we asserted the bound x ≥ q onto each term containing  q and collected like terms together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='  5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' To use Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='1 to obtain explicit knowledge about θ(x, χ), we  require the following simple relationship:  |ψ(x, χ) − θ(x, χ)| ≤  �  2≤k≤  �log x  log 2  � θ(x  1  k ) < log x  log 2θ(√x),  where  θ(x) =  �  p≤x  log p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='  Clearly, we will require [3, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='1], which states that for all x ≥ 0, there is a constant  cθ = 1 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='93378 · 10−8 such that θ(x) ≤ cθx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' It follows that  |ψ(x, χ) − θ(x, χ)|  √x log x  <  cθ  log 2 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='44270.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='  (11)  Therefore, Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='2 is a straightforward consequence of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='1 and (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' Insert Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='1 into (1) to see for all x ≥ max{e10, q}, we have  ����ψ(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' q, a) −  x  ϕ(q)  ���� ≤  1  ϕ(q)  �√x(log x)2  8π  + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='12(log x)2 +  �  χ̸=χ0  |ψ(x, χ)|  �  ≤  �  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='223 +  1  ϕ(q)  � 1  8π + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='12  √x  �� √x(log x)2 + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='446√x log x + 317.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='501  ≤  �  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='223 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='0474  ϕ(q)  � √x(log x)2 + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='446√x log x + 317.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='501,  (12)  since there are ϕ(q) Dirichlet characters modulo q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' Use [3, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='1] in a similar way to how  we used it in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='3 to see  |ψ(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' q, a) − θ(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' q, a)| ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='44270√x log x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='  Now, combine the preceding observations to see that for all x ≥ max{e10, q}, we have  ����θ(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' q, a) −  x  ϕ(q)  ���� ≤  �  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='223 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='0474  ϕ(q)  � √x(log x)2 + 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='889√x log x + 317.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='501.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='  (13)  Finally, partial summation yields the relationship  π(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' q, a) = θ(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' q, a)  log x  +  � x  2  θ(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' q, a)  t(log t)2 dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='  Therefore, the contribution from the main term in (13) is  1  ϕ(q)  �  x  log x +  � x  2  dt  (log t)2  �  =  1  ϕ(q)  �  Li(x) +  2  log 2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='  We also know that for all log x ≥ 10, we have  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='889  � x  2  dt  √  t log t = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='889  �  Ei  �log x  2  �  − Ei  �log 2  2  ��  ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='32718√x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='  6  It follows that for all x ≥ max{e10, q}, we have  ����π(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' q, a) − Li(x)  ϕ(q)  ���� ≤  �  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='223 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='0474  ϕ(q)  � �√x log x +  � x  2  dt  √  t  �  + 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='889  �√x +  � x  2  dt  √  t log t  �  + 317.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='501  �  1  log x +  � x  2  dt  t(log t)2  �  +  2  ϕ(q) log 2  ≤  �  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='223 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='0474  ϕ(q)  � √x log x +  �  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='21618 + 2  �  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='223 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='0474  ϕ(q)  �� √x  +  1  log 2 − 2  √  2  �  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='223 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='0474  ϕ(q)  �  + 317.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='501  log 2  <  �  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='223 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='0474  ϕ(q)  � √x log x + 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='710√x + 460.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='313,  (14)  since ϕ(q) ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' Combine the observations (12), (13), and (14) to retrieve Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' Smoothed difference  The purpose of this section is to prove Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='5, which was an important ingredient  in the derivation of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' Preliminary bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' We will need the following lemmas in what follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' If χ is a primitive, non-principal character modulo q with χ(−1) = 1, then  |bχ| < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='751 log q + 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='878.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' This is [9, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='  □  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' If χ is primitive modulo q ≥ 3 such that χ(−1) = −1, then  ����  L′(0, χ)  L(0, χ)  ���� < f(q) < 316.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='5 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='593 log log q(log q)2 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='0758√q log q + log q,  where  f(q) := γ + log 2 +  ���log q  π  ��� +  \uf8f1  \uf8f4  \uf8f2  \uf8f4  \uf8f3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='027√q log q + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='067√q + 316.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='229  if 3 ≤ q < 4 · 105,  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='715(log q)2  if 4 · 105 ≤ q < 1010,  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='593 log log q + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='205)(log q)2  if q ≥ 1010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' Apply [9, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' 20] to obtain the first inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' For all q ≥ 1010, we have  f(q) < γ + log 2  π + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='593 log log q(log q)2 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='000278√q log q + log q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='  For all 4 · 105 ≤ q < 1010, we have  f(q) < γ + log 2  π + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='0758√q log q + log q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='  For all 3 < q < 4 · 105, we have  f(q) < γ + log 2  π + 316.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='229 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='0754√q log q + log q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='  7  It follows that for all q > 3, we have  f(q) < 316.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='5 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='593 log log q(log q)2 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='0758√q log q + log q  Moreover, if q = 3, then f(q) ≈ 317.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='72, which is also majorised by the preceding bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='  □  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' If x ≥ 2, then  ∞  �  m=1  x1−2m−aχ  (2m + aχ)(2m − 1 + aχ) = aχ + (−x)aχ tanh−1(x−1) − x1−aχ log  √  1 − x−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' If χ(−1) = −1, then aχ = 1 and  ∞  �  m=1  t2m  2m(2m + 1)  ����  t= 1  x  =  � 1/x  0  t−2  ∞  �  m=1  t2m+1  2m + 1 dt =  � 1/x  0  tanh−1 t − t  t2  dt  = 1 − x tanh−1(x−1) − log  √  1 − x−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='  Similarly, if χ(−1) = 1, then aχ = 0 and  ∞  �  m=1  t2m−1  2m(2m − 1)  ����  t= 1  x  =  � 1/x  0  t−2  ∞  �  m=1  t2m  2m dt = −  � 1/x  0  log  √  1 − t2  t2  dt  = tanh−1(x−1) − x log  √  1 − x−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='  □  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' The sum over zeros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' We will need explicit estimates for N(T, χ), which is the number  of non-trivial zeros of L(s, χ) such that |γχ| ≤ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' To this end, we import [1, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='2] below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' If χ is a character with conductor q ≥ 2 and T ≥ 5/7, then  ����N(T, χ) − T  π log qT  2πe  ���� ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='247 log qT  2π + 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='894.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='  Using Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='4, we obtain the following generalisation of [12, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' Let γ denote the ordinates of the non-trivial (and non-exceptional) zeros of  L(s, χ) and φ : [T0, ∞) → [0, ∞) be a monotone, non-increasing function on [T0, ∞) for  some T0 ≥ 5/7 that is continuously differentiable such that φ′(t) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' If 5/7 ≤ U ≤ V , then  �  U≤|γχ|≤V  φ(γ) = log q  π  � V  U  φ(t) dt + 1  π  � V  U  φ(t) log  � t  2π  �  dt + E(U, V ),  (15)  in which  |E(U, V )| ≤ 2φ(U)  �  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='247 log qU  2π + 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='894  �  + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='247  � V  U  φ(t)  t  dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='4 implies  �  U≤|γχ|≤V  φ(γ) =  � V  U  φ(t) dN(t, χ)  = log q  π  � V  U  φ(t) dt + 1  π  � V  U  φ(t) log  � t  2π  �  dt +  � V  U  φ(t) dQ(t, χ),  8  in which |Q(t, χ)| ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='247 log qT  2π + 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='894.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' Therefore, two applications of partial summation  imply that the result holds with  |E(U, V )| =  ����φ(V )Q(V, χ) − φ(U)Q(U, χ) −  � V  U  φ′(t)Q(t, χ) dt  ����  ≤  �  m∈{U,V }  φ(m)  �  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='247 log qm  2π + 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='894  �  −  � V  U  φ′(t)  �  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='247 log qt  2π + 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='894  �  dt  = 2φ(U)  �  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='247 log qU  2π + 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='894  �  + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='247  � V  U  φ(t)  t  dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='  □  Using Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='5, we can prove the following result, which is important later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' If log x ≥ 10 and h = √x log x, then  ������  �  ̺χ  (x + h)̺χ+1 − x̺χ+1  h̺χ(̺χ + 1)  ������  ≤  �  1 + log x  √x  � 1  2 �√x(log x)2  8π  +  √x log q log x  2π  �  + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='74√x log q + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='60√x log x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' For T > 5/7, apply Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='5 with φ(t) = t−1 and φ(t) = t−2 respectively to obtain  �  5  7≤|γχ|<T  1  γχ  ≤ log q log 7T  5  π  + log 7T  5 log  5T  28π2  2π  + 14  5  �  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='247 log 5q  14π + 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='894  �  + 247(7T − 5)  5 000T  =  �  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='7988 + log T  π  �  log q + (log T)2  2π  − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='5850 log T + 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='9303 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='247  T  ,  (16)  �  |γχ|≥T  1  γ2  χ  ≤ log q  πT + 1 + log T  2π  πT  + 2  T 2  �  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='247 log qT  2π + 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='894  �  + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='1235  T 2  =  � 1  π + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='494  T  � log q  T  +  � 1  π + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='494  T  � log T  T  − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='2667  T  + 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='004  T 2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='  (17)  Therefore, to estimate the objective sum over ̺χ, observe that  ������  �  |γχ|≥T  (x + h)̺χ+1 − x̺χ+1  ̺χ(̺χ + 1)h  ������  ≤  �  1 +  �  1 + h  x  � 3  2 �x  3  2  h  �  |γχ|≥T  1  γ2χ  and  ������  �  |γχ|<T  (x + h)̺χ+1 − x̺χ+1  h̺χ(̺χ + 1)  ������  =  ������  �  |γχ|<T  1  h̺χ  � x+h  x  t̺χ dt  ������  ≤  �  |γχ|<T  (x + h)  1  2  γχ  ≤ √x  �  1 + h  x  � 1  2 �  |γχ|<T  1  γχ  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='  9  Combine these observations to see  ������  �  ̺χ  (x + h)̺χ+1 − x̺χ+1  h̺χ(̺χ + 1)  ������  ≤ √x  �  1 + h  x  � 1  2 �  |γχ|<T  1  γχ  +  �  1 +  �  1 + h  x  � 3  2 �x  3  2  h  �  |γχ|≥T  1  γ2χ  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' (18)  Finally, insert the choices h = √x log x, T = √x/ log x (so that h = x/T), and apply  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='4, (16), (17) to yield  ������  �  ̺χ  (x + h)̺χ+1 − x̺χ+1  h̺χ(̺χ + 1)  ������  ≤ √x  � �  1 + h  x  � 1  2 �  N(5/7, χ) +  �  5  7 ≤|γχ|<T  1  γχ  �  +  �  1 +  �  1 + h  x  � 3  2 � �  |γχ|≥T  T  γ2  χ  �  ≤ f1(x)√x(log x)2 + f2(x)√x log q log x + f3(x)√x log x,  where  f1(x) =  �  1 + log x  √x  � 1  2 �  1  8π − log log x  2π log x + 1  2π  �log log x  log x  �2�  ,  f2(x) =  �  1 + log x  √x  � 1  2 � 1  2π − log log x  π log x + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='2732  log x  �  +  1 +  �  1 + log x  √x  � 3  2  log x  � 1  π + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='494 log x  √x  �  ,  f3(x) =  �  1 +  �  1 + log x  √x  � 3  2 � ��1  π + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='494 log x  √x  � �1  2 − log log x  log x  �  − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='266  log x + 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='004  √x  �  −  �  1 + log x  √x  � 1  2 �  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='2925 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='5850 log log x + 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='5655  log x  + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='247  √x  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='  Finally, if log x ≥ 10, then f3(x) < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='5977,  f1(x) ≤ 1  8π  �  1 + log x  √x  � 1  2  ,  and  f2(x) ≤ 1  2π  �  1 + log x  √x  � 1  2  + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='73919  log x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='  □  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' We begin by using the earlier preparatory lemmas to establish  the following ingredients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' If χ is non-principal and primitive modulo q ≥ 3, then  |a(χ)| ≤  �  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='751 log q + 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='878  if aχ = 0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='593 log log q(log q)2 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='0758√q log q + log q + 316.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='5,  if aχ = 1  < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='593 log log q(log q)2 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='0758√q log q + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='751 log q + 316.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' If χ is non-principal, then  a(χ) =  �  1 + bχ  if χ(−1) = 1,  −L′(0,χ)  L(0,χ)  if χ(−1) = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='  Apply Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='1 when χ(−1) = 1 and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='2 when χ(−1) = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='  □  10  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' If log x ≥ 10, h = √x log x, and χ ̸= χ0 modulo q ≥ 3 is primitive, then  ����  c(x + h, χ) − c(x, χ)  h  ���� ≤ aχ  x + (1 − aχ)  �  log x + 1 + log x  √x  �  + 7 · 10−5  h  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' For any 0 < h < x, we have  c(x + h, χ) − c(x, χ)  h  = aχ  h log  �  1 + h  x  �  − (1 − aχ)  �x + h  h  log  �  1 + h  x  �  + log x  �  +  ∞  �  m=1  �  1 −  �  1 + h  x  �1−2m−aχ�  x1−2m−aχ  (2m + aχ)(2m − 1 + aχ)h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='  Apply Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='3, log(1 + x) < x for all x > 0, and h = √x log x to see  ����  c(x + h, χ) − c(x, χ)  h  ���� ≤ aχ  x + (1 − aχ)  �  log x + 1 + log x  √x  �  + aχ + (−x)aχ tanh−1(x−1) − x1−aχ log  √  1 − x−2  h  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='  To complete the proof, observe that the numerator in the final fraction decreases in x whether  aχ = 0 or aχ = 1, so it is is majorised for all log x ≥ 10 by  max{7 · 10−5, 4 · 10−10} = 7 · 10−5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='  □  Insert Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='6, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='7, and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='8 into (7) to bound the difference  ψ1(x + h, χ) − ψ1(x, χ)  h  for non-principal and primitive χ modulo q ≥ 3 and log x ≥ 10 under the choice h = √x log x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='  the result is  ����  ψ1(x + h, χ) − ψ1(x, χ)  h  ����  ≤ |a(χ)| +  ����  c(x + h, χ) − c(x, χ)  h  ���� +  ����  �  ̺χ  (x + h)̺χ+1 − x̺χ+1  h̺χ(̺χ + 1)  ����  ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='593 log log q(log q)2 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='0758√q log q + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='751 log q + 317.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='5  + 1  x +  �  1 + 1  √x  �  log x + 7 · 10−5  h  +  �  1 + log x  √x  � 1  2 �log x  8π  + log q  2π  � √x log x + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='60√x log x + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='74√x log q  <  �  g1(x)  �log x  8π  + log q  2π  �  + g2(x)  � √x log x + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='74√x log q + g3(q),  (19)  in which  g1(x) =  �  1 + log x  √x  � 1  2  ,  g2(x) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='60 + 1  √x + 1  x,  g3(q) = 317.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='50005 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='593 log log q(log q)2 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='0758√q log q + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='751 log q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='  11  Since g1(x) and g2(x) decrease in x, note that g1(x) < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='034 and g2(x) < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='607.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' Furthermore,  if x ≥ max{e10, q}, then  ����  ψ1(x + h, χ) − ψ1(x, χ)  h  ���� < g4(x)√x(log x)2 + g5(x)√x log x + 317.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='50005,  (20)  in which  g4(x) = 5g1(x)  8π  + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='593 log log x  √x  < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='215, g5(x) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='46g1(x)+g2(x)+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='0758+ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='751  √x < 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='442.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='  Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' To see the limiting behaviour of our bounds, suppose x ≥ max{e100, q} and note  that we would have (20) with g4(x) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='199 and g5(x) < 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='416.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' Taking x even larger will  not yield any extra benefits in (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content='  References  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
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+page_content=' 336, 1955–1970.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' MR 4435953  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' Dudek, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' Greni´e, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
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+page_content=' 4, 825–862.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' MR 3943895  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFQT4oBgHgl3EQf_Tct/content/2301.13457v1.pdf'}
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+Empirical Software Engineering manuscript No.
+(will be inserted by the editor)
+Studying Logging Practice in Machine
+Learning-based Applications
+Patrick Loic Foalem · Foutse Khomh ·
+Heng Li
+Received: date / Accepted: date
+Abstract Logging is a common practice in traditional software development.
+Several research works have been done to investigate the different character-
+istics of logging practices in traditional software systems (e.g., Android ap-
+plications, JAVA applications, C/C++ applications). Nowadays, we are wit-
+nessing more and more development of Machine Learning-based applications
+(ML-based applications). Today, there are many popular libraries that facili-
+tate and contribute to the development of such applications, among which we
+can mention: Pytorch, Tensorflow, Theano, MXNet, Scikit-Learn, Caffe, and
+Keras. Despite the popularity of ML, we don’t have a clear understanding of
+logging practices in ML applications. In this paper, we aim to fill this knowl-
+edge gap and help ML practitioners understand the characteristics of logging
+in ML-based applications. In particular, we conduct an empirical study on
+110 open-source ML-based applications. Through a quantitative analysis, we
+find that logging practice in ML-based applications is less pervasive than in
+traditional applications including Android, JAVA, and C/C++ applications.
+Furthermore, the majority of logging statements in ML-based applications are
+in info and warn levels, compared to traditional applications where info is the
+majority of logging statement in C/C++ application and debug, error levels
+constitute the majority of logging statement in Android application. We also
+perform a quantitative and qualitative analysis of a random sample of logging
+statements to understand where ML developers put most of logging statements
+and examine why and how they are using logging. These analyses led to the
+following observations: (i) ML developers put most of the logging statements in
+model training, and in non-ML components. (ii) Data and model management
+*Corresponding author.
+Patrick Loic Foalem, Foutse Khomh, Heng Li
+Department of Computer Engineering and Software Engineering
+Polytechnique Montreal
+Montreal, QC, Canada
+E-mail: {patrick-loic.foalem, foutse.khomh, heng.li}@polymtl.ca
+arXiv:2301.04234v1  [cs.SE]  10 Jan 2023
+
+2
+Patrick Loic Foalem et al.
+appear to be the main reason behind the introduction of logging statements
+in ML-based applications. Indeed, ML developers use logging statements to
+keep track of the different stages of data processing as well as record rele-
+vant statistical information. In addition, they use logging statements to track
+all the experiments performed, to find the best model, and to record all the
+performances achieved by the model on validation or test data. Through this
+work, we hope to shed light on logging practices in ML-based applications.
+Our results also highlight the need for more ML-specific logging libraries to
+record and automate all experiments performed in the pipeline of ML-based
+applications.
+Keywords Logging practices · ML-based applications · Mining software
+repositories · Source code analysis.
+1 Introduction
+Inserting logging statements into software programs is part of good program-
+ming practice (Kernighan and Pike (1999)). Logging statements allow to col-
+lect information about the behavior of a program during its execution. This
+information is often used for various tasks in software maintenance and oper-
+ations, such as diagnosing failures, reporting error, detecting anomalies, and
+automatically generating logging text using log data (Yang and Agrawal, 2016;
+Nagaraj et al., 2012; Xu et al., 2009; Ding et al., 2022). Many research studies
+have been conducted to characterize logging practices in traditional software
+development, for example, Yuan et al. (2012) studied logging practices in open-
+source applications written in C/C++ , Zeng et al. (2019) made similar study
+for android applications, and Chen et al. (2017) studied logging practices in
+JAVA-based applications. However, to the best of our knowledge, no previ-
+ous research work examined logging practices in ML-based applications. This
+paper aims to fill this gap in the literature. Such a study could be useful for
+ML practitioners to understand logging practices in ML applications, identify
+the kind of information that is being logged, and the logging libraries used.
+Logs are generated by logging statements. Listing 1 shows some typical exam-
+ples of logging statements used in ML applications. The logging statements
+from lines 1, 3, and 9 use general logging libraries of the Python program-
+ming language, while the logging statements on lines 5, 7, 11, 13, 15 use ML-
+specific logging libraries. General logging libraries are built from Python mod-
+ule logging. Their logging statement always have log level (INFO, Warn, ERROR,
+DEBUG, CRITICAL) and in general, these libraries are older than ML-specific log-
+ging libraries. They are designed primarily for traditional software systems,
+whereas ML-specific logging libraries are designed for ML applications and
+contain information specific to machine learning and offer a straightforward
+approach for logging performance, hyperparameters, and statistical informa-
+tion about the data. Some ML logging libraries even offer the advantage of
+visualizing the logged elements in a graph, allowing users to browse through
+the learning process of the model.
+
+Studying Logging Practice in Machine Learning-based Applications
+3
+1
+logging.info("*** Reading from input files ***")
+2
+3
+tf.logging.info("[*] num_epochs: %d" % _num_epochs)
+4
+5
+dllogger.log(step=(epoch, steps_per_epoch, batch_idx), data=dllogger_data)
+6
+7
+caffe.log('Using devices %s' % str(gpus))
+8
+9
+rospy.logwarn("Updating Steering PID %s, %s, %s", config["Steer_P"],
+config["Steer_I"], config["Steer_D"])
+�→
+10
+11
+self.hparams.train_logger.log_stats({"Epoch": epoch, "lr": old_lr},
+train_stats={"loss": self.train_loss}, valid_stats=stats, )
+�→
+12
+13
+mlflow.log_metrics(metrics, step=step)
+14
+15
+wandb.log(log_data, step=learner.num_samples_collected + args.start_env_steps)
+Listing 1: Logging statement examples extracted from ML-based open-source
+application.
+To understand logging practices in ML-based applications, in this paper, we
+conduct an empirical study of 110 ML-based software projects from GitHub1;
+examining the prevalence of logging statements, the rationale for their use,
+and the characteristics of the ML components in which logging occur. Our
+research is organized along three following research questions (RQs):
+RQ1: What are the characteristics of ML-based software logging practices?
+To answer this research question, we quantify the logging statements present
+in our studied ML-based systems and compare the result with previous find-
+ings on traditional systems (i.e., JAVA, C/C++, Android). We observe a
+difference between the density of logs contained in ML-based applications
+and the density reported for traditional Android, JAVA, and C/C++ ap-
+plications. In particular, logging in ML-based applications is less pervasive
+than in JAVA, Android, and C/C++ applications. Moreover, the distribu-
+tion of logging levels in ML-based applications are different from those in
+traditional applications.
+RQ2: Which phases of the ML pipeline are more prone to logging?
+The ML pipeline consists of the following steps: data collection, data pro-
+cessing, model training, model evaluation, and model deployment. We manu-
+ally analyzed ∼2K logging statements that corresponds to more than 95% of
+confidence level with a 5% confidence interval (Yamane, 1967). We mapped
+each logging statement to one stage of the ML pipeline and examined their
+content. We observed that the majority of logging statements occurred in
+the model training step and contains information about the models’ hyper-
+parameters and performances.
+1 https://github.com/
+
+4
+Patrick Loic Foalem et al.
+RQ3: Why do developers use logging in ML-based application?
+To understand the rationale behind development decisions for using logging
+statements, we examined the code of file containing 380 logging statements
+from our sample, and assigned a reason behind its use. Results show that
+ML practitioners use logging statements for two mains reasons : data and
+model management.
+These results show a distinct logging practice in ML-based applications
+from widely-studied traditional applications and call for more research work
+to improve our understanding of logging practices in ML-based applications.
+The high proportion of logging statements related to general logging libraries
+(used for code and model management) in ML code may be a sign that ML
+practitioners are not aware of ML-specific logging libraries, such as mlflow,
+wandb. As an example, in Listing 2, ML practitioners continued to use the
+general logging libraries to log the performance of the models during its train-
+ing phase i.e., line 1, instead of using ML-specific logging libraries designed
+to log this type of information despite the advantage offered by these ML-
+specific libraries; e.g., being able to log many ML metrics at once, tracking all
+the experiments performed during the learning phase and summarizing them
+through visualizations, in order to track performance improvements. i.e., line
+3, 5, 6 and 8.
+1
+logger.info('Epoch {} Loss {:.4f} Accuracy {:.4f}'.format( epoch,
+2
+loss_meter.avg, accuracy))
+3
+dllogger.log(step=(epoch, steps per epoch, batch idx),
+4
+data=dllogger data, verbosity=0)
+5
+wandb.log({"RMSE_training": training_RMSE, "epoch": epoch,})
+6
+self.hparams.train_logger.log_stats({"Epoch": epoch, "lr": old_lr},
+7
+train_stats={"loss": self.train_loss}, valid_stats=stats, )
+8
+mlflow.log_metrics(metrics, step=step)
+Listing 2: Logging statement examples use for logging ML performance.
+Paper organization. The remainder of this paper is organized as follows.
+Section 2 describes the design of our study. Section 3 presents the motivation,
+results of each research question. Section 4 discusses threats to the validity of
+our findings. Section 5 discusses the related literature and Section 6 concludes
+the paper.
+2 Study Design
+This section presents the design of our study, which aims to understand logging
+practices in ML-based applications.
+
+Studying Logging Practice in Machine Learning-based Applications
+5
+1
+Code 
+search
+2
+Project filtration 
+& sampling
+ML Project
+Qualitative      
+Analysis
+4
+3
+5
+Extract Logging
+Statement
+RQ1
+Stratified Random 
+Sampling
+Qualitative 
+Analysis
+RQ3
+RQ2
+6
+7
+Fig. 1: Overview of the research workflow.
+2.1 Data collection
+Figure 1 provides an overview of our data extraction and analysis approach.
+We describe each step below.
+x Code search
+To select ML-based software projects for our study, we used the GitHub
+API. GitHub Search API V32 allows us to collect projects by doing code
+search. We searched for projects based on the import statements. To obtain
+ML-based projects, we follow the same approach as Openja et al. (2022),
+which consists in searching for machine learning libraries among the import
+statements of GitHub projects. Specifically, we search for the import state-
+ments corresponding to the following ML libraries: TensorFlow, Theano,
+MXNet, Scikit-Learn, Pytorch, Keras, Caffe. In total, we collect 1,547 ML-
+based projects. We focus on these libraries because of their popularity.
+y Project filtration & sampling.
+In order to retain a relevant quantity of mature projects for our study, we
+apply the filtering approach proposed by previous works (Openja et al.,
+2022; Munaiah et al., 2017), which consists in defining thresholds on the
+following three metrics: number of stars, number of commits, number of
+authors. To identify adequate threshold values, we plot the distributions
+of the three metrics. Figure 2 represents the cumulative frequency curve of
+all projects that we collected from GitHub by number of commits. For this
+metric (i.e., number of commits), we define the threshold value to be 100
+to avoid “toy” projects such as tutorial, student’s class assignments. Using
+this threshold we retained 320 projects. We also plotted the cumulative
+frequency curve for the other two metrics (see the Appendix Figure 8 for
+the corresponding figures) and derived a threshold value of one for the
+number of stars and two for the number of authors. We then filtered out
+all projects in our dataset with less than one star and less than two authors,
+and retained a final set of 110 projects.
+z Collection of ML Projects
+2 https://docs.github.com/en/rest
+
+0
+00
+0
+0
+0
+个
+0000个
+Extracting data6
+Patrick Loic Foalem et al.
+0
+100
+200
+300
+400
+#Commits
+0
+100
+200
+300
+400
+500
+600
+700
+800
+900
+1000
+1100
+1200
+1300
+1400
+Count
+Fig. 2: Cumulative frequency curve base on #Commits
+A total of 110 projects matched our selection criteria and were retained as
+subjects for our study. In order to ensure that no fork projects are present
+among our studied systems, and that at least one ML library mentioned in
+step 1 are covered in our dataset, the first author of this paper manually
+examined the repository of each of the 110 projects, and all 110 study
+projects meet these criteria.
+Table 1 presents an overview of our selected projects. Line SLOC present
+a summary statistic of source lines of code, ranging from the mean to the
+max sources lines of code for our studied subjects. Line #Commits is
+the summary statistic for the number of commits. Line #PythonFiles
+denotes the summary statistic of the number of Python files present in our
+subjects at the time of analysis. The summary statistics of the number of
+contributors in each project is presented at line #Authors. Number of
+stars is an indicator of the popularity of a project, line #Stars is a statis-
+tical description of the number of stars for our subjects with an average of
+1,167 numbers of stars.
+{ Identification of logging libraries contained in the projects
+To obtain information about each logging statement, we built a custom Ab-
+stract Syntax Tree (AST) parser from the Python built-in package AST3
+and used it to extract all import statements contained in all the Python
+files of our selected projects. We extracted a total of 13,270 import state-
+ments (Steps 3 - 4 , Figure 1). At step 4 , we manually analyzed the
+3 https://bit.ly/3gxsOkc
+
+Studying Logging Practice in Machine Learning-based Applications
+7
+Table 1: Overview of studied subjects
+Metric
+Mean
+Min
+25th Quartile
+Median
+75th Quartile
+Max
+SLOC
+53,579
+244
+3,647
+10,662
+31,228
+2,010,902
+# Commits
+759
+42
+172
+323
+782
+9,139
+# PythonFiles
+272
+5
+35
+101
+210
+7,510
+# Authors
+18
+2
+3
+5
+13
+261
+# Stars
+1,167
+1
+2
+20
+284
+32,770
+extracted import statements to identify the logging libraries involved. We
+identified the following 12 libraries in the projects of our dataset: logging4,
+rospy5, hparams6, logger7, warnings8, callbacks9, wandb10, tensorflow11,
+caffe.log12, dllogger13, ml-logger14, mlflow15.
+| Extraction of Logging Statements
+To identify logging statements contained in each file, the following steps
+were taken: (i) For each Python file in the 110 projects, we use the AST
+module from Python 3 to generate an AST for the file. (ii) We use AST
+Python libraries to extract the function call related to the 13 logging li-
+braries mention above. (iii) We analyzed each of the libraries mentioned
+in the previous step, then identified regular expressions that could match
+a logging statement of these libraries. (iv) Accordingly, we used the fol-
+lowing expressions: log, info, debug, error, fatal, warn, callbacks, warnings,
+exception, and critical as a visitor pattern16 to get all the function calls cor-
+responding to a logging statement. In order to minimize false positives in
+our logging statement dataset, we manually removed expressions containing
+“log” that are not logging statements (e.g., “dialog”, “login”, “numpy.log”,
+“tensor.log”, etc) and we also removed expressions corresponding to logging
+configuration (e.g., “caffe.init log()”, “logging.getLogger”, “logging.disable”,
+“logging.config”, etc) and this allowed us to keep a clean dataset17. This
+logging statement extraction process is inspired by the work of (Chen et al.,
+4 https://bit.ly/3gxsOkc
+5 https://bit.ly/3shluMk
+6 https://bit.ly/3sdKpk2
+7 https://bit.ly/3VMYr9K
+8 https://bit.ly/3CVpxCR
+9 https://bit.ly/3SozpLk
+10 https://bit.ly/3Sod80f
+11 https://bit.ly/3guvYoR
+12 https://bit.ly/2QjqYSf
+13 https://bit.ly/3TJyW7m
+14 https://bit.ly/3z0Xz7O
+15 https://bit.ly/2KiwDFd
+16 https://bit.ly/3TKBeD8
+17 https://bit.ly/3DlmGow
+
+8
+Patrick Loic Foalem et al.
+2017; Zhai et al., 2019). We obtained a total of ∼21K logging statements.
+The results for step 5 answer our research question RQ1.
+} Stratified Random Sampling
+In order to answer research questions RQ2 and RQ3, we prepared our
+dataset for qualitative analysis using stratified random sampling. We opted
+for a stratified sampling approach to ensure a proportionate representation
+of projects in our sample. We sampled a total of ∼2K logging statements,
+which corresponds to more than 95% of confidence level with a 5% confi-
+dence interval (Yamane, 1967).
+2.2 Data analysis
+We perform a manual analysis of the logging statements sampled at step
+6, categorizing the logging statements and mapping them to the differ-
+ent phases of the ML pipeline (step 7 on Figure 1). Specifically, for each
+logging statement found in our sample, we identified in the Python file
+and the specific class and function where it appears. Next, we identify the
+phase of the ML pipeline (Amershi et al., 2019), that is concerned by the
+implementation contained in the identified file (including the logging state-
+ment). To map logging statements to ML pipeline phases, we attribute a
+label to each phase of the ML pipeline and assign to each identified logging
+statement, the label corresponding to its phase (as identified previously).
+This allows us to answer
+RQ2 . The manual analysis and the labeling were
+performed in parallel by two authors, and the third author acted as a ref-
+eree for ambiguous cases where opinions diverged. All the diverging cases
+were discussed until reaching a consensus. To understand why developers
+use logging statements in ML-based applications and answer our research
+question
+RQ3 , we manually analyzed the source code where each logging
+statement is placed in order to identify the reason why ML practitioners
+inserted these logging statements in their project.
+3 Case Study Results
+In this section, we present the results of our three research questions. For each
+research question, we describe the motivation, the approach followed to answer
+the research question and our obtained results.
+RQ1: What are the characteristics of ML-based application logging practices?
+Motivation
+Many research studies have been conducted to characterize logging practices
+in traditional software application (Chen et al., 2017; Yuan et al., 2012; Zeng
+
+Studying Logging Practice in Machine Learning-based Applications
+9
+et al., 2019), showing the importance of logging in software systems. The
+conclusion from these researches shows that, logging practice in traditional
+applications is widely used. Many research works have used AI algorithms
+to make prediction and anomaly detection based on log files (Nagaraj et al.,
+2012; Xu et al., 2009). In addition, research has been done to help develop-
+ers in their logging decisions (Zhao et al., 2017; Zhu et al., 2015a). However,
+none of these research efforts have focused on the characteristics of logging
+in ML applications. Intuitively, there exists significant difference in the pro-
+gramming paradigms between ML and traditional application development:
+ML practitioners use a data driven approach, which lets them develop and
+train models with a set of parameters and hyperparameters, on large data, by
+contrast traditional software developers use a logic driven approach which di-
+rectly implements the program logic. Thus, intuitive, the logging of ML-based
+applications would be different from that of traditional applications. There-
+fore, in this research question, we want to study the characteristics of logging
+practices in ML-based applications.
+Approach
+Similar to previous studies on the logging practices of traditional software
+applications (Chen et al., 2017; Yuan et al., 2012; Zeng et al., 2019; Zhu et al.,
+2015a), we have studied the following aspects of the characteristics of logging
+practices in ML-based applications.
+− Logging libraries types. During the data gathering process, we obtained
+a dataset of import statements, from which we performed a qualitative
+analysis in order to identify logging libraries. We also make a distinction
+between ad-hoc logging for Al-based applications, and general-purpose log-
+ging libraries. To achieve this task, the three authors have performed anal-
+ysis of the import dataset in order to identify potential logging libraries.
+We search online materials (e.g., through the GOOGLE 18 search engine)
+in order to have more information about suspected logging libraries. Once
+the libraries were identified, we did additional research to find out more
+information about their usage. The goal of this investigation is to iden-
+tify patterns that can be used further to extract the associated logging
+statements. (Chen et al., 2017) describe general-purpose logging libraries
+as libraries that developers can define a logging level, i.e., INFO, DEBUG,
+WARN, ERROR, FATAL, EXCEPTION, CRITICAL. Then Ad-hoc log-
+ging libraries as libraries designed for the purpose of logging particular
+information which will not be or tangled in case of concurrent logging with
+other type of libraries. We have manually labeled our dataset of logging
+statements obtained during the data gathering according to two categories,
+general-purpose logging and ML-specific logging. The summary of logging
+libraries found is presented in Table 2.
+18 https://bit.ly/3VPCmaM
+
+10
+Patrick Loic Foalem et al.
+Table 2: Summary of logging libraries in ML-based application
+Logging libraries type
+Libraries
+release by
+main purpose
+ML-specific logging
+hparams
+Python software foundation
+Use to log hyperparameter of the model
+wandb
+Weight & Biases
+Use for dataset versioning, track hyperparameter,
+Experiment tracking
+TensorFlow
+Google
+Use to log and summarize ML information when developing with
+TensorFlow
+logger
+NVIDIA
+Use to log ML information at inference stage
+caffe.log
+Berkeley AI Research
+Use to log ML information when developing
+with caffe
+mlflow
+Databricks
+Use to track ML experiment in training and inference stage, logging metric
+and hyperparameter
+ml-logger
+Python software foundation
+Tracking ML experiment
+dllogger
+NVIDIA
+Tracking ML experiment
+callbacks
+Keras
+Tracking ML experiment, do early stopping,
+General logging libraries
+logging
+Python software foundation
+Log all kind of information in python
+rospy
+Open robotics
+Use to log all kind of information when working
+in robot operating system with rospy libraries
+Warnings
+Python software foundation
+Use to log warnings message in all kind of project in Python
+− The density of logging statements. We measure log density on the
+last version of each project at the time we perform this study, through
+the following steps: (i) after cloning the last version of a project, we use
+the cloc19 tool, which allows us to evaluate the total number of lines of
+code present in a project (SLOC), while excluding comments and empty
+lines. (ii) Then, we evaluated the number of logging statements present in
+each project by using a custom-built AST20 parser 2.1, which allowed us
+to extract, then make a quantitative analysis of logging statements present
+in a given project. (iii) Finally, we calculated the number of lines of code
+per logging statement using the SLOC
+NL
+formula. Such a metric has already
+been used in previous works (Yuan et al., 2012).
+− The verbosity levels of logging statements. (Li et al., 2017) shows
+through their studies the importance of assigning the correct log level to a
+logging statement and the difficulty for developers to determine the appro-
+priate log level. Similar to (Zeng et al., 2019) we study the distribution of
+logging statements in ML-based applications. From the logging statement
+dataset extracted during our study design process (Section 2.1), we evalu-
+ate the distribution of the verbosity levels based on the following patterns,
+i.e., info, warn, debug, error, except, critical, fatal, which are the verbosity
+levels suggested in the python’s official documentation21.
+We have studied the three aspects mentioned above on the 110 open-source
+projects, in order to understand the characteristics of logging practices in ML-
+based applications.
+Results
+In this section, we present the results of our first research question, discuss our
+results, findings, and compare them to prior studies. We discover three main
+findings in our RQ1.
+19 https://bit.ly/3DgYhjQ
+20 https://bit.ly/3F5bI7S
+21 https://bit.ly/3TJSpEY
+
+Studying Logging Practice in Machine Learning-based Applications
+11
+Finding 1: Logging statements in ML-based applications is commonly
+used but less pervasive than in JAVA, C#, C/C++ applications and more
+pervasive than in Android applications.
+From the 110 studied projects, 78.18% (86) contain at least one logging state-
+ment in the project. This shows that logging is prevalent in ML-based appli-
+cations. Log density estimates the likelihood for a developer to insert logging
+statements in a project, the smaller it is the more likely that we will find a
+logging statement in that project. We can observe from Figure 3 that there is a
+very high probability that a developer writes a logging statement in a C/C++
+project according to (Yuan et al., 2012) compare to others (on average one
+logging statement per thirty lines of code versus one logging statement per
+fifty-one line of code in JAVA application). For ML-based applications, the log
+density is smaller compared to Android applications (on average one logging
+statement per two hundred and ninety lines of code vs one logging statement
+per four hundred and seventy-nine logging statement) but larger than C#
+applications (290 vs 58). Thus, the probability for an ML practitioner intro-
+ducing a logging statement in an ML application is higher than an Android
+developer introducing a logging statement in an Android application but lower
+than in JAVA, C#, and C/C++ applications.
+Zeng et al. (Android)
+This paper (ML)
+Zhu et al. (C#)
+Chen et al (JAVA)
+Yuan et al. (C/C++)
+Paper
+0
+100
+200
+300
+400
+500
+Log density
+479
+290
+58
+51
+30
+Log density comparaison
+Fig. 3: Comparison of log density between ML-based applications studied in
+our paper and other types of applications studied in prior studies.
+Finding 2: The majority of logging statements in ML-based applications
+use general-purpose logging libraries.
+
+12
+Patrick Loic Foalem et al.
+Figure 4 shows the distribution of logging statements according to whether
+they belong to the general-purpose logging libraries or ad-hoc logging libraries
+for ML-based applications. The general-purpose logging statements are 11
+times more used than the ML component-specific logging statements. This
+domination can be explained by the fact that: (i) general-purpose logging
+statements are used in both ML and non-ML components, while ML-specific
+logging libraries for ML-based applications are used only in the ML compo-
+nents. For example, general-purpose logging statements are used in the ML
+components (data processing), as presented in the following code snippet which
+can be found in AI-training22
+def convert ilsvrc2010(directory , output directory,
+output filename='ilsvrc2010.hdf5',
+shuffle seed =config.default seed):
+with h5py.File(output path, 'w') as f :
+log. info('Creating HDF5 datasets...')
+prepare hdf5 file (f , n train ,
+n valid ,
+n test)
+log. info('Processing training set ... ')
+process train set (f , train , patch, n train , wnid map, shuffle seed)
+log. info('Processing validation set ... ')
+process other set (f ,
+' valid ' ,
+valid , patch, valid groundtruth, n train)
+log. info('Processing test set ... ')
+process other set (f ,
+' test ' ,
+test , patch, test groundtruth,
+n train + n valid)
+log. info('Done.')
+return (output path,)
+def prepare metadata(devkit archive, test groundtruth path):
+# Ascertain the number of filenames to prepare appropriate sized
+# arrays.
+n train = int(synsets[ 'num train images'].sum())
+log. info('Training set : {} images'.format(n train))
+log. info('Validation set : {} images'.format(len(valid groundtruth)))
+log. info('Test set : {} images'.format(len(test groundtruth)))
+n total = n train + len(valid groundtruth) + len(test groundtruth)
+log. info('Total (train/valid/test): {} images'.format(n total))
+return n train , valid groundtruth, test groundtruth, wnid map
+Code Listing 1: Code snippet of general logging statement used in ML
+component
+They are also used in the model training step, as in the following code snippet
+present in cifar10-human-experiments23,
+if
+human tune:
+logger. info('− epoch {}
+c10h train
+:
+{:.4 f} (acc: {:.4 f}) | c10h val
+:
+{:.4 f} (acc:
+{:.4 f})' .format(
+epoch, train loss meter .avg, train accuracy, loss meter.avg, accuracy))
+logger. info('−
+c10h train c10:
+{:.4 f} (acc: {:.4 f}) | c10h val c10:
+{:.4 f} (acc:
+{:.4 f})' .format(
+c10h train c10 loss meter .avg, c10h train c10 accuracy, c10h val c10 loss meter .avg,
+c10h val c10 accuracy))
+else :
+logger. info('Epoch {} Loss {:.4f} Accuracy {:.4f}'.format(
+epoch, loss meter.avg, accuracy))
+elapsed = time.time() − start
+logger. info('Elapsed {:.2f}'.format(elapsed))
+Code Listing 2: Code snippet of general logging statement used in model
+training
+(ii) ML practitioners quite often use general-purpose logging statements in
+ML components instead of specific logging statements for ML components
+22 https://bit.ly/3F1mdsy
+23 https://bit.ly/3ShI2XW
+
+Studying Logging Practice in Machine Learning-based Applications
+13
+in their applications. This is the case for example in Code Listing 1 where
+ML practitioners used a general-purpose logging statement to log the model
+parameters. The same logging task could have been done using the logging
+statement “wandb.log()” which is specifically designed to log model parameters
+during model training. This is the case in the following code snippet, which
+can be found in BPNN24.
+if
+USE WANDB:
+wandb.log({”RMSE val”: np.sqrt(np.mean(val losses)), ”RMSE training”:
+np.sqrt(np.mean(losses))}, commit=(not args.val ood))
+if
+USE WANDB:
+wandb.log({”RMSE valOOD1”: np.sqrt(np.mean(val losses1))})
+print(”root mean squared validation OOD1 error =”, np.sqrt(np.mean(val losses1)))
+if
+USE WANDB:
+wandb.log({”RMSE valOOD3”: np.sqrt(np.mean(val losses3))})
+if
+USE WANDB:
+wandb.log({”RMSE valOOD4”: np.sqrt(np.mean(val losses4))}, commit=True)
+Code Listing 3: Code snippet of ML-specific logging statement used in model
+training
+GENERAL
+ML
+Logging type
+0
+2500
+5000
+7500
+10000
+12500
+15000
+17500
+number of logging
+91.7%
+8.3%
+Log type distribution
+Fig. 4: Logging type distribution.
+Finding 3: The majority of the logging statements in ML-based appli-
+cations are in info and warnings levels, while info level logging statements
+are the majority in C/C++ applications and debug, error level logging
+statements are the majority in Android Applications.
+24 https://bit.ly/3FbDQWC
+
+14
+Patrick Loic Foalem et al.
+Figure 5 shows the distribution of logging levels for the 110 studied projects.
+More than half of the logging statements in ML-based applications are in info
+and warning levels. This can be explained by the fact that ML practitioners
+often use these log levels in data and model management as shown in the
+examples Code Listing 1, Code Listing 2, where the info level is used for
+model and data management (recording model parameters and information
+during data processing). The warnings level is used to alert inconsistencies in
+data preprocessing operations.
+def
+contains nan(a, nan policy='propagate'):
+try:
+contains nan = np.nan in set(a.ravel())
+except TypeError:
+# Don't know what to do. Fall back to omitting nan values and
+# issue a warning.
+contains nan = False
+nan policy = 'omit'
+warnings.warn(”The input array could not be properly checked for nan ”
+”values. nan values will be ignored.”, RuntimeWarning)
+def
+kurtosistest (a, axis=0, nan policy='propagate'):
+if
+n < 20:
+warnings.warn(”kurtosistest only valid for n>=20 ... continuing ”
+”anyway, n=%i” % int(n))
+def dataset from dicts( self ,
+dicts ,
+indices=None, return baskets=False, non initial token=”X”)
+logger.warning(f”[Task: {task name}] Could not convert labels to ids via
+label list !”
+f”\nWe found a problem with labels {str(problematic labels)}”)
+# TODO change this when inference flag is implemented
+except KeyError:
+logger.warning(f”[Task: {task name}] Could not convert labels to ids via
+label list !”
+”\nIf your are running in ∗inference∗ mode: Don't worry!”
+”\nIf you are running in ∗training∗ mode: Verify you are supplying a
+proper label
+list
+to your processor and check that labels in input
+data are correct .”)
+Code Listing 4: Code snippet showing the usage of warning in data processing
+(Air-Pollution25, farn26) Code Listing 4 or model training Code Listing 5
+(DeepPavlov27) in ML component,
+def train evaluate model from config() −> Dict[str, Dict[str ,
+float ]]:
+if
+' train ' not in config :
+log.warning('Train config is missing. Populating with default values')
+train config
+= config.get(' train ')
+if
+' evaluation targets ' not in
+train config
+and (' validate best ' in
+train config
+or ' test best ' in
+train config ):
+log.warning('”validate best” and ”test best” parameters are deprecated.'
+' Please, use ”evaluation targets”
+list
+instead')
+if
+iterator
+is not None:
+if
+to validate
+is not None:
+if
+evaluation targets
+is None:
+log.warning('”to validate” parameter is deprecated and will be removed in future versions. '
+' Please, use ”evaluation targets”
+list
+instead')
+evaluation targets = [' test ' ]
+if
+to validate :
+evaluation targets .append('valid')
+else :
+log.warning('Both ”evaluation targets” and ”to validate” parameters are specified. '
+' ”to validate” is deprecated and will be ignored')
+return res
+Code Listing 5: Code snippet showing the usage of warning in model training
+also its use in non-ML components as follows (Django web28).
+25 https://bit.ly/3SoY9Dh
+26 https://bit.ly/3z0EMJy
+27 https://bit.ly/3TqKwVp
+28 https://bit.ly/3F4EsgT
+
+Studying Logging Practice in Machine Learning-based Applications
+15
+def methods check(req, methods):
+try:
+assert req.method in methods
+except AssertionError:
+message = 'Method Not Allowed ({method}): {path}'.format(
+method=req.method, path=req.path)
+log.warning(message)
+raise ResponseNotAllowed(detail=message)
+Code Listing 6: Code snippet showing the usage of warning in non-ML
+component
+The distribution shown in Figure 5 is different from previous studies. The
+results of (Zeng et al., 2019) show that most logging statements in Android
+applications are at the debug and error level, while (Yuan et al., 2012) show
+that most logging statements in C/C++ applications are at the info level.
+INFO
+WARNING
+DEBUG
+ERROR
+EXCEPTION
+CRITICAL
+FATAL
+Log Level
+0
+1000
+2000
+3000
+4000
+5000
+6000
+7000
+#logging
+35.4%
+32.8%
+15.2%
+5.4%
+0.5%
+0.4%
+0.1%
+Log level distribution
+Fig. 5: Distributions of the logging statement levels.
+
+16
+Patrick Loic Foalem et al.
+Summary: Logging practice in ML-based applications is different from
+logging practices in traditional applications. One of the major differ-
+ences is the existence of multiple logging libraries in ML-based applica-
+tions, which can be categorized into two groups: general-purpose and
+ML-specific logging libraries. General-purpose logging libraries are de-
+signed to be used mainly in traditional applications, and ML-specific
+logging libraries are designed to be used in ML-based applications. Our
+results show that general-purpose logging libraries are still the most
+used in ML-based applications, despite the existence of numerous ML-
+specific logging libraries. This finding suggests that ML practitioners
+might not be aware of the existence of ML-specific logging libraries, and
+might be adopting sub-optimal logging practices. The dominance of info
+and warning in logging levels can be explained by the fact that these
+log levels are most often used initially to log information throughout
+the application, i.e., in the ML and non-ML components of the ap-
+plication. These results are a preliminary step towards understanding
+logging practices in ML-based applications and call for further work to
+investigate good logging practices for ML-based applications.
+RQ2: Which phases of the ML pipeline are more prone to logging?
+Motivation
+The results of our RQ1 indicate that ML practitioners widely use logging
+statements during the development of ML-based applications. This finding
+raises the question of log usage across the pipeline; i.e., which phases of the
+ML pipeline contain the most logging statements? Answering this question will
+help understand developers’ needs for logging during the different phases of
+the ML pipeline, and help support the development of efficient logging libraries
+and recommenders for ML systems engineering.
+Approach
+To select log statements for our analysis, we conduct a stratified random sam-
+pling as described in Subsection 2.1, and obtained ∼2K logging statements in
+our sample. Then, for each project in our sample, we analyzed the portion of
+the code where the logging statements are located and assigned each logging
+statement to one of the following five ML pipeline phases: Model Training,
+Data collection/loading, Data processing, Model evaluation, and Model de-
+ployment. We used the following clues to perform the matching of logging
+statements to the ML pipeline phases: (i) The path to the file containing a
+logging statement, often contains keywords such as “train”, “data”, “load”,
+“load data”, “model deploy”, “train image”, “fit”, “tokenizer”, “evaluator”,
+
+Studying Logging Practice in Machine Learning-based Applications
+17
+“data cleaning”. For example, it is the case for the paths of the following log-
+ging statements: pykale 29, Air-Pollution 30, DeepLearningExamples 31. (ii) In
+the file, we pay particular attention to the name of the function where the
+logging statement is located, to the comment contained in the file, and those
+that surround the function. We also leverage the message written in the log-
+ging statement. For example, for this DeepLearningExamples32 sample shown
+in the following Code Listing 7, we classify the logging statements in line #2,
+#3, #7, #16 as “model training” and those on lines #21, #22, #24, #30,
+#31 as “model evaluation”.
+1
+def train(n token, cutoffs , rank, local rank , num core per host):
+2
+tf .logging. info(”num of batches {}”.format(train record info[”num batch”]))
+3
+tf .logging. info(”step {} | lr
+{:8.9 f} ”
+4
+”| loss
+{:.2 f} | pplx {:>7.2f}, bpc {:>7.4f}, tok/s {:>6.0f}”.format(
+5
+curr step , fetched[−2],
+6
+curr loss , math.exp(curr loss),
+curr loss
+/ math.log(2), throughput))
+7
+dllogger data = {
+8
+' lr ' : fetched[−1],
+9
+' train loss ' :
+curr loss ,
+10
+' train perplexity ' : math.exp(curr loss),
+11
+'train throughput': throughput,
+12
+}
+13
+dllogger .log(step=int(curr step), data=dllogger data)
+14
+tf .logging. info(”Model saved in path: {}”.format(save path))
+15
+if
+rank == 0:
+16
+tf .logging. info(”Training throughput: {:>6.0f} tok/s”.format(meters['train throughput'].avg))
+17
+18
+def evaluate(n token, cutoffs ):
+19
+if
+FLAGS.max eval batch > 0:
+20
+num batch = FLAGS.max eval batch
+21
+tf .logging. info(”num of batches {}”.format(num batch))
+22
+tf .logging. info(”Evaluate {}”.format(eval ckpt path))
+23
+if
+(step+1) % (num batch // 10) == 0:
+24
+tf .logging. info(format str.format(step+1, num batch))
+25
+dllogger data = {
+26
+' eval latency ' : latency,
+27
+'eval throughput': throughput,
+28
+}
+29
+dllogger .log(step=step+1, data=dllogger data)
+30
+tf .logging. info(”Evaluating with: bs {}, math {} ”.format(FLAGS.eval batch size, ”amp” if FLAGS.amp
+else ”fp32”))
+31
+tf .logging. info(”| loss
+{:.2 f} | pplx {:>7.2f}, bpc {:>7.4f}, tok/s {:>6.1f}, ms/batch {:>4.2f}”.format(
+32
+avg loss , math.exp(avg loss), avg loss / math.log(2), meters['eval throughput'].avg,
+meters[' eval latency ' ]. avg))
+Code Listing 7: Code snippet from training and evaluation phase
+The manual classification task was performed by two authors with the kappa
+agreements of 0.71, i.e., substantial agreement. The ambiguous cases were
+discussed during a meeting. If an agreement couldn’t be reached in the meeting,
+the logging statement was labeled as an unclassified case. In total, we were not
+able to classify 66 logging statements in our sample. We attribute this outcome
+to the complexity of the code containing the logging statements and–or the
+absence of the key elements listed above.
+Results
+In this section, we present the results of our second research question and
+highlight our findings.
+29 https://bit.ly/3zioXy8
+30 https://bit.ly/3TXHbwE
+31 https://bit.ly/3zlGQME
+32 https://bit.ly/3zoQXAd
+
+18
+Patrick Loic Foalem et al.
+Table 3: Distribution of logging statement in the different phases of the ML
+pipeline
+ML pipeline
+Description
+Example
+% age
+Model Training
+Model
+training
+includes
+hyper-
+parameter
+tuning,
+optimizing
+cost
+function,
+fitting
+data,
+and
+model
+selection
+logging.info(’Training with a single process on 1 GPU.’)
+tf.logging.info(”[*] num epochs: %d” % num epochs)
+logging.info(f’Training time: (elapsed / 60):.2f minutes’)
+logger.info(’Epoch[%d] Time cost=%.3f’, epoch, (toc - tic))
+logger.info(f”Loading pretrained files for: ’, ’.join(self.loadables)”)
+dllogger.log(step=(epoch,
+steps per epoch,
+batch idx),
+data=dllogger data, verbosity=0)
+dllogger.log(step=”PARAMETER”,
+data=’Scheduled epochs’:
+num epochs, verbosity=0)
+35.83%
+Data
+Collec-
+tion/loading
+Data collection/loading includes col-
+lecting and reading data from different
+sources and formats: online, CSV, Json,
+text, h5, SQL.
+log.info(’Recording driving data to %s’, self.hdf5 dir)
+logging.info(’Load FTP fie without auth ( fromm )’.format(ftpURL,
+ftpFilePath))
+logger.info(”loading model from ”.format(self.path))
+logger.info(”Done loading the dataset”)
+14.47%
+Data Processing
+Data processing step includes normal-
+ization,
+transformation, validation and featur-
+ization of the data
+logger.info(f”Moving label repr(saved label) from index ”f”index, be-
+cause repr(label) was put at its place.”)
+tf.logging.warning(’‘shuffle‘ is false, but the input data stream is ’)
+logger.info(”Frequencies of rankings: ”.format(print dictionary(freq)))
+logger.info(f”Resizing input dimensions of type(self). name ” f”from
+old dims to new dims to match language model”)
+15.84%
+Model
+Evalua-
+tion/validation
+Evaluation of the model on validation or
+testing data, selection of the best model
+and validation of its correctness
+logger.log(”Save the best model into :”.format(final best name))
+logger.log(”The best model has : weights.”.format(best model.numel()))
+logging.info(f’Latency Avg: 1000.0 * latency data.mean():.2f ms’)
+tf.logging.info(’Starting Evaluation.’)
+tf.logging.info(”***** Running prediction*****”)
+logger.info(’Final test accuracy = %.1f%% (N=%d)’ % (test accuracy
+* 100, len(test bottlenecks)))
+6.20%
+Model Deployment
+Deploying the model or the system in
+production
+logger.info(”deploying model ” + self.args.triton model name +” in
+format ” + self.lib.platform)
+logger.info(”model check failed with warning: [”, error, ”]”)
+logger.info(”Warning during onnx.checker.check model in quantized
+model ignored”)
+LOGGER.debug(”Existing
+deployed
+resources:
+%s”,
+exist-
+ing resources)
+3.66%
+From the ∼2K logging statements analyzed, 35.83% of the logging state-
+ments belong to the model training phase and 19.51% belong to the portion of
+code that isn’t part of any ML component of the project. We also found that
+15.84%, 14.47%, 6.20%, and 3.66% of logging statements belong respectively
+to Data Processing, Data Loading, Model Evaluation, and Model Deployment.
+We couldn’t reach an agreement for 4.50% (i.e., 66) of logging statements in
+our sample. Figure 6 presents a distribution of the percentage of logging state-
+ments found in different components of the studied ML-based applications.
+One can see that the majority of logging statements are found in the model
+training component and that the model deployment phase contains the lowest
+percentage of logging statements. This result is not surprising since the model
+training phase produces a larger proportion of code than the other phases;
+there are many parameters to optimize in order to obtain a stable model and
+developers often have to log multiple pieces of information to guide their op-
+timization process (training and re-training). In Table 3, we provide examples
+of logging statements that were found in our studied projects. The examples
+are grouped according to the ML pipeline phase to which they belong. In the
+description column, we present the characteristics of the phase, in particular,
+we list the elements that are considered during the classification of a logging
+statement into that phase.
+
+Studying Logging Practice in Machine Learning-based Applications
+19
+Finding 4: Overall, all five phases of the entire ML pipeline contain
+logging statements. However, the model training phase has the largest
+proportion of logging statements and the model deployment phase has
+the smallest proportion of logging statements.
+Model 
+ Trainning
+Non-ML 
+ Pipeline
+Data 
+ Processing
+Data 
+ Loading
+Model 
+ Evaluation
+Model 
+ Deployment
+Unclassified
+0.0%
+5.0%
+10.0%
+15.0%
+20.0%
+25.0%
+30.0%
+35.0%
+Percentage [%]
+Fig. 6: Percentage of the analyzed sample of logging statements belonging to
+the different phases of the ML pipeline
+RQ3: Why do ML practitioners use logging?
+Motivation
+In RQ1, we observed that logging statements are prevalent in ML-based ap-
+plications but less than in traditional software applications. We also observed
+that ML developers conjointly use ML-specific logging libraries and general-
+purpose logging libraries. In this research question, we aim to get a better
+understanding of ML developers’ need for logging, throughout the life cycle
+of ML-based applications. Specifically, we want to identify the type of infor-
+mation that is being logged at a specific phase of the ML pipeline. A good
+understanding of logging practices in ML-based applications can help design
+efficient tools to support logging. It will also help devise guidelines for practi-
+tioners developing and maintaining ML-based applications.
+
+20
+Patrick Loic Foalem et al.
+Approach
+We performed a qualitative analysis to understand the reasons for using log-
+ging statements in ML-based applications. We randomly sampled 380 logging
+statements from our dataset, which corresponds to a 95% confidence level with
+5% confidence interval. We inspected each logging statement in detail, includ-
+ing the log level, the log text, and the variables in the logging statement, as
+well as the surrounding code. We combined this information with the clues
+mentioned in Section 3, to determine the type(s) of information that has been
+logged and ML developers’ reasons for logging this information. Next, we clas-
+sified the logging statements based on ML developers’ reasons for using them
+and the specific stage(s) of the ML pipeline at which the logging statements
+were introduced in the project. Each logging statement was scrutinized inde-
+pendently by the first and the third author of this paper. All ambiguous cases
+were discussed with the second author until a consensus was reached.
+Results
+In this section, we report and discuss our findings about the types of infor-
+mation being logged in ML-based applications and the specific stages of the
+ML pipeline where the logging events occur. A stage is an important step in
+a phase of the ML pipeline.
+Our analysis reveals two main reasons for logging in ML-based
+applications: data management and model management. We also iden-
+tified a secondary reason which depends on the two previously mentioned
+reasons, namely configuration management. Figure 7 summarizes our derived
+logging reasons. Each block of information in Figure 7 represents a reason for
+using logging in ML-based applications. Each entity in data management and
+model management represents a key stage (i.e., an important step in a phase of
+the ML pipeline), and the attributes are the types of information being logged.
+General purpose management depends on data management and model man-
+agement entities because it allows to log information to ensure that the entire
+environment related to the training process or the data processing process is
+properly set up. Each entity in the general-purpose block represents a specific
+purpose for using a general-purpose logging library, and the attributes are the
+types of information being logged.
+In the following, we elaborate in more detail on each of the reasons for
+logging identified in our analysis.
+Data management concerns the lifecycle of data, from the collection (e.g.,
+via APIs, web scraping, S3, and devices such as cameras, and phones) to the in-
+gestion into the training process which includes the following steps: data load-
+ing, data parallelism (i.e., the distribution of the data across different nodes,
+which operate on the data in parallel), data processing, and data streaming.
+Throughout the lifecycle of the input dataset, ML practitioners often use log-
+ging to extract important information (for example about errors) and track
+
+Studying Logging Practice in Machine Learning-based Applications
+21
+Data management
+Model management
+Data  loading 
+State event
+File name
+Data format
+Data Parallelism
+Memory
+GPU
+Batch  size
+CPU
+Data Processing
+Statistic
+Missing value
+Schema
+Data  Stream
+State event
+Batch  size
+Schema
+Data Collection
+State event
+Data format
+Device
+Model Import
+State event
+Model Evaluation
+State event
+Model
+Memory
+Time
+Epoch
+Loss
+Metric
+Size
+Model Validation
+Metric
+Model Training
+State event
+Time
+Memory
+Metric
+Epoch_step
+Epoch_number
+Epoch_stoping
+Loss
+Scheduler
+Optimizer
+Learning rate
+Model Parallelism
+State event
+GPU
+Memory
+CPU
+Model  Deployement
+State event
+CPU
+GPU
+Time
+inference_optimizer
+Configuration management
+Dependencies
+PyCUDA
+cuDNNA
+Conda
+Environment Setting
+CUDA
+AI_library_version
+Reason for Logging
+Fig. 7: Reason for Logging in ML-based application. Entity represent different
+stage in ML-based system. Attributes represent what type or what information
+is logged.
+the evolution of the data. We have identified five stages in which logging is
+prevalent.
+– Data Collection: When collecting data, ML practitioners often log the
+following information of which some examples are presented in Listing 3: (i)
+State event log -which contains information about a particular event, e.g.,
+recording data, processing erroneous data during data collection. Those
+state event logs are introduced at the beginning, during, or at the end of
+those events, e.g., Line 1, 5, 6, 7 in Listing 3. (ii) Data format log – the
+information logged is related to the type or format of data collected, e.g.,
+Line 4, 1. (iii) Device log – the logged information is related to the data
+acquisition tools or devices, e.g., Line 2, 8. Logging statements present in
+Listing 3 can be identified in the following projects: DeepLearningExam-
+ples33, DeepCamera34.
+1
+log.info('Recording driving data to %s', self.hdf5_dir)
+2
+log.debug('Grabbing images over ZMQ from %s', conn_string)
+3
+log.debug('obz_exists? %r should_record? %r', obz is not
+None,self.should_record)
+�→
+33 https://bit.ly/3GDstYo
+34 https://bit.ly/3XvpQ10
+
+22
+Patrick Loic Foalem et al.
+4
+logging.info("Converted {} frames in {}".format(num_frames, tfrecord))
+5
+logging.info("Processed {} records".format(len(tfrecords)))
+6
+logging.info("Corrupted record {}".format(tfrecord))
+7
+logging.info("Processing record # {}: {}".format(record_id, record))
+8
+logging.info('Camera: using Jetson onboard camera')
+9
+Listing 3: Logging statement examples in the data collection step
+– Data loading: Once the data collection is done, in this step the data is
+loaded from different sources such as: S3, directory, hard disk, in different
+formats such as JSON, CSV, etc. During this step ML practitioners usually
+record : (i) State event log - through this log here ML practitioners
+usually use state event log to monitor the data loading process through
+the records on the beginning of the loading process (e.g. line 4, 6), the
+errors occurred during the data loading (e.g., line 1), information about
+the file (e.g., line 4, 3) and the end of the data loading (e.g., line 5, 2).
+(ii) Data information log - here logged information is related to file
+name and the data formats as mentioned in Figure 7. Listing 4 gives some
+examples of logging statements found in our study subjects, especially in
+the projects: cs-ranking35, deepdrive36
+1
+log.error('Could not load %s - skipping - error was: %r', h5_filename,
+e)
+�→
+2
+log.info('finished loading %s', h5_filename)
+3
+log.info('data name %s', dataset)
+4
+logging.info('loading csv')
+5
+logger.info("Done loading the dataset")
+6
+log.info('loading %s', h5_filename)
+Listing 4: Logging statement examples in the data loading step
+– Data parallelism: This step occurs when the collected data is too heavy
+compared to the available resources such as GPU and memory to be loaded
+directly. Then ML practitioners will perform data parallelization which
+consists in dividing the data according to the number of available GPUs or
+CPUs in order to accelerate the data processing and eventually the training
+process. They usually use: (i) GPU/CPU log - which consists of logging
+resource information related to the number of GPU or CPU used (e.g., line
+3, 4). (ii) Memory consumption log - are logging statement related to
+the memory utilization during the training process by ML practitioners
+(e.g., line 1, 2). (ii) Batch log - Here, data are loaded in batches, so the
+batch details like the batch size are logged (e.g., line 5). Some examples
+of logging statements that we have identified in our study subjects are
+presented in Listing 5 and can be found in projects cs-ranking37, rtg38.
+35 https://bit.ly/3XCGFXQ
+36 https://bit.ly/3U1yhy1
+37 https://bit.ly/3Vkp0lz
+38 https://bit.ly/3EXYwRL
+
+Studying Logging Practice in Machine Learning-based Applications
+23
+1
+log.warning(f"Going to buffer data; this may consume all the memory
+crash. Current usage={max_RSS()[1]}.")
+�→
+2
+log.warning(f"Buffered {self.n_inp_recs:,} records Current memory
+usage={max_RSS()[1]}")
+�→
+3
+logging.info("device_ids%s", gpu)
+4
+logger.info("rank%s", cpu)
+5
+logging.info("sampler size%s", batch_size)
+Listing 5: Logging statement examples in the data parallelism step
+– Data streaming: This step occurs in online learning ML systems where
+the data is fed into ML algorithms in sequential order. This type of system
+is usually implemented when it becomes impossible to train the entire data
+set in a ML model, or when the data arrives in a streaming manner. During
+this stage, it is important for ML practitioners to ensure the consistency
+between different streams of data received by ML algorithms (e.g., line 2)
+we call this (i) Schema log . To this end it is important for ML practi-
+tioners to have information about the size of each stream of data received,
+their schema, monitor the process of receiving each stream of data through
+(ii) State events log (e.g., line 1, 2, 3, 4, 5). all this information will
+allow ML practitioners to have a track in case the distribution of the data
+changes. Listing 6 present some of the relevant logging statements found
+in the projects: DeepCamera 39, attention-lvcsr40, and attention-lvcsr41.
+1
+logger.info("Monitoring on auxiliary data finished")
+2
+logger.info('No changes in schema detected.')
+3
+logger.info("sending StopIteration")
+4
+logger.info("sending {} arrays".format(len(data)))
+5
+logger.info("Monitoring on auxiliary data started")
+Listing 6: Logging statement examples in the data streaming step
+– Data processing: This step involves manipulating attributes and labels of
+data to produce meaningful information from the data. During the handling
+process, ML practitioners usually record (Listing 7) : (i) Missing infor-
+mation log - which are related to missing values, missing labels (e.g., line
+1, 6, 7). (ii) Statistic log - this logging statement contains local estimators
+(mean, median, minimum, maximum), and variable estimators (variance,
+standard deviation) (e.g., line 3, 4). (iii) Schema log - logging statement
+here contains information about the schema of some data attribute (e.g.,
+line 5). These examples can be found in the following subjects: csrank 42,
+Air-Pollution.43
+39 https://bit.ly/3XE6ren
+40 https://bit.ly/3XlSuBD
+41 https://bit.ly/3i8ulOw
+42 https://bit.ly/3Ox7tUZ
+43 https://bit.ly/3Oz2svs
+
+24
+Patrick Loic Foalem et al.
+1
+logger.info("Missing values {}: {}".format( col, np.isnan(arr).sum() /
+len(arr))
+�→
+2
+logger.info("Sampled instances {} objects {}".format(X.shape[0],
+X.shape[1]))
+�→
+3
+log.info(f"mean:{mean}, std:{std:.4f} ;; low:{low:.4f} high:{high:.4f}
+;; invert:{invert}")
+�→
+4
+logger.info("Min {}: Max {}".format(np.nanmin(arr), np.nanmax(arr)))
+5
+logging.info('FORMAT --dateformat: {}'.format(FORMAT))
+6
+tf.logging.fatal('Label does not exist %s.', label_name)
+7
+logger.info("Missing values {}: {}".format( col, np.isnan(arr).sum() /
+len(arr))
+�→
+8
+Listing 7: Logging statement examples in the data processing step
+Model management. Model management concerns the development of the
+model, which consists of developing, evaluating, validating and releasing it
+into production. A major difficulty of this stage is to track all the experiments
+performed in search of the best model and also better generalize on new data.
+ML practitioners often iterate on several experiments before arriving at the
+best model. Knowing which parameters or experiment led to the best model
+is a tedious task and can be time consuming, especially when done manually.
+One approach to solve this problem is to log all relevant information. Logging
+all relevant information allows ML practitioners to reproduce or compare past
+experiments and then select the best model. We have identified six impor-
+tant steps of model management during which ML practitioners log relevant
+information.
+– Model Import: This step includes the import of the models or pre-trained
+weights, in different formats, during which the ML practitioners usually
+record state events log to ensure that the model importation process has
+been carried out correctly, such as the examples shown in Listing 8 from
+the AI-Training44 project.
+1
+logger.info(f"Loading pretrained files for: {',
+'.join(self.loadables)}")
+�→
+2
+log.info('Downloading weights %s', folder)
+3
+logging.info("Model loading done")
+4
+logger.info("loading model from {}".format(self.path))
+Listing 8: Logging statement examples in the model import step
+– Model Parallelism: This step occurs when there are memory constraints
+between the size of the model and the GPU or CPU device, as large mod-
+els can’t be trained on a single GPU. Thus, ML practitioners usually split
+the model onto several GPU devices. During this step, ML practitioners
+44 https://bit.ly/3Vg7BdM
+
+Studying Logging Practice in Machine Learning-based Applications
+25
+usually record (i) State event log - which are logging statements con-
+taining information about the start and end of the training process (e.g.,
+line 5). (ii) Memory consumption log - logging statements are related
+to memory consumed during the training stage. (iii) GPU/CPU log -
+Information about the GPU and CPU devices used (e.g., lines 1, 2, 3, 6,
+7). Listing 9 shows some examples of such logging statements.
+1
+logging.info('Training in distributed mode with multiple processes, 1
+GPU per
+�→
+2
+process. Process %d, total %d.'% (args.rank, args.world_size))
+3
+logger.info('device: {}'.format(device))
+4
+logger.info("device: {} n_gpu: {} distributed training:
+{}".format(device, n_gpu, bool(args.local_rank != -1)))
+�→
+5
+logger.info('begin training on multiple GPU')
+6
+logger.info("device: {} n_gpu: {}".format(device, n_gpu))
+7
+logging.info('Gradient averged for the rank of {}'.format(rank))
+Listing 9: Logging statement examples in the model parallelism step
+– Model Training: In this step, ML practitioners use data to train ML
+models by manipulating many hyperparameters of the models and the op-
+timization functions. This step is very time and resource-consuming and
+non-deterministic hence the need for ML practitioners to track a lot of in-
+formation during this training process and record it through logging state-
+ments (e.g., examples shown in Listing 10). (i) Time log - Evaluation of
+training time or time taken to complete an epoch, e.g., Line 15. (ii) Mem-
+ory consumption log - Evaluation of memory consumption, e.g., Line 14.
+(iii) Hyperparameters log - Recording of hyperparameters during the
+training process, e.g., Line 2, 6, 7. (iv) Metrics log - Recording of metrics
+during training, e.g., Line 18, 13. (v) Model size log - Recording of the
+model size, e.g., : Line 4, 14, 16. (vi) State events log - Recording of states
+event which allows to track the beginning, the possible errors, the update
+of hyperparameters, the end of the training process e.g., Line 5, 8, 13, and
+eventually key information related to the end of the learning process of ML
+algorithms e.g., Line 1, 8. Some logging statements can be found in our fol-
+lowing study subjects: BPNN 45, attention-lvcsr 46, AutoDL-Projects 47,
+AutoDL-Projects 48.
+1
+logger.log("Early stop the pre-training at {:}".format(iepoch))
+2
+log.info(f"New learning rate dividor = {self._learning_rate_cur_div}")
+3
+log.info('pooling_0 shape: %s' % pooling_0.shape)
+4
+logger.log("The base-model has {:}
+weights.".format(base_model.numel()))
+�→
+5
+logger.log("[ONLINE] [{:03d}/{:03d}] loss={:.4f},
+score={:.4f}".format(idx, len(env), future_loss.item(), score )
+�→
+45 https://bit.ly/3V2Y2PE
+46 https://bit.ly/3EWJziG
+47 https://bit.ly/3tRdHpm
+48 https://bit.ly/3Xqnnoo
+
+26
+Patrick Loic Foalem et al.
+6
+logger.log("scheduler
+: {:}".format(scheduler))
+7
+logger.log("optimizer
+: {:}".format(optimizer))
+8
+logger.info("Initializing the training algorithm")
+9
+log.info('Did not improve on the {} of {}'.format(m_name,
+self.score_best))
+�→
+10
+logging.info('rank %s: failing epoch %s batch %s', rank, epoch, batch)
+11
+wandb.log({"RMSE_training": np.sqrt(np.mean(losses))})
+12
+logger.log("TRAIN [{:}] loss = {:.6f}".format(iter_str, loss.item()))
+13
+logger.error("Error occured during training." + error_message)
+14
+logger.log("FLOP = {:} MB, Param = {:} MB".format(flop, param))
+15
+logging.info(f'Training time: {(elapsed / 60):.2f} minutes')
+16
+logger.log("The model size is {:.4f}
+M".format(xmisc.count_parameters(model)))
+�→
+17
+wandb.log({"RMSE_val": val_RMSE, "RMSE_training": training_RMSE})
+Listing 10: Logging statement examples in the model training step
+– Model Evaluation: This step is part of the model development process.
+It includes the search for the best model that generalizes the data, as well
+as the evaluation of its performance on the validation or test data. During
+this stage, ML practitioners usually log information that will allow tracing
+the whole evaluation process through: (i) State events log - that will
+generally allow logging the beginning, the errors that occurred, and the
+end of the evaluation process, e.g., Line 5. (ii) Best model log - log the
+best model information obtained during the fine-tuning phase, e.g., Line
+1. (iii) Memory consumption log - records the memory consumption
+during the evaluation process, e.g., Line 2. (v) Metric log - records the
+performance measures (metric, loss) at each epoch, e.g., Line 8, 3. (iv)
+Model size log - records the model size, e.g., Line 7. Inference time log -
+records the time taken by the model to infer predictions, e.g., Line 6. Listing
+11 are examples of logging statements collected in our study subjects at this
+stage, they can be found in AutoDL-Projects49, DeepLearningExamples50.
+1
+logger.log("Save the best model into {:}".format(final_best_name))
+2
+logger.log( ""Finish training/validation in {:} with Max-GPU-Memory of
+{:.2f} MB, and save final checkpoint into {:}"".format(
+convert_secs2time(epoch_time.sum, True)
+�→
+�→
+3
+tf.logging.info('%s: Step %d: Validation accuracy = %.1f%% (N=%d)'
+%(datetime.now(),
+�→
+4
+i, validation_accuracy * 100,len(validation_bottlenecks)))
+5
+logger.info("***** Running evaluation *****")
+6
+logger.info("time for inference {} perf {}".format(eval_end -
+eval_start, num_examples * 100 / (eval_end - eval_start)))
+�→
+7
+logger.log("FLOP = {:} MB, Param = {:} MB".format(flop, param))
+8
+logger.log("{:} {:} epoch={:03d}/{:03d} :: Train [loss={:.5f},
+acc@1={:.2f}%, acc@5={:.2f}%] Valid [loss={:.5f}, acc@1={:.2f}%,
+acc@5={:.2f}%]".format( time_string(), need_time, epoch,
+total_epoch, train_loss, train_acc1, train_acc5, valid_loss,
+valid_acc1, valid_acc5,)
+�→
+�→
+�→
+�→
+49 https://bit.ly/3GKwJ8J
+50 https://bit.ly/3VmRc7k
+
+Studying Logging Practice in Machine Learning-based Applications
+27
+Listing 11: Logging statement examples in the model evaluation step
+– Model Validation: The purpose of this step is to assess the precision
+and performance of the model obtained during the evaluation phase on
+real data. To do so, ML practitioners record metrics used as a measure of
+performance for their ML system, as presented in Listing 12.
+1
+logger.info('Final test accuracy = %.1f%% (N=%d)' % (test_accuracy *
+100, len(test_bottlenecks)))
+�→
+Listing 12: Logging statement examples in the model validation step
+– Model Deployment: At this stage, in order to keep track of everything
+that happens during this phase ML practitioners often record information
+such as: (i) State events log - which are logging statements related to the
+process of converting the model into a lighter intermediate representation
+by applying graph optimizations, layer merging, e.g., Line 1, 2, 3, 6. (ii)
+GPU/CPU log - Device information such as CPU and GPU are recorded
+during model prediction. (iii) Time latency log - it’s critical in a pro-
+duction environment to deliver inferences quickly. Hence, ML practitioners
+record the execution time of an inference, e.g., Line 7. (iv) Optimizer log
+– there are different optimizers that can be used to convert an ML model
+into a lighter representation optimized for real-time inferences, e.g., Line
+4, 6. (iv) Precision log – the precision enabled by an optimizer is an im-
+portant piece of information that ML practitioners record when the model
+is in production, e.g., Line 5. Listing 13 are examples of logging statements
+collected in our study subjects corresponding to the model deployment
+step.
+1
+logger.info("model check failed with warning: [", error, "]")
+2
+logger.warning("Warning during onnx.checker.check model in quantized
+model ignored")
+�→
+3
+logging.info('Total node count before and after TF-TRT conversion:',
+num_nodes, '->', len(frozen_graph.node))
+�→
+4
+logger.info('TRT node count:',len([1 for n in frozen_graph.node if
+str(n.op)'TRTEngineOp']))
+�→
+5
+tf.compat.v1.logging.info("Precision = %s", "fp16" if FLAGS.amp else
+"fp32")
+�→
+6
+tf.compat.v1.logging.info('Converting graph using
+TensorFlow-TensorRT...')
+�→
+7
+tf.compat.v1.logging.info("Total Inference Time W/O Overhead = %0.2f
+for Sentences = %d", predict_time_wo_overhead, num_sentences)
+�→
+Listing 13: Logging statement examples in the model deployment step
+Configuration management steps are not explicitly represented in the ML
+pipeline but are essential to ensure that the ML components perform in a
+manner consistent with expectations over time. Configuration management
+
+28
+Patrick Loic Foalem et al.
+activities generally include the management of configurations or dependencies
+on special devices and the management of libraries that are essential to en-
+sure that ML components work as expected. Through our analysis, we have
+identified logs related to the following configuration management activities:
+– Dependencies configuration log: This activity is usually implemented
+when the ML component of the application will depend on a specific library
+in order to ensure that the ML module of an application can work properly.
+Hence, the need for ML practitioners to log information related to impor-
+tant libraries used. Listing 14 Presents some logging statements found dur-
+ing this step. They can be found in our studied subjects: DeepLearningEx-
+amples51, deepdrive52.
+1
+log.info('Installed UEPy python dependencies')
+2
+warnings.warn('CVM does not support memory profile, using Stack VM.')
+3
+warnings.warn("PyCUDA import failed in theano.misc.pycuda_init")
+4
+logging.info("Using torch DistributedDataParallel. Install NVIDIA Apex
+for Apex DDP.")
+�→
+Listing 14: Logging statement examples in the dependencies configuration step
+– Environment setting log: The environment in which ML applications
+run is different from traditional applications, in fact, ML applications gen-
+erally need more resources or special devices such as CPU, and GPU in
+order to run properly. ML practitioners will generally ensure that the en-
+vironment in which the application needs to run contains a minimum of
+resources. Therefore, information about critical resources will be logged
+(e.g., examples in Listing 15). Some examples can be found in the follow-
+ing subjects: DeepPavlov53, AutoDL-Projects54, AI-Training55.
+1
+logger.log("cuDNN
+Version
+:
+{:}".format(torch.backends.cudnn.version()))
+�→
+2
+logger.log("PyTorch Version
+: {:}".format(torch.__version__))
+3
+logger.warning("This recipe needs the sox-io backend of torchaudio")
+4
+logger.log("CUDA available
+: {:}".format(torch.cuda.is_available()))
+5
+logger.log("CUDA GPU numbers : {:}".format(torch.cuda.device_count()))
+6
+"logger.log(""CUDA_VISIBLE_DEVICES : {:}""
+7
+.format( os.environ[""CUDA_VISIBLE_DEVICES""] if
+""CUDA_VISIBLE_DEVICES"" in os.environ else ""None"")"
+�→
+8
+caffe.log('Using devices %s' % str(gpus))
+9
+_logger.warning("We are not able to detect the number of CPU cores." "
+We disable openmp by default.")
+�→
+10
+_logger.info('Conda mkl is not available: %s', e)
+Listing 15: Logging statement examples in the environment setting step
+51 https://bit.ly/3Eyc1WB
+52 https://bit.ly/3Ex9tZ5
+53 https://bit.ly/3TZ4mXi
+54 https://bit.ly/3tRawy3
+55 https://bit.ly/3ACmajS
+
+Studying Logging Practice in Machine Learning-based Applications
+29
+Finding 5: ML practitioners use logging statements to record impor-
+tant information in data management (e.g., recording the data format in
+data collection), model management (recording the hyperparameters in
+model training), and configuration management (e.g., recording the used
+resources or libraries). Our observations provide guidance for ML practi-
+tioners to improve their ML logging and insights for future efforts to im-
+prove ML logging practices (e.g., by providing ML logging libraries that
+facilitate the logging of different types of important information through-
+out the life-cycle of ML-based applications).
+4 Threats to validity
+In this section, we discuss the potential threats to the validity of our research
+methodology and findings.
+External validity
+The subjects used in this paper to answer our different research questions are
+open-source ML-based projects from GitHub. The selection of these projects
+may be subject to the following threats:
+Our results and findings may not apply to ML projects written in other
+languages (e.g., JAVA, C# or R) rather than Python since our analysis was
+mainly done on the portions of code written in Python in our study subjects.
+Hence, it is necessary that future works explore ML projects written in other
+programming languages since ML practitioners working with these languages
+may have different logging practices for ML projects. However, since Python
+is considered to be the lingua franca for ML-based application development
+(Dilhara et al., 2021), we believe that our study provides a good understanding
+of the logging practices in ML-based systems.
+Construct validity
+The threats to the construct validity of our research are related to errors that
+may have occurred during the extraction of logging statements.
+To avoid extracting logging statements that have been commented out by
+developers, we have developed a static code analyzer on top of the standard
+Python AST parser (which is widely used in the static analysis of Python code
+(D’Souza et al., 2016; Dilhara et al., 2021)). This static code analyzer which is
+available in our replication package56library extract only uncommented state-
+ments; enabling us to avoid collecting logging statements that were commented
+out by developers.
+56 Scripts and data files used in our research are available online and can be found here:
+http://bitly.ws/yr6c
+
+30
+Patrick Loic Foalem et al.
+Internal validity
+We have manually mapped logging statements to their corresponding ML
+pipeline phase, to answer RQ2. Then, in RQ3, we manually analyzed the log-
+ging statements to identify the type of information logged and create a taxon-
+omy. However, our manual analyses are subject to the subjective judgment of
+the people performing the analysis. This raises a threat to the internal validity
+of our results. To mitigate this threat, manual analyses were performed by the
+three authors of this paper with strong industry and academic backgrounds
+in ML systems engineering. Two authors performed the manual analysis, in
+case of disagreements we had a group discussion with the third author until
+a common agreement was reached. We believe that this approach reduces the
+chance of introducing false positives in our analyses. However, future replica-
+tions and extensions of our work are desirable. All the data and scripts used
+in our study are available in our replication package57
+5 Related work
+In this section, we introduce and discuss two areas of related works on logging
+practices: (i) research done on logging characteristics, and (ii) research done
+on logging decisions.
+(i) Characterizing logging practice: Many research works have been done
+in characterizing logging practices. (Yuan et al., 2012) conducts the first
+empirical study in the characterization of logging practices on four open-
+source projects written in C/C++ and found that logging is pervasive.
+(Chen et al., 2017) conducts a similar study by analyzing 21 projects writ-
+ten in JAVA which is a replication study whose objective is to generalize
+the findings obtained by (Yuan et al., 2012) for projects written in JAVA.
+Their result shows a difference in logging practices between applications
+written in JAVA and those written in C/C++. (Zeng et al., 2019) studied
+the logging practices in 1,444 Android projects and then compared their
+findings with those of previous works. However, none of the previous works
+have focused on logging practices in AI-based applications. Our paper fills
+this knowledge gap.
+(ii) Logging decisions: Logging decision involves research that helps devel-
+opers decide where to introduce a logging statement and what level of
+verbosity should be assigned. These studies use AI algorithms to make
+predictions on where or what to log. (Zhu et al., 2015b) propose a “learn-
+ing to log” framework using machine learning techniques, which aims to
+help developers make decisions on where to add logging statements during
+development. Furthermore, Li et al. (2020) used a deep learning-based ap-
+proach to help developers in their logging decision at the block level. (Li
+et al., 2017) propose an ordinal regression model, which accurately provides
+57 http://bitly.ws/yr6c
+
+Studying Logging Practice in Machine Learning-based Applications
+31
+the suggestion of the logging levels when developers add logging statement
+and (Li et al., 2021) suggest also log level by using Ordinal Based Neu-
+ral Networks. More recently, (Mastropaolo et al., 2022) propose LANCE
+a framework to generate a complete logging statement using deep learn-
+ing. This obviously shows that AI algorithms are used to assist developers
+of traditional applications in their logging decisions but no research has
+been conducted to assist AI practitioners in their logging decisions in AI-
+based applications. Our work is therefore a good starting point for future
+research.
+6 Conclusions
+Logging practices have been adopted by developers as part of good program-
+ming practices. Logs generally allow developers to diagnose their programs at
+runtime in order to reduce maintenance efforts. Logging practices have been
+the subject of numerous studies in traditional software systems such as mobile
+applications, JAVA applications, and open-source applications. To the best
+of our knowledge, this paper presents the first attempt to study the practice
+of logging in ML-based applications. We studied 110 open-source ML-based
+applications from Github. Our research has yielded the following findings:
+Logging in ML-based applications is commonly used but less pervasive than in
+JAVA, C#, C/C++ applications and more pervasive than in Android applica-
+tions. The majority of logging statements are in INFO and WARNING levels. More-
+over, we found that ML-based applications use two kinds of logging libraries:
+general logging libraries and those specific to ML-based applications. However,
+despite the existence of ML-specific libraries, general logging libraries remain
+the most used in ML applications. In order to identify which ML pipeline con-
+tains the most logging statements in ML-based applications, we performed a
+qualitative and quantitative analysis. Our findings indicate that the majority
+of logging statements are found in the model training phase and the model
+deployment phase contains the smallest portion of logging statements. Fur-
+thermore, we observe that ML practitioners use logging statements to record
+important information related to data management (e.g., recording the data
+format in data collection), model management (recording the hyperparame-
+ters in model training), and configuration management (e.g., recording the
+used resources or libraries). The contribution of this paper is as follows:
+– To the best of our knowledge, this is the first study that quantitatively and
+qualitatively analyzes the logging practice in ML-based applications.
+– We identified the general-purpose and ML-specific logging libraries that
+are used in ML-based applications.
+– We identified ML phases in which practitioners use logs and the types of
+information they log. Our findings provide guidance for ML practitioners
+to improve their ML logging, as well as provide insights for future efforts
+to improve ML logging practices (e.g., by providing ML logging libraries
+
+32
+Patrick Loic Foalem et al.
+that facilitate the logging of different types of information in different ML
+phases).
+Overall, our findings highlight the need for more ML-specific libraries to
+support the development of ML-based applications. More research is also
+needed to improve our understanding of logging needs and challenges in the
+context of ML systems engineering.
+Conflict of interest
+The authors declare that they have no conflict of interest.
+Data availability statement
+The datasets generated during and/or analysed during the current study are
+available in the [foalem] repository, [https://github.com/foalem/ML-logging-
+paper].
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+
+34
+Patrick Loic Foalem et al.
+Appendix A
+Figure
+0
+2
+4
+6
+8
+10
+12
+14
+16
+18
+#Authors
+0
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+200
+300
+400
+500
+600
+700
+800
+900
+1000
+1100
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+Count
+(a) Cumulative frequency curve base on #Authors
+0
+2
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+(b) Cumulative frequency curve base on #Stars
+Fig. 8: Cumulative frequency curve of #Stars and #Authors
+
diff --git a/E9E2T4oBgHgl3EQf-Amb/content/tmp_files/load_file.txt b/E9E2T4oBgHgl3EQf-Amb/content/tmp_files/load_file.txt
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf,len=1257
+page_content='Empirical Software Engineering manuscript No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  (will be inserted by the editor)  Studying Logging Practice in Machine  Learning-based Applications  Patrick Loic Foalem · Foutse Khomh ·  Heng Li  Received: date / Accepted: date  Abstract Logging is a common practice in traditional software development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  Several research works have been done to investigate the different character-  istics of logging practices in traditional software systems (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=', Android ap-  plications, JAVA applications, C/C++ applications).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' Nowadays, we are wit-  nessing more and more development of Machine Learning-based applications  (ML-based applications).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' Today, there are many popular libraries that facili-  tate and contribute to the development of such applications, among which we  can mention: Pytorch, Tensorflow, Theano, MXNet, Scikit-Learn, Caffe, and  Keras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' Despite the popularity of ML, we don’t have a clear understanding of  logging practices in ML applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' In this paper, we aim to fill this knowl-  edge gap and help ML practitioners understand the characteristics of logging  in ML-based applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' In particular, we conduct an empirical study on  110 open-source ML-based applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' Through a quantitative analysis, we  find that logging practice in ML-based applications is less pervasive than in  traditional applications including Android, JAVA, and C/C++ applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  Furthermore, the majority of logging statements in ML-based applications are  in info and warn levels, compared to traditional applications where info is the  majority of logging statement in C/C++ application and debug, error levels  constitute the majority of logging statement in Android application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' We also  perform a quantitative and qualitative analysis of a random sample of logging  statements to understand where ML developers put most of logging statements  and examine why and how they are using logging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' These analyses led to the  following observations: (i) ML developers put most of the logging statements in  model training, and in non-ML components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' (ii) Data and model management  Corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  Patrick Loic Foalem, Foutse Khomh, Heng Li  Department of Computer Engineering and Software Engineering  Polytechnique Montreal  Montreal, QC, Canada  E-mail: {patrick-loic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='foalem, foutse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='khomh, heng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='li}@polymtl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='ca  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='04234v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='SE]  10 Jan 2023  2  Patrick Loic Foalem et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  appear to be the main reason behind the introduction of logging statements  in ML-based applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' Indeed, ML developers use logging statements to  keep track of the different stages of data processing as well as record rele-  vant statistical information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' In addition, they use logging statements to track  all the experiments performed, to find the best model, and to record all the  performances achieved by the model on validation or test data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' Through this  work, we hope to shed light on logging practices in ML-based applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  Our results also highlight the need for more ML-specific logging libraries to  record and automate all experiments performed in the pipeline of ML-based  applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  Keywords Logging practices · ML-based applications · Mining software  repositories · Source code analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  1 Introduction  Inserting logging statements into software programs is part of good program-  ming practice (Kernighan and Pike (1999)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' Logging statements allow to col-  lect information about the behavior of a program during its execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' This  information is often used for various tasks in software maintenance and oper-  ations, such as diagnosing failures, reporting error, detecting anomalies, and  automatically generating logging text using log data (Yang and Agrawal, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  Nagaraj et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=', 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=', 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' Ding et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' Many research studies  have been conducted to characterize logging practices in traditional software  development, for example, Yuan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' (2012) studied logging practices in open-  source applications written in C/C++ , Zeng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' (2019) made similar study  for android applications, and Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' (2017) studied logging practices in  JAVA-based applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' However, to the best of our knowledge, no previ-  ous research work examined logging practices in ML-based applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' This  paper aims to fill this gap in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' Such a study could be useful for  ML practitioners to understand logging practices in ML applications, identify  the kind of information that is being logged, and the logging libraries used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  Logs are generated by logging statements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' Listing 1 shows some typical exam-  ples of logging statements used in ML applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' The logging statements  from lines 1, 3, and 9 use general logging libraries of the Python program-  ming language, while the logging statements on lines 5, 7, 11, 13, 15 use ML-  specific logging libraries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' General logging libraries are built from Python mod-  ule logging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' Their logging statement always have log level (INFO, Warn, ERROR,  DEBUG, CRITICAL) and in general, these libraries are older than ML-specific log-  ging libraries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' They are designed primarily for traditional software systems,  whereas ML-specific logging libraries are designed for ML applications and  contain information specific to machine learning and offer a straightforward  approach for logging performance, hyperparameters, and statistical informa-  tion about the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' Some ML logging libraries even offer the advantage of  visualizing the logged elements in a graph, allowing users to browse through  the learning process of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  Studying Logging Practice in Machine Learning-based Applications  3  1  logging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='info("*** Reading from input files ***")  2  3  tf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='logging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='info("[*] num_epochs: %d" % _num_epochs)  4  5  dllogger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='log(step=(epoch, steps_per_epoch, batch_idx), data=dllogger_data)  6  7  caffe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content="log('Using devices %s' % str(gpus))  8  9  rospy." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='logwarn("Updating Steering PID %s, %s, %s", config["Steer_P"],  config["Steer_I"], config["Steer_D"])  �→  10  11  self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='hparams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='train_logger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='log_stats({"Epoch": epoch, "lr": old_lr},  train_stats={"loss": self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='train_loss}, valid_stats=stats, )  �→  12  13  mlflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='log_metrics(metrics, step=step)  14  15  wandb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='log(log_data, step=learner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='num_samples_collected + args.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='start_env_steps)  Listing 1: Logging statement examples extracted from ML-based open-source  application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  To understand logging practices in ML-based applications, in this paper, we  conduct an empirical study of 110 ML-based software projects from GitHub1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  examining the prevalence of logging statements, the rationale for their use,  and the characteristics of the ML components in which logging occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' Our  research is organized along three following research questions (RQs):  RQ1: What are the characteristics of ML-based software logging practices?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  To answer this research question, we quantify the logging statements present  in our studied ML-based systems and compare the result with previous find-  ings on traditional systems (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=', JAVA, C/C++, Android).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' We observe a  difference between the density of logs contained in ML-based applications  and the density reported for traditional Android, JAVA, and C/C++ ap-  plications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' In particular, logging in ML-based applications is less pervasive  than in JAVA, Android, and C/C++ applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' Moreover, the distribu-  tion of logging levels in ML-based applications are different from those in  traditional applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  RQ2: Which phases of the ML pipeline are more prone to logging?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  The ML pipeline consists of the following steps: data collection, data pro-  cessing, model training, model evaluation, and model deployment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' We manu-  ally analyzed ∼2K logging statements that corresponds to more than 95% of  confidence level with a 5% confidence interval (Yamane, 1967).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' We mapped  each logging statement to one stage of the ML pipeline and examined their  content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' We observed that the majority of logging statements occurred in  the model training step and contains information about the models’ hyper-  parameters and performances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  1 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='com/  4  Patrick Loic Foalem et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  RQ3: Why do developers use logging in ML-based application?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  To understand the rationale behind development decisions for using logging  statements, we examined the code of file containing 380 logging statements  from our sample, and assigned a reason behind its use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' Results show that  ML practitioners use logging statements for two mains reasons : data and  model management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  These results show a distinct logging practice in ML-based applications  from widely-studied traditional applications and call for more research work  to improve our understanding of logging practices in ML-based applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  The high proportion of logging statements related to general logging libraries  (used for code and model management) in ML code may be a sign that ML  practitioners are not aware of ML-specific logging libraries, such as mlflow,  wandb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' As an example, in Listing 2, ML practitioners continued to use the  general logging libraries to log the performance of the models during its train-  ing phase i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=', line 1, instead of using ML-specific logging libraries designed  to log this type of information despite the advantage offered by these ML-  specific libraries;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=', being able to log many ML metrics at once, tracking all  the experiments performed during the learning phase and summarizing them  through visualizations, in order to track performance improvements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=', line  3, 5, 6 and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  1  logger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content="info('Epoch {} Loss {:." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='4f} Accuracy {:.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content="4f}'." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='format( epoch,  2  loss_meter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='avg, accuracy))  3  dllogger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='log(step=(epoch, steps per epoch, batch idx),  4  data=dllogger data, verbosity=0)  5  wandb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='log({"RMSE_training": training_RMSE, "epoch": epoch,})  6  self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='hparams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='train_logger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='log_stats({"Epoch": epoch, "lr": old_lr},  7  train_stats={"loss": self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='train_loss}, valid_stats=stats, )  8  mlflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='log_metrics(metrics, step=step)  Listing 2: Logging statement examples use for logging ML performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  Paper organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' The remainder of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  Section 2 describes the design of our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' Section 3 presents the motivation,  results of each research question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' Section 4 discusses threats to the validity of  our findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' Section 5 discusses the related literature and Section 6 concludes  the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  2 Study Design  This section presents the design of our study, which aims to understand logging  practices in ML-based applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  Studying Logging Practice in Machine Learning-based Applications  5  1  Code  search  2  Project filtration  & sampling  ML Project  Qualitative  Analysis  4  3  5  Extract Logging  Statement  RQ1  Stratified Random  Sampling  Qualitative  Analysis  RQ3  RQ2  6  7  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' 1: Overview of the research workflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='1 Data collection  Figure 1 provides an overview of our data extraction and analysis approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  We describe each step below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  x Code search  To select ML-based software projects for our study, we used the GitHub  API.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' GitHub Search API V32 allows us to collect projects by doing code  search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' We searched for projects based on the import statements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' To obtain  ML-based projects, we follow the same approach as Openja et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' (2022),  which consists in searching for machine learning libraries among the import  statements of GitHub projects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' Specifically, we search for the import state-  ments corresponding to the following ML libraries: TensorFlow, Theano,  MXNet, Scikit-Learn, Pytorch, Keras, Caffe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' In total, we collect 1,547 ML-  based projects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' We focus on these libraries because of their popularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  y Project filtration & sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  In order to retain a relevant quantity of mature projects for our study, we  apply the filtering approach proposed by previous works (Openja et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=',  2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' Munaiah et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=', 2017), which consists in defining thresholds on the  following three metrics: number of stars, number of commits, number of  authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' To identify adequate threshold values, we plot the distributions  of the three metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' Figure 2 represents the cumulative frequency curve of  all projects that we collected from GitHub by number of commits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' For this  metric (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=', number of commits), we define the threshold value to be 100  to avoid “toy” projects such as tutorial, student’s class assignments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' Using  this threshold we retained 320 projects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' We also plotted the cumulative  frequency curve for the other two metrics (see the Appendix Figure 8 for  the corresponding figures) and derived a threshold value of one for the  number of stars and two for the number of authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' We then filtered out  all projects in our dataset with less than one star and less than two authors,  and retained a final set of 110 projects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  z Collection of ML Projects  2 https://docs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='com/en/rest  0  00  0  0  0  个  0000个  Extracting data6  Patrick Loic Foalem et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  0  100  200  300  400  #Commits  0  100  200  300  400  500  600  700  800  900  1000  1100  1200  1300  1400  Count  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' 2: Cumulative frequency curve base on #Commits  A total of 110 projects matched our selection criteria and were retained as  subjects for our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' In order to ensure that no fork projects are present  among our studied systems, and that at least one ML library mentioned in  step 1 are covered in our dataset, the first author of this paper manually  examined the repository of each of the 110 projects, and all 110 study  projects meet these criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  Table 1 presents an overview of our selected projects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' Line SLOC present  a summary statistic of source lines of code, ranging from the mean to the  max sources lines of code for our studied subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' Line #Commits is  the summary statistic for the number of commits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' Line #PythonFiles  denotes the summary statistic of the number of Python files present in our  subjects at the time of analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' The summary statistics of the number of  contributors in each project is presented at line #Authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' Number of  stars is an indicator of the popularity of a project, line #Stars is a statis-  tical description of the number of stars for our subjects with an average of  1,167 numbers of stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  { Identification of logging libraries contained in the projects  To obtain information about each logging statement, we built a custom Ab-  stract Syntax Tree (AST) parser from the Python built-in package AST3  and used it to extract all import statements contained in all the Python  files of our selected projects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' We extracted a total of 13,270 import state-  ments (Steps 3 - 4 , Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' At step 4 , we manually analyzed the  3 https://bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='ly/3gxsOkc  Studying Logging Practice in Machine Learning-based Applications  7  Table 1: Overview of studied subjects  Metric  Mean  Min  25th Quartile  Median  75th Quartile  Max  SLOC  53,579  244  3,647  10,662  31,228  2,010,902  # Commits  759  42  172  323  782  9,139  # PythonFiles  272  5  35  101  210  7,510  # Authors  18  2  3  5  13  261  # Stars  1,167  1  2  20  284  32,770  extracted import statements to identify the logging libraries involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' We  identified the following 12 libraries in the projects of our dataset: logging4,  rospy5, hparams6, logger7, warnings8, callbacks9, wandb10, tensorflow11,  caffe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='log12, dllogger13, ml-logger14, mlflow15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  | Extraction of Logging Statements  To identify logging statements contained in each file, the following steps  were taken: (i) For each Python file in the 110 projects, we use the AST  module from Python 3 to generate an AST for the file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' (ii) We use AST  Python libraries to extract the function call related to the 13 logging li-  braries mention above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' (iii) We analyzed each of the libraries mentioned  in the previous step, then identified regular expressions that could match  a logging statement of these libraries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' (iv) Accordingly, we used the fol-  lowing expressions: log, info, debug, error, fatal, warn, callbacks, warnings,  exception, and critical as a visitor pattern16 to get all the function calls cor-  responding to a logging statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' In order to minimize false positives in  our logging statement dataset, we manually removed expressions containing  “log” that are not logging statements (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=', “dialog”, “login”, “numpy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='log”,  “tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='log”, etc) and we also removed expressions corresponding to logging  configuration (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=', “caffe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='init log()”, “logging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='getLogger”, “logging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='disable”,  “logging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='config”, etc) and this allowed us to keep a clean dataset17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' This  logging statement extraction process is inspired by the work of (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=',  4 https://bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='ly/3gxsOkc  5 https://bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='ly/3shluMk  6 https://bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='ly/3sdKpk2  7 https://bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='ly/3VMYr9K  8 https://bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='ly/3CVpxCR  9 https://bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='ly/3SozpLk  10 https://bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='ly/3Sod80f  11 https://bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='ly/3guvYoR  12 https://bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='ly/2QjqYSf  13 https://bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='ly/3TJyW7m  14 https://bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='ly/3z0Xz7O  15 https://bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='ly/2KiwDFd  16 https://bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='ly/3TKBeD8  17 https://bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='ly/3DlmGow  8  Patrick Loic Foalem et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' Zhai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' We obtained a total of ∼21K logging statements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  The results for step 5 answer our research question RQ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  } Stratified Random Sampling  In order to answer research questions RQ2 and RQ3, we prepared our  dataset for qualitative analysis using stratified random sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' We opted  for a stratified sampling approach to ensure a proportionate representation  of projects in our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' We sampled a total of ∼2K logging statements,  which corresponds to more than 95% of confidence level with a 5% confi-  dence interval (Yamane, 1967).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='2 Data analysis  We perform a manual analysis of the logging statements sampled at step  6, categorizing the logging statements and mapping them to the differ-  ent phases of the ML pipeline (step 7 on Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' Specifically, for each  logging statement found in our sample, we identified in the Python file  and the specific class and function where it appears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' Next, we identify the  phase of the ML pipeline (Amershi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=', 2019), that is concerned by the  implementation contained in the identified file (including the logging state-  ment).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' To map logging statements to ML pipeline phases, we attribute a  label to each phase of the ML pipeline and assign to each identified logging  statement, the label corresponding to its phase (as identified previously).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  This allows us to answer  RQ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' The manual analysis and the labeling were  performed in parallel by two authors, and the third author acted as a ref-  eree for ambiguous cases where opinions diverged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' All the diverging cases  were discussed until reaching a consensus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' To understand why developers  use logging statements in ML-based applications and answer our research  question  RQ3 , we manually analyzed the source code where each logging  statement is placed in order to identify the reason why ML practitioners  inserted these logging statements in their project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  3 Case Study Results  In this section, we present the results of our three research questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' For each  research question, we describe the motivation, the approach followed to answer  the research question and our obtained results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  RQ1: What are the characteristics of ML-based application logging practices?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  Motivation  Many research studies have been conducted to characterize logging practices  in traditional software application (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' Yuan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=', 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' Zeng  Studying Logging Practice in Machine Learning-based Applications  9  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=', 2019), showing the importance of logging in software systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' The  conclusion from these researches shows that, logging practice in traditional  applications is widely used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' Many research works have used AI algorithms  to make prediction and anomaly detection based on log files (Nagaraj et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=',  2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=', 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' In addition, research has been done to help develop-  ers in their logging decisions (Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=', 2015a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' However,  none of these research efforts have focused on the characteristics of logging  in ML applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' Intuitively, there exists significant difference in the pro-  gramming paradigms between ML and traditional application development:  ML practitioners use a data driven approach, which lets them develop and  train models with a set of parameters and hyperparameters, on large data, by  contrast traditional software developers use a logic driven approach which di-  rectly implements the program logic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' Thus, intuitive, the logging of ML-based  applications would be different from that of traditional applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' There-  fore, in this research question, we want to study the characteristics of logging  practices in ML-based applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  Approach  Similar to previous studies on the logging practices of traditional software  applications (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' Yuan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=', 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' Zeng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=',  2015a), we have studied the following aspects of the characteristics of logging  practices in ML-based applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  − Logging libraries types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' During the data gathering process, we obtained  a dataset of import statements, from which we performed a qualitative  analysis in order to identify logging libraries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' We also make a distinction  between ad-hoc logging for Al-based applications, and general-purpose log-  ging libraries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' To achieve this task, the three authors have performed anal-  ysis of the import dataset in order to identify potential logging libraries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  We search online materials (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=', through the GOOGLE 18 search engine)  in order to have more information about suspected logging libraries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' Once  the libraries were identified, we did additional research to find out more  information about their usage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' The goal of this investigation is to iden-  tify patterns that can be used further to extract the associated logging  statements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=', 2017) describe general-purpose logging libraries  as libraries that developers can define a logging level, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=', INFO, DEBUG,  WARN, ERROR, FATAL, EXCEPTION, CRITICAL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' Then Ad-hoc log-  ging libraries as libraries designed for the purpose of logging particular  information which will not be or tangled in case of concurrent logging with  other type of libraries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' We have manually labeled our dataset of logging  statements obtained during the data gathering according to two categories,  general-purpose logging and ML-specific logging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' The summary of logging  libraries found is presented in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  18 https://bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='ly/3VPCmaM  10  Patrick Loic Foalem et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  Table 2: Summary of logging libraries in ML-based application  Logging libraries type  Libraries  release by  main purpose  ML-specific logging  hparams  Python software foundation  Use to log hyperparameter of the model  wandb  Weight & Biases  Use for dataset versioning, track hyperparameter,  Experiment tracking  TensorFlow  Google  Use to log and summarize ML information when developing with  TensorFlow  logger  NVIDIA  Use to log ML information at inference stage  caffe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='log  Berkeley AI Research  Use to log ML information when developing  with caffe  mlflow  Databricks  Use to track ML experiment in training and inference stage,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' logging metric  and hyperparameter  ml-logger  Python software foundation  Tracking ML experiment  dllogger  NVIDIA  Tracking ML experiment  callbacks  Keras  Tracking ML experiment,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' do early stopping,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  General logging libraries  logging  Python software foundation  Log all kind of information in python  rospy  Open robotics  Use to log all kind of information when working  in robot operating system with rospy libraries  Warnings  Python software foundation  Use to log warnings message in all kind of project in Python  − The density of logging statements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' We measure log density on the  last version of each project at the time we perform this study, through  the following steps: (i) after cloning the last version of a project, we use  the cloc19 tool, which allows us to evaluate the total number of lines of  code present in a project (SLOC), while excluding comments and empty  lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' (ii) Then, we evaluated the number of logging statements present in  each project by using a custom-built AST20 parser 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='1, which allowed us  to extract, then make a quantitative analysis of logging statements present  in a given project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' (iii) Finally, we calculated the number of lines of code  per logging statement using the SLOC  NL  formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' Such a metric has already  been used in previous works (Yuan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=', 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  − The verbosity levels of logging statements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=', 2017) shows  through their studies the importance of assigning the correct log level to a  logging statement and the difficulty for developers to determine the appro-  priate log level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' Similar to (Zeng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=', 2019) we study the distribution of  logging statements in ML-based applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' From the logging statement  dataset extracted during our study design process (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='1), we evalu-  ate the distribution of the verbosity levels based on the following patterns,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=', info, warn, debug, error, except, critical, fatal, which are the verbosity  levels suggested in the python’s official documentation21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  We have studied the three aspects mentioned above on the 110 open-source  projects, in order to understand the characteristics of logging practices in ML-  based applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  Results  In this section, we present the results of our first research question, discuss our  results, findings, and compare them to prior studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' We discover three main  findings in our RQ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  19 https://bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='ly/3DgYhjQ  20 https://bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='ly/3F5bI7S  21 https://bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='ly/3TJSpEY  Studying Logging Practice in Machine Learning-based Applications  11  Finding 1: Logging statements in ML-based applications is commonly  used but less pervasive than in JAVA, C#, C/C++ applications and more  pervasive than in Android applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  From the 110 studied projects, 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='18% (86) contain at least one logging state-  ment in the project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' This shows that logging is prevalent in ML-based appli-  cations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' Log density estimates the likelihood for a developer to insert logging  statements in a project, the smaller it is the more likely that we will find a  logging statement in that project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' We can observe from Figure 3 that there is a  very high probability that a developer writes a logging statement in a C/C++  project according to (Yuan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=', 2012) compare to others (on average one  logging statement per thirty lines of code versus one logging statement per  fifty-one line of code in JAVA application).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' For ML-based applications, the log  density is smaller compared to Android applications (on average one logging  statement per two hundred and ninety lines of code vs one logging statement  per four hundred and seventy-nine logging statement) but larger than C#  applications (290 vs 58).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' Thus, the probability for an ML practitioner intro-  ducing a logging statement in an ML application is higher than an Android  developer introducing a logging statement in an Android application but lower  than in JAVA, C#, and C/C++ applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  Zeng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' (Android)  This paper (ML)  Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' (C#)  Chen et al (JAVA)  Yuan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' (C/C++)  Paper  0  100  200  300  400  500  Log density  479  290  58  51  30  Log density comparaison  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' 3: Comparison of log density between ML-based applications studied in  our paper and other types of applications studied in prior studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  Finding 2: The majority of logging statements in ML-based applications  use general-purpose logging libraries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  12  Patrick Loic Foalem et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  Figure 4 shows the distribution of logging statements according to whether  they belong to the general-purpose logging libraries or ad-hoc logging libraries  for ML-based applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' The general-purpose logging statements are 11  times more used than the ML component-specific logging statements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' This  domination can be explained by the fact that: (i) general-purpose logging  statements are used in both ML and non-ML components, while ML-specific  logging libraries for ML-based applications are used only in the ML compo-  nents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=" For example, general-purpose logging statements are used in the ML  components (data processing), as presented in the following code snippet which  can be found in AI-training22  def convert ilsvrc2010(directory , output directory,  output filename='ilsvrc2010." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content="hdf5',  shuffle seed =config." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='default seed):  with h5py.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content="File(output path, 'w') as f :  log." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=" info('Creating HDF5 datasets." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content="')  prepare hdf5 file (f , n train ,  n valid ,  n test)  log." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=" info('Processing training set ." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=" ')  process train set (f , train , patch, n train , wnid map, shuffle seed)  log." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=" info('Processing validation set ." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=" ')  process other set (f ,  ' valid ' ,  valid , patch, valid groundtruth, n train)  log." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=" info('Processing test set ." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=" ')  process other set (f ,  ' test ' ,  test , patch, test groundtruth,  n train + n valid)  log." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=" info('Done." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content="')  return (output path,)  def prepare metadata(devkit archive, test groundtruth path):  # Ascertain the number of filenames to prepare appropriate sized  # arrays." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content="  n train = int(synsets[ 'num train images']." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='sum())  log.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=" info('Training set : {} images'." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='format(n train))  log.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=" info('Validation set : {} images'." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='format(len(valid groundtruth)))  log.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=" info('Test set : {} images'." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='format(len(test groundtruth)))  n total = n train + len(valid groundtruth) + len(test groundtruth)  log.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=" info('Total (train/valid/test): {} images'." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='format(n total))  return n train , valid groundtruth, test groundtruth, wnid map  Code Listing 1: Code snippet of general logging statement used in ML  component  They are also used in the model training step, as in the following code snippet  present in cifar10-human-experiments23,  if  human tune:  logger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=" info('− epoch {}  c10h train  :  {:." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='4 f} (acc: {:.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='4 f}) | c10h val  :  {:.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='4 f} (acc:  {:.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content="4 f})' ." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='format(  epoch, train loss meter .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='avg, train accuracy, loss meter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='avg, accuracy))  logger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=" info('−  c10h train c10:  {:." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='4 f} (acc: {:.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='4 f}) | c10h val c10:  {:.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='4 f} (acc:  {:.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content="4 f})' ." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='format(  c10h train c10 loss meter .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='avg, c10h train c10 accuracy, c10h val c10 loss meter .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='avg,  c10h val c10 accuracy))  else :  logger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=" info('Epoch {} Loss {:." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='4f} Accuracy {:.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content="4f}'." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='format(  epoch, loss meter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='avg, accuracy))  elapsed = time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='time() − start  logger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=" info('Elapsed {:." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content="2f}'." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='format(elapsed))  Code Listing 2: Code snippet of general logging statement used in model  training  (ii) ML practitioners quite often use general-purpose logging statements in  ML components instead of specific logging statements for ML components  22 https://bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='ly/3F1mdsy  23 https://bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='ly/3ShI2XW  Studying Logging Practice in Machine Learning-based Applications  13  in their applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' This is the case for example in Code Listing 1 where  ML practitioners used a general-purpose logging statement to log the model  parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' The same logging task could have been done using the logging  statement “wandb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='log()” which is specifically designed to log model parameters  during model training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' This is the case in the following code snippet, which  can be found in BPNN24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  if  USE WANDB:  wandb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='log({”RMSE val”: np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='sqrt(np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='mean(val losses)), ”RMSE training”:  np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='sqrt(np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='mean(losses))}, commit=(not args.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='val ood))  if  USE WANDB:  wandb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='log({”RMSE valOOD1”: np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='sqrt(np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='mean(val losses1))})  print(”root mean squared validation OOD1 error =”, np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='sqrt(np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='mean(val losses1)))  if  USE WANDB:  wandb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='log({”RMSE valOOD3”: np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='sqrt(np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='mean(val losses3))})  if  USE WANDB:  wandb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='log({”RMSE valOOD4”: np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='sqrt(np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='mean(val losses4))}, commit=True)  Code Listing 3: Code snippet of ML-specific logging statement used in model  training  GENERAL  ML  Logging type  0  2500  5000  7500  10000  12500  15000  17500  number of logging  91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='7%  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='3%  Log type distribution  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' 4: Logging type distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  Finding 3: The majority of the logging statements in ML-based appli-  cations are in info and warnings levels, while info level logging statements  are the majority in C/C++ applications and debug, error level logging  statements are the majority in Android Applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  24 https://bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='ly/3FbDQWC  14  Patrick Loic Foalem et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  Figure 5 shows the distribution of logging levels for the 110 studied projects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  More than half of the logging statements in ML-based applications are in info  and warning levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' This can be explained by the fact that ML practitioners  often use these log levels in data and model management as shown in the  examples Code Listing 1, Code Listing 2, where the info level is used for  model and data management (recording model parameters and information  during data processing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' The warnings level is used to alert inconsistencies in  data preprocessing operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content="  def  contains nan(a, nan policy='propagate'):  try:  contains nan = np." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='nan in set(a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content="ravel())  except TypeError:  # Don't know what to do." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' Fall back to omitting nan values and  # issue a warning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content="  contains nan = False  nan policy = 'omit'  warnings." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='warn(”The input array could not be properly checked for nan ”  ”values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=" nan values will be ignored.”, RuntimeWarning)  def  kurtosistest (a, axis=0, nan policy='propagate'):  if  n < 20:  warnings." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='warn(”kurtosistest only valid for n>=20 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' continuing ”  ”anyway, n=%i” % int(n))  def dataset from dicts( self ,  dicts ,  indices=None, return baskets=False, non initial token=”X”)  logger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='warning(f”[Task: {task name}] Could not convert labels to ids via  label list !”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  f”\\nWe found a problem with labels {str(problematic labels)}”)  # TODO change this when inference flag is implemented  except KeyError:  logger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='warning(f”[Task: {task name}] Could not convert labels to ids via  label list !”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content="  ”\\nIf your are running in ∗inference∗ mode: Don't worry!”" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content="  ”\\nIf you are running in ∗training∗ mode: Verify you are supplying a  proper label  list  to your processor and check that labels in input  data are correct .”)  Code Listing 4: Code snippet showing the usage of warning in data processing  (Air-Pollution25, farn26) Code Listing 4 or model training Code Listing 5  (DeepPavlov27) in ML component,  def train evaluate model from config() −> Dict[str, Dict[str ,  float ]]:  if  ' train ' not in config :  log." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content="warning('Train config is missing." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=" Populating with default values')  train config  = config." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content="get(' train ')  if  ' evaluation targets ' not in  train config  and (' validate best ' in  train config  or ' test best ' in  train config ):  log." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content="warning('”validate best” and ”test best” parameters are deprecated." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content="'  ' Please, use ”evaluation targets”  list  instead')  if  iterator  is not None:  if  to validate  is not None:  if  evaluation targets  is None:  log." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content="warning('”to validate” parameter is deprecated and will be removed in future versions." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=" '  ' Please, use ”evaluation targets”  list  instead')  evaluation targets = [' test ' ]  if  to validate :  evaluation targets ." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content="append('valid')  else :  log." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content="warning('Both ”evaluation targets” and ”to validate” parameters are specified." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=" '  ' ”to validate” is deprecated and will be ignored')  return res  Code Listing 5: Code snippet showing the usage of warning in model training  also its use in non-ML components as follows (Django web28)." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  25 https://bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='ly/3SoY9Dh  26 https://bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='ly/3z0EMJy  27 https://bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='ly/3TqKwVp  28 https://bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='ly/3F4EsgT  Studying Logging Practice in Machine Learning-based Applications  15  def methods check(req, methods):  try:  assert req.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content="method in methods  except AssertionError:  message = 'Method Not Allowed ({method}): {path}'." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='format(  method=req.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='method, path=req.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='path)  log.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='warning(message)  raise ResponseNotAllowed(detail=message)  Code Listing 6: Code snippet showing the usage of warning in non-ML  component  The distribution shown in Figure 5 is different from previous studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' The  results of (Zeng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=', 2019) show that most logging statements in Android  applications are at the debug and error level, while (Yuan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=', 2012) show  that most logging statements in C/C++ applications are at the info level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  INFO  WARNING  DEBUG  ERROR  EXCEPTION  CRITICAL  FATAL  Log Level  0  1000  2000  3000  4000  5000  6000  7000  #logging  35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='4%  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='8%  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='2%  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='4%  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='5%  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='4%  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='1%  Log level distribution  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' 5: Distributions of the logging statement levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  16  Patrick Loic Foalem et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  Summary: Logging practice in ML-based applications is different from  logging practices in traditional applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' One of the major differ-  ences is the existence of multiple logging libraries in ML-based applica-  tions, which can be categorized into two groups: general-purpose and  ML-specific logging libraries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' General-purpose logging libraries are de-  signed to be used mainly in traditional applications, and ML-specific  logging libraries are designed to be used in ML-based applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' Our  results show that general-purpose logging libraries are still the most  used in ML-based applications, despite the existence of numerous ML-  specific logging libraries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' This finding suggests that ML practitioners  might not be aware of the existence of ML-specific logging libraries, and  might be adopting sub-optimal logging practices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' The dominance of info  and warning in logging levels can be explained by the fact that these  log levels are most often used initially to log information throughout  the application, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=', in the ML and non-ML components of the ap-  plication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' These results are a preliminary step towards understanding  logging practices in ML-based applications and call for further work to  investigate good logging practices for ML-based applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  RQ2: Which phases of the ML pipeline are more prone to logging?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  Motivation  The results of our RQ1 indicate that ML practitioners widely use logging  statements during the development of ML-based applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' This finding  raises the question of log usage across the pipeline;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=', which phases of the  ML pipeline contain the most logging statements?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' Answering this question will  help understand developers’ needs for logging during the different phases of  the ML pipeline, and help support the development of efficient logging libraries  and recommenders for ML systems engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  Approach  To select log statements for our analysis, we conduct a stratified random sam-  pling as described in Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='1, and obtained ∼2K logging statements in  our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' Then, for each project in our sample, we analyzed the portion of  the code where the logging statements are located and assigned each logging  statement to one of the following five ML pipeline phases: Model Training,  Data collection/loading, Data processing, Model evaluation, and Model de-  ployment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' We used the following clues to perform the matching of logging  statements to the ML pipeline phases: (i) The path to the file containing a  logging statement, often contains keywords such as “train”, “data”, “load”,  “load data”, “model deploy”, “train image”, “fit”, “tokenizer”, “evaluator”,  Studying Logging Practice in Machine Learning-based Applications  17  “data cleaning”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' For example, it is the case for the paths of the following log-  ging statements: pykale 29, Air-Pollution 30, DeepLearningExamples 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' (ii) In  the file, we pay particular attention to the name of the function where the  logging statement is located, to the comment contained in the file, and those  that surround the function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' We also leverage the message written in the log-  ging statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' For example, for this DeepLearningExamples32 sample shown  in the following Code Listing 7, we classify the logging statements in line #2,  #3, #7, #16 as “model training” and those on lines #21, #22, #24, #30,  #31 as “model evaluation”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  1  def train(n token, cutoffs , rank, local rank , num core per host):  2  tf .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='logging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' info(”num of batches {}”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='format(train record info[”num batch”]))  3  tf .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='logging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' info(”step {} | lr  {:8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='9 f} ”  4  ”| loss  {:.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='2 f} | pplx {:>7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='2f}, bpc {:>7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='4f}, tok/s {:>6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='0f}”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='format(  5  curr step , fetched[−2],  6  curr loss , math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='exp(curr loss),  curr loss  / math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content="log(2), throughput))  7  dllogger data = {  8  ' lr ' : fetched[−1],  9  ' train loss ' :  curr loss ,  10  ' train perplexity ' : math." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content="exp(curr loss),  11  'train throughput': throughput,  12  }  13  dllogger ." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='log(step=int(curr step), data=dllogger data)  14  tf .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='logging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' info(”Model saved in path: {}”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='format(save path))  15  if  rank == 0:  16  tf .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='logging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' info(”Training throughput: {:>6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='0f} tok/s”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content="format(meters['train throughput']." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='avg))  17  18  def evaluate(n token, cutoffs ):  19  if  FLAGS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='max eval batch > 0:  20  num batch = FLAGS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='max eval batch  21  tf .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='logging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' info(”num of batches {}”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='format(num batch))  22  tf .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='logging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' info(”Evaluate {}”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='format(eval ckpt path))  23  if  (step+1) % (num batch // 10) == 0:  24  tf .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='logging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' info(format str.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content="format(step+1, num batch))  25  dllogger data = {  26  ' eval latency ' : latency,  27  'eval throughput': throughput,  28  }  29  dllogger ." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='log(step=step+1, data=dllogger data)  30  tf .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='logging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' info(”Evaluating with: bs {}, math {} ”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='format(FLAGS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='eval batch size, ”amp” if FLAGS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='amp  else ”fp32”))  31  tf .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='logging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' info(”| loss  {:.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='2 f} | pplx {:>7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='2f}, bpc {:>7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='4f}, tok/s {:>6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='1f}, ms/batch {:>4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='2f}”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='format(  32  avg loss , math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='exp(avg loss), avg loss / math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content="log(2), meters['eval throughput']." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content="avg,  meters[' eval latency ' ]." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' avg))  Code Listing 7: Code snippet from training and evaluation phase  The manual classification task was performed by two authors with the kappa  agreements of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='71, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=', substantial agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' The ambiguous cases were  discussed during a meeting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' If an agreement couldn’t be reached in the meeting,  the logging statement was labeled as an unclassified case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' In total, we were not  able to classify 66 logging statements in our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' We attribute this outcome  to the complexity of the code containing the logging statements and–or the  absence of the key elements listed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  Results  In this section, we present the results of our second research question and  highlight our findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  29 https://bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='ly/3zioXy8  30 https://bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='ly/3TXHbwE  31 https://bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='ly/3zlGQME  32 https://bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='ly/3zoQXAd  18  Patrick Loic Foalem et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  Table 3: Distribution of logging statement in the different phases of the ML  pipeline  ML pipeline  Description  Example  % age  Model Training  Model  training  includes  hyper-  parameter  tuning,  optimizing  cost  function,  fitting  data,  and  model  selection  logging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='info(’Training with a single process on 1 GPU.’)  tf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='logging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='info(”[*] num epochs: %d” % num epochs)  logging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='info(f’Training time: (elapsed / 60):.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='2f minutes’)  logger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='info(’Epoch[%d] Time cost=%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='3f’, epoch, (toc - tic))  logger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='info(f”Loading pretrained files for: ’, ’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='join(self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='loadables)”)  dllogger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='log(step=(epoch,  steps per epoch,  batch idx),  data=dllogger data, verbosity=0)  dllogger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='log(step=”PARAMETER”,  data=’Scheduled epochs’:  num epochs, verbosity=0)  35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='83%  Data  Collec-  tion/loading  Data collection/loading includes col-  lecting and reading data from different  sources and formats: online, CSV, Json,  text, h5, SQL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  log.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='info(’Recording driving data to %s’, self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='hdf5 dir)  logging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='info(’Load FTP fie without auth ( fromm )’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='format(ftpURL,  ftpFilePath))  logger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='info(”loading model from ”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='format(self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='path))  logger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='info(”Done loading the dataset”)  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='47%  Data Processing  Data processing step includes normal-  ization,  transformation, validation and featur-  ization of the data  logger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='info(f”Moving label repr(saved label) from index ”f”index, be-  cause repr(label) was put at its place.”)  tf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='logging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='warning(’‘shuffle‘ is false, but the input data stream is ’)  logger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='info(”Frequencies of rankings: ”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='format(print dictionary(freq)))  logger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='info(f”Resizing input dimensions of type(self).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' name ” f”from  old dims to new dims to match language model”)  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='84%  Model  Evalua-  tion/validation  Evaluation of the model on validation or  testing data, selection of the best model  and validation of its correctness  logger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='log(”Save the best model into :”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='format(final best name))  logger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='log(”The best model has : weights.”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='format(best model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='numel()))  logging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='info(f’Latency Avg: 1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='0 * latency data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='mean():.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='2f ms’)  tf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='logging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='info(’Starting Evaluation.’)  tf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='logging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='info(”***** Running prediction*****”)  logger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='info(’Final test accuracy = %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='1f%% (N=%d)’ % (test accuracy  100, len(test bottlenecks)))  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='20%  Model Deployment  Deploying the model or the system in  production  logger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='info(”deploying model ” + self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='args.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='triton model name +” in  format ” + self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='lib.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='platform)  logger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='info(”model check failed with warning: [”, error, ”]”)  logger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='info(”Warning during onnx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='checker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='check model in quantized  model ignored”)  LOGGER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='debug(”Existing  deployed  resources:  %s”,  exist-  ing resources)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='66%  From the ∼2K logging statements analyzed, 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='83% of the logging state-  ments belong to the model training phase and 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='51% belong to the portion of  code that isn’t part of any ML component of the project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' We also found that  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='84%, 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='47%, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='20%, and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='66% of logging statements belong respectively  to Data Processing, Data Loading, Model Evaluation, and Model Deployment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  We couldn’t reach an agreement for 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='50% (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=', 66) of logging statements in  our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' Figure 6 presents a distribution of the percentage of logging state-  ments found in different components of the studied ML-based applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  One can see that the majority of logging statements are found in the model  training component and that the model deployment phase contains the lowest  percentage of logging statements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' This result is not surprising since the model  training phase produces a larger proportion of code than the other phases;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  there are many parameters to optimize in order to obtain a stable model and  developers often have to log multiple pieces of information to guide their op-  timization process (training and re-training).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' In Table 3, we provide examples  of logging statements that were found in our studied projects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' The examples  are grouped according to the ML pipeline phase to which they belong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' In the  description column, we present the characteristics of the phase, in particular,  we list the elements that are considered during the classification of a logging  statement into that phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  Studying Logging Practice in Machine Learning-based Applications  19  Finding 4: Overall, all five phases of the entire ML pipeline contain  logging statements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' However, the model training phase has the largest  proportion of logging statements and the model deployment phase has  the smallest proportion of logging statements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  Model  Trainning  Non-ML  Pipeline  Data  Processing  Data  Loading  Model  Evaluation  Model  Deployment  Unclassified  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='0%  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='0%  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='0%  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='0%  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='0%  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='0%  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='0%  35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='0%  Percentage [%]  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' 6: Percentage of the analyzed sample of logging statements belonging to  the different phases of the ML pipeline  RQ3: Why do ML practitioners use logging?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  Motivation  In RQ1, we observed that logging statements are prevalent in ML-based ap-  plications but less than in traditional software applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' We also observed  that ML developers conjointly use ML-specific logging libraries and general-  purpose logging libraries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' In this research question, we aim to get a better  understanding of ML developers’ need for logging, throughout the life cycle  of ML-based applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' Specifically, we want to identify the type of infor-  mation that is being logged at a specific phase of the ML pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' A good  understanding of logging practices in ML-based applications can help design  efficient tools to support logging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' It will also help devise guidelines for practi-  tioners developing and maintaining ML-based applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  20  Patrick Loic Foalem et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  Approach  We performed a qualitative analysis to understand the reasons for using log-  ging statements in ML-based applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' We randomly sampled 380 logging  statements from our dataset, which corresponds to a 95% confidence level with  5% confidence interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' We inspected each logging statement in detail, includ-  ing the log level, the log text, and the variables in the logging statement, as  well as the surrounding code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' We combined this information with the clues  mentioned in Section 3, to determine the type(s) of information that has been  logged and ML developers’ reasons for logging this information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' Next, we clas-  sified the logging statements based on ML developers’ reasons for using them  and the specific stage(s) of the ML pipeline at which the logging statements  were introduced in the project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' Each logging statement was scrutinized inde-  pendently by the first and the third author of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' All ambiguous cases  were discussed with the second author until a consensus was reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  Results  In this section, we report and discuss our findings about the types of infor-  mation being logged in ML-based applications and the specific stages of the  ML pipeline where the logging events occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' A stage is an important step in  a phase of the ML pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  Our analysis reveals two main reasons for logging in ML-based  applications: data management and model management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' We also iden-  tified a secondary reason which depends on the two previously mentioned  reasons, namely configuration management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' Figure 7 summarizes our derived  logging reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' Each block of information in Figure 7 represents a reason for  using logging in ML-based applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' Each entity in data management and  model management represents a key stage (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=', an important step in a phase of  the ML pipeline), and the attributes are the types of information being logged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  General purpose management depends on data management and model man-  agement entities because it allows to log information to ensure that the entire  environment related to the training process or the data processing process is  properly set up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' Each entity in the general-purpose block represents a specific  purpose for using a general-purpose logging library, and the attributes are the  types of information being logged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  In the following, we elaborate in more detail on each of the reasons for  logging identified in our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  Data management concerns the lifecycle of data, from the collection (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=',  via APIs, web scraping, S3, and devices such as cameras, and phones) to the in-  gestion into the training process which includes the following steps: data load-  ing, data parallelism (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=', the distribution of the data across different nodes,  which operate on the data in parallel), data processing, and data streaming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  Throughout the lifecycle of the input dataset,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' ML practitioners often use log-  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='ging to extract important information (for example about errors) and track  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='Studying Logging Practice in Machine Learning-based Applications  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='21  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='Data management  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='Model management  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='Data  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='loading  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='State event  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='File name  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='Data format  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='Data Parallelism  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='Memory  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='GPU  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='Batch  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='size  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='CPU  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='Data Processing  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='Statistic  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='Missing value  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='Schema  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='Data  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='Stream  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='State event  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='Batch  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='size  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='Schema  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='Data Collection  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='State event  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='Data format  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='Device  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='Model Import  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='State event  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='Model Evaluation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='State event  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='Model  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='Memory  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='Time  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='Epoch  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='Loss  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='Metric  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='Size  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='Model Validation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='Metric  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='Model Training  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='State event  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='Time  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='Memory  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='Metric  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='Epoch_step  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='Epoch_number  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='Epoch_stoping  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='Loss  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='Scheduler  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='Optimizer  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='Learning rate  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='Model Parallelism  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='State event  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='GPU  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='Memory  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='CPU  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='Model  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='Deployement  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='State event  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='CPU  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='GPU  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='Time  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='inference_optimizer  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='Configuration management  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='Dependencies  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='PyCUDA  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='cuDNNA  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='Conda  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='Environment Setting  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='CUDA  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='AI_library_version  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='Reason for Logging  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' 7: Reason for Logging in ML-based application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' Entity represent different  stage in ML-based system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' Attributes represent what type or what information  is logged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  the evolution of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' We have identified five stages in which logging is  prevalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  – Data Collection: When collecting data, ML practitioners often log the  following information of which some examples are presented in Listing 3: (i)  State event log -which contains information about a particular event, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=',  recording data, processing erroneous data during data collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' Those  state event logs are introduced at the beginning, during, or at the end of  those events, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=', Line 1, 5, 6, 7 in Listing 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' (ii) Data format log – the  information logged is related to the type or format of data collected, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=',  Line 4, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' (iii) Device log – the logged information is related to the data  acquisition tools or devices, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=', Line 2, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' Logging statements present in  Listing 3 can be identified in the following projects: DeepLearningExam-  ples33, DeepCamera34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  1  log.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content="info('Recording driving data to %s', self." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='hdf5_dir)  2  log.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content="debug('Grabbing images over ZMQ from %s', conn_string)  3  log." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content="debug('obz_exists?" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' %r should_record?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=" %r', obz is not  None,self." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='should_record)  �→  33 https://bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='ly/3GDstYo  34 https://bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='ly/3XvpQ10  22  Patrick Loic Foalem et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  4  logging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='info("Converted {} frames in {}".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='format(num_frames, tfrecord))  5  logging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='info("Processed {} records".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='format(len(tfrecords)))  6  logging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='info("Corrupted record {}".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='format(tfrecord))  7  logging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='info("Processing record # {}: {}".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='format(record_id, record))  8  logging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content="info('Camera: using Jetson onboard camera')  9  Listing 3: Logging statement examples in the data collection step  – Data loading: Once the data collection is done, in this step the data is  loaded from different sources such as: S3, directory, hard disk, in different  formats such as JSON, CSV, etc." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' During this step ML practitioners usually  record : (i) State event log - through this log here ML practitioners  usually use state event log to monitor the data loading process through  the records on the beginning of the loading process (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' line 4, 6), the  errors occurred during the data loading (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=', line 1), information about  the file (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=', line 4, 3) and the end of the data loading (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=', line 5, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  (ii) Data information log - here logged information is related to file  name and the data formats as mentioned in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' Listing 4 gives some  examples of logging statements found in our study subjects, especially in  the projects: cs-ranking35, deepdrive36  1  log.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content="error('Could not load %s - skipping - error was: %r', h5_filename,  e)  �→  2  log." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content="info('finished loading %s', h5_filename)  3  log." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content="info('data name %s', dataset)  4  logging." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content="info('loading csv')  5  logger." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='info("Done loading the dataset")  6  log.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content="info('loading %s', h5_filename)  Listing 4: Logging statement examples in the data loading step  – Data parallelism: This step occurs when the collected data is too heavy  compared to the available resources such as GPU and memory to be loaded  directly." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' Then ML practitioners will perform data parallelization which  consists in dividing the data according to the number of available GPUs or  CPUs in order to accelerate the data processing and eventually the training  process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' They usually use: (i) GPU/CPU log - which consists of logging  resource information related to the number of GPU or CPU used (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=', line  3, 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' (ii) Memory consumption log - are logging statement related to  the memory utilization during the training process by ML practitioners  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=', line 1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' (ii) Batch log - Here, data are loaded in batches, so the  batch details like the batch size are logged (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=', line 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' Some examples  of logging statements that we have identified in our study subjects are  presented in Listing 5 and can be found in projects cs-ranking37, rtg38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  35 https://bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='ly/3XCGFXQ  36 https://bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='ly/3U1yhy1  37 https://bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='ly/3Vkp0lz  38 https://bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='ly/3EXYwRL  Studying Logging Practice in Machine Learning-based Applications  23  1  log.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='warning(f"Going to buffer data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' this may consume all the memory  crash.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' Current usage={max_RSS()[1]}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='")  �→  2  log.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='warning(f"Buffered {self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='n_inp_recs:,} records Current memory  usage={max_RSS()[1]}")  �→  3  logging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='info("device_ids%s", gpu)  4  logger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='info("rank%s", cpu)  5  logging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='info("sampler size%s", batch_size)  Listing 5: Logging statement examples in the data parallelism step  – Data streaming: This step occurs in online learning ML systems where  the data is fed into ML algorithms in sequential order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' This type of system  is usually implemented when it becomes impossible to train the entire data  set in a ML model, or when the data arrives in a streaming manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' During  this stage, it is important for ML practitioners to ensure the consistency  between different streams of data received by ML algorithms (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=', line 2)  we call this (i) Schema log .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' To this end it is important for ML practi-  tioners to have information about the size of each stream of data received,  their schema, monitor the process of receiving each stream of data through  (ii) State events log (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=', line 1, 2, 3, 4, 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' all this information will  allow ML practitioners to have a track in case the distribution of the data  changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' Listing 6 present some of the relevant logging statements found  in the projects: DeepCamera 39, attention-lvcsr40, and attention-lvcsr41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  1  logger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='info("Monitoring on auxiliary data finished")  2  logger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content="info('No changes in schema detected." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content="')  3  logger." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='info("sending StopIteration")  4  logger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='info("sending {} arrays".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='format(len(data)))  5  logger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='info("Monitoring on auxiliary data started")  Listing 6: Logging statement examples in the data streaming step  – Data processing: This step involves manipulating attributes and labels of  data to produce meaningful information from the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' During the handling  process, ML practitioners usually record (Listing 7) : (i) Missing infor-  mation log - which are related to missing values, missing labels (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=', line  1, 6, 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' (ii) Statistic log - this logging statement contains local estimators  (mean, median, minimum, maximum), and variable estimators (variance,  standard deviation) (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=', line 3, 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' (iii) Schema log - logging statement  here contains information about the schema of some data attribute (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=',  line 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' These examples can be found in the following subjects: csrank 42,  Air-Pollution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='43  39 https://bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='ly/3XE6ren  40 https://bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='ly/3XlSuBD  41 https://bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='ly/3i8ulOw  42 https://bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='ly/3Ox7tUZ  43 https://bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='ly/3Oz2svs  24  Patrick Loic Foalem et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  1  logger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='info("Missing values {}: {}".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='format( col, np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='isnan(arr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='sum() /  len(arr))  �→  2  logger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='info("Sampled instances {} objects {}".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='format(X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='shape[0],  X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='shape[1]))  �→  3  log.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='info(f"mean:{mean}, std:{std:.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='4f} ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' low:{low:.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='4f} high:{high:.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='4f}  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' invert:{invert}")  �→  4  logger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='info("Min {}: Max {}".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='format(np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='nanmin(arr), np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='nanmax(arr)))  5  logging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content="info('FORMAT --dateformat: {}'." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='format(FORMAT))  6  tf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='logging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content="fatal('Label does not exist %s." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=" ', label_name)  7  logger." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='info("Missing values {}: {}".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='format( col, np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='isnan(arr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='sum() /  len(arr))  �→  8  Listing 7: Logging statement examples in the data processing step  Model management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' Model management concerns the development of the  model, which consists of developing, evaluating, validating and releasing it  into production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' A major difficulty of this stage is to track all the experiments  performed in search of the best model and also better generalize on new data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  ML practitioners often iterate on several experiments before arriving at the  best model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' Knowing which parameters or experiment led to the best model  is a tedious task and can be time consuming, especially when done manually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  One approach to solve this problem is to log all relevant information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' Logging  all relevant information allows ML practitioners to reproduce or compare past  experiments and then select the best model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' We have identified six impor-  tant steps of model management during which ML practitioners log relevant  information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  – Model Import: This step includes the import of the models or pre-trained  weights, in different formats, during which the ML practitioners usually  record state events log to ensure that the model importation process has  been carried out correctly, such as the examples shown in Listing 8 from  the AI-Training44 project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  1  logger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='info(f"Loading pretrained files for: {\',  \'.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='join(self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='loadables)}")  �→  2  log.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content="info('Downloading weights %s', folder)  3  logging." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='info("Model loading done")  4  logger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='info("loading model from {}".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='format(self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='path))  Listing 8: Logging statement examples in the model import step  – Model Parallelism: This step occurs when there are memory constraints  between the size of the model and the GPU or CPU device, as large mod-  els can’t be trained on a single GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' Thus, ML practitioners usually split  the model onto several GPU devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' During this step, ML practitioners  44 https://bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='ly/3Vg7BdM  Studying Logging Practice in Machine Learning-based Applications  25  usually record (i) State event log - which are logging statements con-  taining information about the start and end of the training process (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=',  line 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' (ii) Memory consumption log - logging statements are related  to memory consumed during the training stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' (iii) GPU/CPU log -  Information about the GPU and CPU devices used (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=', lines 1, 2, 3, 6,  7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' Listing 9 shows some examples of such logging statements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  1  logging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content="info('Training in distributed mode with multiple processes, 1  GPU per  �→  2  process." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' Process %d, total %d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=" '% (args." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='rank, args.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='world_size))  3  logger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content="info('device: {}'." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='format(device))  4  logger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='info("device: {} n_gpu: {} distributed training:  {}".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='format(device, n_gpu, bool(args.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='local_rank !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='= -1)))  �→  5  logger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content="info('begin training on multiple GPU')  6  logger." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='info("device: {} n_gpu: {}".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='format(device, n_gpu))  7  logging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content="info('Gradient averged for the rank of {}'." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='format(rank))  Listing 9: Logging statement examples in the model parallelism step  – Model Training: In this step, ML practitioners use data to train ML  models by manipulating many hyperparameters of the models and the op-  timization functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' This step is very time and resource-consuming and  non-deterministic hence the need for ML practitioners to track a lot of in-  formation during this training process and record it through logging state-  ments (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=', examples shown in Listing 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' (i) Time log - Evaluation of  training time or time taken to complete an epoch, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=', Line 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' (ii) Mem-  ory consumption log - Evaluation of memory consumption, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=', Line 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  (iii) Hyperparameters log - Recording of hyperparameters during the  training process, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=', Line 2, 6, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' (iv) Metrics log - Recording of metrics  during training, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=', Line 18, 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' (v) Model size log - Recording of the  model size, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=', : Line 4, 14, 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' (vi) State events log - Recording of states  event which allows to track the beginning, the possible errors, the update  of hyperparameters, the end of the training process e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=', Line 5, 8, 13, and  eventually key information related to the end of the learning process of ML  algorithms e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=', Line 1, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' Some logging statements can be found in our fol-  lowing study subjects: BPNN 45, attention-lvcsr 46, AutoDL-Projects 47,  AutoDL-Projects 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  1  logger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='log("Early stop the pre-training at {:}".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='format(iepoch))  2  log.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='info(f"New learning rate dividor = {self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='_learning_rate_cur_div}")  3  log.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content="info('pooling_0 shape: %s' % pooling_0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='shape)  4  logger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='log("The base-model has {:}  weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' ".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='format(base_model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='numel()))  �→  5  logger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='log("[ONLINE] [{:03d}/{:03d}] loss={:.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='4f},  score={:.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='4f}".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='format(idx, len(env), future_loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='item(), score )  �→  45 https://bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='ly/3V2Y2PE  46 https://bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='ly/3EWJziG  47 https://bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='ly/3tRdHpm  48 https://bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='ly/3Xqnnoo  26  Patrick Loic Foalem et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  6  logger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='log("scheduler  : {:}".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='format(scheduler))  7  logger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='log("optimizer  : {:}".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='format(optimizer))  8  logger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='info("Initializing the training algorithm")  9  log.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content="info('Did not improve on the {} of {}'." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='format(m_name,  self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='score_best))  �→  10  logging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content="info('rank %s: failing epoch %s batch %s', rank, epoch, batch)  11  wandb." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='log({"RMSE_training": np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='sqrt(np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='mean(losses))})  12  logger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='log("TRAIN [{:}] loss = {:.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='6f}".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='format(iter_str, loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='item()))  13  logger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='error("Error occured during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='" + error_message)  14  logger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='log("FLOP = {:} MB, Param = {:} MB".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='format(flop, param))  15  logging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content="info(f'Training time: {(elapsed / 60):." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content="2f} minutes')  16  logger." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='log("The model size is {:.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='4f}  M".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='format(xmisc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='count_parameters(model)))  �→  17  wandb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='log({"RMSE_val": val_RMSE, "RMSE_training": training_RMSE})  Listing 10: Logging statement examples in the model training step  – Model Evaluation: This step is part of the model development process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  It includes the search for the best model that generalizes the data, as well  as the evaluation of its performance on the validation or test data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' During  this stage, ML practitioners usually log information that will allow tracing  the whole evaluation process through: (i) State events log - that will  generally allow logging the beginning, the errors that occurred, and the  end of the evaluation process, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=', Line 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' (ii) Best model log - log the  best model information obtained during the fine-tuning phase, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=', Line  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' (iii) Memory consumption log - records the memory consumption  during the evaluation process, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=', Line 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' (v) Metric log - records the  performance measures (metric, loss) at each epoch, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=', Line 8, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' (iv)  Model size log - records the model size, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=', Line 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' Inference time log -  records the time taken by the model to infer predictions, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=', Line 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' Listing  11 are examples of logging statements collected in our study subjects at this  stage, they can be found in AutoDL-Projects49, DeepLearningExamples50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  1  logger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='log("Save the best model into {:}".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='format(final_best_name))  2  logger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='log( ""Finish training/validation in {:} with Max-GPU-Memory of  {:.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='2f} MB, and save final checkpoint into {:}"".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='format(  convert_secs2time(epoch_time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='sum, True)  �→  �→  3  tf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='logging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content="info('%s: Step %d: Validation accuracy = %." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content="1f%% (N=%d)'  %(datetime." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='now(),  �→  4  i, validation_accuracy * 100,len(validation_bottlenecks)))  5  logger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='info("***** Running evaluation *****")  6  logger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='info("time for inference {} perf {}".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='format(eval_end -  eval_start, num_examples * 100 / (eval_end - eval_start)))  �→  7  logger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='log("FLOP = {:} MB, Param = {:} MB".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='format(flop, param))  8  logger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='log("{:} {:} epoch={:03d}/{:03d} :: Train [loss={:.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='5f},  acc@1={:.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='2f}%, acc@5={:.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='2f}%] Valid [loss={:.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='5f}, acc@1={:.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='2f}%,  acc@5={:.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='2f}%]".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='format( time_string(), need_time, epoch,  total_epoch, train_loss, train_acc1, train_acc5, valid_loss,  valid_acc1, valid_acc5,)  �→  �→  �→  �→  49 https://bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='ly/3GKwJ8J  50 https://bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='ly/3VmRc7k  Studying Logging Practice in Machine Learning-based Applications  27  Listing 11: Logging statement examples in the model evaluation step  – Model Validation: The purpose of this step is to assess the precision  and performance of the model obtained during the evaluation phase on  real data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' To do so, ML practitioners record metrics used as a measure of  performance for their ML system, as presented in Listing 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  1  logger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content="info('Final test accuracy = %." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content="1f%% (N=%d)' % (test_accuracy *  100, len(test_bottlenecks)))  �→  Listing 12: Logging statement examples in the model validation step  – Model Deployment: At this stage, in order to keep track of everything  that happens during this phase ML practitioners often record information  such as: (i) State events log - which are logging statements related to the  process of converting the model into a lighter intermediate representation  by applying graph optimizations, layer merging, e." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=', Line 1, 2, 3, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' (ii)  GPU/CPU log - Device information such as CPU and GPU are recorded  during model prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' (iii) Time latency log - it’s critical in a pro-  duction environment to deliver inferences quickly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' Hence, ML practitioners  record the execution time of an inference, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=', Line 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' (iv) Optimizer log  – there are different optimizers that can be used to convert an ML model  into a lighter representation optimized for real-time inferences, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=', Line  4, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' (iv) Precision log – the precision enabled by an optimizer is an im-  portant piece of information that ML practitioners record when the model  is in production, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=', Line 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' Listing 13 are examples of logging statements  collected in our study subjects corresponding to the model deployment  step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  1  logger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='info("model check failed with warning: [", error, "]")  2  logger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='warning("Warning during onnx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='checker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='check model in quantized  model ignored")  �→  3  logging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content="info('Total node count before and after TF-TRT conversion:',  num_nodes, '->', len(frozen_graph." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='node))  �→  4  logger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content="info('TRT node count:',len([1 for n in frozen_graph." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='node if  str(n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content="op)'TRTEngineOp']))  �→  5  tf." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='compat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='logging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='info("Precision = %s", "fp16" if FLAGS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='amp else  "fp32")  �→  6  tf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='compat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='logging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content="info('Converting graph using  TensorFlow-TensorRT." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content="')  �→  7  tf." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='compat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='logging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='info("Total Inference Time W/O Overhead = %0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='2f  for Sentences = %d", predict_time_wo_overhead, num_sentences)  �→  Listing 13: Logging statement examples in the model deployment step  Configuration management steps are not explicitly represented in the ML  pipeline but are essential to ensure that the ML components perform in a  manner consistent with expectations over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' Configuration management  28  Patrick Loic Foalem et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  activities generally include the management of configurations or dependencies  on special devices and the management of libraries that are essential to en-  sure that ML components work as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' Through our analysis, we have  identified logs related to the following configuration management activities:  – Dependencies configuration log: This activity is usually implemented  when the ML component of the application will depend on a specific library  in order to ensure that the ML module of an application can work properly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  Hence, the need for ML practitioners to log information related to impor-  tant libraries used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' Listing 14 Presents some logging statements found dur-  ing this step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' They can be found in our studied subjects: DeepLearningEx-  amples51, deepdrive52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  1  log.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content="info('Installed UEPy python dependencies')  2  warnings." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content="warn('CVM does not support memory profile, using Stack VM." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content="')  3  warnings." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='warn("PyCUDA import failed in theano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='misc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='pycuda_init")  4  logging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='info("Using torch DistributedDataParallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' Install NVIDIA Apex  for Apex DDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='")  �→  Listing 14: Logging statement examples in the dependencies configuration step  – Environment setting log: The environment in which ML applications  run is different from traditional applications, in fact, ML applications gen-  erally need more resources or special devices such as CPU, and GPU in  order to run properly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' ML practitioners will generally ensure that the en-  vironment in which the application needs to run contains a minimum of  resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' Therefore, information about critical resources will be logged  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=', examples in Listing 15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' Some examples can be found in the follow-  ing subjects: DeepPavlov53, AutoDL-Projects54, AI-Training55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  1  logger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='log("cuDNN  Version  :  {:}".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='format(torch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='backends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='cudnn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='version()))  �→  2  logger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='log("PyTorch Version  : {:}".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='format(torch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='__version__))  3  logger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='warning("This recipe needs the sox-io backend of torchaudio")  4  logger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='log("CUDA available  : {:}".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='format(torch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='cuda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='is_available()))  5  logger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='log("CUDA GPU numbers : {:}".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='format(torch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='cuda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='device_count()))  6  "logger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='log(""CUDA_VISIBLE_DEVICES : {:}""  7  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='format( os.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='environ[""CUDA_VISIBLE_DEVICES""] if  ""CUDA_VISIBLE_DEVICES"" in os.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='environ else ""None"")"  �→  8  caffe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content="log('Using devices %s' % str(gpus))  9  _logger." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='warning("We are not able to detect the number of CPU cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='" "  We disable openmp by default.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='")  �→  10  _logger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content="info('Conda mkl is not available: %s', e)  Listing 15: Logging statement examples in the environment setting step  51 https://bit." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='ly/3Eyc1WB  52 https://bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='ly/3Ex9tZ5  53 https://bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='ly/3TZ4mXi  54 https://bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='ly/3tRawy3  55 https://bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='ly/3ACmajS  Studying Logging Practice in Machine Learning-based Applications  29  Finding 5: ML practitioners use logging statements to record impor-  tant information in data management (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=', recording the data format in  data collection), model management (recording the hyperparameters in  model training), and configuration management (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=', recording the used  resources or libraries).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' Our observations provide guidance for ML practi-  tioners to improve their ML logging and insights for future efforts to im-  prove ML logging practices (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=', by providing ML logging libraries that  facilitate the logging of different types of important information through-  out the life-cycle of ML-based applications).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  4 Threats to validity  In this section, we discuss the potential threats to the validity of our research  methodology and findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  External validity  The subjects used in this paper to answer our different research questions are  open-source ML-based projects from GitHub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' The selection of these projects  may be subject to the following threats:  Our results and findings may not apply to ML projects written in other  languages (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=', JAVA, C# or R) rather than Python since our analysis was  mainly done on the portions of code written in Python in our study subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  Hence, it is necessary that future works explore ML projects written in other  programming languages since ML practitioners working with these languages  may have different logging practices for ML projects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' However, since Python  is considered to be the lingua franca for ML-based application development  (Dilhara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=', 2021), we believe that our study provides a good understanding  of the logging practices in ML-based systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  Construct validity  The threats to the construct validity of our research are related to errors that  may have occurred during the extraction of logging statements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  To avoid extracting logging statements that have been commented out by  developers, we have developed a static code analyzer on top of the standard  Python AST parser (which is widely used in the static analysis of Python code  (D’Souza et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' Dilhara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=', 2021)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' This static code analyzer which is  available in our replication package56library extract only uncommented state-  ments;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' enabling us to avoid collecting logging statements that were commented  out by developers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  56 Scripts and data files used in our research are available online and can be found here:  http://bitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='ws/yr6c  30  Patrick Loic Foalem et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  Internal validity  We have manually mapped logging statements to their corresponding ML  pipeline phase, to answer RQ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' Then, in RQ3, we manually analyzed the log-  ging statements to identify the type of information logged and create a taxon-  omy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' However, our manual analyses are subject to the subjective judgment of  the people performing the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' This raises a threat to the internal validity  of our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' To mitigate this threat, manual analyses were performed by the  three authors of this paper with strong industry and academic backgrounds  in ML systems engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' Two authors performed the manual analysis, in  case of disagreements we had a group discussion with the third author until  a common agreement was reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' We believe that this approach reduces the  chance of introducing false positives in our analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' However, future replica-  tions and extensions of our work are desirable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' All the data and scripts used  in our study are available in our replication package57  5 Related work  In this section, we introduce and discuss two areas of related works on logging  practices: (i) research done on logging characteristics, and (ii) research done  on logging decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  (i) Characterizing logging practice: Many research works have been done  in characterizing logging practices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' (Yuan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=', 2012) conducts the first  empirical study in the characterization of logging practices on four open-  source projects written in C/C++ and found that logging is pervasive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=', 2017) conducts a similar study by analyzing 21 projects writ-  ten in JAVA which is a replication study whose objective is to generalize  the findings obtained by (Yuan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=', 2012) for projects written in JAVA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  Their result shows a difference in logging practices between applications  written in JAVA and those written in C/C++.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' (Zeng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=', 2019) studied  the logging practices in 1,444 Android projects and then compared their  findings with those of previous works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' However, none of the previous works  have focused on logging practices in AI-based applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' Our paper fills  this knowledge gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  (ii) Logging decisions: Logging decision involves research that helps devel-  opers decide where to introduce a logging statement and what level of  verbosity should be assigned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' These studies use AI algorithms to make  predictions on where or what to log.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' (Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=', 2015b) propose a “learn-  ing to log” framework using machine learning techniques, which aims to  help developers make decisions on where to add logging statements during  development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' Furthermore, Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' (2020) used a deep learning-based ap-  proach to help developers in their logging decision at the block level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' (Li  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=', 2017) propose an ordinal regression model, which accurately provides  57 http://bitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='ws/yr6c  Studying Logging Practice in Machine Learning-based Applications  31  the suggestion of the logging levels when developers add logging statement  and (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=', 2021) suggest also log level by using Ordinal Based Neu-  ral Networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' More recently, (Mastropaolo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=', 2022) propose LANCE  a framework to generate a complete logging statement using deep learn-  ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' This obviously shows that AI algorithms are used to assist developers  of traditional applications in their logging decisions but no research has  been conducted to assist AI practitioners in their logging decisions in AI-  based applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' Our work is therefore a good starting point for future  research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  6 Conclusions  Logging practices have been adopted by developers as part of good program-  ming practices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' Logs generally allow developers to diagnose their programs at  runtime in order to reduce maintenance efforts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' Logging practices have been  the subject of numerous studies in traditional software systems such as mobile  applications, JAVA applications, and open-source applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' To the best  of our knowledge, this paper presents the first attempt to study the practice  of logging in ML-based applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' We studied 110 open-source ML-based  applications from Github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' Our research has yielded the following findings:  Logging in ML-based applications is commonly used but less pervasive than in  JAVA, C#, C/C++ applications and more pervasive than in Android applica-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' The majority of logging statements are in INFO and WARNING levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' More-  over, we found that ML-based applications use two kinds of logging libraries:  general logging libraries and those specific to ML-based applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' However,  despite the existence of ML-specific libraries, general logging libraries remain  the most used in ML applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' In order to identify which ML pipeline con-  tains the most logging statements in ML-based applications, we performed a  qualitative and quantitative analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' Our findings indicate that the majority  of logging statements are found in the model training phase and the model  deployment phase contains the smallest portion of logging statements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' Fur-  thermore, we observe that ML practitioners use logging statements to record  important information related to data management (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=', recording the data  format in data collection), model management (recording the hyperparame-  ters in model training), and configuration management (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=', recording the  used resources or libraries).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' The contribution of this paper is as follows:  – To the best of our knowledge, this is the first study that quantitatively and  qualitatively analyzes the logging practice in ML-based applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  – We identified the general-purpose and ML-specific logging libraries that  are used in ML-based applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  – We identified ML phases in which practitioners use logs and the types of  information they log.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' Our findings provide guidance for ML practitioners  to improve their ML logging, as well as provide insights for future efforts  to improve ML logging practices (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=', by providing ML logging libraries  32  Patrick Loic Foalem et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  that facilitate the logging of different types of information in different ML  phases).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  Overall, our findings highlight the need for more ML-specific libraries to  support the development of ML-based applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' More research is also  needed to improve our understanding of logging needs and challenges in the  context of ML systems engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  Conflict of interest  The authors declare that they have no conflict of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='  Data availability statement  The datasets generated during and/or analysed during the current study are  available in the [foalem] repository, [https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content='com/foalem/ML-logging-  paper].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
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+page_content='  Appendix A  Figure  0  2  4  6  8  10  12  14  16  18  #Authors  0  100  200  300  400  500  600  700  800  900  1000  1100  1200  1300  1400  Count  (a) Cumulative frequency curve base on #Authors  0  2  4  6  8  10  12  14  16  18  #Stars  0  100  200  300  400  500  600  700  800  900  1000  1100  1200  1300  1400  Count  (b) Cumulative frequency curve base on #Stars  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
+page_content=' 8: Cumulative frequency curve of #Stars and #Authors' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQf-Amb/content/2301.04234v1.pdf'}
diff --git a/EdE0T4oBgHgl3EQfgwF8/content/tmp_files/2301.02422v1.pdf.txt b/EdE0T4oBgHgl3EQfgwF8/content/tmp_files/2301.02422v1.pdf.txt
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+Non-parametric Multi-Partitions Clustering
+Marie du Roy de Chaumaray and Vincent Vandewalle
+January 9, 2023
+Abstract
+In the framework of model-based clustering, a model, called multi-partitions clustering, allowing
+several latent class variables has been proposed. This model assumes that the distribution of the
+observed data can be factorized into several independent blocks of variables, each block following
+its own mixture model. In this paper, we assume that each block follows a non parametric latent
+class model, i.e. independence of the variables in each component of the mixture with no parametric
+assumption on their class conditional distribution. The purpose is to deduce, from the observation
+of a sample, the number of blocks, the partition of the variables into the blocks and the number of
+components in each block, which characterise the proposed model. By following recent literature on
+model and variable selection in non-parametric mixture models, we propose to discretize the data
+into bins. This permits to apply the classical multi-partition clustering procedure for parametric
+multinomials, which are based on a penalized likelihood method (e.g. BIC). The consistency of the
+procedure is obtained and an efficient optimization is proposed. The performances of the model are
+investigated on simulated data.
+1
+Introduction
+Finite mixture models allows to perform clustering by modelling the distribution of the variable and
+identifying each component of the mixture as a cluster (McLachlan and Peel, 2000; McNicholas, 2016;
+Bouveyron et al., 2019). This has permitted to perform the clustering of a wide range of data (continuous,
+categorical, functional, mixed) by adapting the class conditional model to each kind of data. Most of
+the development of mixture models have been performed in a parametric framework where it is possible
+to perform consistent model selection such has choosing the number of cluster (Keribin, 2000). However
+assuming a parametric model can be too restrictive to encompass the variety of cluster shapes. Thus
+non-parametric models have been developed (see Chauveau et al. (2015) for a review), they typically only
+assume class conditional independence of the variables of the cluster making no parametric assumption
+on the class conditional univariate distribution. Then they are able to perform estimation with an EM-
+like algorithm (Benaglia et al., 2009) or by maximizing the smoothed log-likelihood (Levine et al., 2011).
+In this setting the choice of the number of cluster is a difficult issue for which Du Roy de Chaumaray and
+Marbac (2021) have recently proposed a consistent solution based on the discretization of the variables.
+One limitation of finite mixture models is that they try to summarize the heterogeneity of the data
+by only one categorical variable. However, in the area of massive data, with individuals described by
+possibly thousands of variables the whole heterogeneity in the data cannot be described by only one
+latent variable. It can for instance be the case if variables related to some focus are more present than
+variables related to another one. Thus model-based clustering approaches has been developed to possibly
+handle several latent class variables, in what we latter call multiple partitions clustering (see Rodriguez-
+Sanchez et al. (2022) for a recent review on the subject).
+Assuming several latent variables in the
+model can be performed in three principal ways. It can be performed by assuming that the distribution
+factorises in independent blocks of variables, the heterogeneity in each block being explained by a latent
+clustering variable (Galimberti and Soffritti, 2007; Marbac and Vandewalle, 2018). It can be performed
+by assuming several classifying linear projections of the variables, each one being explained by a latent
+cluster variable (Attias, 1999; Vandewalle, 2020). Finally, several latent variables can be considered in
+more complex dependence structure such as trees (Poon et al., 2013), Bayesian networks (Rodriguez-
+Sanchez et al., 2022), or multilayer (potentially deep) discrete latent structure (Gu and Dunson, 2021).
+Multiple partition clustering induces an additional complexity compared to standard clustering, since a
+1
+arXiv:2301.02422v1  [stat.ME]  6 Jan 2023
+
+lot of structure parameters need to be learned (number of blocks, repartition of the variables in blocks,
+number of modalities of each latent variables, structure of the network).
+Since the above multiple
+partitions models fall in the parametric framework all these parameters can be selected using penalized
+(such as BIC) or integrated likelihood based criteria (such as MICL). The difficulty of this search depends
+on the complexity of models which is assumed. For instance in the continuous setting Galimberti et al.
+(2018) propose a extension of Galimberti and Soffritti (2007) where many possible roles of the variables
+need to be considered, thus needing a lot of computation even for the re-affectation of only one variable.
+Contrarily Marbac and Vandewalle (2018) propose a very simple model in which variables are grouped
+into independent blocks and each block of variables is a assumed to follow a mixture model with the
+class conditional independence assumption in a parametric setting, they were able to propose a modified
+EM algorithm allowing to update independently the affectation of the variables to the block at each step
+of the algorithm. Let notice that this model is an extension of the approaches proposed by Marbac and
+Sedki (2017); Marbac et al. (2018, 2020) in the framework of variable selection in clustering, where only
+two blocks are considered, i.e. one block of classifying variables assuming conditional independence, and
+one block of non classifying variables assuming total independence.
+In this paper we propose a non parametric extension of the approach proposed by Marbac and
+Vandewalle (2018). In order to solve the difficulty of the model choice, we first discretize the data, where
+the granularity of the discretization depends on the number of data as in Du Roy de Chaumaray and
+Marbac (2021). In this case the non parametric multipartition mixture model becomes a parametric one
+where each block follows a mixture of product of multinomial distributions. This model is a particular
+case of the model proposed by Marbac and Vandewalle (2018), thus we are able to perform an efficient
+model search and parameters estimation by optimizing a penalized likelihood. Moreover following the
+same lines as Du Roy de Chaumaray and Marbac (2021) we are able to prove the consistency of this
+procedure. Once the issue of model selection is solved, we propose to refit a non parametric latent class
+model on each block of variables with fixed number of clusters in order to limit the loss of information
+caused by the discretization of variables and thus to obtain a more accurate estimation of the partitions.
+The outline of the paper is the following: Section 2 introduces the non-parametric multiple partitions
+model, Section 3 explains how the data are discretized, while Section 4 presents the estimator of the
+model along with its consistency and with the modified EM algorithm used for model selection. Finally,
+Section 5 illustrates the good performances of our procedure on simulated data.
+2
+Multiple partitions mixture model
+2.1
+The underlying model
+Data to cluster x = (x1, . . . , xn) are composed of n observations xi = (xi1, . . . , xid) described by d vari-
+ables potentially of different types (i.e., each variable can be continuous or categorical). Observations are
+assumed to independently arise from a multiple partitions model (MPM) which considers that variables
+are grouped into B independent blocks. The blocks of variables are defined by ω = (ω1, . . . , ωd), where
+ωj = b indicates that variable j belongs to block b. The set of the indexes of variables which belong
+to the same block b is denoted by Ωb = {j : ωj = b}. Moreover, MPM considers that the variables of
+block b follow a Gb-components mixture assuming within-component independence. This assumption
+permits to write the conditional density of those variables as the product of |Ωb| =card(Ωb) univariate
+densities. A model m is thus given by the number of blocks B, the repartition of the variables in each
+block ω and the numbers of components G = (G1, . . . , GB) in the mixture-model driving each block.
+Let xi{b} = (xij; j ∈ Ωb) be the vector of observed variables of block b. The probability distribution
+function (pdf) of xi, for a model m = (B, G, ω), is thus given by
+p(xi|m, θ) =
+B
+�
+b=1
+pb(xi{b}|m, θ) with pb(xi{b}|m, θ) =
+Gb
+�
+g=1
+πbg
+�
+j∈Ωb
+ηgj(xij),
+(1)
+where θ = (π, η) groups the model parameters, with π = (πb; b = 1, . . . , B) being the proportions of
+the components in each mixture, with πb = (πb1, . . . , πbGb), πbg > 0 and �Gb
+g=1 πbg = 1, and η grouping
+the univariate densities ηgj, which are infinite dimensional parameters for the continuous variables. We
+denote by Θm the set of all possible parameters θ associated with a given model m. And we denote by
+p0 the pdf under the true model m0 and the true parameter θ0, i.e. p0 = p(.|m0, θ0).
+2
+
+2.2
+Resulting partition of the data
+MPM provides B partitions among the observations (one partition per block of variables). The partition
+of block b is denoted by zb = (z1b, . . . , znb) ∈ ZGb, where ZGb is the set of all partitions of n elements
+into Gb clusters, and zib = (zib1, . . . , zibGb) with zibg = 1 if observation i belongs to cluster g for
+block b and zibg = 0 otherwise. The multiple partitions z = (z1, . . . , zB) for model m thus belongs to
+Zm = ZG1 × . . . × ZGB.
+2.3
+Model and parameters identifiability
+In this part, we explain why model (1) is identifiable up to a switching of the component labels and a
+change in the order of the blocks. We need the following Assumptions.
+Assumption 1.
+(i) The number of variables is at least three in each block and all the proportions in
+the mixtures are not zero, i.e. for any b = 1, . . . B, |Ωb| ≥ 3 and, for any g = 1, . . . , Gb, πbg > 0.
+(ii) For each b = 1, . . . , B, there exists Υb ⊆ Ωb such that |Υb| = 3 and for any j ∈ Υb the univariate
+densities {ηgj; g = 1, . . . , Gb} are linearly independent.
+(iii) For each b = 1, . . . , B, pb cannot be decomposed in a product of densities.
+Suppose B and ω fixed and known, which means that the block structure of the variables is given,
+then, under Assumptions 1(i) and (ii), the model parameters θ and the numbers of components G are
+identifiable (see Allman et al. (2009)).
+We define B as the greatest integer which permits to decompose p(·|m, θ) as the product �B
+b=1 pb(·|m, θ).
+Assumption 1(iii) thus ensures the identifiability of B and of the associated partition of the variables ω.
+Note that Assumption 1(iii) means that the variables belonging to the same block are not independent.
+3
+Model selection via bin estimation
+3.1
+The discretized model used for estimation
+If all the variables are assumed to be continuous, the method is based on the discretization of each
+variable j into R non-overlapping bins IRj1, . . . , IRjR such that ∪R
+r=1IRjr = Xj and for any (r, r′) with
+r ̸= r′, IRjr ∩ IRjr′ = ∅. We denote by σRjr with r ∈ {1, . . . , R}, the indicator functions of each bin,
+defined by σRjr(xij) = 1 if xij ∈ IRjr and σRjr(xij) = 0 if xij /∈ IRjr, and we denote by |IRjr| the
+Lebesgue measure of the bin IRjr. Alternatively, one could use different numbers of bins per variables,
+which is not done in the following for ease of reading. The number of bins as well as their location is
+known and fixed upstream. Conditions to ensure the good property of the procedure and ways to choose
+those items will be given in the next section.
+The discretized variables of block b follow a latent class model where each component is a product of
+|Ωb| multinomial distributions each having R levels. Therefore, the discretized pdf of the subject i for
+the variables in block b is given by
+pRb(xi{b}|m, θRb) =
+Gb
+�
+g=1
+πbg
+�
+j∈Ωb
+R
+�
+r=1
+�αRgjr
+|IRjr|
+�σRjr(xij)
+,
+(2)
+where θRb = (πb, αRb) groups the component proportions πbg and the probabilities αRgjr that one
+subject arisen from component g takes level r for the variable j when this variable is discretized into R
+bins. The division by |IRjr| stands for the histogram approximation of the class conditional univariate
+densities. The parameter space is given by the product of simplexes SGb × SGb|Ωb|
+R
+. Note that pRb is
+an approximation of pb and that this approximation becomes more accurate as R tends to infinity. We
+deduce a discretized version pR of the pdf p of subject i, which is given by
+pR(xi|m, θR) =
+B
+�
+b=1
+pRb(xi{b}|m, θRb),
+3
+
+where pRb is given by Equation (2) and θR groups all parameters of each block θR1, . . . , θRB. The
+discretized version of the true density p(.|m0, θ0) will be denoted by p0,R. Note that, by construction,
+the discretization induces a loss of identifiability concerning the parameters αRgjr.
+3.2
+Dealing with mixed-type data
+The model permits to deal with mixed-type data also, where some of the variables are categorical. Indeed,
+we only discretize the variables which are continuous.
+4
+Model selection via penalized likelihood
+The set of competing models M is given by
+M = {m = (B, G, ω); B ≤ Bmax, ∀b, j Gb ≤ Gmax, ωj ∈ {1, . . . , B}, |Ωb| ≥ 3},
+(3)
+where Bmax is the maximum number of blocks and Gmax is the maximum number of components within
+a block. For a given model m, the parameters θR, constituted by all the πbg and all the αRgjr, are
+unknown and must be estimated from the observations. The parameter space for a given model m is
+denoted by ΘR,m and is the following product space
+ΘR,m =
+B
+�
+b=1
+SGb × SGb|Ωb|
+R
+,
+(4)
+where SK = {u ∈ [0, 1]K :
+�K
+k=1 uk = 1} designates the simplex of size K. We decide to choose
+the model which maximizes the penalized log-likelihood of the discretized version, for some well-chosen
+penalty an,m,R. For instance, it corresponds to the BIC (Schwarz, 1978) if an,m,R is equal to νm ln(n)/2
+where νm is the complexity of model m. According to Equation (4),
+νm =
+B
+�
+b=1
+(Gb − 1) + (R − 1)Gb|Ωb|.
+4.1
+Model inference
+For a sample x and a model m, the observed-data log-likelihood is defined by
+ℓ(θR|m, x) =
+n
+�
+i=1
+B
+�
+b=1
+ln pRb(xi{b}|m, θRb),
+(5)
+and we obtain its penalized version ℓpen by subtracting the penalty term
+ℓpen(θR|m, x) = ℓ(θR|m, x) − an,m,R
+(6)
+Model selection with penalized likelihood consists in maximizing ℓpen with respect to θR and then with
+respect to m. Thus, the selected model is given by
+�
+mn,R = arg max
+m∈M
+max
+θR∈ΘR,m ℓpen(θR|m, x).
+(7)
+Note that, thanks to the conditional independence, we can sum over i before summing over b in equation
+(5), which means that the maximization can be done separately in each block.
+In practice, to avoid numerical issues, we introduce a threshold ε such that the parameter space
+becomes ΘR,m,ε = �B
+b=1 SGb,ε × SGb|Ωb|
+R,ε
+, with ε > 0 being the minimal value of all the elements defined
+in the simplexes, i.e., SK,ε = {u ∈]ε, 1]K :
+�K
+k=1 uk = 1}. Note that the use of such a threshold is quite
+usual in this framework (see Toussile and Gassiat (2009); Bontemps and Toussile (2013); Du Roy de
+Chaumaray and Marbac (2021)). Under the condition that Rε tends to zero as R goes to infinity and ε
+to zero, the parameter space ΘR,m,ε converges to the whole parameter space ΘR,m as ε tends to zero.
+According to the assumptions on the growth rate of B which will be stated by Assumption 4(i) in the
+next section, it is sufficient to set ε−1 = O(nα+1) for some α > 0.
+4
+
+4.2
+Asymptotic convergence in probability
+The consistency of the estimator is established in Du Roy de Chaumaray and Marbac (2021) under
+four sets of assumptions which are recalled here for completeness.
+Assumption 1 has been given in
+Section 2.3 to ensure the identifiability of the underlying model. Assumption 2 state the constraints on
+the distribution of the components. Assumption 3 gives some conditions on the penalty term. Finally,
+Assumption 4 gives some conditions on the discretization.
+Assumption 2.
+(i) There exists some function τ in L1(p0ν) such that, for any model m ∈ M and
+any parameter θ ∈ Θm,
+| ln p(.|m, θ)| < τ ν-a.e..
+(ii) Each variable j is defined on a compact space Xj and its densities for each component g, denoted
+by ηgj, are strictly positive except on a set of Lebesgue measure zero.
+(iii) There exists some positive constant L < ∞ which, for any block b and any variable j ∈ Ωb, bounds
+the derivative of the densities ηgj over Xj:
+∀xj ∈ Xj, |η′
+gj(xj)| ≤ L.
+Assumption 3.
+(i) For any model m, an,m,R is an increasing function of R, G1, . . . , GB, |Ω1|, . . . , and |ΩB|.
+(ii) For any model m, an,m,R/n tends to 0 as n tends to infinity.
+(iii) For any model m, R/an,m,R tends to 0 as n tends to infinity.
+(iv) For any models m and �
+m with m ⊂ �
+m, the limit inferior of an,�
+m,R/an,m,R is strictly larger than
+one as n tends to infinity.
+Assumption 4.
+(i) The number of bins R tends to infinity with n in the following way limn→∞ R = ∞
+and limn→∞ R(ln2 n)/n1/2 = 0.
+(ii) The length of each bin is not zero and satisfies, for any variable j and any bin r, |IRjr|−1 = O(R).
+(iii) For any variable j, let IjR be the set of the upper bounds of the R intervals, then, for any xj ∈ Xj,
+d(xj, IjR) tends to zero as R tends to infinity.
+Theorem 1. Assume that independent data arise from (1) with true model m0 = {B0, G0, ω0}, and
+that the set of competing models M is given by (3). If, in addition, Assumptions 1, 2, 3 and 4 hold true,
+then, the estimator �
+mn,R defined by (7) converges in probability to m0, as n goes to infinity.
+The proof follows the same lines as Theorem 1 in Du Roy de Chaumaray and Marbac (2021), by
+noticing that we deal with each block separately and we have to distinguish in the set of variables which
+are involved in the considered block and which are not.
+4.3
+EM algorithm for model selection
+In order to compute �
+mn,R, we need to maximize the penalized log-likelihood over both θR and m. We
+will make use of the complete-data log-likelihood, which is based on the supposed observation of the
+component membership z and is thus defined by
+ℓ(θR|m, x, z) =
+B
+�
+b=1
+ln p(zb|Gb, πb) +
+B
+�
+b=1
+�
+j∈Ωb
+ln p(xj|Gb, zb, αRbj),
+(8)
+where xj = (x1j, . . . , xnj), αRbj = {αRgjr, g = 1, . . . , Gb, r = 1, . . . , R},
+ln p(zb|Gb, πb) =
+n
+�
+i=1
+Gb
+�
+g=1
+zibg ln πbg,
+5
+
+and
+ln p(xj|Gb, zb, αRj) =
+n
+�
+i=1
+Gb
+�
+g=1
+zibg
+R
+�
+r=1
+σRjr(xij) ln
+�αRgjr
+|IRjr|
+�
+.
+The maximum likelihood estimates (MLE) can be obtained by an EM algorithm (Dempster et al., 1977;
+McLachlan and Krishnan, 1997). Independence between the B blocks of variables permits to maximize
+the observed-data log-likelihood on each block independently.
+Due to the number of competing models, an exhaustive approach which consists in computing BIC
+for each competing models is not doable in practice.
+We will use the same idea as in Marbac and
+Vandewalle (2018) to circumvent the combinatorial issue. Holding (B, G) fixed, model selection with
+BIC and maximum likelihood inference implies to maximize the penalized likelihood with respect to
+(ω, θR). This maximization can be carried out simultaneously by using a modified version of the EM
+algorithm (Green, 1990; Marbac et al., 2018) and, then, �
+mn,R can be found by running this algorithm for
+each value of (B, G) allowed by M. Therefore, less than �Bmax
+B=1 GB
+max calls of the EM algorithm should
+be done. Note that the number of calls of EM algorithm does not depend on the number of variables and
+this not intensive if one considers Bmax small (i.e., Bmax < 5). This can seem restrictive, but note that
+classical clustering methods consider Bmax = 1. Moreover, if Bmax is wanted to be more than five, then
+the model remains well defined but the proposed method for model selection suffers from combinatorial
+issues. Then, in this case, other algorithms (like forward/backward search) should be used for model
+estimation.
+To implement this modified EM algorithm, we introduce the penalized complete-data log-likelihood
+ℓpen(θR|m, x, z) = ℓ(θR|m, x, z) − an,m,R.
+We need to assume that the penalty can be decomposed as a sum of penalties concerning each type of
+parameter as follows: an,m,R = �B
+b=1
+�
+an,πb,R + �
+j∈Ωb an,αRbj,R
+�
+. This assumption is not stringent and
+is satisfied, for instance, by the BIC with 2an,πb,R = (Gb − 1) ln n and 2an,αRbj,R = (R − 1)(Gb − 1) ln n.
+The penalized complete-data log-likelihood thus rewrites as
+ℓpen(θR|m, x, z) =
+B
+�
+b=1
+(ln p(zb|πb) − an,πb,R) +
+B
+�
+b=1
+�
+j∈Ωb
+�
+ln p(xj|zb, αRbj) − an,αRbj,R
+�
+.
+(9)
+Holding (B, G) fixed and starting from (ω[0], θ[0]
+R ), the iteration [s] of the algorithm is composed of
+three steps:
+E-step Computation of the fuzzy partitions t[s]
+ibg := E[Zibg|xi, m[s−1], θ[s−1]
+R
+], hence for b = 1, . . . , B, for
+g = 1, . . . , Gb, for i = 1, . . . , n
+t[s]
+ibg =
+π[s−1]
+bg
+�
+j∈Ω[s−1]
+b
+�
+α[s−1]
+Rgjr
+�σRjr(xij)
+�Gb
+k=1 π[s−1]
+bk
+�
+j∈Ω[s−1]
+b
+�
+α[s−1]
+Rkjr
+�σRjr(xij) .
+M-step1 Updating the affectation of the variables to blocks, for j = 1, . . . , d,
+ω[s]
+j
+= arg max
+b∈{1,...,B}
+� Gb
+�
+g=1
+max
+αRgj∈SR Q(αRgj|xj, t[s]
+bg) − an,αRbj,R
+�
+,
+where αRgj = (αRgj1, . . . , αRgjR) and Q(αRgj|xj, t) = �n
+i=1 ti
+�R
+r=1 σRjr(xij) ln αRgjr. Thus Ω[s]
+b
+=
+{j : ω[s]
+j
+= b}.
+M-step2 Updating the model parameters: for b = 1, . . . , B, for g = 1, . . . , Gb,
+π[s]
+bg = 1
+n
+n
+�
+i=1
+t[s]
+ibg
+and for j ∈ Ω[s]
+b ,
+α[s]
+Rgj = arg max
+αRgj∈SR
+Q(αjg|xj, t[s]
+ω[s]
+j g),
+6
+
+which implies that for any r = 1, . . . , R,
+α[s]
+Rgjr =
+�n
+i=1 tiω[s]
+j gσRjr(xij)
+�n
+i=1 tiω[s]
+j g
+.
+Like for the standard EM algorithm, the objective function ℓpen(θR|m, x, z) increases at each iteration
+but the global optimum is not achieved in general. Hence, different random initializations must be done.
+Finally, note that the algorithm can return empty blocks. Indeed, M-step1 is done without constraining
+each block to contain at least one variable. Thus, each ω[s]
+j
+can be obtained independently.
+This algorithm permits to estimate the densities of the underlying model. However, the bin-based
+density estimators are generally proven to be outperformed by kernel-based density estimators. Thus,
+we advise to use the proposed method only for selecting the model, and then to use kernel-based density
+estimates for the selected model, for instance EM-like algorithm (Benaglia et al., 2009) or MM-algorithm
+(Levine et al., 2011). This will be illustrated in the numerical experiments.
+5
+Numerical experiments
+5.1
+Simulation setup
+Data are generated from a mixture with three blocks of variables (B = 3), each block being a mixture
+of three components (G = (3, 3, 3) ) with well-balanced clusters (i.e., with equal proportions πbg = 1/3)
+and with the same number of involved variables (i.e., |Ω1| = |Ω2| = |Ω3|). In block b, the marginal
+density of Xi given the component membership is a product of |Ωb| univariate densities such that
+Xij =
+Gb
+�
+g=1
+zibgδgj + ξij
+where all the ξij are independent and where δgj = τ if j is equal to g modulo Gb and δgj = 0 otherwise.
+The value of τ is tuned in order to obtain a chosen theoretical misclassification rate.
+We consider three distributions for the ξij (standard Gaussian, Student with three degrees of freedom
+and Laplace), four sample sizes (n = 50, 100, 200, 400), different numbers of variables in each block ( for
+each block b, |Ωb| equals successively 6, 9 and 12) while the value of τ is defined to obtain a theoretical
+misclassification rates of 5% and of 10%. For each situation ( sample size, number of variables, law and
+misclassification rate), 100 replicates are generated.
+The discretization step is conducted with R bins given by the empirical quantiles. We investigate
+four number of bins: R = [n1/4], [n1/5], [n1/6] and [n1/7], for each previous replicate.
+The model selection is done with the proposed modified EM algorithm, with Bmax = 3 and Gmax = 3,
+and with a BIC penalty. Then, for the selected model, parameters and density estimation is conducted
+with the package mixtools, leading to a more accurate estimated partition.
+5.2
+Performances of the proposed method.
+We compute in each situation the Adjusted Rand Index (ARI) between the obtained blocks of variables
+and the true ones, as well as the ARI between the obtained partition of the individuals and the true
+partition, in each block.
+Figure 1 displays the boxplot of the ARI between the true and estimated blocks of variables obtained
+on each situation over the 100 replicates. The misclassification rate is fixed equal to 5%. It shows that
+the proposed method is able to perfectly recover the different blocks of variables, for each family of law,
+as soon as the sample size and the number of bins are sufficiently large. We recall that the number of
+bins should not be too large compared to the sample size, in order to satisfy Assumption 4. The method
+is more accurate as n grows, while it deteriorates a little when the number of variables in each blocks
+grows, for small values of n. This is not surprising as it makes the classification problem more complex.
+This will be further illustrated on the last figures.
+Figure 2 displays the boxplots over of the mean of the ARI obtained in each block between the true
+and estimated partitions. The misclassification rate is fixed equal to 5%. Here, one can notice that the
+performance are good as well.
+7
+
+Gauss
+Laplace
+Student
+6
+9
+12
+50
+100
+200
+400
+50
+100
+200
+400
+50
+100
+200
+400
+0.00
+0.25
+0.50
+0.75
+1.00
+0.00
+0.25
+0.50
+0.75
+1.00
+0.00
+0.25
+0.50
+0.75
+1.00
+Sample size
+Adjusted Rand Index
+Bandwith
+4
+5
+6
+7
+Figure 1: ARI between the true and the estimated blocks of variables, for different sample size, different
+families of components, different number of variable, and for different number of bins [n1/7], [n1/6], [n1/5],
+[n1/4] indicated by grey levels.
+Gauss
+Laplace
+Student
+6
+9
+12
+50
+100
+200
+400
+50
+100
+200
+400
+50
+100
+200
+400
+0.00
+0.25
+0.50
+0.75
+0.00
+0.25
+0.50
+0.75
+0.00
+0.25
+0.50
+0.75
+Sample size
+Adjusted Rand Index
+Bandwith
+4
+5
+6
+7
+Figure 2: mean ARI between the true and the estimated partition of individuals, for different sample size,
+different families of components, different number of variable, and for different number of bins [n1/7],
+[n1/6], [n1/5], [n1/4] indicated by grey levels.
+8
+
+In Figures 3 and 4, we investigate the behavior of the proposed method, for different values of the
+theoretical misclassification rate. The number of bins is fixed equal to [n1/6]. Again, we note that the
+methods is more accurate as n grows, while it deteriorates with the increase of the number of variables
+in each block.
+Gauss
+Laplace
+Student
+0.05
+0.1
+50
+100
+200
+400
+50
+100
+200
+400
+50
+100
+200
+400
+0.00
+0.25
+0.50
+0.75
+0.00
+0.25
+0.50
+0.75
+Sample size
+Adjusted Rand Index
+Variables
+6
+9
+12
+Figure 3: mean ARI between the true and the estimated partition of individuals, for different sample
+size, different families of components, different misclassification rate and different number of variables
+indicated by grey levels.
+Gauss
+Laplace
+Student
+0.05
+0.1
+50
+100
+200
+400
+50
+100
+200
+400
+50
+100
+200
+400
+0.00
+0.25
+0.50
+0.75
+1.00
+0.00
+0.25
+0.50
+0.75
+1.00
+Sample size
+Adjusted Rand Index
+Variables
+6
+9
+12
+Figure 4: mean ARI between the true and the estimated blocks of variables, for different sample size, dif-
+ferent families of components, different misclassification rate and different number of variables indicated
+by grey levels.
+9
+
+6
+Conclusion
+We have proposed a new method for performing clustering with multiple partitions, when no parametric
+assumptions are made on the conditional distributions of the variables given the component memberships.
+This methods permits do deal with continuous data. It is based on the discretization of the continuous
+data which permits to reuse previous methods which are able to deal with multinomials distributions.
+For mixed-type data, the method can be straightforwardly extended by discretizing only the continuous
+ones. Model selection is conducted on the discretized data by using a modified EM algorithm, which
+estimates simultaneously the partition of the variables and the parameters of the model. The procedure
+is consistent for a range of penalty including the classical BIC.
+Bibliography
+References
+Allman, E., Matias, C., and Rhodes, J. (2009). Identifiability of parameters in latent structure models
+with many observed variables. The Annals of Statistics, 37(6A):3099–3132.
+Attias, H. (1999). Independent factor analysis. Neural computation, 11(4):803–851.
+Benaglia, T., Chauveau, D., and Hunter, D. R. (2009). An EM-like algorithm for semi-and nonparametric
+estimation in multivariate mixtures. Journal of Computational and Graphical Statistics, 18(2):505–526.
+Bontemps, D. and Toussile, W. (2013). Clustering and variable selection for categorical multivariate
+data. Electronic Journal of Statistics, 7(none):2344 – 2371.
+Bouveyron, C., Celeux, G., Murphy, T. B., and Raftery, A. E. (2019).
+Model-based clustering and
+classification for data science: with applications in R, volume 50. Cambridge University Press.
+Chauveau, D., Hunter, D. R., and Levine, M. (2015). Semi-parametric estimation for conditional inde-
+pendence multivariate finite mixture models. Statistics Surveys, 9:1–31.
+Dempster, A., Laird, N., and Rubin, D. (1977). Maximum likelihood from incomplete data via the EM
+algorithm. Journal of the Royal Statistical Society. Series B (Methodological), 39(1):1–38.
+Du Roy de Chaumaray, M. and Marbac, M. (2021). Full model estimation for non-parametric multivariate
+finite mixture models.
+Galimberti, G., Manisi, A., and Soffritti, G. (2018). Modelling the role of variables in model-based cluster
+analysis. Statistics and Computing, 28(1):145–169.
+Galimberti, G. and Soffritti, G. (2007). Model-based methods to identify multiple cluster structures in
+a data set. Computational Statistics & Data Analysis, 52(1):520 – 536.
+Green, P. J. (1990). On use of the em for penalized likelihood estimation. Journal of the Royal Statistical
+Society. Series B (Methodological), pages 443–452.
+Gu, Y. and Dunson, D. B. (2021). Bayesian pyramids: Identifiable multilayer discrete latent structure
+models for discrete data. arXiv preprint arXiv:2101.10373.
+Keribin, C. (2000). Consistent estimation of the order of mixture models. Sankhy¯a: The Indian Journal
+of Statistics, Series A, pages 49–66.
+Levine, M., Hunter, D. R., and Chauveau, D. (2011). Maximum smoothed likelihood for multivariate
+mixtures. Biometrika, pages 403–416.
+Marbac, M., Patin, E., and Sedki, M. (2018). Variable selection for mixed data clustering: a model-based
+approach. Journal of Classification, to appear.
+Marbac, M. and Sedki, M. (2017). Variable selection for model-based clustering using the integrated
+complete-data likelihood. Statistics and Computing, 27(4):1049–1063.
+10
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+Marbac, M., Sedki, M., and Patin, T. (2020). Variable selection for mixed data clustering: application
+in human population genomics. Journal of Classification, 37(1):124–142.
+Marbac, M. and Vandewalle, V. (2018). A tractable Multi-Partitions Clustering. Computational Statistics
+and Data Analysis.
+McLachlan, G. and Krishnan, T. (1997). The EM algorithm. Wiley Series in Probability and Statistics:
+Applied Probability and Statistics, Wiley-Interscience, New York.
+McLachlan, G. and Peel, D. (2000). Finite mixutre models. Wiley Series in Probability and Statistics:
+Apllied Probability and Statistics, Wiley-Interscience, New York.
+McNicholas, P. D. (2016). Mixture model-based classification. CRC press.
+Poon, L. K., Zhang, N. L., Liu, T., and Liu, A. H. (2013). Model-based clustering of high-dimensional
+data: Variable selection versus facet determination. International Journal of Approximate Reasoning,
+54(1):196–215.
+Rodriguez-Sanchez, F., Bielza, C., and Larra˜naga, P. (2022). Multipartition clustering of mixed data
+with bayesian networks. International Journal of Intelligent Systems, 37(3):2188–2218.
+Schwarz, G. (1978). Estimating the dimension of a model. The Annals of Statistics, 6(2):461–464.
+Toussile, W. and Gassiat, E. (2009).
+Variable selection in model-based clustering using multilocus
+genotype data. Advances in Data Analysis and Classification, 3(2):109–134.
+Vandewalle, V. (2020). Multi-partitions subspace clustering. Mathematics, 8(4):597.
+11
+
diff --git a/EdE0T4oBgHgl3EQfgwF8/content/tmp_files/load_file.txt b/EdE0T4oBgHgl3EQfgwF8/content/tmp_files/load_file.txt
new file mode 100644
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@@ -0,0 +1,538 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf,len=537
+page_content='Non-parametric Multi-Partitions Clustering  Marie du Roy de Chaumaray and Vincent Vandewalle  January 9, 2023  Abstract  In the framework of model-based clustering, a model, called multi-partitions clustering, allowing  several latent class variables has been proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' This model assumes that the distribution of the  observed data can be factorized into several independent blocks of variables, each block following  its own mixture model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' In this paper, we assume that each block follows a non parametric latent  class model, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' independence of the variables in each component of the mixture with no parametric  assumption on their class conditional distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' The purpose is to deduce, from the observation  of a sample, the number of blocks, the partition of the variables into the blocks and the number of  components in each block, which characterise the proposed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' By following recent literature on  model and variable selection in non-parametric mixture models, we propose to discretize the data  into bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' This permits to apply the classical multi-partition clustering procedure for parametric  multinomials, which are based on a penalized likelihood method (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' BIC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' The consistency of the  procedure is obtained and an efficient optimization is proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' The performances of the model are  investigated on simulated data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='  1  Introduction  Finite mixture models allows to perform clustering by modelling the distribution of the variable and  identifying each component of the mixture as a cluster (McLachlan and Peel, 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' McNicholas, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='  Bouveyron et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' This has permitted to perform the clustering of a wide range of data (continuous,  categorical, functional, mixed) by adapting the class conditional model to each kind of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' Most of  the development of mixture models have been performed in a parametric framework where it is possible  to perform consistent model selection such has choosing the number of cluster (Keribin, 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' However  assuming a parametric model can be too restrictive to encompass the variety of cluster shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' Thus  non-parametric models have been developed (see Chauveau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' (2015) for a review), they typically only  assume class conditional independence of the variables of the cluster making no parametric assumption  on the class conditional univariate distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' Then they are able to perform estimation with an EM-  like algorithm (Benaglia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=', 2009) or by maximizing the smoothed log-likelihood (Levine et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=', 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='  In this setting the choice of the number of cluster is a difficult issue for which Du Roy de Chaumaray and  Marbac (2021) have recently proposed a consistent solution based on the discretization of the variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='  One limitation of finite mixture models is that they try to summarize the heterogeneity of the data  by only one categorical variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' However, in the area of massive data, with individuals described by  possibly thousands of variables the whole heterogeneity in the data cannot be described by only one  latent variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' It can for instance be the case if variables related to some focus are more present than  variables related to another one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' Thus model-based clustering approaches has been developed to possibly  handle several latent class variables, in what we latter call multiple partitions clustering (see Rodriguez-  Sanchez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' (2022) for a recent review on the subject).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='  Assuming several latent variables in the  model can be performed in three principal ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' It can be performed by assuming that the distribution  factorises in independent blocks of variables, the heterogeneity in each block being explained by a latent  clustering variable (Galimberti and Soffritti, 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' Marbac and Vandewalle, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' It can be performed  by assuming several classifying linear projections of the variables, each one being explained by a latent  cluster variable (Attias, 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' Vandewalle, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' Finally, several latent variables can be considered in  more complex dependence structure such as trees (Poon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=', 2013), Bayesian networks (Rodriguez-  Sanchez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=', 2022), or multilayer (potentially deep) discrete latent structure (Gu and Dunson, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='  Multiple partition clustering induces an additional complexity compared to standard clustering, since a  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='02422v1  [stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='ME]  6 Jan 2023  lot of structure parameters need to be learned (number of blocks, repartition of the variables in blocks,  number of modalities of each latent variables, structure of the network).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='  Since the above multiple  partitions models fall in the parametric framework all these parameters can be selected using penalized  (such as BIC) or integrated likelihood based criteria (such as MICL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' The difficulty of this search depends  on the complexity of models which is assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' For instance in the continuous setting Galimberti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='  (2018) propose a extension of Galimberti and Soffritti (2007) where many possible roles of the variables  need to be considered, thus needing a lot of computation even for the re-affectation of only one variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='  Contrarily Marbac and Vandewalle (2018) propose a very simple model in which variables are grouped  into independent blocks and each block of variables is a assumed to follow a mixture model with the  class conditional independence assumption in a parametric setting, they were able to propose a modified  EM algorithm allowing to update independently the affectation of the variables to the block at each step  of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' Let notice that this model is an extension of the approaches proposed by Marbac and  Sedki (2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' Marbac et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' (2018, 2020) in the framework of variable selection in clustering, where only  two blocks are considered, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' one block of classifying variables assuming conditional independence, and  one block of non classifying variables assuming total independence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='  In this paper we propose a non parametric extension of the approach proposed by Marbac and  Vandewalle (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' In order to solve the difficulty of the model choice, we first discretize the data, where  the granularity of the discretization depends on the number of data as in Du Roy de Chaumaray and  Marbac (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' In this case the non parametric multipartition mixture model becomes a parametric one  where each block follows a mixture of product of multinomial distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' This model is a particular  case of the model proposed by Marbac and Vandewalle (2018), thus we are able to perform an efficient  model search and parameters estimation by optimizing a penalized likelihood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' Moreover following the  same lines as Du Roy de Chaumaray and Marbac (2021) we are able to prove the consistency of this  procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' Once the issue of model selection is solved, we propose to refit a non parametric latent class  model on each block of variables with fixed number of clusters in order to limit the loss of information  caused by the discretization of variables and thus to obtain a more accurate estimation of the partitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='  The outline of the paper is the following: Section 2 introduces the non-parametric multiple partitions  model, Section 3 explains how the data are discretized, while Section 4 presents the estimator of the  model along with its consistency and with the modified EM algorithm used for model selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' Finally,  Section 5 illustrates the good performances of our procedure on simulated data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='  2  Multiple partitions mixture model  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='1  The underlying model  Data to cluster x = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' , xn) are composed of n observations xi = (xi1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' , xid) described by d vari-  ables potentially of different types (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=', each variable can be continuous or categorical).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' Observations are  assumed to independently arise from a multiple partitions model (MPM) which considers that variables  are grouped into B independent blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' The blocks of variables are defined by ω = (ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' , ωd), where  ωj = b indicates that variable j belongs to block b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' The set of the indexes of variables which belong  to the same block b is denoted by Ωb = {j : ωj = b}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' Moreover, MPM considers that the variables of  block b follow a Gb-components mixture assuming within-component independence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' This assumption  permits to write the conditional density of those variables as the product of |Ωb| =card(Ωb) univariate  densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' A model m is thus given by the number of blocks B, the repartition of the variables in each  block ω and the numbers of components G = (G1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' , GB) in the mixture-model driving each block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='  Let xi{b} = (xij;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' j ∈ Ωb) be the vector of observed variables of block b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' The probability distribution  function (pdf) of xi, for a model m = (B, G, ω), is thus given by  p(xi|m, θ) =  B  �  b=1  pb(xi{b}|m, θ) with pb(xi{b}|m, θ) =  Gb  �  g=1  πbg  �  j∈Ωb  ηgj(xij),  (1)  where θ = (π, η) groups the model parameters, with π = (πb;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' b = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' , B) being the proportions of  the components in each mixture, with πb = (πb1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' , πbGb), πbg > 0 and �Gb  g=1 πbg = 1, and η grouping  the univariate densities ηgj, which are infinite dimensional parameters for the continuous variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' We  denote by Θm the set of all possible parameters θ associated with a given model m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' And we denote by  p0 the pdf under the true model m0 and the true parameter θ0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' p0 = p(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='|m0, θ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='2  Resulting partition of the data  MPM provides B partitions among the observations (one partition per block of variables).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' The partition  of block b is denoted by zb = (z1b, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' , znb) ∈ ZGb, where ZGb is the set of all partitions of n elements  into Gb clusters, and zib = (zib1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' , zibGb) with zibg = 1 if observation i belongs to cluster g for  block b and zibg = 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' The multiple partitions z = (z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' , zB) for model m thus belongs to  Zm = ZG1 × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' × ZGB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='3  Model and parameters identifiability  In this part, we explain why model (1) is identifiable up to a switching of the component labels and a  change in the order of the blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' We need the following Assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='  Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='  (i) The number of variables is at least three in each block and all the proportions in  the mixtures are not zero, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' for any b = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' B, |Ωb| ≥ 3 and, for any g = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' , Gb, πbg > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='  (ii) For each b = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' , B, there exists Υb ⊆ Ωb such that |Υb| = 3 and for any j ∈ Υb the univariate  densities {ηgj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' g = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' , Gb} are linearly independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='  (iii) For each b = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' , B, pb cannot be decomposed in a product of densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='  Suppose B and ω fixed and known, which means that the block structure of the variables is given,  then, under Assumptions 1(i) and (ii), the model parameters θ and the numbers of components G are  identifiable (see Allman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' (2009)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='  We define B as the greatest integer which permits to decompose p(·|m, θ) as the product �B  b=1 pb(·|m, θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='  Assumption 1(iii) thus ensures the identifiability of B and of the associated partition of the variables ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='  Note that Assumption 1(iii) means that the variables belonging to the same block are not independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='  3  Model selection via bin estimation  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='1  The discretized model used for estimation  If all the variables are assumed to be continuous, the method is based on the discretization of each  variable j into R non-overlapping bins IRj1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' , IRjR such that ∪R  r=1IRjr = Xj and for any (r, r′) with  r ̸= r′, IRjr ∩ IRjr′ = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' We denote by σRjr with r ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' , R}, the indicator functions of each bin,  defined by σRjr(xij) = 1 if xij ∈ IRjr and σRjr(xij) = 0 if xij /∈ IRjr, and we denote by |IRjr| the  Lebesgue measure of the bin IRjr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' Alternatively, one could use different numbers of bins per variables,  which is not done in the following for ease of reading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' The number of bins as well as their location is  known and fixed upstream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' Conditions to ensure the good property of the procedure and ways to choose  those items will be given in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='  The discretized variables of block b follow a latent class model where each component is a product of  |Ωb| multinomial distributions each having R levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' Therefore, the discretized pdf of the subject i for  the variables in block b is given by  pRb(xi{b}|m, θRb) =  Gb  �  g=1  πbg  �  j∈Ωb  R  �  r=1  �αRgjr  |IRjr|  �σRjr(xij)  ,  (2)  where θRb = (πb, αRb) groups the component proportions πbg and the probabilities αRgjr that one  subject arisen from component g takes level r for the variable j when this variable is discretized into R  bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' The division by |IRjr| stands for the histogram approximation of the class conditional univariate  densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' The parameter space is given by the product of simplexes SGb × SGb|Ωb|  R  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' Note that pRb is  an approximation of pb and that this approximation becomes more accurate as R tends to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' We  deduce a discretized version pR of the pdf p of subject i, which is given by  pR(xi|m, θR) =  B  �  b=1  pRb(xi{b}|m, θRb),  3  where pRb is given by Equation (2) and θR groups all parameters of each block θR1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' , θRB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' The  discretized version of the true density p(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='|m0, θ0) will be denoted by p0,R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' Note that, by construction,  the discretization induces a loss of identifiability concerning the parameters αRgjr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='2  Dealing with mixed-type data  The model permits to deal with mixed-type data also, where some of the variables are categorical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' Indeed,  we only discretize the variables which are continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='  4  Model selection via penalized likelihood  The set of competing models M is given by  M = {m = (B, G, ω);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' B ≤ Bmax, ∀b, j Gb ≤ Gmax, ωj ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' , B}, |Ωb| ≥ 3},  (3)  where Bmax is the maximum number of blocks and Gmax is the maximum number of components within  a block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' For a given model m, the parameters θR, constituted by all the πbg and all the αRgjr, are  unknown and must be estimated from the observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' The parameter space for a given model m is  denoted by ΘR,m and is the following product space  ΘR,m =  B  �  b=1  SGb × SGb|Ωb|  R  ,  (4)  where SK = {u ∈ [0, 1]K :  �K  k=1 uk = 1} designates the simplex of size K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' We decide to choose  the model which maximizes the penalized log-likelihood of the discretized version, for some well-chosen  penalty an,m,R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' For instance, it corresponds to the BIC (Schwarz, 1978) if an,m,R is equal to νm ln(n)/2  where νm is the complexity of model m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' According to Equation (4),  νm =  B  �  b=1  (Gb − 1) + (R − 1)Gb|Ωb|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='1  Model inference  For a sample x and a model m, the observed-data log-likelihood is defined by  ℓ(θR|m, x) =  n  �  i=1  B  �  b=1  ln pRb(xi{b}|m, θRb),  (5)  and we obtain its penalized version ℓpen by subtracting the penalty term  ℓpen(θR|m, x) = ℓ(θR|m, x) − an,m,R  (6)  Model selection with penalized likelihood consists in maximizing ℓpen with respect to θR and then with  respect to m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' Thus, the selected model is given by  �  mn,R = arg max  m∈M  max  θR∈ΘR,m ℓpen(θR|m, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='  (7)  Note that, thanks to the conditional independence, we can sum over i before summing over b in equation  (5), which means that the maximization can be done separately in each block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='  In practice, to avoid numerical issues, we introduce a threshold ε such that the parameter space  becomes ΘR,m,ε = �B  b=1 SGb,ε × SGb|Ωb|  R,ε  , with ε > 0 being the minimal value of all the elements defined  in the simplexes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=', SK,ε = {u ∈]ε, 1]K :  �K  k=1 uk = 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' Note that the use of such a threshold is quite  usual in this framework (see Toussile and Gassiat (2009);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' Bontemps and Toussile (2013);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' Du Roy de  Chaumaray and Marbac (2021)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' Under the condition that Rε tends to zero as R goes to infinity and ε  to zero, the parameter space ΘR,m,ε converges to the whole parameter space ΘR,m as ε tends to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='  According to the assumptions on the growth rate of B which will be stated by Assumption 4(i) in the  next section, it is sufficient to set ε−1 = O(nα+1) for some α > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='  4  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='2  Asymptotic convergence in probability  The consistency of the estimator is established in Du Roy de Chaumaray and Marbac (2021) under  four sets of assumptions which are recalled here for completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='  Assumption 1 has been given in  Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='3 to ensure the identifiability of the underlying model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' Assumption 2 state the constraints on  the distribution of the components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' Assumption 3 gives some conditions on the penalty term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' Finally,  Assumption 4 gives some conditions on the discretization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='  Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='  (i) There exists some function τ in L1(p0ν) such that, for any model m ∈ M and  any parameter θ ∈ Θm,  | ln p(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='|m, θ)| < τ ν-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='.  (ii) Each variable j is defined on a compact space Xj and its densities for each component g, denoted  by ηgj, are strictly positive except on a set of Lebesgue measure zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='  (iii) There exists some positive constant L < ∞ which, for any block b and any variable j ∈ Ωb, bounds  the derivative of the densities ηgj over Xj:  ∀xj ∈ Xj, |η′  gj(xj)| ≤ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='  Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='  (i) For any model m, an,m,R is an increasing function of R, G1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' , GB, |Ω1|, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' , and |ΩB|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='  (ii) For any model m, an,m,R/n tends to 0 as n tends to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='  (iii) For any model m, R/an,m,R tends to 0 as n tends to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='  (iv) For any models m and �  m with m ⊂ �  m, the limit inferior of an,�  m,R/an,m,R is strictly larger than  one as n tends to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='  Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='  (i) The number of bins R tends to infinity with n in the following way limn→∞ R = ∞  and limn→∞ R(ln2 n)/n1/2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='  (ii) The length of each bin is not zero and satisfies, for any variable j and any bin r, |IRjr|−1 = O(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='  (iii) For any variable j, let IjR be the set of the upper bounds of the R intervals, then, for any xj ∈ Xj,  d(xj, IjR) tends to zero as R tends to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' Assume that independent data arise from (1) with true model m0 = {B0, G0, ω0}, and  that the set of competing models M is given by (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' If, in addition, Assumptions 1, 2, 3 and 4 hold true,  then, the estimator �  mn,R defined by (7) converges in probability to m0, as n goes to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='  The proof follows the same lines as Theorem 1 in Du Roy de Chaumaray and Marbac (2021), by  noticing that we deal with each block separately and we have to distinguish in the set of variables which  are involved in the considered block and which are not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='3  EM algorithm for model selection  In order to compute �  mn,R, we need to maximize the penalized log-likelihood over both θR and m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' We  will make use of the complete-data log-likelihood, which is based on the supposed observation of the  component membership z and is thus defined by  ℓ(θR|m, x, z) =  B  �  b=1  ln p(zb|Gb, πb) +  B  �  b=1  �  j∈Ωb  ln p(xj|Gb, zb, αRbj),  (8)  where xj = (x1j, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' , xnj), αRbj = {αRgjr, g = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' , Gb, r = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' , R},  ln p(zb|Gb, πb) =  n  �  i=1  Gb  �  g=1  zibg ln πbg,  5  and  ln p(xj|Gb, zb, αRj) =  n  �  i=1  Gb  �  g=1  zibg  R  �  r=1  σRjr(xij) ln  �αRgjr  |IRjr|  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='  The maximum likelihood estimates (MLE) can be obtained by an EM algorithm (Dempster et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=', 1977;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='  McLachlan and Krishnan, 1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' Independence between the B blocks of variables permits to maximize  the observed-data log-likelihood on each block independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='  Due to the number of competing models, an exhaustive approach which consists in computing BIC  for each competing models is not doable in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='  We will use the same idea as in Marbac and  Vandewalle (2018) to circumvent the combinatorial issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' Holding (B, G) fixed, model selection with  BIC and maximum likelihood inference implies to maximize the penalized likelihood with respect to  (ω, θR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' This maximization can be carried out simultaneously by using a modified version of the EM  algorithm (Green, 1990;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' Marbac et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=', 2018) and, then, �  mn,R can be found by running this algorithm for  each value of (B, G) allowed by M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' Therefore, less than �Bmax  B=1 GB  max calls of the EM algorithm should  be done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' Note that the number of calls of EM algorithm does not depend on the number of variables and  this not intensive if one considers Bmax small (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=', Bmax < 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' This can seem restrictive, but note that  classical clustering methods consider Bmax = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' Moreover, if Bmax is wanted to be more than five, then  the model remains well defined but the proposed method for model selection suffers from combinatorial  issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' Then, in this case, other algorithms (like forward/backward search) should be used for model  estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='  To implement this modified EM algorithm, we introduce the penalized complete-data log-likelihood  ℓpen(θR|m, x, z) = ℓ(θR|m, x, z) − an,m,R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='  We need to assume that the penalty can be decomposed as a sum of penalties concerning each type of  parameter as follows: an,m,R = �B  b=1  �  an,πb,R + �  j∈Ωb an,αRbj,R  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' This assumption is not stringent and  is satisfied, for instance, by the BIC with 2an,πb,R = (Gb − 1) ln n and 2an,αRbj,R = (R − 1)(Gb − 1) ln n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='  The penalized complete-data log-likelihood thus rewrites as  ℓpen(θR|m, x, z) =  B  �  b=1  (ln p(zb|πb) − an,πb,R) +  B  �  b=1  �  j∈Ωb  �  ln p(xj|zb, αRbj) − an,αRbj,R  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='  (9)  Holding (B, G) fixed and starting from (ω[0], θ[0]  R ), the iteration [s] of the algorithm is composed of  three steps:  E-step Computation of the fuzzy partitions t[s]  ibg := E[Zibg|xi, m[s−1], θ[s−1]  R  ], hence for b = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' , B, for  g = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' , Gb, for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' , n  t[s]  ibg =  π[s−1]  bg  �  j∈Ω[s−1]  b  �  α[s−1]  Rgjr  �σRjr(xij)  �Gb  k=1 π[s−1]  bk  �  j∈Ω[s−1]  b  �  α[s−1]  Rkjr  �σRjr(xij) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='  M-step1 Updating the affectation of the variables to blocks, for j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' , d,  ω[s]  j  = arg max  b∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=',B}  � Gb  �  g=1  max  αRgj∈SR Q(αRgj|xj, t[s]  bg) − an,αRbj,R  �  ,  where αRgj = (αRgj1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' , αRgjR) and Q(αRgj|xj, t) = �n  i=1 ti  �R  r=1 σRjr(xij) ln αRgjr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' Thus Ω[s]  b  =  {j : ω[s]  j  = b}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='  M-step2 Updating the model parameters: for b = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' , B, for g = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' , Gb,  π[s]  bg = 1  n  n  �  i=1  t[s]  ibg  and for j ∈ Ω[s]  b ,  α[s]  Rgj = arg max  αRgj∈SR  Q(αjg|xj, t[s]  ω[s]  j g),  6  which implies that for any r = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' , R,  α[s]  Rgjr =  �n  i=1 tiω[s]  j gσRjr(xij)  �n  i=1 tiω[s]  j g  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='  Like for the standard EM algorithm, the objective function ℓpen(θR|m, x, z) increases at each iteration  but the global optimum is not achieved in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' Hence, different random initializations must be done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='  Finally, note that the algorithm can return empty blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' Indeed, M-step1 is done without constraining  each block to contain at least one variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' Thus, each ω[s]  j  can be obtained independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='  This algorithm permits to estimate the densities of the underlying model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' However, the bin-based  density estimators are generally proven to be outperformed by kernel-based density estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' Thus,  we advise to use the proposed method only for selecting the model, and then to use kernel-based density  estimates for the selected model, for instance EM-like algorithm (Benaglia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=', 2009) or MM-algorithm  (Levine et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=', 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' This will be illustrated in the numerical experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='  5  Numerical experiments  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='1  Simulation setup  Data are generated from a mixture with three blocks of variables (B = 3), each block being a mixture  of three components (G = (3, 3, 3) ) with well-balanced clusters (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=', with equal proportions πbg = 1/3)  and with the same number of involved variables (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=', |Ω1| = |Ω2| = |Ω3|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' In block b, the marginal  density of Xi given the component membership is a product of |Ωb| univariate densities such that  Xij =  Gb  �  g=1  zibgδgj + ξij  where all the ξij are independent and where δgj = τ if j is equal to g modulo Gb and δgj = 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='  The value of τ is tuned in order to obtain a chosen theoretical misclassification rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='  We consider three distributions for the ξij (standard Gaussian, Student with three degrees of freedom  and Laplace), four sample sizes (n = 50, 100, 200, 400), different numbers of variables in each block ( for  each block b, |Ωb| equals successively 6, 9 and 12) while the value of τ is defined to obtain a theoretical  misclassification rates of 5% and of 10%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' For each situation ( sample size, number of variables, law and  misclassification rate), 100 replicates are generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='  The discretization step is conducted with R bins given by the empirical quantiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' We investigate  four number of bins: R = [n1/4], [n1/5], [n1/6] and [n1/7], for each previous replicate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='  The model selection is done with the proposed modified EM algorithm, with Bmax = 3 and Gmax = 3,  and with a BIC penalty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' Then, for the selected model, parameters and density estimation is conducted  with the package mixtools, leading to a more accurate estimated partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='2  Performances of the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='  We compute in each situation the Adjusted Rand Index (ARI) between the obtained blocks of variables  and the true ones, as well as the ARI between the obtained partition of the individuals and the true  partition, in each block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='  Figure 1 displays the boxplot of the ARI between the true and estimated blocks of variables obtained  on each situation over the 100 replicates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' The misclassification rate is fixed equal to 5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' It shows that  the proposed method is able to perfectly recover the different blocks of variables, for each family of law,  as soon as the sample size and the number of bins are sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' We recall that the number of  bins should not be too large compared to the sample size, in order to satisfy Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' The method  is more accurate as n grows, while it deteriorates a little when the number of variables in each blocks  grows, for small values of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' This is not surprising as it makes the classification problem more complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='  This will be further illustrated on the last figures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='  Figure 2 displays the boxplots over of the mean of the ARI obtained in each block between the true  and estimated partitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' The misclassification rate is fixed equal to 5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' Here, one can notice that the  performance are good as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='  7  Gauss  Laplace  Student  6  9  12  50  100  200  400  50  100  200  400  50  100  200  400  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='75  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='75  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='75  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='00  Sample size  Adjusted Rand Index  Bandwith  4  5  6  7  Figure 1: ARI between the true and the estimated blocks of variables, for different sample size, different  families of components, different number of variable, and for different number of bins [n1/7], [n1/6], [n1/5],  [n1/4] indicated by grey levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='  Gauss  Laplace  Student  6  9  12  50  100  200  400  50  100  200  400  50  100  200  400  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='75  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='75  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='75  Sample size  Adjusted Rand Index  Bandwith  4  5  6  7  Figure 2: mean ARI between the true and the estimated partition of individuals, for different sample size,  different families of components, different number of variable, and for different number of bins [n1/7],  [n1/6], [n1/5], [n1/4] indicated by grey levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='  8  In Figures 3 and 4, we investigate the behavior of the proposed method, for different values of the  theoretical misclassification rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' The number of bins is fixed equal to [n1/6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' Again, we note that the  methods is more accurate as n grows, while it deteriorates with the increase of the number of variables  in each block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='  Gauss  Laplace  Student  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='1  50  100  200  400  50  100  200  400  50  100  200  400  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='75  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='75  Sample size  Adjusted Rand Index  Variables  6  9  12  Figure 3: mean ARI between the true and the estimated partition of individuals, for different sample  size, different families of components, different misclassification rate and different number of variables  indicated by grey levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='  Gauss  Laplace  Student  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='1  50  100  200  400  50  100  200  400  50  100  200  400  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='75  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='75  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='00  Sample size  Adjusted Rand Index  Variables  6  9  12  Figure 4: mean ARI between the true and the estimated blocks of variables, for different sample size, dif-  ferent families of components, different misclassification rate and different number of variables indicated  by grey levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='  9  6  Conclusion  We have proposed a new method for performing clustering with multiple partitions, when no parametric  assumptions are made on the conditional distributions of the variables given the component memberships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='  This methods permits do deal with continuous data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' It is based on the discretization of the continuous  data which permits to reuse previous methods which are able to deal with multinomials distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content='  For mixed-type data, the method can be straightforwardly extended by discretizing only the continuous  ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' Model selection is conducted on the discretized data by using a modified EM algorithm, which  estimates simultaneously the partition of the variables and the parameters of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
+page_content=' The procedure  is consistent for a range of penalty including the classical BIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfgwF8/content/2301.02422v1.pdf'}
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+Wenkang Zhu et al.: In-situ monitoring additive manufacturing process with AI edge computing 
+1 
+Abstract—In-situ monitoring system can be used to 
+monitor the quality of additive manufacturing (AM) 
+processes. In the case of digital image correlation (DIC) 
+based in-situ monitoring systems, high-speed cameras 
+were used to capture images of high resolutions. This 
+paper proposed a novel in-situ monitoring system to 
+accelerate the process of digital images using artificial 
+intelligence (AI) edge computing board. It built a visual 
+transformer based video super resolution (ViTSR) network 
+to reconstruct high resolution (HR) videos frames. Fully 
+convolutional network (FCN) was used to simultaneously 
+extract the geometric characteristics of molten pool and 
+plasma arc during the AM processes. Compared with 6 
+state-of-the-art super resolution methods, ViTSR ranks first 
+in terms of peak signal to noise ratio (PSNR). The PSNR of 
+ViTSR for 4× super resolution reached 38.16 dB on test 
+data with input size of 75 pixels × 75 pixels. Inference time 
+of ViTSR and FCN was optimized to 50.97 ms and 67.86 ms 
+on AI edge board after operator fusion and model pruning. 
+The total inference time of the proposed system was 118.83 
+ms, which meets the requirement of real-time quality 
+monitoring with low cost in-situ monitoring equipment 
+during AM processes. The proposed system achieved an 
+accuracy of 96.34% on the multi-objects extraction task 
+and can be applied to different AM processes. 
+ 
+Index Terms—Plasma arc additive manufacturing, AI edge 
+computing, In-situ monitoring, Super resolution. 
+I. Introduction 
+dditive manufacturing (AM) is the process of creating a 
+part by joining material, typically by adding news layers 
+over a substrate, in order to obtain a final product from data in a 
+computer-aided design model [1]. The AM technology has 
+gained tremendous interest from industry and academia 
+because of its potential to manufacture complex components in 
+a single stage [2]. However, it has limited acceptance in 
+industries owing to its quality uncertainty including 
+ 
+This work was supported by the National Key Research 
+and 
+Development 
+Program 
+of 
+China 
+(No. 
+2022YFB4600800). Corresponding author: Hui Li. 
+Wenkang Zhu, Hui Li, Yikai Zhang, and Yuqing Hou are 
+with the Institute of Technological Sciences, Wuhan 
+University, 
+430072 
+Wuhan, 
+China 
+(e-mail: 
+wenkang_zhu@whu.edu.cn; 
+li_hui@whu.edu.cn, 
+zhang_yikai@whu.edu.cn, houyuqing@whu.edu.cn).  
+Liwei Chen is with Department of Mechanical 
+Engineering, Graduate School of Engineering, The 
+University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 
+113-8656, Japan (email: chen@hnl.t.u-tokyo.ac.jp). 
+microstructural defects and residual stresses. Plasma arc 
+additive manufacturing (PAM) process, a significant part of 
+AM, is highly valued because of its potential for large-scaling 
+manufacturing. Nevertheless, it is associated with dimensional 
+inaccuracies and defects that hinder its further application in 
+terms of high quality assurance. Meanwhile, the in-situ 
+monitoring system is used widely for monitoring the quality of 
+AM. To extract the features related to the product quality 
+during manufacturing, Yang et al. [3] captured the shapes of 
+spatters using max entropy method and revealed the 
+relationship with number of spatters and laser power. Fang et al. 
+[4] used a U-Net-based convolutional neural network (CNN) 
+with lightweight architecture to accurately extract the molten 
+pool signature. Similarly, Tan et al. [5] built a novel image 
+segmentation network for spatter extraction involved with 
+CNN-based selection and thresholding. Mi et al. [6] proposed a 
+deep CNN to extract the geometric shape of molten pool and 
+spatters simultaneously as molten pool and spatters are 
+intra-related. Zhang et al. [7] used a fully convolutional 
+network (FCN) to extract geometric properties of molten pool 
+and plasma arc simultaneously in PAM. Up to 89% of full-field 
+optical measurements use digital image correlation (DIC) [8], 
+which relies heavily on the quality of images shot by the 
+high-speed camera during the AM process. 
+With respect to the monitoring of the AM process, several 
+image preprocessing methods are used to improve the image 
+quality with the development of DIC. Luck et al. [9] got a series 
+of normalized images by applying distortion correction to raw 
+images using fully-constrained Homography matrix and light 
+levelization process. Zhan et al. [10] constructed a multi-step 
+image preprocessing method including graying, stretching 
+method, 
+histogram 
+equalization, 
+binarization, 
+and 
+morphological filtering to enhance the visual appearance of 
+images. Scime and Beuth’s work [11] used a Homography 
+matrix and baseline intensity mask generated from an 
+anomaly-free powder bed image to correct the distortion and 
+remedy uneven lighting conditions. The above-mentioned 
+image preprocessing methods can enhance the quality of raw 
+images, but the fixed resolution of images limits the 
+achievement of more accurate results. 
+In this paper, a novel in-situ monitoring system is proposed 
+to extract the geometric characteristics of plasma arc and 
+molten pool in PAM process from low resolution to high 
+resolution. This system adopts the efficient and low-cost AI 
+edge computing board as its computing center, instead of the 
+traditional computer workstation. The high-resolution frames 
+are reconstructed using a visual transformer based video super 
+resolution model (ViTSR). The FCN takes the reconstruction of 
+Wenkang Zhu, Hui Li, Yikai Zhang, Yuqing Hou, Liwei Chen  
+In-situ monitoring additive manufacturing process 
+with AI edge computing 
+A 
+
+Wenkang Zhu et al.: In-situ monitoring additive manufacturing process with AI edge computing 
+2 
+ViTSR as input to simultaneously extract geometric 
+characteristics of multiple objects. In Section 2, the details of 
+the proposed system, such as including system architecture, AI 
+edge computing board, video super resolution algorithm used in 
+in-situ monitoring system, are described. In Section 3, the 
+results of the in-situ monitoring system are shown; these results 
+include the quality of 4× super resolution reconstruction and the 
+extraction of geometric characteristics of molten pool and 
+plasma arc during the AM processes. In Section 4, the 
+performance of the proposed system is discussed. Finally, the 
+conclusions are presented in Section 5.  
+II. IN-SITU MONITORING SYSTEM WITH AI EDGE COMPUTING 
+A. System architecture 
+Fig. 1 illustrates the architecture of the proposed in-situ 
+monitoring system. As mentioned earlier, this system can 
+extract the high-resolution geometric characteristics in PAM 
+process. The system includes the powder feed PAM equipment 
+(ABB IRB 2600, Guangzhou LeiJia Additive Manufacturing 
+Technology Co., Ltd., China), AI edge computing board 
+(Jetson Xavier NX, NVIDIA Co., Ltd., USA), and a high-speed 
+camera 
+with 
+maximum 
+sample 
+rate 
+of 
+30000 
+fps 
+(MEMRECAM ACS-1, NAC Image Technology Inc., Japan).  
+The resolution and sample rate of the high-speed camera can 
+be adjusted manually. The video sequence from the high-speed 
+camera is first sent to the AI edge computing board via 
+universal serial bus (USB) interface, and then the super 
+resolution 
+frames 
+are 
+exported 
+to 
+the 
+monitor 
+via 
+high-definition multimedia interface (HDMI) interface. The 
+video sequence is acquired and encoded by OpenCV [12]. 
+Neural networks are programmed by Python 3.8 using 
+TensorFlow 2.5 and then optimized by TensorRT to accelerate 
+the inference process on the neural process unit (NPU) of AI 
+edge computing board. 
+ 
+ 
+ 
+B. AI edge computing board 
+This AI edge computing board uses Jetson Xavier NX as 
+system-on-a-chip (Soc), which integrates CPU, GPU, and NPU 
+into a single circuit by NVIDIA Co., Ltd., USA. Jetson Xavier 
+NX is designed for high-performance and energy-efficient 
+usage at a low cost. The technical specification of Jetson Xavier 
+NX is given in Table I. It runs Jetson operating system based on 
+Ubuntu 18.04 with JetPack software development kit (SDK). 
+The system also contains Linux driver packages, CUDA 
+libraries, and related application programming interfaces. 
+ 
+ 
+ 
+Fig. 2 shows the overall layout and components of the AI 
+edge computing board. This board contains two CMOS serial 
+interfaces for interface. It also contains an HDMI, multiple 
+USB 3.0 interfaces, and micro USB interface for display and 
+data interaction. A wide area network port and a Bluetooth chip 
+are equipped on the board for wireless transfer of data. The 
+fixed 128 GB solid state disk meets the requirements of space 
+for storing different AI models, SDK, and deep learning 
+framework. 
+ 
+ 
+ 
+C. Video super resolution algorithm for low resolution 
+video sequence 
+Traditional single image super resolution [14]-[16] 
+algorithms often lead to blurry effects and motion artifacts for 
+the cause of excessive destruction of ground truth textures and 
+missing consideration of temporal relationship [17]. A previous 
+study [18] used interaction-learning strategy to reduce 
+computation, although the complex architecture is unsuitable to 
+parallel acceleration. Most of the popular VSR methods [19], 
+[20] adopt the fixed pipeline of motion estimation, motion 
+compensation, fusion, and upsampling. The optical flows 
+between frames are first estimated and then used to align 
+features so as to eliminate the motion effects. However, these 
+methods heavily rely on optical flow estimation, which is 
+complex and time-consuming. Thus, more deep neural 
+networks [21], [22] are proposed to compensate the motion by 
+implicit estimation.  
+Super resolution methods are commonly used to enhance the 
+quality of raw images during preprocessing. Shen et al. [23] 
+applied smooth and sparse tensor completion in data preprocess 
+of AM to propose a super resolution method for multi-sources 
+image stream data. Walecki et al. [24] improved the confidence 
+of object surface by using multiple images to tighten the line 
+Super-resolution 
+Results
+High Speed 
+Camera
+Machining Head
+AI Edge 
+Computing Board
+ 
+Fig. 1. In-situ monitoring system integrated with AI edge computing board 
+during PAM process. 
+Fan
+Camera
+Interface
+DC Port
+HDMI USB 3.0
+WAN Mirco USB
+SSD
+Bluetooth Chip
+ 
+Fig. 2. AI edge computing board for video decoding and inference of AI 
+models during in-situ monitoring of PAM. 
+TABLE I 
+TECHNICAL SPECIFICATION OF JETSON XAVIER NX [13] 
+Feature 
+Description 
+AI performance 
+21 TOPS (INT8) 
+GPU 
+384-core VoltaTM GPU 
+CPU 
+6-core Nvidia Cameral CPU ARM®v8.2 
+RAM 
+8 GB 128-bit LPDDR4x 
+Tensor cores 
+48 
+Power 
+10 W | 15 W | 20 W 
+ 
+
+OOOAATWenkang Zhu et al.: In-situ monitoring additive manufacturing process with AI edge computing 
+3 
+segments along the camera ray. However, these traditional 
+mathematical-based methods are limited by complex task 
+settings. 
+In this paper, a super resolution model based on a visual 
+transformer is proposed to upscale the input. Fig. 3 shows the 
+overall pipeline of ViTSR. The ViTSR considers a sequence of 
+low resolution (LR) frames {
+,...,
+,...
+}
+t N
+t
+t N
+X
+X
+X
+−
++
+ as its input, 
+where 
+t
+X  represents the reference frame and the others stand 
+for neighboring frames. The output of ViTSR is 
+tY , which 
+refers to the high resolution (HR) version of the reference frame 
+t
+X .  
+ 
+ 
+ 
+LR frames are downscaled from the corresponding ground 
+truth (GT) frames by nearest neighbor interpolation with scale 
+of r. The height and width of single input frame are represented 
+as H and W. As per the standard practice, RGB images are 
+converted into Y-Cb-Cr color space, and only the Y channel is 
+used for super resolution [25].  
+In addition, temporary and spatial information is explicitly 
+encoded and concatenated to the input frames at first. Given a 
+pixel with coordinates (x, y, i) in the frame Xi, where x, y, and i 
+imply the horizontal, vertical, and temporal codes of this pixel, 
+temporary, and spatial encoding of this pixel can be formulated 
+as： 
+1
+1
+1
+2
+[ ( ), ( ), ( )]
+[sin(
+),[sin(
+),[sin(
+)
+2 ]
+2
+h x v
+x
+y
+i
+W
+y t i
+H
+T
+−
+−
+−
+=
+ 
+(1) 
+where h(x), v(x), and t(i) rapidly stand for horizontal code, 
+vertical code, and temporal code of this pixel, respectively. 
+A residual block with n cells serves as a feature extractor, 
+inspired by [26]. Each cell receives the concatenation of the 
+internal outputs before itself and finally exports a feature map 
+with 32 channels. As shown in Fig. 3, every single cell consists 
+of convolutional layers with batch normalization. The residual 
+cell i (1≤i≤n) consists of 1 × 1 convolutional layers, as 
+described previously [27], and 3 × 3 convolutional layers to 
+enhance feature interaction. 
+After the residual block, a visual transformer block is 
+followed. The visual transformer block is composed of 3-way 
+parallel cells with an inner structure similar to residual cell. Q 
+(Query), K (Key), and V (Value) indicate three essential tensors 
+in the attention mechanism [28] that will be used to calculate 
+the fused feature map. In particular, the last way in the block 
+applies no padding in the temporary axis of feature map, which 
+leads the shape of Q tensor to 1 × H × W × r2. In addition, the K 
+tensor includes all key features of neighboring frames with the 
+shape of 2N × H × W × r2, and V tensor includes all value 
+features of frames with the shape of (2N+1) × H × W × r2. The 
+fused feature map Ht can be calculated as 
+t
+,
+(
+,
+)
+t
+N
+t
+t
+t
+i
+i
+i t
+N i
+H
+V
+Softmax Q K V
++
+= −
+=
++ 
+≠
+ 
+(2) 
+where Ki and Vi refer to the slices of K and V on the temporary 
+axis, and Qt equals to Q. Then a pixel shuffle, which is a 
+periodic shuffling operator [13], is applied to rearrange the 
+tensor 
+t
+H  to ˆ
+t
+H . The reconstructed frame 
+tY  is produced by 
+the sum of bicubic interpolation 
+tB  and ˆ
+t
+H . 
+III. RESULTS 
+A. Video sequence captured by in-situ monitoring 
+system 
+During the PAM process, video sequences containing molten 
+pool and plasma arc are captured by the high-speed camera at a 
+current intensity of 40 A and scanning speed of 10 mm/s. Fig. 4 
+shows the original images captured by the high-speed camera 
+with no preprocess in 1 ms. Frames in Fig. 4 show the temporal 
+continuity of motion, indicating the correlation of neighboring 
+frames that can be used to enhance the feature of the reference 
+frame. Our findings showed that the plasma arc sways over 
+time and molten pool flows slowly.  
+ 
+ 
+ 
+B. In-situ video super resolution reconstruction of 
+monitored videos 
+The proposed ViTSR was used to reconstruct the frames 
+obtained from the high-speed camera to obtain high resolution 
+video sequence. Fig. 5 shows the high-resolution reconstruction 
+of local regions in video sequence by ViTSR, where N is set to 
+1 and upscaling ratio r is 4. The input of the super resolution 
+model includes one reference frame and two neighbor frames. 
+Here, t0 is the start time of video sequence, and T is the interval 
+time of shooting. In the figure, the images on the left show the 
+position of the regions to be reconstructed, while the images on 
+the right are coupled with low-resolution reference frames and 
+corresponding reconstructed frames. The top part of images 
+shows regions of powder bed, the middle part shows regions of 
+plasma arc, and the bottom part shows molten pool. In the 
+proposed VSR model, the resolution of the images is enlarged 
+from 25 pixels × 25 pixels to 100 pixels × 100 pixels. The 
+motion between high-resolution frames is natural and 
+reasonable because the visual transformer enables implicit 
+motion estimation and pixel-wide fusion. 
+t0 ms - t0+0.5 ms
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+ 
+Fig. 4. Original video of molten pool and plasma arc captured at a current 
+intensity of 40 A and scanning speed of 10 mm/s. 
+Vt
+...
+i
+n
+......
+...
+Xt-N
+Xt
+Xt+N
+1×1 Conv3d
+Kernels: 32i
+BN
+3×3 Conv3d
+Kernels: 32
+BN
+1×1 Conv3d
+Kernels: 32n
+BN
+3×3 Conv3d
+Kernels: r2
+BN
+·
+·
+Softmax
++
+Residual Block
+Visual Transformer
+Pixel 
+Shuffle
+Qt
+Kt
+Temporary and Spatial Encoding
+......
+Yt
+Input: 75 pixels × 75 pixels
+Output: 300 pixels × 300 pixels
+Bt
+1×1 Conv3d
+Kernels: 32n
+BN
+3×3 Conv3d
+Kernels: r2
+BN
+1×1 Conv3d
+Kernels: 32n
+BN
+3×3 Conv3d
+Kernels: r2
+BN
+ 
+Fig. 3. Video super-resolution using visual-transformer. 
+
+Wenkang Zhu et al.: In-situ monitoring additive manufacturing process with AI edge computing 
+4 
+ 
+ 
+In Fig. 6, critical internal feature maps were considered for 
+visualization to evaluate the visual transformer block with 
+respect to feature fusion. The corresponding internal feature 
+maps of input frames are 
+1
+1
+{
+,
+,
+}
+i
+i
+i
+X
+X X
+−
++
+. Meanwhile, Q, K, 
+and V are sliced as 
+i
+Q ， {
+,
+}
+i T
+i T
+K
+K
+−
++
+, and {
+,
+,
+}
+i T
+i
+i T
+V
+V V
+−
++
+ by the 
+axis of channel, which is also the last channel of feature maps. 
+In Fig. 6, the visualization of feature maps Q, K, and V have 
+profiles similar to the corresponding input frames, including the 
+topography of the plasma arc and molten pool, and the 
+distribution of metal powder. The fused feature map Hi shows 
+details of the fusion of Q, K, and V on the way of attention 
+mechanism, which keeps the regions that can be used for the 
+reconstruction of the reference frame and suppresses those 
+regions that are different from the reference frame. The 
+visualization of Hi shows great similarity of geometric shape 
+with reference frame. This indicates that high attention is paid 
+to similar regions of neighbor frames and reference frame, 
+while low attention is paid to different regions, indicating the 
+effectiveness of this block. When applying visual transformer 
+block instead of motion estimation and feature alignment, the 
+beneficial features from neighbor frames can be fused into the 
+correct regions of reference frame by using parallel neural 
+nodes and nonlinear combination.  
+ 
+ 
+The proposed model was compared with 6 state-of-the-art 
+super resolution methods, including Bicubic, super resolution 
+convolution neural network (SRCNN) [15], fast SRCNN 
+(FSRCNN) [16], efficient sub-pixel convolutional neural 
+network (ESPCN) [14], dynamic upsampling filters (DUF) [21], 
+and temporary group attention (TGA) [22]. Bicubic is a simple 
+math-based method that is widely used for its low cost. The 
+SRCNN 
+applied 
+convolutional 
+layers 
+behind 
+bicubic 
+interpolation to seek higher quality of HR reconstruction. 
+Deconvolution is used in FSRCNN to upscale the output 
+instead of interpolating the input image in the beginning so as to 
+make full use of feature maps. Pixel shuffle is used in ESPCN 
+to obtain surprising results. DUF applies dynamic upsampling 
+filters to compensate for motion between frames implicitly 
+instead of traditional optic-flow motion estimation. The 
+reconstruction in TGA can be improved using temporary group 
+attention for the inner relationship between long-term frames. 
+Bicubic interpolation realized by OpenCV is directly used for 
+its high efficiency. Fig. 7 shows comparisons between 4× super 
+resolution reconstruction by the state-of-the-arts models and 
+the proposed model. The input size of models was set to 75 
+i = 0
+i = 1
+i = 2
+X-T
+XT
+X0
+X1-T
+X1
+X1+T
+X2-T
+X2
+X2+T
+K-T
+Q0
+KT
+K1-T
+Q1
+K1+T
+K2-T
+Q2
+K2+T
+V-T
+V0
+VT
+V1-T
+V1
+V1+T
+V2-T
+V2
+V2+T
+ 
+Fig. 6. The attention map H decoded from query Q, key K and value V. 
+Video Sequence
+From t0 ~ t0 + 4T
+t0 + 4T
+t0 + 2T
+t0 + T
+t0
+t0 + 3T
+t0 + 4T
+t0 + 2T
+t0 + T
+t0
+t0 + 3T
+t0 + 4T
+t0 + 2T
+t0 + T
+t0
+t0 + 3T
+ 
+Fig. 5. Super-resolution for molten pool and spatter areas in a video sequence 
+by ViTSR, T = 1/30 ms. 
+ 
+LR
+GT
+Bicubic
+SRCNN
+FSRCNN
+ESPCN
+DUF
+TGA
+ViTSR
+75 × 75
+300 × 300
+300 × 300
+300 × 300
+300 × 300
+300 × 300
+300 × 300
+300 × 300
+300 × 300
+From video
+SSIM=0.9883
+SSIM=0.9873
+SSIM=0.9885
+PSNR=27.7576
+PSNR=27.7757
+PSNR=27.3202
+SSIM=0.9959
+SSIM=0.9955
+SSIM=0.9959
+PSNR=32.4473
+PSNR=32.3423
+PSNR=31.9197
+SSIM=0.9980
+SSIM=0.9974
+SSIM=0.9979
+PSNR=35.5085
+PSNR=35.2222
+PSNR=34.2179
+SSIM=0.9985
+SSIM=0.9980
+SSIM=0.9985
+PSNR=36.7854
+PSNR=36.5366
+PSNR=35.4501
+SSIM=0.9991
+SSIM=0.9987
+SSIM=0.9990
+PSNR=38.7048
+PSNR=38.5372
+PSNR=37.1383
+SSIM=0.9990
+SSIM=0.9988
+SSIM=0.9991
+PSNR=38.5202
+PSNR=38.6073
+PSNR=37.5364
+SSIM=0.9991
+SSIM=0. 9988
+SSIM=0.9991
+PSNR=39.3199
+PSNR=39.3843
+PSNR=37.6522
+t0
+t0 + Ts
+t0 + 2Ts
+Input: low resolution 
+samples
+Output: high resolution results
+ 
+Fig. 7. Performance of super-resolution results by various methods. 
+
+Wenkang Zhu et al.: In-situ monitoring additive manufacturing process with AI edge computing 
+5 
+pixels × 75 pixels, which is LR in the figure. Three reference 
+frames were selected with interval time (Ts) of 1/3 ms. Finally, 
+ViTSR ranks first in terms of PSNR and was 0.7770 dB higher 
+than the second highest model. 
+The top 3 models (ViTSR, TGA and DUF) were selected in 
+Fig. 8 to visualize the subtracting of reconstructed frames with 
+GT frames and to clearly show the difference between 
+reconstructed frames and GT frames. Negative values of 
+subtraction were set to zero, and the values of subtraction were 
+scaled to 0–255. As shown in Fig. 8, most areas of DUF were 
+different with GT frames for its dynamic filters, and the results 
+of DUF showed lower brightness than GT frames. For TGA, 
+the different pixels were concentrated around molten pool and 
+plasma arc. In the case of ViTSR, the different pixels were 
+concentrated around molten pool, and the number is the least. 
+Fig. 9 shows the average PSNR and inference time of various 
+methods on the AI computing board with the same setting. A 
+total of 58 video frames were tested. As shown in Fig. 9, ViTSR 
+achieved the best performance in terms of PSNR but only 
+half-time cost compared with the TGA. After time optimizing 
+using operator fusion and model pruning, ViTSR performs the 
+best by making most of the NPU. Inference time of ViTSR was 
+233.01 ms on CPU that was optimized to 50.97 ms on AI edge 
+board. The PSNR of ViTSR was reduced to 37.67 dB, which 
+indicates saved inference time. 
+ 
+ 
+The effectiveness of core components of this model, 
+including the temporal and spatial encoding, visual transformer, 
+and 3D convolution, was verified. The core components of 
+ViTSR were eliminated to generate three degenerated models, 
+named as ViTSR_d1, ViTSR_d2, and ViTSR_d3. In brief, 
+ViTSR_d1 removed temporal and spatial encoding from the 
+proposed model, ViTSR_d2 used DUF instead of the visual 
+transformer, and ViTSR_d3 replaced all 3D convolution with 
+2D convolution. Table II shows the PSNR of ViTSR and 
+degenerated models on the test data after training. The ViTSR 
+showed the highest PSNR (38.16 dB) with 0.96 dB higher than 
+the second-best model.  
+ 
+ 
+ 
+C. In-situ extraction of geometric characteristics of 
+molten pool and plasma arc 
+In our previous work [6], plasma arc and molten pool were 
+simultaneously extracted in the process of PAM. The present 
+paper used the well-trained FCN to extract the geometric 
+characteristics of molten pool and plasma arc after ViTSR. Fig. 
+10 shows the segmentation of the proposed in-situ monitoring 
+system. The resolution of input video sequence was 75 pixels × 
+75 pixels, while the output resolution of segmentation was 300 
+pixels × 300 pixels after ViTSR and FCN.  
+ 
+ 
+ 
+The AI edge computing used ViTSR and FCN to fetch the 
+areas of real-time super resolution segmentation. The ViTSR 
+27.5444
+32.0692
+34.7968
+35.9899
+37.8276
+38.1486
+38.1600
+37.6700
+27
+29
+31
+33
+35
+37
+39
+0
+100
+200
+300
+400
+500
+PSNR (dB)
+Inference time (ms)
+Bicubic
+SRCNN
+FSRCNN
+ESPCN
+DUF
+TGA
+ViTSR
+ViTSR'
+Time Optimized
+ 
+Fig. 9. Performance of various models in terms of PSNR and inference time on 
+AI computing board. 
+Output: segmentation with 300 pixels × 300 pixels
+Input: video sequence with 75 pixels × 75 pixels
+ 
+Fig. 10. Extraction of geometric characteristic of in-situ monitoring video. 
+TABLE II 
+PSNR OF VITSR AND DEGENERATED MODELS ON TEST DATA AFTER TRAINING 
+Models 
+Description 
+PSNR 
+ViTSR 
+the proposed model 
+38.16 dB 
+ViTSR_d1 
+remove temporal and spatial encoding 
+36.42 dB 
+ViTSR_d2 
+replace VIT with DUF 
+36.29 dB 
+ViTSR_d3 
+replace 3D-Conv with 2D 
+37.20 dB 
+ 
+GT
+DUF
+DUF - GT
+GT - DUF
+GT - TGA
+TGA
+TGA - GT
+ViTSR
+ViTSR - GT
+t0
+GT - ViTSR
+t0 + Ts
+t0 + 2Ts
+ 
+Fig. 8. Visualization of the difference between super-resolution reconstruction and GT, Ts = 1/3 ms. 
+
+Wenkang Zhu et al.: In-situ monitoring additive manufacturing process with AI edge computing 
+6 
+was used to reconstruct HR frames, while the FCN was used to 
+extract the geometric shapes of molten pool and plasma arc. Fig. 
+11 shows the extracted pixels of molten pool and plasma arc in 
+PAM process with time. The area of molten pool and plasma 
+arc was found to be in the range of 3383–4135 pixels and 
+7474–8709 pixels, respectively. This shows the temporal 
+coherence of the target areas. As shown in this figure, the AI 
+edge computing board was used to process 58 frames with a 
+resolution of 75 pixels × 75 pixels to 300 pixels × 300 pixels 
+results. The video sequence frames with a total number of 58 
+were processed for super resolution reconstruction and frame 
+segmentation in 6.89 s. The average inference time for 
+processing a single frame is 118.83 ms, where the inference 
+time of frame segmentation is 67.86 ms. The in-situ monitoring 
+system showed its immense future for high-resolution and 
+low-cost quality monitoring of AM process. 
+ 
+ 
+ 
+IV. DISCUSSION 
+Traditional 
+image 
+segmentation 
+algorithms 
+were 
+successfully applied in many fields, whereas the in-situ 
+monitoring task of AM could not fit because of its halo effects 
+and high dynamic range imaging. Fig. 12 shows the comparison 
+of extraction performance using traditional and proposed 
+methods with input size of 75 pixels × 75 pixels in PAM 
+process. As shown in Fig. 12(a), plasma arc, molten pool, and 
+part of powder bed were segmented together when using 
+triangle algorithm. As shown in Fig. 12(b), the maximum 
+entropy algorithm extracted the general shapes of plasma arc, 
+molten pool, and bright metal powders. The region of plasma 
+arc extracted by the two methods was severely enlarged for 
+interference of halo. With respect to the extraction of targets 
+and the suppression of bright metal powders, watershed in Fig. 
+12(c), Otsu in Fig. 12(d), and basic global thresholding 
+algorithm in Fig. 12(e) performed better than the former 
+segmentation algorithms. The three algorithms were not 
+critically affected by the halo and reflected light, while the 
+extraction of plasma arc was still enlarged and the molten pool 
+was smaller than the fact. Traditional algorithms could not 
+extract target areas accurately due to the interference of halo 
+and reflection. As shown in Fig. 12(f), FCN extracted the 
+plasma arc and molten pool efficiently owing to its nonlinear 
+fitting ability, where the halo was excluded from the extraction 
+and the low brightness showed less interference of the results. 
+Fig. 12(g) showed the extracted result of 4× super resolution 
+frame using ViTSR and FCN. Our findings showed that the 
+proposed method extracted plasma arc and molten pool 
+accurately, eliminating the interference of light conditions and 
+bright powders. Table III shows the performance of traditional 
+methods, FCN method, and the proposed method. The table 
+also showed that the proposed method upscaled the resolution 
+of input frames from 75 pixels × 75 pixels to 300 pixels × 300 
+pixels, thus, achieving the highest accuracy. 
+ 
+ 
+ 
+ 
+ 
+Table IV showed the performance of AI methods for the 
+in-situ monitoring of multiple AM process. Tan’s method [4] 
+was used to process the image tiles with a resolution of 200 
+pixels × 200 pixels so as to detect the spatters in the process of 
+laser power bed fusion (LPBF). Despite the acceptable 
+inference time, the accuracy may not be suitable for real-world 
+applications. Fang reported high accuracy and inference speed 
+after he adopted U-Net to extract the molten pool with a 
+resolution of 224 pixels × 224 pixels. Mi et al. [5] achieved an 
+accuracy of 94.71% in the process of laser based directed 
+energy deposition (L-DED) when the proposed D-CNN 
+architecture was used to extract the spatters and molten pool 
+simultaneously. Zhang et al. [6] extracted plasma arc and 
+molten pool simultaneously in the PAM process. Although the 
+inference time of Zhang’s method [6] was longer than that of 
+Mi’s method [5], the accuracy achieved by using Zhang’s 
+method [6] was much higher. In this paper, the proposed 
+method was used to reconstruct the 4× super resolution frames 
+and extract the geometric characteristics of plasma arc and 
+molten pool during PAM process. The proposed method 
+achieved an accuracy of 96.34% with inference time of 15 ms 
+TABLE III 
+PERFORMANCE OF TRADITIONAL SEGMENTATION METHODS AND PROPOSED 
+METHOD 
+Methods 
+Extracted 
+objects 
+Resolution 
+(pixel) 
+Upscaling 
+Accuracy 
+Triangle 
+Bright area & 
+Dark area 
+75 × 75 
+1 × 1 
+27.57% 
+Maximum 
+Entropy 
+Bright area & 
+Dark area 
+75 × 75 
+1 × 1 
+85.46% 
+Watershed 
+Bright area & 
+Dark area 
+75 × 75 
+1 × 1 
+92.11% 
+Otsu 
+Bright area & 
+Dark area 
+75 × 75 
+1 × 1 
+90.57% 
+Basic 
+Global 
+Thresholding 
+Bright area & 
+Dark area 
+75 × 75 
+1 × 1 
+92.68% 
+FCN 
+Plasma arc & 
+Molten pool 
+75 × 75 
+1×1 
+96.29% 
+Proposed: 
+ViTSR + 
+FCN 
+Plasma arc & 
+Molten pool 
+Input:  
+75 × 75 
+Output:  
+300 × 300 
+4×4 
+96.34% 
+ 
+300 pixels × 300 pixels
+(a)
+(b)
+(c)
+(d)
+(e)
+(f)
+(g)
+75 pixels ×75 pixels
+75 pixels ×75 pixels
+75 pixels ×75 pixels
+75 pixels ×75 pixels
+75 pixels ×75 pixels
+75 pixels ×75 pixels
+ 
+Fig. 12 Extraction of molten pool and plasma arc using different image 
+segmentation methods: (a) triangle, (b) maximum entropy, (c) watershed, (d) 
+Otsu, (e) basic global thresholding, (f) FCN, and (g) ViTSR+FCN. 
+5000
+6000
+7000
+8000
+9000
+3000
+4000
+5000
+6000
+7000
+0.12
+0.96
+1.79
+2.62
+3.45
+4.28
+5.11
+5.94
+Pixels of Plasma Arc
+Pixels of Molten Pool
+Time (s)
+Molten Pool
+Plasma Arc
+ 
+Fig. 11. Extracted pixels of molten pool and plasma arc with time. 
+
+Wenkang Zhu et al.: In-situ monitoring additive manufacturing process with AI edge computing 
+7 
+and 118.83 ms on Nvidia RTX 3070 Laptop GPU and Nvidia 
+Jetson Xavier NX, respectively. The sum of two models 
+denoted the number of parameters; this is one of the reasons of 
+the higher inference time. In the case of less power, the AI edge 
+computing board was equipped with 48 tensor cores and only 
+26% of GPU, which is the reason of higher inference time. The 
+inference time of the proposed method using GPU was found to 
+be the smallest in Table IV even when the GPU had more 
+parameters, which shows the efficiency of the proposed method. 
+The inference time can be optimized in the future by integrating 
+the FCN and ViTSR into one step. The proposed method shows 
+its low requirement of pixel resolution for high-speed camera 
+used in the video capture of the AM process. This reduced the 
+cost of camera and contributed to the industrial applications of 
+in-situ monitoring system for AM.  
+ 
+ 
+ 
+V. CONCLUSIONS 
+In this paper, a novel in-situ monitoring system was 
+proposed to extract the high-resolution features of the AM 
+process. For this, high-speed camera with variable resolutions 
+was used to capture the video data during the AM processes. 
+The captured video sequences were processed using AI edge 
+computing because of its low cost and high efficiency. The 
+maximum power of the AI edge computing board was found to 
+be 20 W, which is lower than the power consumption of PCs 
+and computing servers. The proposed system used 118.83 ms 
+for total inference of video sequences. The inference time of 
+video super resolution is 50.97 ms from a resolution of 75 
+pixels × 75 pixels to 300 pixels × 300 pixels, and frame 
+segmentation is 67.86 ms, respectively. Two-stage strategy is 
+used to reconstruct the input with high resolution and extracts 
+the key features with high accuracy. To the best of our 
+knowledge, this is the first study that used a video super 
+resolution algorithm before image segmentation to seek high 
+resolution geometric characteristics of AM processes. The 
+output of the proposed system demonstrated greater tolerance 
+to halo and shadow of captured video and finally achieved an 
+accuracy of 96.34%, which is similar to our previous work. 
+Considering that the proposed system needed 1/4 resolution as 
+input with same results, the system will sharply reduce the cost 
+of high-speed cameras used in in-situ monitoring systems. Thus, 
+this paper provides a way to lower the cost of DIC-based 
+methods and may finally improve the quality of products 
+manufactured by various AM process including PAM, LPBF, 
+and L-DED. 
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+[10] Q. Zhan, Y. Liang, J. Ding, and S. Williams, “A wire deflection detection 
+method based on image processing in wire + arc additive manufacturing,” Int. J. 
+Adv. Manuf. Technol., vol. 89, no. 1–4, pp. 755–763, Mar. 2017. 
+[11] L. Scime and J. Beuth, “Anomaly detection and classification in a laser 
+powder bed additive manufacturing process using a trained computer vision 
+algorithm,” Addit. Manuf., vol. 19, pp. 114–126, Jan. 2018. 
+[12] OpenCV, “OpenCV,” 2022. Accessed: Apr. 24, 2022. [Online]. Available: 
+https://opencv.org/ 
+[13] Nvidia Developer, “Jetson Xavier NX,” 2022. Accessed: Apr. 24, 2022. 
+[Online]. https://developer.nvidia.com/embedded/jetson-xavier-nx 
+[14] W. Shi, J. Caballero, F. Huszar, J. Totz, A. Aitken, R. Bishop, D. Rueckert, 
+and Z. Wang, “Real-time single image and video super-resolution using an 
+efficient sub-pixel convolutional neural network,” Presented at 2016 IEEE 
+Conference on Computer Vision and Pattern Recognition. [Online]. Available: 
+http://ieeexplore.ieee.org/document/7780576/ 
+[15] C. Dong, C. C. Loy, K. He, and X. Tang, “Learning a deep convolutional 
+network for image super-resolution,” Presented at European conference on 
+computer 
+vision. 
+[Online]. 
+Available: 
+http://ieeexplore.ieee.org/document/7780576/ 
+[16] C. Dong, C. C. Loy, and X. Tang, “Accelerating the super-resolution 
+convolutional neural network,” Presented at European conference on computer 
+vision. [Online]. Available: http://ieeexplore.ieee.org/document/7780576/ 
+[17] F. Yang, H. Yang, J. Fu, H. Lu, and B. Guo, “Learning texture transformer 
+network for image super-resolution,” Presented at 2020 IEEE/CVF Conference 
+on Computer Vision and Pattern Recognition (CVPR). [Online]. Available: 
+https://ieeexplore.ieee.org/document/9157061/ 
+[18] J. Qin, L. Chen, S. Jeon, and X. Yang, “Progressive interaction-learning 
+network for 
+lightweight single-image super-resolution in industrial 
+applications,” IEEE Trans. Industr. Inform., pp. 1-10, Jul. 2022. 
+[19] T. Xue, B. Chen, J. Wu, D. Wei, and W. T. Freeman, “Video enhancement 
+with task-oriented flow,” Int. J. Comput. Vis., vol. 127, no. 8, pp. 1106–1125, 
+Aug. 2019. 
+[20] D. Liu, Z. Wang, Y. Fan, X. Liu, Z. Wang, S. Chang, and T. Huang, 
+“Robust video super-resolution with learned temporal dynamics,” Presented at 
+TABLE IV 
+PERFORMANCE OF OUR PREVIOUS METHODS AND THE PROPOSED METHOD IN 
+THIS WORK FOR IN-SITU MONITORING OF PAM 
+Methods 
+Extracted 
+objects 
+Resolution 
+(pixel) 
+Upscaling 
+Inference time 
+Accuracy 
+Tan’s [4] 
+Spatters 
+200 × 200 
+1 × 1 
+70 ms 
+80.48% 
+Fang’s [3] 
+Molten 
+pool 
+224 × 224  
+1 × 1 
+37 ms 
+98.06% 
+Mi’s [5] 
+Spatters & 
+Molten 
+pool 
+450 × 512 
+1 × 1 
+63 ms 
+94.71% 
+Zhang’s 
+[6] 
+Plasma arc 
+& Molten 
+pool 
+450 × 512 
+1 × 1 
+84 ms 
+95.10% 
+Proposed: 
+ViTSR + 
+FCN 
+Plasma arc 
+& Molten 
+pool 
+Input: 
+75 × 75 
+Output: 
+300 × 300 
+4 × 4 
+15 ms (GPU)  
+118.83 ms 
+(AI edge 
+computing board)  
+96.34% 
+ 
+
+Wenkang Zhu et al.: In-situ monitoring additive manufacturing process with AI edge computing 
+8 
+2017 IEEE International Conference on Computer Vision (ICCV). [Online]. 
+Available: http://ieeexplore.ieee.org/document/8237536/ 
+[21] Y. Jo, S. W. Oh, J. Kang, and S. J. Kim, “Deep Video super-resolution 
+network using dynamic upsampling filters without explicit motion 
+compensation,” Presented at 2018 IEEE/CVF Conference on Computer Vision 
+and 
+Pattern 
+Recognition. 
+[Online]. 
+Available: 
+https://ieeexplore.ieee.org/document/8578438/ 
+[22] T. Isobe, S. Li, X. Jia, S. Yuan, G. Slabaugh, C. Xu, Y-L. Li, S. Wang, and 
+Q. Tian, “Video super-resolution with temporal group attention,” Presented at 
+2020 IEEE/CVF Conference on Computer Vision and Pattern Recognition 
+(CVPR). [Online]. Available: https://ieeexplore.ieee.org/document/9157178/ 
+[23] B. Shen, R. Wang, A. C. C. Law, R. Kamath, H. Choo, and Z. Kong, 
+“Super resolution for multi-sources image stream data using smooth and sparse 
+tensor completion and its applications in data acquisition of additive 
+manufacturing,” Technometrics, vol. 64, no. 1, pp. 2–17, Jan. 2022. 
+[24] P. Walecki and G. Taubin, “Super-resolution 3-D laser scanning based on 
+interval arithmetic,” IEEE Trans. Instrum. Meas., vol. 69, no. 10, pp. 10, Apr. 
+2020. 
+[25] J. A. M. Basilio, G. A. Torres, S. Pérez, L. K. T. Medina, and H. M. P. 
+Meana, “Explicit image detection using YCbCr space color model as skin 
+detection,” Presented at Proceedings of the 2011 American conference on 
+applied mathematics and the 5th WSEAS international conference on 
+Computer 
+engineering 
+and 
+applications. 
+[Online]. 
+Available: 
+https://www.researchgate.net/publication/262371199_Explicit_image_detectio
+n_using_YCbCr_space_color_model_as_skin_detection 
+[26] K. He, X. Zhang, S. Ren, and J. Sun, “Deep residual learning for image 
+recognition,” Presented at 2016 IEEE Conference on Computer Vision and 
+Pattern 
+Recognition 
+(CVPR). 
+[Online]. 
+Available: 
+http://ieeexplore.ieee.org/document/7780459/ 
+[27] C. Szegedy, W. Liu, Y. Jia, P. Sermanet, S. Reed, D. Anguelov, D.  Erhan, 
+V. Vanhoucke, and A. Rabinovich, “Going deeper with convolutions,” 
+Presented at 2015 IEEE Conference on Computer Vision and Pattern 
+Recognition 
+(CVPR). 
+[Online]. 
+Available: 
+http://ieeexplore.ieee.org/document/7298594/ 
+[27] A. Vaswani, N. Shazeer, N. Parmar, J. Uszkoreit, L. Jones, A. N Gomez, Ł. 
+Kaiser, and I. Polosukhin, “Attention is all you need,” Advances in Neural 
+Information 
+Processing 
+Systems. 
+[Online]. 
+Available: 
+https://proceedings.neurips.cc/paper/2017/hash/3f5ee243547dee91fbd053c1c4
+a845aa-Abstract.html 
+ 
+Wenkang Zhu received the B.S. degree in 
+mechanical engineering and automation from Wuhan 
+University, Wuhan, China in 2017. 
+He is studying for master's degree in Institute of 
+Technological Sciences, Wuhan University, Wuhan, 
+China. His research interests include additive 
+manufacturing and artificial intelligence. 
+ 
+ 
+ 
+ 
+ 
+ 
+Hui Li received the B.S. degree from Huazhong 
+University of Science and Technology, Wuhan, China, 
+and the Ph.D. degree in electrical & computer 
+engineering from National University of Singapore, 
+Singapore, in 1999 and 2007, respectively. 
+He is currently a Professor at Wuhan University, 
+China. His research interests include electronics 
+manufacturing and additive manufacturing. 
+ 
+ 
+ 
+ 
+ 
+Yikai Zhang received his B.S. degree in electronic 
+information engineering from Hefei University of 
+Technology in 2020 and is currently studying for a 
+master's degree at the Institute of Industrial Science of 
+Wuhan University.  
+His current research interests include image 
+processing, 
+deep 
+learning, 
+and 
+additive 
+manufacturing. 
+ 
+Yuqing Hou received his M.S. degree from Xian 
+University of Technology, Xian, China. He is 
+currently pursuing the Ph.D. degree with Institute of 
+Technology Science, Wuhan University, Wuhan, 
+China.  
+His research focuses on artificial intelligence and 
+computational 
+fluid 
+dynamics 
+for 
+additive 
+manufacturing.  
+ 
+ 
+ 
+ 
+ 
+Liwei Chen received his M.S degree form Wuhan 
+University, Wuhan, China, and the Ph.D. degree in 
+mechanical 
+system 
+engineering 
+from 
+Tohoku 
+University, Sendai, Japan, in 2019 and 2022, 
+respectively. 
+He is currently a Postdoctoral Fellow at the 
+University of Tokyo, Japan. His research interests 
+include image processing and laser manufacturing. 
+ 
+ 
+ 
+ 
+ 
+
diff --git a/EtAyT4oBgHgl3EQfq_nZ/content/tmp_files/load_file.txt b/EtAyT4oBgHgl3EQfq_nZ/content/tmp_files/load_file.txt
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf,len=755
+page_content='Wenkang Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' : In-situ monitoring additive manufacturing process with AI edge computing  1  Abstract—In-situ monitoring system can be used to  monitor the quality of additive manufacturing (AM)  processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' In the case of digital image correlation (DIC)  based in-situ monitoring systems, high-speed cameras  were used to capture images of high resolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' This  paper proposed a novel in-situ monitoring system to  accelerate the process of digital images using artificial  intelligence (AI) edge computing board.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' It built a visual  transformer based video super resolution (ViTSR) network  to reconstruct high resolution (HR) videos frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' Fully  convolutional network (FCN) was used to simultaneously  extract the geometric characteristics of molten pool and  plasma arc during the AM processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' Compared with 6  state-of-the-art super resolution methods, ViTSR ranks first  in terms of peak signal to noise ratio (PSNR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' The PSNR of  ViTSR for 4× super resolution reached 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='16 dB on test  data with input size of 75 pixels × 75 pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' Inference time  of ViTSR and FCN was optimized to 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='97 ms and 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='86 ms  on AI edge board after operator fusion and model pruning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='  The total inference time of the proposed system was 118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='83  ms, which meets the requirement of real-time quality  monitoring with low cost in-situ monitoring equipment  during AM processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' The proposed system achieved an  accuracy of 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='34% on the multi-objects extraction task  and can be applied to different AM processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='  Index Terms—Plasma arc additive manufacturing, AI edge  computing, In-situ monitoring, Super resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' Introduction  dditive manufacturing (AM) is the process of creating a  part by joining material, typically by adding news layers  over a substrate, in order to obtain a final product from data in a  computer-aided design model [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' The AM technology has  gained tremendous interest from industry and academia  because of its potential to manufacture complex components in  a single stage [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' However, it has limited acceptance in  industries owing to its quality uncertainty including  This work was supported by the National Key Research  and  Development  Program  of  China  (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='  2022YFB4600800).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' Corresponding author: Hui Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='  Wenkang Zhu, Hui Li, Yikai Zhang, and Yuqing Hou are  with the Institute of Technological Sciences, Wuhan  University,  430072  Wuhan,  China  (e-mail:  wenkang_zhu@whu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='cn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='  li_hui@whu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='cn,  zhang_yikai@whu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='cn, houyuqing@whu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='  Liwei Chen is with Department of Mechanical  Engineering, Graduate School of Engineering, The  University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo  113-8656, Japan (email: chen@hnl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='u-tokyo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='jp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='  microstructural defects and residual stresses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' Plasma arc  additive manufacturing (PAM) process, a significant part of  AM, is highly valued because of its potential for large-scaling  manufacturing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' Nevertheless, it is associated with dimensional  inaccuracies and defects that hinder its further application in  terms of high quality assurance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' Meanwhile, the in-situ  monitoring system is used widely for monitoring the quality of  AM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' To extract the features related to the product quality  during manufacturing, Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' [3] captured the shapes of  spatters using max entropy method and revealed the  relationship with number of spatters and laser power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' Fang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='  [4] used a U-Net-based convolutional neural network (CNN)  with lightweight architecture to accurately extract the molten  pool signature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' Similarly, Tan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' [5] built a novel image  segmentation network for spatter extraction involved with  CNN-based selection and thresholding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' Mi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' [6] proposed a  deep CNN to extract the geometric shape of molten pool and  spatters simultaneously as molten pool and spatters are  intra-related.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' [7] used a fully convolutional  network (FCN) to extract geometric properties of molten pool  and plasma arc simultaneously in PAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' Up to 89% of full-field  optical measurements use digital image correlation (DIC) [8],  which relies heavily on the quality of images shot by the  high-speed camera during the AM process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='  With respect to the monitoring of the AM process, several  image preprocessing methods are used to improve the image  quality with the development of DIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' Luck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' [9] got a series  of normalized images by applying distortion correction to raw  images using fully-constrained Homography matrix and light  levelization process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' Zhan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' [10] constructed a multi-step  image preprocessing method including graying, stretching  method,  histogram  equalization,  binarization,  and  morphological filtering to enhance the visual appearance of  images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' Scime and Beuth’s work [11] used a Homography  matrix and baseline intensity mask generated from an  anomaly-free powder bed image to correct the distortion and  remedy uneven lighting conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' The above-mentioned  image preprocessing methods can enhance the quality of raw  images, but the fixed resolution of images limits the  achievement of more accurate results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='  In this paper, a novel in-situ monitoring system is proposed  to extract the geometric characteristics of plasma arc and  molten pool in PAM process from low resolution to high  resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' This system adopts the efficient and low-cost AI  edge computing board as its computing center, instead of the  traditional computer workstation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' The high-resolution frames  are reconstructed using a visual transformer based video super  resolution model (ViTSR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' The FCN takes the reconstruction of  Wenkang Zhu, Hui Li, Yikai Zhang, Yuqing Hou, Liwei Chen  In-situ monitoring additive manufacturing process  with AI edge computing  A  Wenkang Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' : In-situ monitoring additive manufacturing process with AI edge computing  2  ViTSR as input to simultaneously extract geometric  characteristics of multiple objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' In Section 2, the details of  the proposed system, such as including system architecture, AI  edge computing board, video super resolution algorithm used in  in-situ monitoring system, are described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' In Section 3, the  results of the in-situ monitoring system are shown;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' these results  include the quality of 4× super resolution reconstruction and the  extraction of geometric characteristics of molten pool and  plasma arc during the AM processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' In Section 4, the  performance of the proposed system is discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' Finally, the  conclusions are presented in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' IN-SITU MONITORING SYSTEM WITH AI EDGE COMPUTING  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' System architecture  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' 1 illustrates the architecture of the proposed in-situ  monitoring system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' As mentioned earlier, this system can  extract the high-resolution geometric characteristics in PAM  process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' The system includes the powder feed PAM equipment  (ABB IRB 2600, Guangzhou LeiJia Additive Manufacturing  Technology Co.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=', Ltd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=', China), AI edge computing board  (Jetson Xavier NX, NVIDIA Co.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=', Ltd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=', USA), and a high-speed  camera  with  maximum  sample  rate  of  30000  fps  (MEMRECAM ACS-1, NAC Image Technology Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=', Japan).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='  The resolution and sample rate of the high-speed camera can  be adjusted manually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' The video sequence from the high-speed  camera is first sent to the AI edge computing board via  universal serial bus (USB) interface, and then the super  resolution  frames  are  exported  to  the  monitor  via  high-definition multimedia interface (HDMI) interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' The  video sequence is acquired and encoded by OpenCV [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='  Neural networks are programmed by Python 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='8 using  TensorFlow 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='5 and then optimized by TensorRT to accelerate  the inference process on the neural process unit (NPU) of AI  edge computing board.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' AI edge computing board  This AI edge computing board uses Jetson Xavier NX as  system-on-a-chip (Soc), which integrates CPU, GPU, and NPU  into a single circuit by NVIDIA Co.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=', Ltd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=', USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' Jetson Xavier  NX is designed for high-performance and energy-efficient  usage at a low cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' The technical specification of Jetson Xavier  NX is given in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' It runs Jetson operating system based on  Ubuntu 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='04 with JetPack software development kit (SDK).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='  The system also contains Linux driver packages, CUDA  libraries, and related application programming interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' 2 shows the overall layout and components of the AI  edge computing board.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' This board contains two CMOS serial  interfaces for interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' It also contains an HDMI, multiple  USB 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='0 interfaces, and micro USB interface for display and  data interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' A wide area network port and a Bluetooth chip  are equipped on the board for wireless transfer of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' The  fixed 128 GB solid state disk meets the requirements of space  for storing different AI models, SDK, and deep learning  framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' Video super resolution algorithm for low resolution  video sequence  Traditional single image super resolution [14]-[16]  algorithms often lead to blurry effects and motion artifacts for  the cause of excessive destruction of ground truth textures and  missing consideration of temporal relationship [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' A previous  study [18] used interaction-learning strategy to reduce  computation, although the complex architecture is unsuitable to  parallel acceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' Most of the popular VSR methods [19],  [20] adopt the fixed pipeline of motion estimation, motion  compensation, fusion, and upsampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' The optical flows  between frames are first estimated and then used to align  features so as to eliminate the motion effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' However, these  methods heavily rely on optical flow estimation, which is  complex and time-consuming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' Thus, more deep neural  networks [21], [22] are proposed to compensate the motion by  implicit estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='  Super resolution methods are commonly used to enhance the  quality of raw images during preprocessing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' Shen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' [23]  applied smooth and sparse tensor completion in data preprocess  of AM to propose a super resolution method for multi-sources  image stream data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' Walecki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' [24] improved the confidence  of object surface by using multiple images to tighten the line  Super-resolution  Results  High Speed  Camera  Machining Head  AI Edge  Computing Board  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' In-situ monitoring system integrated with AI edge computing board  during PAM process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='  Fan  Camera  Interface  DC Port  HDMI USB 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='0  WAN Mirco USB  SSD  Bluetooth Chip  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' AI edge computing board for video decoding and inference of AI  models during in-situ monitoring of PAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='  TABLE I  TECHNICAL SPECIFICATION OF JETSON XAVIER NX [13]  Feature  Description  AI performance  21 TOPS (INT8)  GPU  384-core VoltaTM GPU  CPU  6-core Nvidia Cameral CPU ARM®v8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='2  RAM  8 GB 128-bit LPDDR4x  Tensor cores  48  Power  10 W | 15 W | 20 W  OOOAATWenkang Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' : In-situ monitoring additive manufacturing process with AI edge computing  3  segments along the camera ray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' However, these traditional  mathematical-based methods are limited by complex task  settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='  In this paper, a super resolution model based on a visual  transformer is proposed to upscale the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' 3 shows the  overall pipeline of ViTSR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' The ViTSR considers a sequence of  low resolution (LR) frames {  ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=',  ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='  }  t N  t  t N  X  X  X  −  +  as its input,  where  t  X  represents the reference frame and the others stand  for neighboring frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' The output of ViTSR is  tY , which  refers to the high resolution (HR) version of the reference frame  t  X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='  LR frames are downscaled from the corresponding ground  truth (GT) frames by nearest neighbor interpolation with scale  of r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' The height and width of single input frame are represented  as H and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' As per the standard practice, RGB images are  converted into Y-Cb-Cr color space, and only the Y channel is  used for super resolution [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='  In addition, temporary and spatial information is explicitly  encoded and concatenated to the input frames at first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' Given a  pixel with coordinates (x, y, i) in the frame Xi, where x, y, and i  imply the horizontal, vertical, and temporal codes of this pixel,  temporary, and spatial encoding of this pixel can be formulated  as：  1  1  1  2  [ ( ), ( ), ( )]  [sin(  ),[sin(  ),[sin(  )  2 ]  2  h x v  x  y  i  W  y t i  H  T  −  −  −  =  (1)  where h(x), v(x), and t(i) rapidly stand for horizontal code,  vertical code, and temporal code of this pixel, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='  A residual block with n cells serves as a feature extractor,  inspired by [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' Each cell receives the concatenation of the  internal outputs before itself and finally exports a feature map  with 32 channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' 3, every single cell consists  of convolutional layers with batch normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' The residual  cell i (1≤i≤n) consists of 1 × 1 convolutional layers, as  described previously [27], and 3 × 3 convolutional layers to  enhance feature interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='  After the residual block, a visual transformer block is  followed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' The visual transformer block is composed of 3-way  parallel cells with an inner structure similar to residual cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' Q  (Query), K (Key), and V (Value) indicate three essential tensors  in the attention mechanism [28] that will be used to calculate  the fused feature map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' In particular, the last way in the block  applies no padding in the temporary axis of feature map, which  leads the shape of Q tensor to 1 × H × W × r2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' In addition, the K  tensor includes all key features of neighboring frames with the  shape of 2N × H × W × r2, and V tensor includes all value  features of frames with the shape of (2N+1) × H × W × r2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' The  fused feature map Ht can be calculated as  t  ,  (  ,  )  t  N  t  t  t  i  i  i t  N i  H  V  Softmax Q K V  +  = −  =  + \uf0e5  ≠  (2)  where Ki and Vi refer to the slices of K and V on the temporary  axis, and Qt equals to Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' Then a pixel shuffle, which is a  periodic shuffling operator [13], is applied to rearrange the  tensor  t  H  to ˆ  t  H .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' The reconstructed frame  tY  is produced by  the sum of bicubic interpolation  tB  and ˆ  t  H .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' RESULTS  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' Video sequence captured by in-situ monitoring  system  During the PAM process, video sequences containing molten  pool and plasma arc are captured by the high-speed camera at a  current intensity of 40 A and scanning speed of 10 mm/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' 4  shows the original images captured by the high-speed camera  with no preprocess in 1 ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' Frames in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' 4 show the temporal  continuity of motion, indicating the correlation of neighboring  frames that can be used to enhance the feature of the reference  frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' Our findings showed that the plasma arc sways over  time and molten pool flows slowly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' In-situ video super resolution reconstruction of  monitored videos  The proposed ViTSR was used to reconstruct the frames  obtained from the high-speed camera to obtain high resolution  video sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' 5 shows the high-resolution reconstruction  of local regions in video sequence by ViTSR, where N is set to  1 and upscaling ratio r is 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' The input of the super resolution  model includes one reference frame and two neighbor frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='  Here, t0 is the start time of video sequence, and T is the interval  time of shooting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' In the figure, the images on the left show the  position of the regions to be reconstructed, while the images on  the right are coupled with low-resolution reference frames and  corresponding reconstructed frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' The top part of images  shows regions of powder bed, the middle part shows regions of  plasma arc, and the bottom part shows molten pool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' In the  proposed VSR model, the resolution of the images is enlarged  from 25 pixels × 25 pixels to 100 pixels × 100 pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' The  motion between high-resolution frames is natural and  reasonable because the visual transformer enables implicit  motion estimation and pixel-wide fusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='  t0 ms - t0+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='5 ms  1  2  3  4  5  6  7  8  9  10  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' Original video of molten pool and plasma arc captured at a current  intensity of 40 A and scanning speed of 10 mm/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='  Vt  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='  i  n  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='.  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='  Xt-N  Xt  Xt+N  1×1 Conv3d  Kernels: 32i  BN  3×3 Conv3d  Kernels: 32  BN  1×1 Conv3d  Kernels: 32n  BN  3×3 Conv3d  Kernels: r2  BN Softmax  +  Residual Block  Visual Transformer  Pixel  Shuffle  Qt  Kt  Temporary and Spatial Encoding  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='.  Yt  Input: 75 pixels × 75 pixels  Output: 300 pixels × 300 pixels  Bt  1×1 Conv3d  Kernels: 32n  BN  3×3 Conv3d  Kernels: r2  BN  1×1 Conv3d  Kernels: 32n  BN  3×3 Conv3d  Kernels: r2  BN  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' Video super-resolution using visual-transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='  Wenkang Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' : In-situ monitoring additive manufacturing process with AI edge computing  4  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' 6, critical internal feature maps were considered for  visualization to evaluate the visual transformer block with  respect to feature fusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' The corresponding internal feature  maps of input frames are  1  1  {  ,  ,  }  i  i  i  X  X X  −  +  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' Meanwhile, Q, K,  and V are sliced as  i  Q ， {  ,  }  i T  i T  K  K  −  +  , and {  ,  ,  }  i T  i  i T  V  V V  −  +  by the  axis of channel, which is also the last channel of feature maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' 6, the visualization of feature maps Q, K, and V have  profiles similar to the corresponding input frames, including the  topography of the plasma arc and molten pool, and the  distribution of metal powder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' The fused feature map Hi shows  details of the fusion of Q, K, and V on the way of attention  mechanism, which keeps the regions that can be used for the  reconstruction of the reference frame and suppresses those  regions that are different from the reference frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' The  visualization of Hi shows great similarity of geometric shape  with reference frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' This indicates that high attention is paid  to similar regions of neighbor frames and reference frame,  while low attention is paid to different regions, indicating the  effectiveness of this block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' When applying visual transformer  block instead of motion estimation and feature alignment, the  beneficial features from neighbor frames can be fused into the  correct regions of reference frame by using parallel neural  nodes and nonlinear combination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='  The proposed model was compared with 6 state-of-the-art  super resolution methods, including Bicubic, super resolution  convolution neural network (SRCNN) [15], fast SRCNN  (FSRCNN) [16], efficient sub-pixel convolutional neural  network (ESPCN) [14], dynamic upsampling filters (DUF) [21],  and temporary group attention (TGA) [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' Bicubic is a simple  math-based method that is widely used for its low cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' The  SRCNN  applied  convolutional  layers  behind  bicubic  interpolation to seek higher quality of HR reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='  Deconvolution is used in FSRCNN to upscale the output  instead of interpolating the input image in the beginning so as to  make full use of feature maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' Pixel shuffle is used in ESPCN  to obtain surprising results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' DUF applies dynamic upsampling  filters to compensate for motion between frames implicitly  instead of traditional optic-flow motion estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' The  reconstruction in TGA can be improved using temporary group  attention for the inner relationship between long-term frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='  Bicubic interpolation realized by OpenCV is directly used for  its high efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' 7 shows comparisons between 4× super  resolution reconstruction by the state-of-the-arts models and  the proposed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' The input size of models was set to 75  i = 0  i = 1  i = 2  X-T  XT  X0  X1-T  X1  X1+T  X2-T  X2  X2+T  K-T  Q0  KT  K1-T  Q1  K1+T  K2-T  Q2  K2+T  V-T  V0  VT  V1-T  V1  V1+T  V2-T  V2  V2+T  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' The attention map H decoded from query Q, key K and value V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='  Video Sequence  From t0 ~ t0 + 4T  t0 + 4T  t0 + 2T  t0 + T  t0  t0 + 3T  t0 + 4T  t0 + 2T  t0 + T  t0  t0 + 3T  t0 + 4T  t0 + 2T  t0 + T  t0  t0 + 3T  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' Super-resolution for molten pool and spatter areas in a video sequence  by ViTSR, T = 1/30 ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='  LR  GT  Bicubic  SRCNN  FSRCNN  ESPCN  DUF  TGA  ViTSR  75 × 75  300 × 300  300 × 300  300 × 300  300 × 300  300 × 300  300 × 300  300 × 300  300 × 300  From video  SSIM=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='9883  SSIM=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='9873  SSIM=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='9885  PSNR=27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='7576  PSNR=27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='7757  PSNR=27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='3202  SSIM=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='9959  SSIM=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='9955  SSIM=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='9959  PSNR=32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='4473  PSNR=32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='3423  PSNR=31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='9197  SSIM=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='9980  SSIM=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='9974  SSIM=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='9979  PSNR=35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='5085  PSNR=35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='2222  PSNR=34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='2179  SSIM=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='9985  SSIM=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='9980  SSIM=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='9985  PSNR=36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='7854  PSNR=36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='5366  PSNR=35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='4501  SSIM=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='9991  SSIM=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='9987  SSIM=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='9990  PSNR=38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='7048  PSNR=38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='5372  PSNR=37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='1383  SSIM=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='9990  SSIM=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='9988  SSIM=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='9991  PSNR=38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='5202  PSNR=38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='6073  PSNR=37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='5364  SSIM=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='9991  SSIM=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' 9988  SSIM=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='9991  PSNR=39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='3199  PSNR=39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='3843  PSNR=37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='6522  t0  t0 + Ts  t0 + 2Ts  Input: low resolution  samples  Output: high resolution results  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' Performance of super-resolution results by various methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='  Wenkang Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' : In-situ monitoring additive manufacturing process with AI edge computing  5  pixels × 75 pixels, which is LR in the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' Three reference  frames were selected with interval time (Ts) of 1/3 ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' Finally,  ViTSR ranks first in terms of PSNR and was 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='7770 dB higher  than the second highest model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='  The top 3 models (ViTSR, TGA and DUF) were selected in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' 8 to visualize the subtracting of reconstructed frames with  GT frames and to clearly show the difference between  reconstructed frames and GT frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' Negative values of  subtraction were set to zero, and the values of subtraction were  scaled to 0–255.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' 8, most areas of DUF were  different with GT frames for its dynamic filters, and the results  of DUF showed lower brightness than GT frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' For TGA,  the different pixels were concentrated around molten pool and  plasma arc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' In the case of ViTSR, the different pixels were  concentrated around molten pool, and the number is the least.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' 9 shows the average PSNR and inference time of various  methods on the AI computing board with the same setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' A  total of 58 video frames were tested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' 9, ViTSR  achieved the best performance in terms of PSNR but only  half-time cost compared with the TGA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' After time optimizing  using operator fusion and model pruning, ViTSR performs the  best by making most of the NPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' Inference time of ViTSR was  233.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='01 ms on CPU that was optimized to 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='97 ms on AI edge  board.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' The PSNR of ViTSR was reduced to 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='67 dB, which  indicates saved inference time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='  The effectiveness of core components of this model,  including the temporal and spatial encoding, visual transformer,  and 3D convolution, was verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' The core components of  ViTSR were eliminated to generate three degenerated models,  named as ViTSR_d1, ViTSR_d2, and ViTSR_d3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' In brief,  ViTSR_d1 removed temporal and spatial encoding from the  proposed model, ViTSR_d2 used DUF instead of the visual  transformer, and ViTSR_d3 replaced all 3D convolution with  2D convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' Table II shows the PSNR of ViTSR and  degenerated models on the test data after training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' The ViTSR  showed the highest PSNR (38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='16 dB) with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='96 dB higher than  the second-best model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' In-situ extraction of geometric characteristics of  molten pool and plasma arc  In our previous work [6], plasma arc and molten pool were  simultaneously extracted in the process of PAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' The present  paper used the well-trained FCN to extract the geometric  characteristics of molten pool and plasma arc after ViTSR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='  10 shows the segmentation of the proposed in-situ monitoring  system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' The resolution of input video sequence was 75 pixels ×  75 pixels, while the output resolution of segmentation was 300  pixels × 300 pixels after ViTSR and FCN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='  The AI edge computing used ViTSR and FCN to fetch the  areas of real-time super resolution segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' The ViTSR  27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='5444  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='0692  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='7968  35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='9899  37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='8276  38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='1486  38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='1600  37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content="6700  27  29  31  33  35  37  39  0  100  200  300  400  500  PSNR (dB)  Inference time (ms)  Bicubic  SRCNN  FSRCNN  ESPCN  DUF  TGA  ViTSR  ViTSR'  Time Optimized  Fig." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' Performance of various models in terms of PSNR and inference time on  AI computing board.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='  Output: segmentation with 300 pixels × 300 pixels  Input: video sequence with 75 pixels × 75 pixels  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' Extraction of geometric characteristic of in-situ monitoring video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='  TABLE II  PSNR OF VITSR AND DEGENERATED MODELS ON TEST DATA AFTER TRAINING  Models  Description  PSNR  ViTSR  the proposed model  38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='16 dB  ViTSR_d1  remove temporal and spatial encoding  36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='42 dB  ViTSR_d2  replace VIT with DUF  36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='29 dB  ViTSR_d3  replace 3D-Conv with 2D  37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='20 dB  GT  DUF  DUF - GT  GT - DUF  GT - TGA  TGA  TGA - GT  ViTSR  ViTSR - GT  t0  GT - ViTSR  t0 + Ts  t0 + 2Ts  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' Visualization of the difference between super-resolution reconstruction and GT, Ts = 1/3 ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='  Wenkang Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' : In-situ monitoring additive manufacturing process with AI edge computing  6  was used to reconstruct HR frames, while the FCN was used to  extract the geometric shapes of molten pool and plasma arc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='  11 shows the extracted pixels of molten pool and plasma arc in  PAM process with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' The area of molten pool and plasma  arc was found to be in the range of 3383–4135 pixels and  7474–8709 pixels, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' This shows the temporal  coherence of the target areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' As shown in this figure, the AI  edge computing board was used to process 58 frames with a  resolution of 75 pixels × 75 pixels to 300 pixels × 300 pixels  results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' The video sequence frames with a total number of 58  were processed for super resolution reconstruction and frame  segmentation in 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='89 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' The average inference time for  processing a single frame is 118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='83 ms, where the inference  time of frame segmentation is 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='86 ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' The in-situ monitoring  system showed its immense future for high-resolution and  low-cost quality monitoring of AM process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' DISCUSSION  Traditional  image  segmentation  algorithms  were  successfully applied in many fields, whereas the in-situ  monitoring task of AM could not fit because of its halo effects  and high dynamic range imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' 12 shows the comparison  of extraction performance using traditional and proposed  methods with input size of 75 pixels × 75 pixels in PAM  process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' 12(a), plasma arc, molten pool, and  part of powder bed were segmented together when using  triangle algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' 12(b), the maximum  entropy algorithm extracted the general shapes of plasma arc,  molten pool, and bright metal powders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' The region of plasma  arc extracted by the two methods was severely enlarged for  interference of halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' With respect to the extraction of targets  and the suppression of bright metal powders, watershed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='  12(c), Otsu in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' 12(d), and basic global thresholding  algorithm in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' 12(e) performed better than the former  segmentation algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' The three algorithms were not  critically affected by the halo and reflected light, while the  extraction of plasma arc was still enlarged and the molten pool  was smaller than the fact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' Traditional algorithms could not  extract target areas accurately due to the interference of halo  and reflection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' 12(f), FCN extracted the  plasma arc and molten pool efficiently owing to its nonlinear  fitting ability, where the halo was excluded from the extraction  and the low brightness showed less interference of the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' 12(g) showed the extracted result of 4× super resolution  frame using ViTSR and FCN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' Our findings showed that the  proposed method extracted plasma arc and molten pool  accurately, eliminating the interference of light conditions and  bright powders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' Table III shows the performance of traditional  methods, FCN method, and the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' The table  also showed that the proposed method upscaled the resolution  of input frames from 75 pixels × 75 pixels to 300 pixels × 300  pixels, thus, achieving the highest accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='  Table IV showed the performance of AI methods for the  in-situ monitoring of multiple AM process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' Tan’s method [4]  was used to process the image tiles with a resolution of 200  pixels × 200 pixels so as to detect the spatters in the process of  laser power bed fusion (LPBF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' Despite the acceptable  inference time, the accuracy may not be suitable for real-world  applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' Fang reported high accuracy and inference speed  after he adopted U-Net to extract the molten pool with a  resolution of 224 pixels × 224 pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' Mi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' [5] achieved an  accuracy of 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='71% in the process of laser based directed  energy deposition (L-DED) when the proposed D-CNN  architecture was used to extract the spatters and molten pool  simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' [6] extracted plasma arc and  molten pool simultaneously in the PAM process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' Although the  inference time of Zhang’s method [6] was longer than that of  Mi’s method [5], the accuracy achieved by using Zhang’s  method [6] was much higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' In this paper, the proposed  method was used to reconstruct the 4× super resolution frames  and extract the geometric characteristics of plasma arc and  molten pool during PAM process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' The proposed method  achieved an accuracy of 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='34% with inference time of 15 ms  TABLE III  PERFORMANCE OF TRADITIONAL SEGMENTATION METHODS AND PROPOSED  METHOD  Methods  Extracted  objects  Resolution  (pixel)  Upscaling  Accuracy  Triangle  Bright area &  Dark area  75 × 75  1 × 1  27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='57%  Maximum  Entropy  Bright area &  Dark area  75 × 75  1 × 1  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='46%  Watershed  Bright area &  Dark area  75 × 75  1 × 1  92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='11%  Otsu  Bright area &  Dark area  75 × 75  1 × 1  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='57%  Basic  Global  Thresholding  Bright area &  Dark area  75 × 75  1 × 1  92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='68%  FCN  Plasma arc &  Molten pool  75 × 75  1×1  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='29%  Proposed:  ViTSR +  FCN  Plasma arc &  Molten pool  Input:  75 × 75  Output:  300 × 300  4×4  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='34%  300 pixels × 300 pixels  (a)  (b)  (c)  (d)  (e)  (f)  (g)  75 pixels ×75 pixels  75 pixels ×75 pixels  75 pixels ×75 pixels  75 pixels ×75 pixels  75 pixels ×75 pixels  75 pixels ×75 pixels  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' 12 Extraction of molten pool and plasma arc using different image  segmentation methods: (a) triangle, (b) maximum entropy, (c) watershed, (d)  Otsu, (e) basic global thresholding, (f) FCN, and (g) ViTSR+FCN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='  5000  6000  7000  8000  9000  3000  4000  5000  6000  7000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='12  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='96  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='79  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='62  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='45  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='28  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='11  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='94  Pixels of Plasma Arc  Pixels of Molten Pool  Time (s)  Molten Pool  Plasma Arc  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' Extracted pixels of molten pool and plasma arc with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='  Wenkang Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' : In-situ monitoring additive manufacturing process with AI edge computing  7  and 118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='83 ms on Nvidia RTX 3070 Laptop GPU and Nvidia  Jetson Xavier NX, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' The sum of two models  denoted the number of parameters;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' this is one of the reasons of  the higher inference time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' In the case of less power, the AI edge  computing board was equipped with 48 tensor cores and only  26% of GPU, which is the reason of higher inference time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' The  inference time of the proposed method using GPU was found to  be the smallest in Table IV even when the GPU had more  parameters, which shows the efficiency of the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='  The inference time can be optimized in the future by integrating  the FCN and ViTSR into one step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' The proposed method shows  its low requirement of pixel resolution for high-speed camera  used in the video capture of the AM process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' This reduced the  cost of camera and contributed to the industrial applications of  in-situ monitoring system for AM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' CONCLUSIONS  In this paper, a novel in-situ monitoring system was  proposed to extract the high-resolution features of the AM  process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' For this, high-speed camera with variable resolutions  was used to capture the video data during the AM processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='  The captured video sequences were processed using AI edge  computing because of its low cost and high efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' The  maximum power of the AI edge computing board was found to  be 20 W, which is lower than the power consumption of PCs  and computing servers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' The proposed system used 118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='83 ms  for total inference of video sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' The inference time of  video super resolution is 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='97 ms from a resolution of 75  pixels × 75 pixels to 300 pixels × 300 pixels, and frame  segmentation is 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='86 ms, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' Two-stage strategy is  used to reconstruct the input with high resolution and extracts  the key features with high accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' To the best of our  knowledge, this is the first study that used a video super  resolution algorithm before image segmentation to seek high  resolution geometric characteristics of AM processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' The  output of the proposed system demonstrated greater tolerance  to halo and shadow of captured video and finally achieved an  accuracy of 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='34%, which is similar to our previous work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='  Considering that the proposed system needed 1/4 resolution as  input with same results, the system will sharply reduce the cost  of high-speed cameras used in in-situ monitoring systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' Thus,  this paper provides a way to lower the cost of DIC-based  methods and may finally improve the quality of products  manufactured by various AM process including PAM, LPBF,  and L-DED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
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+page_content=' Inform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
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+page_content='  [19] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
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+page_content=' Chen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' Wu, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' Wei, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
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+page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' Vis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' 127, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
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+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
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+page_content='48%  Fang’s [3]  Molten  pool  224 × 224  1 × 1  37 ms  98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='06%  Mi’s [5]  Spatters &  Molten  pool  450 × 512  1 × 1  63 ms  94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='71%  Zhang’s  [6]  Plasma arc  & Molten  pool  450 × 512  1 × 1  84 ms  95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='10%  Proposed:  ViTSR +  FCN  Plasma arc  & Molten  pool  Input:  75 × 75  Output:  300 × 300  4 × 4  15 ms (GPU)  118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
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+page_content=' Reed, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' Anguelov, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='  Erhan,  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' Vanhoucke, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' Rabinovich, “Going deeper with convolutions,”  Presented at 2015 IEEE Conference on Computer Vision and Pattern  Recognition  (CVPR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='  [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='  Available:  http://ieeexplore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='ieee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='org/document/7298594/  [27] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' Vaswani, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' Shazeer, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' Parmar, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' Uszkoreit, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' Jones, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' N Gomez, Ł.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='  Kaiser, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' Polosukhin, “Attention is all you need,” Advances in Neural  Information  Processing  Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='  [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='  Available:  https://proceedings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='neurips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='cc/paper/2017/hash/3f5ee243547dee91fbd053c1c4  a845aa-Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='html  Wenkang Zhu received the B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' degree in  mechanical engineering and automation from Wuhan  University, Wuhan, China in 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content="  He is studying for master's degree in Institute of  Technological Sciences, Wuhan University, Wuhan,  China." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' His research interests include additive  manufacturing and artificial intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='  Hui Li received the B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' degree from Huazhong  University of Science and Technology, Wuhan, China,  and the Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' degree in electrical & computer  engineering from National University of Singapore,  Singapore, in 1999 and 2007, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='  He is currently a Professor at Wuhan University,  China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' His research interests include electronics  manufacturing and additive manufacturing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='  Yikai Zhang received his B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=" degree in electronic  information engineering from Hefei University of  Technology in 2020 and is currently studying for a  master's degree at the Institute of Industrial Science of  Wuhan University." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='  His current research interests include image  processing,  deep  learning,  and  additive  manufacturing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='  Yuqing Hou received his M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' degree from Xian  University of Technology, Xian, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' He is  currently pursuing the Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' degree with Institute of  Technology Science, Wuhan University, Wuhan,  China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='  His research focuses on artificial intelligence and  computational  fluid  dynamics  for  additive  manufacturing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='  Liwei Chen received his M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='S degree form Wuhan  University, Wuhan, China, and the Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' degree in  mechanical  system  engineering  from  Tohoku  University, Sendai, Japan, in 2019 and 2022,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content='  He is currently a Postdoctoral Fellow at the  University of Tokyo, Japan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
+page_content=' His research interests  include image processing and laser manufacturing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfq_nZ/content/2301.00554v1.pdf'}
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+arXiv:2301.04237v1  [quant-ph]  10 Jan 2023
+Solving the semidefinite relaxation of QUBOs in matrix
+multiplication time, and faster with a quantum computer
+Brandon Augustino ∗†, Giacomo Nannicini ‡, Tam´as Terlaky†, and Luis F. Zuluaga†
+January 12, 2023
+Abstract
+Recent works on quantum algorithms for solving semidefinite optimization (SDO) problems have
+leveraged a quantum-mechanical interpretation of positive semidefinite matrices to develop methods
+that obtain quantum speedups with respect to the dimension n and number of constraints m. While
+their dependence on other parameters suggests no overall speedup over classical methodologies, some
+quantum SDO solvers provide speedups in the low-precision regime. We exploit this fact to our advan-
+tage, and present an iterative refinement scheme for the Hamiltonian Updates algorithm of Brand˜ao et
+al. (Quantum 6, 625 (2022)) to exponentially improve the dependence of their algorithm on precision. As
+a result, we obtain a classical algorithm to solve the semidefinite relaxation of Quadratic Unconstrained
+Binary Optimization problems (QUBOs) in matrix multiplication time.
+Provided access to a quan-
+tum read/classical write random access memory (QRAM), a quantum implementation of our algorithm
+exhibits a worst case running time of O
+�
+ns + n1.5 · polylog
+�
+n, ∥C∥F , 1
+ǫ
+��
+.
+1
+Introduction
+We consider optimization problems of the form:
+max x⊤Cx
+s.t. x ∈ {−1, 1}n,
+(1)
+where C ∈ Sn is the problem data and Sn is the space of symmetric matrices in Rn×n. Solving (1) can
+be viewed as computing the ∞ → 1 norm of the coefficient matrix C. This particular norm is intrinsically
+related the cut norm of a matrix, which plays a crucial role in developing efficient approximation algorithms
+for dense graph and matrix problems [1, 21], with perhaps the most well-known application being the task
+of finding the largest cut in a graph (MaxCut). These problems also play an important role in quantum
+information sciences; the Ising model belongs to this class of problems [52], and quantum algorithms such as
+the Quantum Approximate Optimization Algorithm (QAOA) [18] and quantum annealing [19] can address
+its solution.
+Computing the cut norm corresponds to replacing x ∈ {−1, 1}n with z ∈ {0, 1}n in (1), giving rise to
+quadratic unconstrained binary optimization (QUBO) problems. A standard QUBO is of the form
+max z⊤Cz
+s.t. z ∈ {0, 1}n.
+(2)
+∗Corresponding Author: bra216@lehigh.edu
+†Department of Industrial and Systems Engineering, Quantum Computing and Optimization Lab, Lehigh University
+‡Department of Industrial and Systems Engineering, University of Southern California
+1
+
+Provided that we allow for linear terms (in both formulations), it is well known that solutions to (1) can
+be used to compute a solution to (2) which differs only by a constant factor, and vice-versa, due to the
+equivalence z = x+e
+2
+if z ∈ {0, 1}n and x ∈ {−1, 1}n, where e ∈ Rn is the all ones vector of dimension n.
+Although (1) and (2) cover many applications of interest, they are intrinsically difficult to solve; computing
+optimal solutions to either (1) or (2) is NP-Hard in general. Following the seminal work of Lov´asz [43] and the
+theoretical and practical development of Interior Point Methods (IPMs) for solving semidefinite optimization
+(SDO) problems [46, 47, 48, 49, 50, 55, 56], a prevailing approach has been to obtain approximate solutions
+to (1) and (2) by relaxing integrality and lifting the problem from a vector space of dimension n, to the
+space of n × n symmetric matrices. The quadratic form x⊤Cx can be equivalently expressed by tr (Cxx⊤),
+where tr (U) denotes the sum of the diagonal elements (or, trace) of a matrix U ∈ Rn×n. To deal with the
+bilinear term xx⊤, we introduce a matrix variable X ∈ Rn×n, and require that X satisfies the following:
+diag(X) = e,
+X ⪰ 0,
+rank(X) = 1,
+where the notation U ⪰ V means that the matrix U − V is a symmetric positive semidefinite matrix. Under
+these requirements, X is guaranteed to be of the form X = xx⊤ for x ∈ {−1, 1}n. The rank constraint,
+however, is not convex, and thus dropping it yields the following (convex) SDO relaxation of (1):
+max
+tr (CX)
+s.t.
+diag (X) = e,
+X ⪰ 0.
+(3)
+Although the optimal solution X∗ to (3) is no longer guaranteed to satisfy X∗ = x∗x∗⊤ and may not be
+integral in general, the approximation of x∗ provided by X∗ is of sufficient quality to justify its use. In
+fact, SDO approximations cover some of the most celebrated results in optimization, such as the 0.878-
+approximation guarantee of Goemans and Williamson for MaxCut [28] and the Lov´asz-ϑ number [43].
+1.1
+Literature Review
+More generally, a (primal) SDO problem involving n × n matrices and m constraints is of the form
+max
+X
+tr (CX)
+s.t.
+tr (AiX) = bi
+for i ∈ [m],
+X ⪰ 0,
+where [m] = {1, . . . , m} and A1, . . . , Am, C ∈ Sn, and b ∈ Rm are the (given) problem data. The dual SDO
+problem associated with the primal is given by
+min
+(u,S)
+b⊤u
+s.t.
+S =
+m
+�
+i=1
+uiAi − C ⪰ 0.
+where S is the dual slack matrix.1 The classical literature on algorithms for solving SDO problems is rich
+and can be categorized into two classes; algorithms that depend poly-logarithimically on the inverse precision
+to which we solve the problem and the size of the minimally inscribed ellipsoid, and algorithms that depend
+polynomially on these quantities but exhibit an advantage with respect to n and m. For instances with
+m ≤ √n, the cutting plane methods (CPMs) of [35, 42] are the best performing classical algorithms,2 and
+can solve SDO problems in time
+O
+�
+m(mns + m2 + nω) · polylog
+�
+m, n, R, 1
+ǫ
+��
+,
+1While the dual variable is typically denoted by y rather than u, it is also customary in the literature to use y to denote a
+certain state preparation pair, and we do so later in this paper.
+2We remark that the running time in [35] does however exhibit improved dependence with respect to poly-logarithmic factors
+compared to the running time of [42].
+2
+
+where ω ∈ [2, 2.38] is the matrix multiplication exponent, R is an upper bound on the trace of a primal
+optimal solution X (which can be exponentially large), ǫ is the precision parameter, s denotes the maximum
+number of nonzeros per row of the input matrices and hence, O(mns) is the total number of nonzeros in the
+constraints of SDO problem. However, we typically have m ∈ [Ω(n), O(n2)], in which case the CPMs given
+in [35, 42] are outperformed by the IPM for SDO from Jiang et al. [34]. Their IPM exhibits a worst case
+running time of
+O
+�√n(mns + mω + nω) · polylog
+�
+m, n, 1
+ǫ
+��
+,
+where the term mω + nω represents the per-iteration cost of inverting the Hessian and matrices of the
+variables.
+While quantum SDO solvers could also be categorized in a somewhat similar fashion, it is perhaps more
+natural to do so according to how they attempt to obtain quantum speedups. In this case we also have two
+classes; at a high level, all proposed quantum SDO solution methodologies quantize a classical algorithm by
+either using quantum linear system algorithms (QLSAs) [12, 14, 31], or a quantum mechanical interpretation
+of normalized positive semidefinite matrices. We now review these works in detail.
+The former class is comprised of algorithms that quantize IPMs, giving rise to quantum IPMs (QIPMs).
+QIPMs attempt to speedup the bottleneck of the classical IPM by substituting the classical solution of the
+Newton linear system with the combined use of QLSA and quantum state tomography (with some classical
+computation between iterates).
+Augustino et al. [6] present a convergent QIPM for SDO, avoiding the
+shortcomings prevalent in early works on QIPMs (see, e.g., [39]), by properly symmetrizing the Newton linear
+system, and utilizing an orthogonal subspace representation of the search directions. This representation
+guarantees that primal and dual feasibility are satisfied exactly by all the iterates generated by inexact
+solutions of the Newton linear system obtained via quantum subroutines. The worst case complexity of their
+algorithm is
+�On,κ, 1
+ǫ
+�√n
+�n3κ2
+ǫ
++ n4
+��
+,
+where κ is an upper bound on the condition numbers of the intermediate Newton linear system coefficient ma-
+trices that arise over the course of the algorithm. Here, the notation �Oa,b(f(x)) suppresses poly-logarithmic
+factors in f(x), a and b that appear in the overall running time, i.e., �
+Oa,b(f(x)) ≡ O(f(x)·polylog(a, b, f(x))).
+While this QIPM achieves a speedup in n over the IPM from [34] when m = O(n2), its dependence on κ and
+ǫ suggest no quantum advantage overall: the complexity of the classical IPM does not depend on κ and its
+dependence on ǫ−1 is logarithmic. As the authors in [6] note, dependence on the condition number bound κ
+is particularly problematic in the context of IPMs.
+The second class of quantum SDO solvers are those that quantize algorithms based on matrix exponentials
+and Gibbs states. The most prominent example is the Matrix Multiplicative Weights Update (MMWU)
+Method of Arora and Kale [3], which can solve SDO problems in time
+�On,R, 1
+ǫ
+�
+nms
+�Rr
+ǫ
+�4
++ ns
+�Rr
+ǫ
+�7�
+.
+where r is a known ℓ1-norm upper bound3 on a dual optimal solution u. Unlike IPMs, the MMWU framework
+does not involve the solution of linear systems; rather, these algorithms alternate between candidate solutions
+to the primal and dual SDO problems. IPMs and MMWUs also employ different definitions of optimality;
+for IPMs, ǫ-optimality implies that the primal and dual feasible solutions exhibit a normalized duality gap
+bounded by ǫ, i.e.:
+tr (XS)
+n
+≤ ǫ,
+whereas an ǫ-optimal solution obtained using an MMWU approximates the optimal objective value to addi-
+tive error ǫ (via binary search). Finally, we point out a distinction between these algorithms with respect to
+3It is also assumed that R, r ≥ 1.
+3
+
+output. While primal-dual IPMs return the primal-dual optimal solution (X, u, S), MMWUs report u, but
+may avoid explicitly reporting X and S to maintain the speedups they offer with respect to n. Reporting X
+or S under the MMWU framework necessitates the computation of matrix exponentials, which may impose
+a considerable overhead because it generally resorts to matrix multiplication.
+The MMWU framework has been specialized to solve SDO problems of the form in (3) (see, e.g., [4]),
+and the current state of the art is attributed to Lee and Padmanabhan [41], who give an algorithm that can
+solve (3) to additive error ∥C∥ℓ1ǫ with overall complexity
+�On, 1
+ǫ
+�
+nsǫ−3.5�
+,
+where ∥C∥ℓ1 = �
+i,j |Cij|. It is important to note however, that to achieve the stated complexity their
+methodology does not explicitly report the solution X4 and the authors assume �
+i,j |Cij| = n. Thus, to
+achieve the same error scaling as the algorithms we present in this work, the algorithm found in [41] would
+incur overall cost �
+On, 1
+ǫ
+�
+n4.5sǫ−3.5�
+.
+Brand˜ao and Svore [11] and van Apeldoorn et al. [61] were the first to quantize the MMWU framework,
+utilizing a clever interpretation of the primal variables: Gibbs states, which can be efficiently prepared on a
+quantum computer, naturally correspond to trace-normalized positive definite matrices. The running time of
+these MMWU-based algorithms was subsequently improved [30, 60], and the current state of the art running
+time of the quantum MMWU (QMMWU) algorithm for SDO problems is:
+�
+On,s,R, 1
+ǫ
+��√m + √nRr
+ǫ
+�
+s
+�Rr
+ǫ
+�4�
+.
+Similar to the complexity of QIPMs, QMMWU algorithms are faster with respect to m and n when compared
+to their classical counterparts, but these algorithms still exhibit a non-polynomial running time, due to their
+polynomial dependence on the scale invariant parameter Rr
+ǫ , whereas the natural input size depends on the
+logarithm of this quantity.
+Seeking to improve the performance of quantum SDO solvers, Brand˜ao et al. [10] present an algorithm,
+which they call Hamiltonian Updates (HU), for solving the SDO approximation (3) of (1). The HU method
+is a primal-only algorithm closely related to the QMMWU framework, in that it leverages a Gibbs state
+representation of the primal variable and progression towards the optimal solution is made via matrix-
+exponentiated gradient updates. Specifically, the authors in [10] are interested in solving an SDO feasibility
+problem that arises upon renormalizing and relaxing (3):
+find
+X
+s.t.
+tr
+� C
+∥C∥X
+�
+≥ γ − ǫ
+�
+i∈[n]
+����⟨i|X|i⟩ − 1
+n
+���� ≤ ǫ
+tr (X) = 1,
+X ⪰ 0.
+(4)
+Here, γ is an upper bound on the absolute value of the optimal objective value of (3) when the cost matrix
+C is normalized, obtained via binary search over [−1, 1], and |i⟩ for i ∈ {1, . . . , n} are the computational
+basis states. Since any log(n)-qubit Gibbs state is an element of the set {X ∈ Rn×n : tr(X) = 1, X ⪰ 0} by
+definition, solutions to (4) can be naturally be expressed as a Gibbs state
+ρ =
+exp(−H)
+tr(exp(−H)),
+where H is the Hamiltonian associated with ρ. The key observation in [10] is that upon using the Gibbs
+state change of variables in (4), one can model the n constraints on the diagonal elements as single constraint
+4Alternatively, they report a “gradient” G ∈ Sn such that X = W exp(G)W for a diagonal matrix W .
+4
+
+which requires that the distribution on the diagonal elements of a feasible solution ρ to (4) be at most ǫ in
+total variation distance to the uniform distribution. In other words, the task of solving (4) reduces to finding
+a log(n)-qubit mixed quantum state that upon measurement in the computational basis is approximately
+indistinguishable from the maximally-mixed state, and whose trace inner product with the normalized cost
+matrix C∥C∥−1 is at least γ − ǫ.
+Using a quantum computer, the HU method of [10] solves (3) to additive error O (n∥C∥ǫ) in time
+�On, 1
+ǫ
+�
+n1.5√s
+1+o(1)ǫ−28+o(1) exp
+�
+1.6
+�
+log(ǫ−1)
+��
+.
+The authors in [10] also provide an analysis of essentially the same algorithm when using a classical computer,
+and show that the classical algorithm has a complexity of
+�
+On
+�
+min{n2s, nω}ǫ−12�
+.
+The quantum algorithm yields a speedup in n over classical algorithms, for a specific class of SDO problems.
+However, as we have already seen with QIPMs and QMMWU algorithms, its dependence on other parameters
+(in this case the inverse precision) is prohibitive unless a very low precision solution is acceptable. This
+raises the question as to whether the poor scaling in the inverse precision can be mitigated without incurring
+additional cost in n and s. We answer this question in the affirmative using iterative refinement techniques.
+Iterative Refinement (IR) is a methodology for computing high-precision solutions to linear system of
+equations [29], as well as linear [25, 26, 27] and mixed integer optimization problems [2, 17]. We summarize
+the methodology at a high level as follows, and present a detailed discussion for the case of convex feasibility
+problems later in the paper. Given an initial solution x(0) ∈ Rd, at each iteration k IR produces a refined
+solution x(k+1) ← x(k) + u(k), where u(k) acts as a correction of the error r(k) associated with x(k), and
+is determined by solving a refining problem induced by the current solution. These operations can all be
+carried out using the same level of accuracy, called the fixed-precision approach. Alternatively, one may
+increase the accuracy with which the residuals r(k) are computed as compared to u(k), and this approach is
+called a mixed precision approach [29, 62]. In this paper, we utilize the fixed precision approach.
+1.2
+Contributions
+In this paper we develop an iterative refinement scheme for SDO approximations of QUBO problems that uses
+the HU algorithm of [10] as a subroutine. We show that proceeding in this way allows one to exponentially
+improve the dependence on the inverse precision for both the quantum and classical algorithms.
+With the proposed IR scheme, the classical algorithm solves the SDO problem (3) up to additive error
+O(ǫ) with a worst-case overall complexity of
+O
+�
+min{n2s, nω} · polylog
+�
+n, ∥C∥F, 1
+ǫ
+��
+.
+This is a significant speedup compared to general-purpose SDO solvers, such as IPMs. This algorithm can
+be quantized following a similar strategy to [10]. When provided access to quantum random access memory
+(QRAM), the quantum algorithm requires at most
+O
+�
+n1.5 · polylog
+�
+n, ∥C∥F , 1
+ǫ
+��
+accesses to the QRAM data structure in the worst case, and O(ns) classical arithmetic operations (to load
+and normalize the cost matrix C).
+Summarizing, the combination of HU with IR described in this paper provides exponential speedups over
+the methodology proposed in [10] with respect to the precision parameter ǫ. To the best of our knowledge, our
+classical and quantum algorithms are the fastest known algorithms in their respective model of computation
+for this class of problems, and our quantum algorithm provides a genuine asymptotic speedup over known
+5
+
+classical solution methodologies, provided that we have access to QRAM. Without access to QRAM, one can
+bound the running time of the quantum algorithm using the sparse-access input model, in which case the
+algorithm requires O
+��
+n1.5s0.5+o(1)�
+· polylog
+�
+n, ∥C∥F, 1
+ǫ
+��
+accesses to an oracle describing the coefficient
+matrix C and O
+��
+n2.5s0.5+o(1)�
+· polylog
+�
+n, ∥C∥F , 1
+ǫ
+��
+additional gates.
+The remainder of this paper is organized in the following manner. Section 2 introduces notation, as
+well as the relevant input models and quantum subroutines. In Section 3 we introduce the Hamiltonian
+Updates (HU) algorithm from [10], and our Iterative Refinement scheme for SDO approximations of QUBOs
+is presented in Section 4. The running time analysis is performed in Section 5, and Section 6 concludes the
+manuscript.
+2
+Preliminaries
+We write [n] to represent the set of elements {1, . . . , n}. We denote the i-th element of a vector x ∈ Rn by
+xi for i ∈ [n], and the ij-th element of a matrix A ∈ Rm×n by Aij for i ∈ [m] and j ∈ [n]. To refer to the
+i-th row of a matrix A, we write Ai,· and write A·,j when referring to its j-th column. We distinguish the
+quantity a to the k-th power and the value of a at iterate k using round brackets, writing ak and a(k) to
+denote these quantities, respectively.
+The smallest and largest singular values of a matrix A are denoted σmin(A), σmax(A), and if A ∈ Sn, then
+the smallest and largest eigenvalues are denoted λmin(A), λmax(A). We let Sn
++ and Sn
+++ represent the cones
+of symmetric positive semidefinite, and symmetric positive definite matrices, respectively. For A, B ∈ Sn,
+we write A ⪰ B (A ≻ B) to indicate that the matrix A − B is symmetric positive semidefinite (symmetric
+positive definite), i.e., A − B ∈ Sn
++ (A − B ∈ Sn
+++). The matrix exponential exp(A), which is defined by the
+power series
+exp(A) = I + A + 1
+2!A2 + 1
+3!A3 + · · · ,
+maps symmetric matrices to the space of symmetric positive definite matrices. Given the spectral decompo-
+sition A = V ΛV ⊤, then exp(A) = V exp(Λ)V ⊤, where exp(Λ) = diag(exp(Λ11), exp(Λ22) . . . , exp(Λnn)).
+We let A ◦ B denote the Hadamard (or element-wise) product of two matrices, and A ⊗ B denotes their
+tensor product. Later in this work, we make use of the following facts regarding Hadamard products.
+Lemma 1 (Lemma 5.1.4 in [33]). Let E, F and G be m × n matrices. Then, the i-th diagonal entry of the
+matrix (E ◦ F)G⊤ coincides with the i-th diagonal entry of the matrix (E ◦ G)F ⊤. That is,
+[(E ◦ F)G⊤)]ii = [(E ◦ G)F ⊤)]ii
+∀i ∈ [m].
+Lemma 2 (Theorem 5.3.4 in [33]). Let A and B be n×n Hermitian matrices. If A ∈ Sn
++, then any eigenvalue
+λ(A ◦ B) of A ◦ B satisfies
+λmin(A) · λmin(B) ≤ λ(A ◦ B) ≤ λmax(A) · λmax(B).
+Corollary 1. Suppose A and B are n × n Hermitian matrices. If A ∈ Sn
++ and B ∈ Sn with B having at
+least one negative eigenvalue, then
+λmax(A) · λmin(B) ≤ λmin(A ◦ B).
+Proof. First, note that by Lemma 2, we have λmin(A ◦ B) ≥ λmin(A) · λmin(B). Moreover, since B has at
+least one negative eigenvalue, we have λmin(B) < 0, which combined with the fact that λmin(A) ≥ 0 yields
+λmax(A) · λmin(B) ≤ λmin(A) · λmin(B). Hence,
+λmax(A) · λmin(B) ≤ λmin(A ◦ B),
+and the proof is complete.
+6
+
+We write e to refer to the vector of all ones in Rn, and use the notation ei to refer to the i-th unit
+vector in the standard orthonormal basis {e1, . . . , en} for Rn. Analogously, the computational basis states
+are denoted by |i⟩ for i ∈ [n]. Hence, for x ∈ Rn, we denote its amplitude encoding by |x⟩, defined as
+|x⟩ =
+1
+∥x∥
+�
+i∈[n]
+xi |i⟩ .
+Observe that |x⟩ is a log(n)-qubit state; for simplicity, we assume that the dimensions of all spaces are powers
+of 2. All logarithms are base 2.
+Where appropriate, our analysis makes use of the Schatten p-norm, defined for a bounded linear operator
+A as
+∥A∥p = [tr (|A|p)]
+1
+p ,
+where |A| = (A†A)
+1
+2 with A† denoting the conjugate transpose of A. Notice that the trace and operator
+norms ∥ · ∥tr and ∥ · ∥ are the Schatten-1 and Schatten-∞ norms, respectively, and the Frobenius norm ∥ · ∥F
+corresponds to the Schatten-2 norm.
+For a scalar x ∈ R define the sign function sign(x) as
+sign(x) =
+
+
+
+
+
+−1
+if x < 0
+0
+if x = 0
+1
+if x > 0.
+When x ∈ Rn, sign(x) = (sign(x1), . . . , sign(xn))⊤.
+For any positive integer q, and binary strings j, k ∈ {0, 1}q, we denote by j ⊕ k the bitwise modulo 2
+addition of q-digit strings, defined as
+j ⊕ k = h
+where h ∈ {0, 1}q is the bitstring whose elements hp are defined for p ∈ [q] as
+hp =
+�
+0
+if jp = kp,
+1
+otherwise.
+“Big-O” notation
+We define O(·) as
+f(x) = O(g(x)) ⇐⇒ ∃ℓ ∈ R, c ∈ R+, such that f(x) ≤ cg(x)
+∀x > ℓ.
+We write f(x) = Ω(g(x)) ⇐⇒ g(x) = O(f(x)). We also define �
+O(f(x)) = O(f(x)·polylog(f(x))) and when
+the function depends poly-logarithmically on other variables we write
+�
+Oa,b (f(x)) = O(f(x) · polylog(a, b, f(x))).
+2.1
+Input models and subroutines
+For our quantum algorithm, we provide analyses for two distinct models of input. One model considers
+a quantum-read/classical-write RAM (QRAM), and the other is the sparse-access model, which we use to
+bound the running time without access to QRAM.
+7
+
+2.1.1
+Sparse-access model
+In the sparse-access model, the input matrix C is assumed to be s-row sparse for some known bound s ∈ [n].
+In other words, C has at most s nonzero entries per row. The sparse-access model is closely related to the
+classical notion, in that we assume access to an oracle Osparse, which upon being queried with input (i, j)
+returns the index of the j-th nonzero entry of the i-th row of C by calculating the index function:
+index : [n] × [s] → [n].
+That is, for i ∈ [n] and j ∈ [s], Osparse computes the position in place:
+Osparse |i, j⟩ = |i, index(i, j)⟩ .
+We also assume access to an oracle that returns a bitstring representation of the individual entries of the
+normalized cost matrix C∥C∥−1
+F
+for every i, j ∈ [n]:
+OC |i, j, z⟩ =
+��i, j, z ⊕ (Cij∥C∥−1
+F )
+�
+.
+2.1.2
+Quantum random access memory
+We consider a quantum-read/classical-write RAM (QRAM), which enables us to store classical data that
+our quantum algorithms can make oracle calls to. Note that while the QRAM we consider does not need to
+be able to store a quantum state, the data is addressable in a superposition. Accessing a QRAM of size n
+requires O(n) gates [5, 24], however, one can arrange these gates in parallel in order to ensure that the circuit
+depth remains O(polylog(n)). Therefore, when we analyze the complexity of our quantum algorithms, we
+make the standard assumption that the cost of accessing a QRAM of size n is O(polylog(n)).
+The next result from Chakraborty et al. [12], is adapted from an earlier result of Kerenidis and Prakash
+[38] and summarizes the aspects of the data structure we utilize.
+Theorem 1 (Theorem 1 in [12]). (Implementing quantum operators using an efficient data structure) Let
+A ∈ Rm×n be a matrix. If w is the number of non-zero entries of A, then there exists a data structure
+of size O
+�
+w log2(mn)
+�
+that, given the entries (i, j, Aij) in an arbitrary order, stores them such that time
+taken to store each entry of A is O(log(mn)). Once this data structure has been initiated with all non-zero
+entries of A, there exists a quantum algorithm that can perform the following maps with ξ-precision in time
+O
+�
+polylog
+�
+mn
+ξ
+��
+:
+�U : |i⟩ |0⟩ �→ |i⟩
+1
+∥Ai,·∥
+n
+�
+j=1
+Aij |j⟩ = |i, Ai,·⟩ ,
+�V : |0⟩ |j⟩ �→
+1
+∥A∥F
+m
+�
+i=1
+∥Ai,·∥ |i⟩ |j⟩ =
+��� �A, j
+�
+,
+where |Ai,·⟩ is the normalized quantum state corresponding to the i-th row of A and
+��� �A
+�
+is a normalized
+quantum state such that ⟨i| �A⟩ = ∥Ai,·∥, i.e., the norm of the i-th row of A.
+2.1.3
+Working with block-encoded matrices
+We now give a formal definition of a block-encoding from [12].
+Definition 1 (Block-encoding). Let A ∈ C2w×2w be a w-qubit operator. Then, a (w + a)-qubit unitary U is
+an (α, a, ξ)-block-encoding of A if U =
+� �A
+·
+·
+·
+�
+, with the property that
+∥α �A − A∥ ≤ ξ.
+8
+
+It was shown by Kerenidis and Prakash [38] and Chakraborty et al. [12] how to efficiently implement
+block-encodings of matrices that are stored in a QRAM data structure, which is formalized in the next result.
+Lemma 3 (Lemma 3.3.7 in [22]). Let A ∈ C2w×2w and ξ > 0.
+(i) Fix q ∈ [0, 2] and define µq(A) =
+�
+nq(A)n(2−q)(A⊤) where nq(A) = maxi ∥Ai,·∥q
+q is the q-th power of
+the maximum q-norm of the rows of A. Defining A{q} to be the matrix with elements A{q}
+ij
+=
+�
+Aq
+ij, if
+A{q} and (A{2−q})† are both stored in QRAM data structures, then there exist unitaries UR and UL that
+can be implemented in time O(poly(w log 1
+ξ )) and such that U †
+RUL is a (µq(A), w + 2, ξ)-block-encoding
+of A.
+(ii) If A is stored in a QRAM data structure, then there exist unitaries UR and UL that can be implemented
+in time O(poly(w log 1
+ξ )) and such that U †
+RUL is an (∥A∥F , w + 2, ξ)-block-encoding of A.
+Linear combinations of block-encodings can also be constructed at cost that is merely logarithmic in the
+dimension.
+Definition 2 (Definition 3.3.8 in [22]). (State preparation pair) Let y ∈ Cm and ∥y∥1 ≤ β.
+The pair
+of unitaries (PL, PR) is called a (β, p, ξ)-state-preparation-pair if PL |0⟩⊗p = �2p−1
+j=0 cj |j⟩ and PR |0⟩⊗p =
+�2p−1
+j=1 dj |j⟩ such that �m−1
+j=0 |β(c∗
+jdj) − yj| ≤ ξ and for all j ∈ m, . . . , 2p − 1 we have c∗
+jdj = 0.
+Proposition 1 (Lemma 52 in [23]). (Linear combination of block-encoded matrices, with weights given
+by a state preparation pair) Let A = �m−1
+j=0 yjAj be a w-qubit operator, where Aj are matrices. Suppose
+PL, PR is a (β, p, ξ1)-state-preparation pair for y, W = �m−1
+j=0 |j⟩ ⟨j| ⊗ Uj + ((I − �m−1
+j=0 |j⟩ ⟨j|) ⊗ Ia ⊗ Is)
+is an (w + a + p)-qubit unitary with the property that Uj is an (α, a, ξ2)-block-encoding of Aj. Then we can
+implement a (αβ, a + p, αξ1 + αβξ2)-block-encoding of A with a single use of W, PR and P †
+L.
+It turns out that the sparse-access model reduces to the quantum operator model upon choosing α = s (if
+row and column sparsity are the same). The next result from [23] describes how to implement block-encodings
+using the sparse-access input model, and the associated costs.
+Lemma 4 (Lemma 48 in [23]). Let A ∈ C2w×2w be a matrix that is sr-row-sparse and sc-column-sparse, and
+each element of A has absolute value at most 1. Suppose that we have access to the following sparse-access
+oracles acting on two (w + 1) qubit registers:
+Or : |i⟩ |k⟩ �→ |i⟩ |rik⟩
+∀i ∈ [2w] − 1, k ∈ [sr], and
+Oc : |ℓ⟩ |j⟩ �→ |cℓj⟩ |j⟩
+∀ℓ ∈ [sc], j ∈ [2w] − 1, where
+rij is the index for the j-th non-zero entry of the i-th row of A, or if there are less than i non-zero entries,
+then it is j + 2w, and similarly cij is the index for the i-th non-zero entry of the j-th column of A, or if there
+are less than j non-zero entries, then it is i + 2w. Additionally, assume that we have access to an oracle OA
+that returns the entries of A in a binary description:
+OA : |i⟩ |j⟩ |0⟩⊗p �→ |i⟩ |j⟩ |aij⟩ ,
+∀i, j ∈ [2w] − 1,
+where aij is a p-bit binary description of the ij-matrix element of A. Then, we can implement a (√srsc, w +
+3, ξ)-block-encoding of A with a single use of Or, Oc and two uses of OA, and additionally using O
+�
+w + log2.5 �
+srsc
+ξ
+��
+one and two qubit gates while using O
+�
+p + log2.5 �
+srsc
+ξ
+��
+ancilla qubits.
+The block-encoding framework will be useful in speeding up the overall running time found in [10], as it
+allows us to perform matrix computations and Hamiltonian simulation efficiently.
+9
+
+Theorem 2 (Corollary 3.4.7 in [22]). (Optimal block-Hamiltonian simulation) Suppose that U is an (α, a, ξ/|2t|)-
+block-encoding of the Hamiltonian H. Then, we can implement a ξ-precise Hamiltonian simulation unitary V
+which is an (1, a + 2, ξ)-block-encoding of eitH, with O
+�
+|αt| +
+log(1/ξ)
+log log(1/ξ)
+�
+uses of controlled-U or its inverse
+and with O
+�
+a|αt| + a
+log(1/ξ)
+log log(1/ξ)
+�
+two-qubit gates.
+Additionally, one can easily take the product of block-encodings.
+Proposition 2 (Lemma 4 in [12]). (Product of block-encoded matrices) If UA is an (α1, a1, ξA)-block-encoding
+of an s-qubit operator A, and UB is an (α2, a2, ξB)-block-encoding of an s-qubit operator B, then (Ia2 ⊗
+UA)(Ia1 ⊗ UB) is an (α1α2, a1 + a2, α1ξB + α2ξA)-block-encoding of AB.
+Relevant to our work in the quantum operator input model is the idea of block-encoding the Hadamard,
+or element-wise product of two matrices. We will demonstrate how one can carry out the Hadamard product
+of block-encodings of matrices A and B as a reduction of the Kronecker product of block-encodings, which
+is straightforward to construct given block encodings of A and B.
+Proposition 3. (Kronecker product of block-encoded matrices) Suppose that UA is an (α1, a1, ξA)-block-
+encoding of A ∈ Rn×n, and UB is an (α2, a2, ξB)-block-encoding of B ∈ Rn×n. Then, taking the tensor
+product of UA and UB, we obtain a (α1α2, a1 + a2, ξA + ξB)-block-encoding of A ⊗ B.
+We do not give a formal proof here as the result directly follows from the definition of a block-encoding;
+to obtain the tensor product of two block-encoded matrices, it suffices to take the tensor product of their
+block-encodings while keeping the ancilla qubits separate.
+Proposition 4 (Hadamard product of block-encoded matrices). Suppose that UA is an (α1, a1, ξA)-block-
+encoding of A ∈ Rn×n, and UB is a (α2, a2, ξB)-block-encoding B ∈ Rn×n. Then, using UA and UB, we can
+implement an (α1α2, a1 + a2 + 8 log(n) + 12, 5(ξA + ξB))-block-encoding of A ◦ B using one application of UA
+and UB, and �On(1) additional gates.
+Proof. First, note that
+A ◦ B = (A ⊗ B)[ιA, ιB],
+where ιA = ιB = {1, n + 2, 2n + 3, . . . , n2} are index sets of cardinality n (see, e.g., Lemma 5.1.1 in [33]).
+Our goal is to use the index sets ιA and ιB along with a block encoding of A ⊗ B to construct a unitary
+which block-encodes M ∈ Rn2×n2, a matrix which contains the elements of A◦B in its upper left-most n×n
+block, while all other entries are 0:
+Mij =
+�
+Aij · Bij
+for i, j = 1, . . . , n,
+0
+otherwise,
+i.e.,
+M =
+�
+A ◦ B
+0n×(n2−n)
+0(n2−n)×n
+0(n2−n)×(n2−n)
+�
+.
+We will first show how one can use ιA and ιB to construct sparse matrices that map A ⊗ B to M, and then
+subsequently analyze the cost of constructing the corresponding unitary block-encoding.
+Consider the matrix Z ∈ Rn2×n2, whose elements are defined as
+Zij =
+�
+1
+if i = j = (k − 1)n + k,
+k = 1, . . . , n,
+0
+otherwise.
+Multiplying A ⊗ B on the left by Z sets the rows of A ⊗ B which do not contain elements of A ◦ B to zero,
+and subsequently multiplying Z(A ⊗ B) on the right by Z will set the columns of Z(A ⊗ B) which do not
+10
+
+appear in A ◦ B to zero. As a result, a block-encoding of Z(A ⊗ B)Z corresponds to block-encoding A ⊗ B,
+and setting all terms not appearing in A ◦ B to zero:
+[Z(A ⊗ B)Z]ij =
+�
+[A ⊗ B]ij
+if i = (k − 1)n + k and j = (ℓ − 1)n + ℓ
+k, ℓ = 1, . . . , n,
+0
+otherwise.
+Next, let G ∈ Rn2×n2 be a matrix whose elements are defined as follows:
+Gij =
+
+
+
+
+
+1
+if i ∈ [n2] and i = j = (k − 1)n + k,
+k = 1, . . . , n,
+1
+if i ∈ [n2] \ {1, n + 2, 2n + 3, . . . , n2} and j = (i − 1)n + i,
+0
+otherwise.
+We will now establish that GZ(A⊗B)Z)G⊤ is precisely the matrix we seek to block-encode, by demonstrating
+that G(Z(A⊗B)Z)G⊤ = M. First, observe that G is a (partial) permutation matrix: multiplying Z(A⊗B)Z
+on the left by G performs the necessary row-exchanges, as the elements of G(Z(A ⊗ B)Z) are given by
+[G (Z(A ⊗ B)Z)]ik =
+�
+Aij · Bij
+for k = (j − 1)n + j,
+i, j = 1, . . . , n,
+0
+otherwise.
+On the other hand, multiplying Z(A ⊗ B)Z) on the right by G⊤ performs this transformation with respect
+to the columns such that
+[(Z(A ⊗ B)Z) G]kj =
+�
+Aij · Bij
+for k = (i − 1)n + i,
+i, j = 1, . . . , n,
+0
+otherwise.
+Hence, multiplying G (Z(A ⊗ B)Z) on the right by G⊤ conducts the column exchanges to move A◦ B to the
+top left n-dimensional block of Z(A ⊗ B)Z, i.e.,
+[G (Z(A ⊗ B)Z) G]ij =
+�
+Aij · Bij
+for i, j = 1, . . . , n,
+0
+otherwise.
+Therefore, G(Z(A ⊗ B)Z)G⊤ = M as desired.
+We now analyze the cost associated with block-encoding M.
+Under the stated hypothesis, we have
+access to an (α1, a1, ξA)-block-encoding UA of A, and an (α2, a2, ξB)-block-encoding UB of B, and thus
+applying Proposition 3 we can construct an (α1α2, a1 + a2, ξA + ξB)-block-encoding UA⊗B of A ⊗ B using
+one application of UA and of UB, and no additional gates.
+Using the description of Z, we can construct the sparse-access oracles Or and Oc as defined in Lemma 4
+(which act on two (2 log n + 1) qubit registers). Additionally, from the definition of Z, we can construct an
+oracle OZ, which returns the entries of Z in a binary description:
+OZ : |i⟩ |j⟩ |0⟩⊗p �→ |i⟩ |j⟩ |zij⟩ ,
+∀i, j ∈ [22 log n] − 1,
+where zij is a p-bit binary description of the ij-matrix element of Z. Note that the circuit for the position
+and value of the nonzero elements of Z using �
+On(1) gates because they admit an efficient description: their
+value is 1 and we have a compact description of their position. By construction the matrix Z is 1-row sparse
+and 1-column sparse, and hence an application of Lemma 4 with sr = sc = 1 asserts that one can construct a
+(1, 2 log(n)+3, ξZ)-block-encoding UZ of Z. Given block-encodings UZ and UA⊗B, we can apply Proposition
+2 with
+ξZ = ξA + ξB
+α1α2
+,
+ξA⊗B = ξA + ξB,
+yielding an (α1α2, a1 + a2 + 2 log(n) + 3, 2(ξA + ξB))-block-encoding of Z(A ⊗ B). Applying Proposition 2
+once more with
+ξZ = ξA + ξB
+α1α2
+,
+ξZ(A⊗B) = 2(ξA + ξB),
+11
+
+we obtain an (α1α2, a1 + a2 + 4 log(n) + 6, 3(ξA + ξB))-block-encoding of Z(A ⊗ B)Z.
+Just as was the case with Z, we can use the description of G to construct the sparse-access oracles Or
+and Oc as defined in Lemma 4 (which again, act on two (2 log n + 1) qubit registers), as well as an oracle
+OG using �
+On(1) gates, that returns the entries of G in a binary description:
+OG : |i⟩ |j⟩ |0⟩⊗p �→ |i⟩ |j⟩ |gij⟩ ,
+∀i, j ∈ [22 log n] − 1,
+where gij is a p-bit binary description of Gij (the ij-matrix element of G). Noting that G is 1-row sparse
+and 1-column sparse (and hence, so its transpose); applying Lemma 4 twice more allows us to construct
+a (1, 2 log(n) + 3, ξG)-block-encoding UG of G, as well as a (1, 2 log(n) + 3, ξ⊤
+G)-block-encoding UG⊤ of the
+transpose G⊤. We can then use UG and our (α1α2, a1+a2+4 log(n)+6, 3(ξA+ξB))-block-encoding UZ(A⊗B)Z
+of Z(A ⊗ B)Z to construct an (α1α2, a1 + a2 + 6 log(n) + 9, 4(ξA + ξB))-block-encoding of G(Z(A ⊗ B)Z)
+by applying Proposition 2 with
+ξG = ξA + ξB
+α1α2
+,
+ξZ(A⊗B)Z = 3(ξA + ξB).
+Applying Proposition 2 a final time, with
+ξG⊤ = ξA + ξB
+α1α2
+,
+ξG(Z(A⊗B)Z) = 4(ξA + ξB),
+produces an (α1α2, a1 + a2 + 8 log(n) + 12, 5(ξA + ξB))-block-encoding UM of M = G(Z(A ⊗ B)Z)G⊤.
+The stated complexity result follows upon noting that the steps required to construct the unitary
+UM = UGUZUA⊗BUZUG⊤
+requires one application of UA⊗B and one application of each of the other matrices. In turn, this amounts
+to 1 application of UA and UB each, plus the �On(1) gate cost of the remaining matrices UG, UZ and UG⊤,
+and the proof is complete.
+We remark that a similar result to Proposition 4 was independently derived and discussed in the recent
+paper [13].
+2.1.4
+Gibbs Samplers and Trace Estimators
+For clarity, we begin with a formal definition of a subnormalized density operators and their purifications.
+Definition 3 (Definition 6.3.1 in [22]). (Subnormalized density operators & Purification) A subnormalized
+density operator ρ is a positive semidefinite matrix of trace at most 1. A purification ̺ of a subnormalized
+density operator ρ is a 3-register pure state such that tracing out the third register and projecting on the
+subspace where the second register is |0⟩ yields ρ.
+The frameworks introduced later in this paper require that we implement a Gibbs sampler and a trace
+estimator, which we define next.
+Definition 4 (Definition 4.11 in [58]). (Gibbs Sampler) A θ-precise Gibbs-sampler for the input matrix H,
+is a unitary that takes as input a data structure storing a Hamiltonian H and creates as output a purification
+of a θ-approximation (in trace distance) of the Gibbs state
+ρ =
+exp(−H)
+tr(exp(−H)).
+We will use these approximate Gibbs states in order to check the diagonal entries of our solutions, as well
+as compute the trace inner products of matrices (or, expectation values), i.e., quantities of the form tr(Aρ).
+12
+
+Definition 5 (Definition 4.12 in [58]). (Trace Estimator) A θ-precise trace estimator is a unitary that as
+input takes a state ρ and a matrix A. It outputs a sample from a random variable x ∈ R such that x is an
+estimator for tr(Aρ) that is at most θ/4 biased.
+These implementations require polynomial approximations of the exponential function, which can be
+obtained using quantum singular value transformation techniques introduced in [22, 23].
+Lemma 5 (Lemma 4.14 in [58]). Let ξ ∈ (0, 1/6] and β ≥ 1. There exists a polynomial P(x) such that
+• For all x ∈ [−1, 0], we have |P(x) − exp(2βx)/4| ≤ ξ.
+• For all x ∈ [−1, 1], we have |P(x)| ≤ 1/2.
+• deg(P) = �O 1
+ξ (β).
+Lemma 6 (Lemma 4.15 in [58]). Let θ ∈ (0, 1/3], β > 1, and let d be the degree of the polynomial from Lemma
+5 when we let ξ =
+θ
+128n. Let U be a (β, a,
+θ2β
+10242d2n2 )-block-encoding of a Hermitian operator H ∈ Rn×n, i.e,,
+a (β, a, �
+O(θ/βn2))-block-encoding. Then, we can create a purification of a state ˜ρ such that
+����˜ρ −
+exp(H)
+tr (exp(H))
+����
+tr
+≤ θ
+using �O 1
+θ (√nβ) applications of U and �O 1
+θ (√nβa) elementary operations.
+Provided access to a unitary that prepares a purification of a density operator, we can also construct a
+block-encoding of it. This is formalized in the following lemma from [22], which was based on ideas found
+in [45, Corollary 9].
+Lemma 7 (Lemma 6.4.4 in [22]). (Block-encoding of a (subnormalized) density operator) Let G be a (w +a)
+unitary which on the input state |0⟩w |0⟩a prepares a purification |̺⟩ of the subnormalized w-qubit density
+operator ρ. Then we can implement a (1, w + a, 0)-block-encoding of ρ with a single use of G and its inverse
+and with w + 1 two-qubit gates.
+We are now in a position to define a trace estimator using the quantum operator input model.
+Lemma 8 (Lemma 4.18 in [58]). Let ρ be an n-dimensional quantum state and U an (α, a, θ/2)-block-
+encoding of a matrix A ∈ Rn×n with ∥A∥ ≤ 1.
+A trace estimator for tr (Aρ) with bias at most θ and
+σ = O(1) can be implemented using �
+O(α) uses of U and U † and �
+O 1
+θ (α) elementary operations.
+2.1.5
+Computational complexity
+When discussing the computational complexity of quantum algorithms we generally express the cost in terms
+of the number of calls to some input oracles. Unless otherwise specified, the gate complexity is at most a
+poly-logarithmic factor larger than the stated oracle complexity. The meaning of “input oracles access”
+depends on the input model:
+• For the sparse-oracle access model, it refers to a query to the oracle describing C/∥C∥F.
+• For the QRAM model, it refers to the number of accesses to QRAM. A QRAM of size O
+�
+ns log2(n)
+�
+is sufficient for our algorithms, and in particular, we only need classical write access to the QRAM,
+i.e., we do not write in superposition.
+It is straightforward to translate each of these oracle costs into a running time in the gate model, by
+considering the cost of implementing each oracle.
+13
+
+3
+Hamiltonian Updates
+In this section, we present the algorithm from [10] and relevant results required to prove its convergence and
+analyze its cost.
+3.1
+Convex Feasibility Problems
+In order to avoid any normalization issues for the problems that arise over the course of our IR scheme,
+we deviate slightly from [10] and renormalize the problem (3) using the Frobenius norm of the cost matrix
+rather than use its operator norm:
+find
+X
+s.t.
+tr
+�
+C
+∥C∥F
+X
+�
+≥ γ − ǫ
+�
+i∈[n]
+����⟨i|X|i⟩ − 1
+n
+���� ≤ ǫ
+tr (X) = 1,
+X ⪰ 0.
+(5)
+The relaxed renormalized SDO problem (5) is a specific example of the convex optimization problem
+max
+f(X)
+s.t.
+X ∈ P1 ∩ P2 ∩ · · · ∩ Pm,
+tr(X) = 1, X ⪰ 0,
+(6)
+where P1, . . . , Pm are convex sets.
+In this context, the trace constraint enforces normalization, but also allows us to obtain a bound on
+the optimal objective value. Letting �C = C∥C∥−1
+F
+and invoking the tracial matrix H¨older inequality [8], it
+follows that any X∗ that solves (6) satisfies the following relation:
+���tr( �CX∗)
+��� ≤ ∥ �C∥∥X∗∥tr = ∥ �C∥.
+It is well known in the optimization literature that performing binary search over the range of values
+γ ∈
+�
+−∥ �C∥, ∥ �C∥
+�
+⊆ [−1, 1]
+that the objective can take reduces the task of solving (6) to solving a sequence of feasibility problems of
+the form:
+find
+X ∈ Sn
++ ∩ {X : tr(X) = 1}
+s.t.
+tr( �CX) ≥ γ
+X ∈ P1 ∩ P2 ∩ · · · ∩ Pm.
+(7)
+In particular, log(∥ �C∥ǫ−1) queries to (7) are sufficient to estimate the optimal objective value of of (6) up
+to additive error ǫ.
+3.2
+Solving Convex Feasibility Problems via Hamiltonian Updates
+Hamiltonian Updates (HU) is a meta-algorithm for solving convex feasibility problems of the form (7), which
+can be viewed as an adaptation of the work of Tsdua, R¨atsch and Warmuth [57] as well as [4, 9, 32, 42].
+At a high level, HU can be viewed as a mirror descent algorithm with von Neumann entropy as the mirror
+map. In each iteration, the method makes use of subroutines that can be used to test ǫ-closeness to convex
+sets P1, P2, . . . , Pm, which we formally define next.
+14
+
+Definition 6 (Definition 2.1 in [10]). Let P ⊂ {X ∈ Sn
++ : tr(X) = 1} be a closed, convex subset of quantum
+states, and �P ⊂ {X ∈ Cn×n : X = X†, ∥X∥ ≤ 1} be a closed, convex subset of observables of operator norm
+at most 1. For ǫ > 0, an ǫ-separation oracle with respect to �P is a subroutine that either accepts a state ρ
+(in the sense that observables from �P cannot distinguish ρ from the elements of P), or provides a normal
+vector (in the matrix space) P of a hyperplane that separates ρ from the set P using a test from �P:
+OP,ǫ(ρ) =
+�
+accept ρ
+if minY ∈P maxP ∈ �
+P tr(P(ρ − Y )) ≤ ǫ,
+output P ∈ �P s.t. tr(P(ρ − Y )) ≥ ǫ
+2 for all Y ∈ P
+otherwise.
+The authors in [10] point out that the above oracle construction is well defined, as we can always choose
+some hyperplane P ∈ �P such that
+tr (P(ρ − Y )) ≥ ǫ
+2,
+holds for all Y ∈ P whenever
+min
+Y ∈P max
+P ∈ �
+P
+tr(P(ρ − Y )) > ǫ.
+From Sion’s min-max theorem [54], it follows that
+max
+P ∈ �
+P
+min
+Y ∈P tr(P(ρ − Y )) = min
+Y ∈P max
+P ∈ �
+P
+tr(P(ρ − Y )) > ǫ,
+and hence there exists a hyperplane which separates ρ from P by ǫ. By relaxing the requirement to
+ǫ
+2-
+separation, the algorithm is able to reconcile with the errors that result from approximating quantities
+computed with ρ, or estimating its entries.
+The Hamiltonian Updates (HU) algorithm of Brand˜ao et al. [10] is provided in full detail in Algorithm
+1. The algorithm takes as input the precision parameter ǫ, and m ǫ-separation oracles O1,ǫ, O2,ǫ, . . . , Om,ǫ.
+In the initialization steps, the starting point is defined to be the maximally mixed state ρ ← n−1I. This is
+critical to ensuring the convergence of mirror descent-based approaches such as Algorithm 1 and the works
+in [4, 9, 32, 42, 57]; initialization to the maximally mixed state ensures that the quantum relative entropy
+between any feasible state and the initial state is bounded by log(n) (see, e.g., Theorem 11.8 pt. 2 [51]), and
+is reduced at every iteration. Consequently, Algorithm 1 terminates in a finite number of iterations.
+As noted in [10], how we define �P determines the number of closeness conditions that need to be tested.
+By using the Gibbs state change of variables, we do not need to test if our candidate solution is trace
+normalized or positive semidefinite; any Gibbs state
+ρH =
+exp(−H)
+tr(exp(−H))
+is an element of the set {X ∈ Sn
++ : tr(X) = 1} by definition.
+Our task therefore reduces to finding a
+log(n)-qubit mixed state state ρ which is ǫ-close to the convex sets Pi that arise from any other constraints
+included in the feasibility problem. At each iteration, ǫ-closeness is tested by querying ǫ-separation oracles
+which are constructed using observables in �Pi. If each of our oracles accepts the candidate state, the algorithm
+terminates and reports (ρ, H) as an ǫ-precise solution. Otherwise, upon detecting infeasibility the matrix
+exponent is updated to penalize the infeasible directions using the rule
+H ← H + ǫ
+16P,
+where P is a normal vector in the matrix space of a hyperplane that witnesses infeasibility.
+15
+
+Algorithm 1 Hamiltonian Updates for Convex Feasibility Problems
+Input: Query access to m ǫ-separation oracles O1,ǫ(·), . . . , Om,ǫ(·)
+Initialize ρ ← n−1I and H ← 0n×n
+for t = 1, . . . , T do
+for i = 1, . . . , m do
+if Oi,ǫ(ρ) = P then
+H ← H +
+ǫ
+16P
+ρ ←
+exp(−H)
+tr(exp(−H))
+break
+end
+end
+return (ρ, H) and exit
+end
+The following result establishes the iteration complexity of Algorithm 1.
+Theorem 3 (Theorem 2.1 in [10]). Algorithm 1 requires at most T = ⌈64 log(n)ǫ−2⌉ + 1 iterations to certify
+that (7) is infeasible or output a state ρ satisfying
+for all 1 ≤ i ≤ m :
+max
+Pi∈ �
+Pi
+min
+Yi∈Pi tr(Pi(ρ − Yi)) ≤ ǫ.
+Note that Theorem 3 applies to any convex feasibility problem (on density operators, i.e., trace-normalized
+positive semidefinite matrices) for which we have separation oracles as outlined in Definition 6. This is crucial
+for the development of an iterative refinement scheme.
+There is an important distinction with respect to output across the models of computation we study.
+A classical implementation of Algorithm 1 outputs an explicit description of an ǫ-precise solution ρ∗ to (5)
+and its associated Hamiltonian H∗, whereas a quantum implementation reports a real valued vector y ∈ R2
+along with a diagonal matrix D (with ∥D∥ ≤ 1) such that H∗ = y1
+C
+∥C∥F + y2D. The vector y = (y1, y2)⊤ is
+the state preparation pair of ρ∗, in particular:
+ρ∗ =
+exp
+�
+−
+�
+y1
+C
+∥C∥F + y2D
+��
+tr
+�
+exp
+�
+−
+�
+y1
+C
+∥C∥F + y2D
+���,
+and we refer to this type of output as a state preparation pair description of ρ. This choice of output is
+used in all quantum SDO solvers based on Gibbs sampling techniques (see, e.g., [9, 10, 11, 60, 61]), and is
+motivated by the fact that it is difficult to develop quantum algorithms that are substantially faster than
+classical algorithms if we still have to output each entry of the solution (an n × n matrix).
+The Gibbs sampling approaches that we apply later exhibit a cost that depends on a norm bound for y.
+Observe that we initialize y to the all zeros vector of appropriate dimension, and in every iteration, at most
+one entry of y changes by a magnitude of
+ǫ
+16 (specifically, an entry yi, where the oracle Oi,ǫ has detected
+infeasibility). As a consequence, the vector y satisfies the inequality
+���y(t+1) − y(t)��� ≤ ǫ
+16
+(8)
+for each iteration t. In view of the iteration bound for Algorithm 1 provided in Theorem 3, it is easy to see
+that for any y obtained from Algorithm 1 we have
+∥y∥1 ≤ ⌈64 log(n)ǫ−2⌉
+���y(t+1) − y(t)��� ≤ ⌈64 log(n)ǫ−2⌉ ǫ
+16 ≤ 4 log(n)ǫ−1.
+(9)
+To instantiate the algorithm to solve problem (3) we need to choose the sets Pi, and provide separation
+oracles for them. This is what we do in the following section.
+16
+
+3.2.1
+Oracle Construction
+The goal of Hamiltonian Updates is to solve, for fixed γ ∈ [−1, 1], the following feasibility problem:
+find
+ρ ∈ {X ∈ Sn
++ : tr(X) = 1} ∩ Cγ ∩ Dn
+where
+Cγ =
+�
+X : tr
+�
+C
+∥C∥F
+X
+�
+≥ γ
+�
+,
+Dn =
+�
+X : ⟨i|X|i⟩ = 1
+n, i ∈ [n]
+�
+.
+(10)
+One can observe that the set Cγ constitutes a halfspace, while Dn is an affine space of codimension n. The
+sets of observables for Cγ and Dn are given by �Cγ and �Dn respectively, with
+�Cγ = {−C∥C∥−1
+F }, and �Dn = {D ∈ Rn×n : ∥D∥ ≤ 1, D is diagonal}.
+As noted in [10], it follows
+max
+P ∈ �
+Cγ
+min
+Y ∈Cγ tr(P(ρ − Y )) ≤ ǫ ⇐⇒ − tr
+�
+C∥C∥−1
+F (ρ − Y )
+�
+≤ ǫ
+for some Y ∈ Cγ,
+which in turn implies tr (C∥C∥−1
+F ρ) ≥ γ − ǫ.
+Given the structure of Cγ and Dn, the authors in [10] suggest the following two separation oracles:
+OCγ : compute an approximation ˜c of tr
+�
+C∥C∥−1
+F ρ
+�
+up to additive error ǫ
+4. Check if ˜c ≥ γ − 3ǫ
+4 and
+output P = −C∥C∥−1
+F
+if the inequality is violated.
+ODn : compute an approximation ˜p ∈ Rn of pi = ⟨i|ρ|i⟩ satisfying
+n
+�
+i=1
+|pi − ˜pi| ≤ ǫ
+4.
+Check if
+n
+�
+i=1
+����˜pi − 1
+n
+���� ≤ 3ǫ
+4 and output P =
+n
+�
+i=1
+�
+I
+�
+˜pi > 1
+n
+�
+− I
+�
+˜pi < 1
+n
+��
+|i⟩ ⟨i|
+if the inequality is violated.
+For any given
+ρH =
+exp(−H)
+tr(exp(−H)),
+the required separation oracles are straightforward to implement on a classical computer that has access to
+ρH. Thus, classically we only need to prepare ρH once and store it to build the separation oracles. The next
+result from [10] establishes that computing an O(log(n)ǫ−1)-degree Taylor series suffices to produce accurate
+approximations.
+Lemma 9 (Lemma 3.2 in [10]). Fix a Hermitian n×n matrix H, an accuracy ǫ, and let ℓ be the smallest even
+number satisfying (ℓ + 1)(log(ℓ + 1) − 1) ≥ 2∥H∥ + log(n) + log
+� 1
+ǫ
+�
+. Then, the truncated matrix exponential
+Tℓ = �ℓ
+k=0
+1
+k!(−H)k satisfies
+����
+exp(−H)
+tr (exp(−H)) −
+Tℓ
+tr(Tℓ)
+����
+tr
+≤ ǫ.
+The task of implementing our separation oracles and testing feasibility on a quantum computer reduces
+to preparing Gibbs states [10], which are subsequently used to test closeness to the sets Cγ and Dn via
+quantum measurements. The next result can be viewed as the quantum analogue to Lemma 9; contrary to
+bounding the number of required Taylor series steps for computing ρ via a matrix exponential, it bounds
+the number of copies of ρ required to estimate its diagonal entries and expectation values tr(Aρ) using the
+QRAM model.
+17
+
+Lemma 10. Fix ǫ ∈ (0, 1). Let ρ be a log(n)-qubit quantum state and U a (1, log(n) + 2, ǫ/(2n))-block-
+encoding of C∥C∥−1
+F . Then, in the QRAM model, we can implement the oracle OCγ on a quantum computer
+given access to O(ǫ−1) copies of a state that is an ǫ
+8-approximation of the input state ρ in trace distance and
+O(ǫ−1) applications of U and U †. The oracle ODn can be implemented using O(nǫ−2) ǫ
+8-approximate copies
+of the input, and the classical post-processing time needed to implement the oracle is O(nǫ−2).
+Proof. First, note that we can obtain an estimate ˜p of the diagonal elements of ρ whose total variation
+distance from p is no more than
+ǫ
+8 using �
+On
+�
+nǫ−2�
+copies of ρ to measure ρ in the computational basis.
+Further, provided accesses to ρ and a (1, log(n) + 2, ǫ/(2n))-block-encoding U of C∥C∥−1
+F , by Lemma 8, a
+trace estimator for tr
+�
+C∥C∥−1
+F ρ
+�
+with bias at most ǫ
+n can be implemented using �
+O(1) uses of U and U † and
+�
+O n
+ǫ (1) elementary operations. From here, applying amplitude estimation using O(ǫ−1) samples from the
+trace estimator to suffice to compute an approximation tr
+�
+C∥C∥−1
+F ρ
+�
+up to additive ǫ
+8 to implement OCγ.
+The rest of the proof exactly follows the proof of [10, Lemma 3.3].
+We remark that in the presence of QRAM, utilizing multidimensional phase estimation techniques from
+[59] improves the dependence on ǫ−1 for estimating the diagonal elements of ρ to linear, which is a factor
+ǫ−1 better than a n¨aive application of computational basis measurements. However, in the context of the
+iterative refinement scheme we present later, the improvement would only reduce the amount of constant
+overhead in the overall running time. There are also numerous ways to prepare Gibbs states using a quantum
+computer [16, 20, 37, 53, 60, 61, 63]. Following [10], we utilize the Gibbs sampler from [53] when working
+with the sparse-access input model, and for the QRAM input model we consider Gibbs sampling techniques
+introduced in [60].
+3.3
+Complexity
+Having understood the cost of constructing the oracles in both the classical and quantum settings, we are
+now in a position to analyze the complexity associated with using Algorithm 1 to obtain solutions to (5)
+and approximations to (3).
+Relevant to this discussion is the following result, which imposes precision
+requirements on solving (3) to an additive error of the order O (n∥C∥F ǫ) using Algorithm 1.
+Proposition 5 (Proposition 3.1 in [10]). Let ρ be an ǫ4-accurate solution to the relaxed SDO problem (5)
+with input matrix C. Let γǫ4 = tr (Cρ) be the value attained by ρ. Then, there is a quantum state ρ∗ at trace
+distance O(ǫ) of ρ such that nρ∗ is a feasible point of SDO problem (3). In particular
+|γǫ4n∥C∥F − tr (nρ∗C)| = O (n∥C∥Fǫ) .
+Moreover, it is possible to construct ρ∗ in time O(n2) given the entries of ρ.
+We do not provide a proof of this result here, as later we will provide an improved approximation
+guarantee and a proof of the improved statement.
+3.3.1
+Classical running time
+Using Lemma 9 in combination with Theorem 3, we can bound the running time required to solve (5) to
+additive error ǫ using a classical implementation of Algorithm 1.
+Proposition 6. Suppose that C has row sparsity s. Then, the classical cost of solving (5) up to additive
+error ǫ using Algorithm 1 is O
+�
+min{n2s, nω} log2(n)ǫ−3�
+.
+Proof. The result follows directly from the proof of Corollary 3.1 in [10], but we repeat the argument here
+for completeness.
+First, observe that over the course of the iterations t = 0, . . . , T , the operator norms ∥H(t)∥ do not
+become prohibitively large. This follows from initializing H(0) = 0n×n, and that by (8), the inequality
+���H(t+1) − H(t)��� ≤ ǫ
+16
+���P (t)��� ≤ ǫ
+16
+18
+
+holds for all t. By Theorem 3, Algorithm 1 requires at most T = ⌈64 log(n)ǫ−2⌉ iterations, which implies
+∥H(t)∥ ≤ 4 log(n)ǫ−1 for all t.
+By Lemma 9, it suffices to compute O(log(n)ǫ−1) steps of the Taylor series corresponding to exp(−H(t))
+in order to obtain a matrix ˜ρ(t) that is at most a trace distance of ǫ
+4 from ρ(t). Moreover, given that H(t)
+is defined as a linear combination of C∥C∥−1
+F
+with a diagonal matrix, matrix multiplication involving H(t)
+can be carried out in O(min{n2s, nω}) arithmetic operations. Given classical access to ˜ρ(t), the diagonal
+constraints comprising Dn can be checked in time O(n), whereas computing tr
+�
+C
+∥C∥F ˜ρ(t)�
+requires O(ns)
+arithmetic operations. Thus, the dominant operation at each iteration is computing the matrix exponential
+and the classical per-iteration cost of Algorithm 1 is given by
+O
+�
+min{n2s, nω} log(n)ǫ−1�
+.
+Taking into account the iteration bound O(log(n)ǫ−2) provided in Theorem 3, we arrive at an overall running
+time of
+O
+�
+min{n2s, nω} log2(n)ǫ−3�
+.
+The proof is complete.
+The next corollary from [10] follows from Proposition 5 in the context of the previous result, and provides
+the overall running time of Algorithm 1 to solve (3) to additive error O (n∥C∥Fǫ) in the classical setting.
+Corollary 2. Suppose that C has row-sparsity s. Then, the classical cost of solving (3) up to an additive
+error O (n∥C∥Fǫ) using Algorithm 1 is O
+�
+min{n2s, nω} log2(n)ǫ−12�
+.
+Proof. By Proposition 6, Algorithm 1 requires time
+O
+�
+min{n2s, nω} log2(n)˜ǫ−3�
+,
+to solve (5) up to additive error ˜ǫ. In order to satisfy the approximation guarantee for (3) given in Proposition
+5, it suffices to solve (5) to error ˜ǫ = ǫ4. Plugging in this value for the precision parameter, the total cost
+required to solve (3) up to an additive error O (n∥C∥Fǫ) using Algorithm 1 is
+O
+�
+min{n2s, nω} log2(n)˜ǫ−3�
+= O
+�
+min{n2s, nω} log2(n)(ǫ4)−3�
+= O
+�
+min{n2s, nω} log2(n)ǫ−12�
+.
+3.3.2
+Quantum running time
+Combining the sampling requirements provided in Lemma 10 with the cost of preparing a single Gibbs state
+and the iteration bound from Theorem 3 gives the complexity of Algorithm 1 when run on a quantum com-
+puter. However, Gibbs samplers based on the block-encoding framework depend only poly-logarithmically
+on the inverse precision, therefore they are exponentially faster (in the parameter ǫ−1) compared to the
+Gibbs sampling algorithm from [53] utilized in [10]. It thus makes sense to analyze the running time in the
+more efficient model. This will require an efficient data structure for storing y so that we can efficiently
+prepare linear combinations of block-encodings.
+Lemma 11 (Lemma 15 in [60]). There is a data structure that can store an m-dimensional χ-sparse vector
+y with θ-precision using a QRAM of size �O m
+θ (χ). Furthermore:
+• Given a classical O(1)-sparse vector, adding it to the stored vector has classical cost �
+O m
+θ (1).
+• Given that β ≥ ∥y∥1, we can implement a (symmetric) (β, �
+O m
+θ (1), θ)-state preparation pair for y with
+�O m
+θ (1) queries to the QRAM.
+19
+
+Corollary 3 (Corollary 16 in [60]). Suppose A1, . . . , Am are Hermitian matrices with operator norm at most
+1, and that y ∈ Rm satisfies ∥y∥1 ≤ β. Having access to the above data structure for y, we can prepare one
+copy of the Gibbs state
+ρ =
+exp (− �m
+i=1 yiAi)
+tr (exp (− �m
+i=1 yiAi))
+using �Oθ(√nαβ) accesses to the data structure for y and block-encodings of A1, . . . , Am.
+We can now use Corollary 3 in combination with results from Sections 2.1.3 and 2.1.4 to establish the
+running time of Algorithm 1 in the QRAM input model.
+Proposition 7. Let
+C
+∥C∥F ∈ Sn be stored in QRAM. Then, the complexity of solving (5) up to additive error
+ǫ with Algorithm 1 using the QRAM input model is
+�
+O n
+ǫ
+�
+n1.5ǫ−5�
+.
+Here, the complexity corresponds to the number of accesses to the QRAM.
+Proof. Given that
+C
+∥C∥F is stored in QRAM, Lemma 3(ii) asserts that when constructing a block-encoding
+of
+C
+∥C∥F , one can set the subnormalization factor to be αC =
+���
+C
+∥C∥F
+���
+F = 1. Hence, one can construct a
+(1, log(n) + 2, ǫ/(2n))-block-encoding of C∥C∥−1
+F
+in time �O n
+ǫ (1).
+Next, recall that in iteration t ∈ [T ] of Algorithm 1, our Hamiltonian is defined as
+H(t) = y(t)
+1
+C
+∥C∥F
++ y(t)
+2 D(t),
+where D(t) is a diagonal matrix with the diagonal entries taking value −1, 0 or 1. Now, the diagonal elements
+of D change in each iteration, and therefore, a new D must be block-encoded in each iteration. For this,
+we can utilize the QRAM model described in Section 2.1.2, which allows for insertions to be made in time
+�
+On(1) to keep the cost of this step negligible. In this case, provided a classical description of D, we can store
+D in the QRAM in time O(n log(n)). Thus, applying Lemma 4, a (1, log(n) + 3, ǫ)-block-encoding of D(t)
+can be constructed in time �O n
+ǫ (1).
+In an earlier discussion we saw that any y obtained from a call to Algorithm 1 will satisfy ∥y∥1 = �On(ǫ−1)
+if we call Algorithm 1 using precision ǫ (see, e.g., equation (9)). Hence, an application of Corollary 3 with
+β = �
+On(ǫ−1) implies that we can prepare one copy of our Gibbs state using
+�
+O n
+ǫ
+�√nαǫ−1�
+accesses to the data structure for y and the block-encodings of C∥C∥−1
+F
+and D, where α is defined as the
+maximum over the subnormalization factors used to block-encode
+C
+∥C∥F and D. Since α = max{αC, αD} = 1,
+it follows
+�
+O n
+ǫ
+�√nαǫ−1�
+= �O n
+ǫ
+�√nǫ−1�
+.
+Now, one can see from Lemma 10 that the cost of constructing ODn dominates that of constructing OCγ.
+Noting that ODn can be implemented using O(nǫ−2) copies of a state that is an
+ǫ
+8-approximation of the
+input state ρ in trace distance and its inverse, the per-iteration cost of Algorithm 1 in the QRAM input
+model is given by
+�
+O n
+ǫ
+�
+n1.5ǫ−3�
+.
+Factoring in the iteration bound of �
+On(ǫ−2) from Theorem 3, it follows that when provided access to QRAM,
+Algorithm 1 solves (5) up to additive error ǫ using
+T quantum
+HU
+= �O n
+ǫ
+�
+n1.5ǫ−5�
+accesses to the QRAM. The proof is complete.
+20
+
+Corollary 4. Let
+C
+∥C∥F ∈ Sn be stored in QRAM. Then, the complexity of solving (3) up to additive error
+O(n∥C∥F ǫ) with Algorithm 1 using the QRAM input model is
+�O n
+ǫ
+�
+n1.5ǫ−20�
+.
+Here, the complexity corresponds to the number of accesses to the QRAM.
+Proof. By Proposition 7, Algorithm 1 requires
+�
+O n
+˜ǫ
+�
+n1.5˜ǫ−5�
+,
+accesses to the QRAM to solve (5) up to additive error ˜ǫ. In order to satisfy the approximation guarantee
+for (3) given in Proposition 5, it suffices to solve (5) to error ˜ǫ = ǫ4. Plugging in this value for the precision
+parameter, the total cost required to solve (3) up to an additive error O (n∥C∥F ǫ) using Algorithm 1 is
+�O n
+˜ǫ
+�
+n1.5˜ǫ−5�
+= �O n
+ǫ
+�
+n1.5(ǫ4)−5�
+= �O n
+ǫ
+�
+n1.5ǫ−20�
+.
+The proof is complete.
+Corollary 4 establishes that utilizing Gibbs samplers and trace estimators based on the block-encoding
+framework for our oracle construction in Algorithm 1 leads to an
+O
+�√s
+1+o(1)ǫ−8+o(1) exp
+�
+1.6
+�
+log(ǫ−4)
+��
+speedup over the running time result provided in [10, Corollary 3.2] when applied to solving (3). Yet, the
+costly accuracy requirements for the rounding procedure (see, e.g., Proposition 5) lead to a prohibitive scaling
+in the inverse precision for the overall running time. Given the advantageous dependence on the dimension,
+as compared to classical algorithms, we study how to improve the dependence on the precision parameter.
+This is discussed next.
+4
+Iterative Refinement for SDO approximations of QUBOs
+In this section, we introduce an iterative refinement method for obtaining accurate solutions to the renor-
+malized relaxed SDO problem (5), that at a high level can be viewed as solving a series of problems related
+to the feasibility problem (10) associated with (5). We then discuss how to test ǫ-closeness to the convex sets
+which comprise the feasible regions of the intermediate refining problems before presenting our algorithm in
+full detail. We conclude the section by proving our algorithm’s correctness and iteration complexity, and use
+these results to provide an improved approximation guarantee.
+4.1
+The refining problem
+To develop an iterative refinement scheme for (5), we need to design a problem whose solution can be used
+to improve the quality of solutions to (5). Suppose we run Algorithm 1 and obtain an ǫ-precise solution ˆρ
+to (5). Letting ˆγ = tr
+�
+C∥C∥−1
+F ˆρ
+�
+, ˆρ must satisfy
+tr
+�
+C∥C∥−1
+F ˆρ
+�
+= ˆγ ≥ γ − ǫ,
+n
+�
+i=1
+����⟨i|ˆρ|i⟩ − 1
+n
+���� ≤ ǫ.
+In refining our solution to (5), we should aim to reduce the trace distance to the maximally mixed state
+n−1I, while also improving the precision to which the optimal objective value is approximated. Thus, an
+21
+
+improved solution ρ′ should obey
+tr
+�
+C∥C∥−1
+F ρ′�
+≥ γ − ǫ′,
+n
+�
+i=1
+����⟨i|ρ′|i⟩ − 1
+n
+���� ≤ ǫ′,
+with ǫ′ < ǫ. The basic idea behind constructing the refining problem is to use our current solution ˆρ to first
+shift the renormalized relaxed SDO problem (5) to the origin, and then scale the shifted problem back to
+the domain of the original problem. In particular, we solve a series of problems related to the feasibility
+problem (10).
+Let ε ∈ Rn be a vector whose elements are the residuals along the diagonal εi = ˆρii − 1
+n for i ∈ [n], and
+η ≥ 1 to be a scalar defined as
+η =
+1
+max
+�
+γ − tr
+�
+C∥C∥−1
+F ˆρ
+�
+, �n
+i=1 |εi|
+� =
+1
+max
+�
+γ − tr
+�
+C∥C∥−1
+F ˆρ
+�
+, ∥ˆρ − n−1I∥tr
+�.
+Using these quantities, the refining problem is given by:
+find
+ρr ∈ {X ∈ Sn
++ : tr(X) = 1} ∩ Cη(γ−ˆγ) ∩ Dηε
+where
+Cη(γ−ˆγ) =
+�
+X : tr
+�
+C
+∥C∥F
+(Q ◦ X)
+�
+≥ η(γ − ˆγ)
+�
+,
+Dηε = {X : ⟨i|X|i⟩ = η|εi|, ∀i ∈ [n]} ,
+(11)
+where Q ∈ Sn is a matrix whose diagonal elements are chosen such that for any X ∈ Dηε, we have
+(Q ◦ X)ii = ηεi
+for i ∈ [n]. Further details and requirements on the structure of Q are specified later in this section. We
+refer to solutions ρr to (11) as refining solutions, which we use to update our current solution ˆρ to (5).
+The set Dηε is comprised of the diagonal constraints
+⟨i|X|i⟩ = η|εi|,
+∀i ∈ [n],
+and similar to Dn, is an affine space with codimension n. Our use of the absolute value function of the
+residuals and scaling by η ensures the viability of applying Gibbs sampling techniques to solve the refining
+problem (11); the diagonal terms of any density matrix must be nonnegative and sum to 1. Whenever
+n
+�
+i=1
+|εi| > γ − tr
+�
+C∥C∥−1
+F ˆρ
+�
+,
+then η∥ε∥1 = 1, and the parameter η therefore scales the shifted problem back to the space of the log(n)-qubit
+mixed states, ensuring that any solution ρr to (11) is indeed a (trace normalized) Gibbs state.
+On the other hand, should it be the case that
+n
+�
+i=1
+|εi| ≤ γ − tr
+�
+C∥C∥−1
+F ˆρ
+�
+,
+then for any X ∈ Dηε we have tr(X) ≤ 1, rather than tr(X) = 1. Our primal SDO oracle in Algorithm 1
+solves feasibility problems in which the trace upper bound is tight, i.e., tr(X) = 1. The authors in [60] note
+that this can be dealt with adding one extra variable w such that
+¯ρr :=
+�ρr
+0
+0
+w
+�
+.
+22
+
+Then, tr (¯ρr) = 1 and ¯ρr ⪰ 0 imply that tr(ρr) ≤ 1, and as a result we obtain an SDO problem that is
+equivalent to (11). Since we know exactly the amount of subnormalization, we can also get rid of the extra
+variable in subsequent calculations and rescale the trace back to 1 when necessary (e.g., when combining
+solutions from multiple iterative refinement iterations for trace estimations).
+Crucially, using the input
+models described in Section 2.1, these modifications do not introduce more than constant overhead in the
+overall complexity, as the problem data in this case is simply given by
+C =
+�
+C
+∥C∥F
+0
+0
+0
+�
+,
+Q =
+�
+Q
+0
+0
+0
+�
+,
+with (C, Q) ∈ Sn+1 × Sn+1.
+The Hadamard product Q ◦ ρr that appears in the definition of Cη(γ−ˆγ) is required for similar reasons;
+properly setting Q allows us to drive the trace distance to the maximally mixed state to zero using the
+solutions to the refining problem. Later, in Section 4.3 we demonstrate that this can be achieved by updating
+the current solution ˆρ using the rule
+ˆρ = ˆρ + 1
+η Q ◦ ρr,
+(12)
+with a suitable choice for Q being
+Q = (ee⊤ − I) + diag (sign(−ε)) =
+
+
+
+
+
+
+sign(−ε1)
+1
+. . .
+1
+1
+sign(−ε2)
+...
+...
+...
+...
+...
+1
+1
+. . .
+1
+sign(−εn)
+
+
+
+
+
+
+.
+(13)
+Choosing Q in this manner also implies that the Hadamard product Q ◦ A can be carried out classically
+using O(n) arithmetic operations for any A ∈ Rn×n, as the element-wise products QijAij = Aij for i ̸= j.
+We also point out that updating Q at each iterate only requires updating its diagonal elements, which again
+is an O(n) operation.
+It is important to note that the update we propose in (12) does not preserve positive semidefiniteness
+in general. However, later in our analysis, we demonstrate that the eigenvalues of the final solution ˆρ are
+only slightly negative in the worst case, i.e., λmin(ˆρ) ≥ −δ for a small constant δ; one can restore positive
+semidefiniteness by adding δ to the diagonal elements of the final solution, and we renormalize by (1 + nδ)
+to obtain unit trace. It turns out that the trace distance from resulting matrix to ˆρ is bounded by the final
+precision tolerance parameter of our refining scheme. We show that these modifications required to restore
+positive semidefiniteness have only a mild (in fact, constant) impact on feasibility. In order to accomplish
+this, we will need to utilize bounds on the eigenvalues of Q. This will require use of the next result, which
+is a special instance of Weyl’s inequality.
+Lemma 12. Suppose that A ∈ Rn×n and B ∈ Rn×n are Hermitian matrices. Then
+λmin(A + B) ≥ λmin(A) + λmin(B).
+Using the preceding lemma, the following result bounds the minimum eigenvalue of Q.
+Lemma 13. Suppose that Q ∈ Sn is defined according to Equation (13). Then, λmin(Q) ≥ −2.
+Proof. Let A = (ee⊤ − I) and B = diag (sign(−ε)), such that Q = A + B. Now, it can be easily seen
+from the definition of A that A + I is an all-ones matrix of dimension n. Upon performing row-reduction
+(via, e.g., Guassian elimination) on A, it is trivial to observe that the resulting row-echelon form will have
+n − 1 zero rows, and as a consequence, A has the eigenvalue −1, repeated (at least) n − 1 times. Further,
+since tr (A) = 0, the other eigenvalue is n − 1. Therefore, we have λmin(A) ≥ −1. On the other hand, B
+is a diagonal matrix whose diagonal elements can take value −1, 0, or 1, from which λmin(B) ≥ −1 readily
+follows.
+23
+
+Applying Lemma 12, we obtain
+λmin(Q) = λmin(A + B) ≥ λmin(A) + λmin(B) ≥ −2.
+The proof is complete.
+4.2
+Oracle construction for the refining problem
+In order to construct separation oracles for testing closeness to Cη(γ−ˆγ), we rely on the following result.
+Lemma 14. Let E, F and G ∈ Sn. We have
+tr (G(E ◦ F)) = tr ((E ◦ G)F).
+Proof. Applying Lemma 1 with m = n, we have
+[(E ◦ F)G]ii = [(E ◦ G)F]ii
+∀i ∈ [n].
+Note that we have dropped the transpose terms, as E, F and G are symmetric matrices, and hence, so are
+E ◦ F and E ◦ G. It follows
+tr (G(E ◦ F)) = tr ((E ◦ F)G) =
+�
+i∈[n]
+[(E ◦ F)G]ii =
+�
+i∈[n]
+[(E ◦ G)F]ii = tr ((E ◦ G)F).
+In addition to Q ∈ Sn, we also require maxi,j∈[n]{|Qij|} ≤ 1 to avoid any normalization issues with
+respect to Q ◦
+C
+∥C∥F . Note that defining of Q according to equation (13) satisfies both of these properties
+trivially, as each of the diagonal elements are 1, 0, or −1, while the off-diagonal elements are all set to 1.
+This idea is formalized next.
+Lemma 15. Let A, Q ∈ Sn be matrices satisfying maxi,j∈[n]{|Qij|} ≤ 1 and ∥A∥F ≤ 1. Then,
+∥Q ◦ A∥ ≤ ∥Q ◦ A∥F ≤ 1.
+Proof. Under the stated conditions for Q, it follows
+∥Q ◦ A∥2
+F =
+�
+i∈[n]
+�
+j∈[n]
+�
+[Q ◦ A]ij
+�2
+=
+�
+i∈[n]
+�
+j∈[n]
+(Qij · Aij)2 =
+�
+i∈[n]
+�
+j∈[n]
+(Qij)2 (Aij)2
+≤
+�
+i∈[n]
+�
+j∈[n]
+(Aij)2 = ∥A∥2
+F ,
+and applying the square root throughout the above we obtain ∥Q ◦ A∥F ≤ ∥A∥F . From here, the result
+follows upon noting ∥A∥F ≤ 1 and ∥A∥ ≤ ∥A∥F is true for any A ∈ Rn×n.
+Although the sets Cγ and Dn differ from their refining counterparts Cη(γ−ˆγ) and Dηε, their dissimilarity
+merely affects the right hand side of the inequality defining the sets, and are thus no more difficult to
+construct. Just as in the case of (10), the task of obtaining separation oracles for the refining problem (11)
+in the quantum regime reduces to preparing many copies of Gibbs states. Likewise, these oracles can also
+be implemented on a classical computer, given access to ρr.
+The similarities between (10) and (11) become transparent when we demonstrate that they are specific
+instances of the same problem. In particular, it is easy to see that solving (10) corresponds to solving
+find
+ρ ∈ {X ∈ Sn
++ : tr(X) = 1} ∩ Cη(γ−ˆγ) ∩ Dηε
+where
+Cη(ˆγ−γ) =
+�
+X : tr
+�
+C
+∥C∥F
+Q ◦ X
+�
+≥ η(γ − ˆγ)
+�
+,
+Dηε = {X : ⟨i|X|i⟩ = η|εi|, ∀i ∈ [n]} ,
+(14)
+24
+
+with εi = 1
+n, η = 1, Q = ee⊤, and ˆγ = 0. In view of this relationship, we can unify the oracle construction
+for (10) and (11) as follows:
+OCη(γ−ˆγ) : Compute an approximation ˜c of tr
+�
+Q ◦ C∥C∥−1
+F ρ
+�
+up to additive error ǫ
+4.
+Check if ˜c ≥ η(γ − ˆγ) + 3ǫ
+4 and output P = −Q ◦ C∥C∥−1
+F
+if the inequality is violated.
+ODηε : Compute an approximation ˜p ∈ Rn of pi = ⟨i|ρ|i⟩ satisfying
+�
+i∈[n]
+|pi − ˜pi| ≤ ǫ
+4.
+Check if
+�
+i∈[n]
+|˜pi − η|εi|| ≤ 3ǫ
+4 and output P =
+�
+i∈[n]
+(I{˜pi > η|εi|} − I{˜pi < η|εi|}) |i⟩ ⟨i|
+if the inequality is violated.
+Again, the sets of observables for Cη(γ−ˆγ) and Dηε are given by
+�Cη(γ−ˆγ) = {−Q ◦ C∥C∥−1
+F }, and �Dηε = {D ∈ Rn×n : ∥D∥ ≤ 1, D is diagonal}.
+Although these observations are straightforward, they justify our use of Algorithm 1 as a semidefinite opti-
+mization oracle that solves a convex feasibility problem at hand in every iteration for different values of Q.
+In particular, these facts, along with Lemmas 14 and 15 ensure that the complexity results in Propositions
+6 and 7 hold when applying Algorithm 1 to solve (14).
+Proposition 8. Let Q ◦
+C
+∥C∥F ∈ Sn be stored in QRAM. Then, the complexity of solving (14) up to additive
+error ǫ with Algorithm 1 using the QRAM input model is
+�
+O n
+ǫ
+�
+n1.5ǫ−5�
+.
+Here, the complexity corresponds to the number of accesses to the QRAM.
+Proof. Given that Q◦
+C
+∥C∥F is stored in QRAM, Lemma 3(ii) asserts that when constructing a block-encoding
+of Q◦
+C
+∥C∥F , one can set the subnormalization factor to be αC =
+���Q ◦
+C
+∥C∥F
+���
+F . In particular, one can always
+choose αC = 1, as it can be seen from the proof of Lemma 15 that the inequality
+����Q ◦
+C
+∥C∥F
+����
+F
+≤
+����
+C
+∥C∥F
+����
+F
+= 1
+always holds for any Q defined according to equation (13).
+Collecting these facts, one can construct a
+(1, O(log(n)), ǫ/(2n))-block-encoding of Q◦C∥C∥−1
+F
+in time �
+O n
+ǫ (1). Note that the quantity Q◦
+C
+∥C∥F remains
+unchanged for the duration of Algorithm 1. From here, the rest of the proof follows exactly that of Proposition
+7 upon replacing
+C
+∥C∥F , OCγ and ODn with Q ◦
+C
+∥C∥F , OCη(γ−ˆγ) and ODηε, respectively, in what remains.
+4.3
+Iterative Refinement using Hamiltonian Updates
+We are now in a position to provide our iterative refinement method for SDO approximations of QUBOs
+presented in full detail in Algorithm 2.
+The algorithm takes three parameters as input; (i) ξ, the fixed precision used to test closeness to the sets
+Cη(γ−ˆγ) and Dηε in every iteration, (ii) ζ, the precision to which the final solution satisfies the functional
+constraints of (5), and (iii) ǫ, the additive error to which we seek to solve (3). In our initialization steps we
+set the values of Q, ε and η such that the first iteration corresponds to solving the feasibility problem (10).
+In each iteration k, Algorithm 2 calls Algorithm 1 with separation oracles OCη(γ−ˆγ) and ODηε using
+fixed precision ξ such that every call to Algorithm 1 produces a ξ-precise classical solution ρ(k) to (14). If
+ˆρ is indistinguishable up to precision ζ from the maximally mixed state n−1I upon measurement in the
+computational basis, and satisfies tr
+�
+C
+∥C∥F ˆρ
+�
+≥ γ − ζ, the algorithm terminates and reports ˆρ. Otherwise,
+we construct the refining problem associated with our current solution, and proceed to the next iteration.
+25
+
+Algorithm 2 Iterative Refinement for SDO Approximations of QUBOs
+Input: Error tolerances ǫ ∈ (0, 1) and ζ =
+�
+ǫ
+n∥C∥F
+�4
+, upper bound on objective value γ ∈ [−1, 1]
+Output: A matrix ˆρ ∈ Sn satisfying max
+�
+γ − tr
+�
+C
+∥C∥F ˆρ
+�
+, ∥ˆρ − n−1I∥tr
+�
+≤ ζ
+Initialize: ˆρ ← 0n×n, Q ← ee⊤, εi = 1
+n for i ∈ [n], ˆγ ← 0, η(0) ← 1, k ← 0
+while max
+�
+γ − tr
+�
+C
+∥C∥F ρ
+�
+, ∥ε∥1
+�
+> ζ do
+1. Solve (14) to precision ξ
+4 for ρ(k) using Algorithm 1 with oracles OCη(γ−ˆγ) and ODηε
+2. Update Solution:
+ˆρ ← ˆρ +
+1
+η(k) Q ◦ ρ(k),
+ˆγ ← ˆγ +
+1
+η(k) tr
+�
+C
+∥C∥F
+Q ◦ ρ(k)
+�
+3. Compute element-wise deviations from the maximally mixed state:
+εi ← ˆρii − 1
+n for i ∈ [n]
+4. Update refining problem parameters:
+Qii ← sign(−εi) for i ∈ [n],
+η(k+1) ←
+1
+max
+�
+γ − tr
+�
+C
+∥C∥ρ
+�
+, ∥ε∥1
+�
+5. k ← k + 1
+end
+26
+
+To define the parameters for the next refining problem, we first calculate the deviation of the diagonal
+elements from 1
+n, and the violation with respect to satisfying our objective value. Then, we define our scaling
+factor to be the maximum over the ℓ1-norm of the diagonal deviations, and the objective violation. We stress
+that ξ is a (chosen) constant, and does not change throughout the algorithm.
+We now state a series of results in order to bound the iteration complexity of Algorithm 2, and use our
+findings to improve the approximation guarantee given in Proposition 5. We begin by proving that the
+iterates generated by Algorithm 2 satisfy the constraints in (5) with increasing accuracy.
+Theorem 4. Let ˆρ be the current overall solution, and let ρ(k) be a solution to (14) obtained from running
+Algorithm 1 using fixed precision ξ ∈ (0, 1) in iteration k of Algorithm 2. Then, the following hold:
+(a) For k ≥ 0, η(k) ≥
+1
+ξk .
+(b) For k ≥ 0, ρ = ˆρ +
+1
+η(k) Q ◦ ρ(k) satisfies
+max
+�
+γ − tr
+�
+C
+∥C∥F
+ρ
+�
+, ∥ρ − n−1I∥tr
+�
+≤ ξk+1.
+Proof. We begin by establishing that for k ≥ 0, the current solution satisfies
+max
+�
+γ − tr
+�
+C
+∥C∥F
+ρ
+�
+, ∥ρ − n−1I∥tr
+�
+≤
+ξ
+η(k) .
+(15)
+First, observe that when k = 0, we have εi = 1
+n for i ∈ [n], ˆρ = 0n×n, η(0) = 1 and Q = ee⊤. Under these
+conditions, one can observe that if ρ(0) solves (14) to precision ξ, then
+n
+�
+i=1
+����⟨i|ρ|i⟩ − 1
+n
+���� =
+n
+�
+i=1
+����⟨i|ˆρ +
+1
+η(0) ρ(0)|i⟩ − 1
+n
+���� =
+n
+�
+i=1
+����
+1
+η(0) ρ(0) − εi
+���� =
+1
+η(0)
+n
+�
+i=1
+���ρ(0) − η(0)εi
+��� ≤
+ξ
+η(0) .
+In other words, ρ = ρ(0) satisfies
+∥ρ − n−1I∥tr ≤
+ξ
+η(0) .
+Next, by the definition of OCη(γ−ˆγ) we have
+tr
+�
+C∥C∥−1
+F ρ
+�
+= tr
+�
+C∥C∥−1
+F
+�
+ˆρ +
+1
+η(0) Q ◦ ρ(0)
+��
+= tr
+�
+C∥C∥−1
+F ˆρ
+�
++
+1
+η(0) tr
+�
+C∥C∥−1
+F
+�
+Q ◦ ρ(0)��
+= tr
+�
+C∥C∥−1
+F 0n×n�
+�
+��
+�
+=0
++ 1
+η(0) tr
+�
+C∥C∥−1
+F
+�
+Q ◦ ρ(0)��
+=
+1
+η(0) tr
+�
+C∥C∥−1
+F
+�
+Q ◦ ρ(0)��
+=
+1
+η(0) tr
+�
+C∥C∥−1
+F ρ(0)�
+≥
+1
+η(0)
+�
+[η(0)(γ − ˆγ(0))] − ξ
+�
+= γ −
+ξ
+η(0) ,
+where we used the fact that ˆγ(0) = 0.
+Next, letting ˆρ be our current solution and ˆγ = tr
+�
+C
+∥C∥F ˆρ
+�
+be the objective value attained at this
+solution. When k ≥ 1, we have εi = ˆρii − 1
+n for i ∈ [n] and Q = (ee⊤ − I) + diag(sign(−ε)). For this choice
+of parameters, the general feasibility problem (14) reduces to the refining problem (11) and the solution
+ρ(k) obtained via Algorithm 1 is therefore a ξ-precise solution to (11).
+Accordingly, for k ≥ 1, setting
+27
+
+ρ = ˆρ +
+1
+η(k) Q ◦ ρ(k) improves the precision to which the maximally mixed state is approximated:
+n
+�
+i=1
+����⟨i|ρ|i⟩ − 1
+n
+���� =
+n
+�
+i=1
+����⟨i|ˆρ +
+1
+η(k) Q ◦ ρ(k)|i⟩ − 1
+n
+����
+=
+n
+�
+i=1
+����
+�
+ˆρii +
+1
+η(k)
+�
+sign(−εi) · ρ(k)
+ii
+��
+− 1
+n
+����
+=
+n
+�
+i=1
+����
+�
+ˆρii − 1
+n
+�
++
+1
+η(k) sign(−εi)ρ(k)
+ii
+����
+=
+n
+�
+i=1
+����εi +
+1
+η(k) sign(−εi)ρ(k)
+ii
+���� =
+1
+η(k)
+n
+�
+i=1
+���η(k)εi + sign(−εi)ρ(k)
+ii
+��� ≤
+ξ
+η(k) .
+Consequently, we can conclude that at iteration k ≥ 1, the trace distance from our solution to the maximally
+mixed state is
+∥ρ − n−1I∥tr ≤
+ξ
+η(k) .
+(16)
+Next, letting ˜γ(k) = tr
+�
+C
+∥C∥F Q ◦ ρ(k)�
+, one can observe
+tr
+�
+C
+∥C∥F
+ρ
+�
+= tr
+�
+C
+∥C∥F
+�
+ˆρ +
+1
+η(k) Q ◦ ρ(k)
+��
+= tr
+�
+C
+∥C∥F
+ˆρ
+�
++
+1
+η(k) tr
+�
+C
+∥C∥F
+Q ◦ ρ(k)
+�
+= ˆγ + ˜γ(k)
+η(k) .
+For any ρ(k) which is ξ-close to the set Cη(γ−ˆγ) we must have
+˜γ(k) ≥ η(k)(γ − ˆγ) − ξ.
+It follows:
+tr
+�
+C
+∥C∥F
+ρ
+�
+= ˆγ + ˜γ(k)
+η(k) ≥ ˆγ +
+1
+η(k)
+�
+η(k)(γ − ˆγ) − ξ
+�
+= γ −
+ξ
+η(k) .
+(17)
+It therefore follows from (16) and (17) that ρ = ˆρ +
+1
+η(k) Q ◦ ρ(k) satisfies inequality (15) for all k ≥ 0, and
+we can now use this fact to establish the lower bound on η(k), which we prove by induction.
+For k = 0, we have η(0) = 1, for which η(k) ≥
+1
+ξk trivially holds. By the induction hypothesis, it assumed
+that η(ℓ) ≥
+1
+ξℓ is true for ℓ = 1, . . . , k. From here, applying (15) yields
+η(k+1) =
+1
+max
+�
+γ − tr
+�
+C
+∥C∥F ρ
+�
+, ∥ρ − n−1I∥tr
+� ≥
+1
+ξ
+η(k)
+≥
+1
+ξk+1 ,
+which completes the proof of (a).
+Having demonstrated that (a) holds, to prove (b), we can simply combine inequality (15) with the lower
+bound η(k) ≥
+1
+ξk , which together imply
+max
+�
+γ − tr
+�
+C
+∥C∥F
+ρ
+�
+, ∥ρ − n−1I∥tr
+�
+≤
+ξ
+η(k) ≤ ξk+1.
+That is, (b) holds, and the proof is complete.
+The next result establishes polynomial convergence of Algorithm 2.
+Corollary 5. Let 0 < ζ ≪ ξ < 1, and η(0) = 1. Then, Algorithm 2 terminates in at most
+K = O
+�
+log
+�1
+ζ
+��
+iterations.
+28
+
+Proof. The result follows from Theorem 4(b).
+Observe that in Theorem 4, we do not consider whether the updated solution remains positive semidef-
+inite. It turns out that, by the nature of our update scheme, the minimum eigenvalue of the final solution
+obtained via Algorithm 2 will never fall significantly below zero. Moreover, we employ a rounding procedure
+to modify the entries of the solution we obtain from Algorithm 2 so that we arrive at an exactly feasible
+solution to (3). Before proceeding further, we formalize this notion, and derive a lower bound on the small-
+est eigenvalue of the final solution. We utilize Lemma 13 to bound the minimum eigenvalue of the terms
+1
+η(k) Q(k) ◦ ρ(k) that are used to update the overall solution in each iteration, from which a lower bound
+minimum eigenvalue bound for the final solution obtained by Algorithm 2 readily follows upon applying
+Lemma 12.
+Proposition 9. Let ρ(k) be a solution to (14) obtained from running Algorithm 1 using fixed precision
+ξ ∈ (0, 1) in iteration k of Algorithm 2. Then, the following hold:
+(a) For k ≥ 1,
+1
+η(k) Q(k) ◦ ρ(k) ⪰ −2 · ξkI.
+(b) Suppose Algorithm 2 is run with final precision ζ, and terminates after K iterations. Then, the solution
+ρ output by Algorithm 2 satisfies
+λmin(ρ) ≥ λmin
+�
+ρ(0)�
+− 2 ·
+K
+�
+k=1
+ξk.
+Proof. We begin with a proof of (a). In what follows, we assume without loss of generality that Q(k) has at
+least one negative eigenvalue (otherwise, Q(k) ◦ ρ(k) ⪰ 0 trivially holds). Hence, a combined application of
+Corollary 1 and Lemma 13 yields
+λmin
+�
+Q(k) ◦ ρ(k)�
+≥ λmax
+�
+ρ(k)�
+· λmin
+�
+Q(k)�
+≥ −2.
+Therefore, recalling that by Theorem 4(a) we have η(k) ≥
+1
+ξk , it follows
+λmin
+� 1
+η(k) Q ◦ ρ(k)
+�
+=
+1
+η(k) λmin
+�
+Q ◦ ρ(k)�
+≥ −2
+η(k) ≥ −2 · ξk,
+which completes the proof of (a).
+Noting that the final solution can be expressed as
+ρ =
+K
+�
+k=0
+1
+η(k) Q(k) ◦ ρ(k) = ρ(0) +
+K
+�
+k=1
+1
+η(k) Q(k) ◦ ρ(k),
+the result in (b) follows from (a) and repeated application of Lemma 12.
+The next corollary computes bounds for the geometric series that appears in Proposition 9(b) for different
+values of the fixed precision parameter ξ, by computing the value of the series to ∞.
+Corollary 6. Let ξ =
+˜ξ
+4. Then, for any positive integer K, the following hold.
+(a) If ˜ξ = 10−2, then −2 · �K
+k=1 ξk ≥ −0.005.
+(b) If ˜ξ = 10−4, then −2 · �K
+k=1 ξk ≥ −0.00005.
+The goal of Corollary 6 is simply to show that with fixed precision we obtain matrices with eigenvalues that
+are, in the worst case, slightly negative. A shift of the spectrum suffices to restore positive semidefiniteness,
+and it does not change the constraint violation or the objective function value by a large amount, as we show
+next.
+29
+
+Proposition 10. Suppose Algorithm 2 is run with final precision ζ, and terminates after K iterations. Let
+ρ be the solution output by Algorithm 2, and let ξ be the fixed precision used in every iteration. Then, letting
+δ = 2 ·
+K
+�
+k=1
+ξk,
+(18)
+it follows that
+˜ρ =
+1
+1 + nδ (ρ + δI)
+is a positive semidefinite matrix at trace distance ζ from ρ. Moreover, ˜ρ satisfies:
+max
+�
+γ − tr
+�
+C
+∥C∥F
+˜ρ
+�
+,
+��˜ρ − n−1I
+��
+tr
+�
+≤ 2ζ.
+In other words, ˜ρ is a 2ζ-precise solution to (5).
+Proof. First, observe that ˜ρ ⪰ 0 by the definition of ˜ρ; applying Proposition 9(b), we have λmin(ρ) ≥ −δ,
+which implies that ρ + δI ⪰ 0. From the definition of ˜ρ, we also have:
+∥ρ − ˜ρ∥tr =
+����ρ −
+�
+1
+1 + nδ (ρ + δI)
+�����
+tr
+=
+����
+1 + nδ − 1
+1 + nδ
+ρ −
+δ
+1 + nδ I
+����
+tr
+=
+δ
+1 + nδ ∥nρ − I∥tr
+=
+nδ
+1 + nδ
+��ρ − n−1I
+��
+tr
+≤
+nδ
+1 + nδζ
+< ζ,
+where the second to last inequality follows from the fact that ρ is obtained from running Algorithm 2 with
+ζ as the final precision parameter.
+Next, we leverage our bound on the trace distance from ˜ρ to ρ to establish that ˜ρ is indeed an accurate
+solution to the renormalized SDO problem (5). First, note that
+��˜ρ − n−1I
+��
+tr =
+��˜ρ − n−1I + (ρ − ρ)
+��
+tr
+=
+��(˜ρ − ρ) +
+�
+ρ − n−1I
+���
+tr ≤ ∥˜ρ − ρ∥tr +
+��ρ − n−1I
+��
+tr ≤ 2ζ.
+Further, applying a matrix H¨older inequality, one can observe:
+����tr
+�
+C
+∥C∥F
+ρ
+�
+− tr
+�
+C
+∥C∥F
+˜ρ
+����� ≤
+����
+C
+∥C∥F
+���� ∥ρ − ˜ρ∥tr ≤ ∥ρ − ˜ρ∥tr < ζ,
+from which we can conclude
+γ − tr
+�
+C
+∥C∥F
+˜ρ
+�
+= γ − tr
+�
+C
+∥C∥F
+˜ρ
+�
++
+�
+tr
+�
+C
+∥C∥F
+ρ
+�
+− tr
+�
+C
+∥C∥F
+ρ
+��
+=
+�
+γ − tr
+�
+C
+∥C∥F
+ρ
+��
++
+�
+tr
+�
+C
+∥C∥F
+ρ
+�
+− tr
+�
+C
+∥C∥F
+˜ρ
+��
+≤ 2ζ,
+as γ − tr
+�
+C
+∥C∥F ρ
+�
+≤ ζ.
+It is important at this point for us to remark that fixing ξ ∈ (0, 1) does not limit us with respect to how
+accurately we can solve (3). We can always make the final precision parameter arbitrarily small using only
+�
+O 1
+ζ (1) iterations, as the overall running time depends only poly-logarithmically on ζ−1. Accordingly, we
+take advantage of this fact and revisit the approximation guarantee provided in Proposition 5.
+30
+
+Proposition 11. Let ρ be a ζ-accurate solution to the renormalized and relaxed SDO problem (5) with input
+matrix C and ζ =
+�
+ǫ
+n∥C∥F
+�4
+. Let γζ = tr (Cρ) be the value attained by ρ. Then, there is a quantum state
+ρ∗ at trace distance O
+�
+ǫ
+n∥C∥F
+�
+of ρ such that nρ∗ is a feasible point of SDO problem (3). In particular
+|γζn∥C∥F − tr (nρ∗C)| = O (ǫ) .
+Moreover, it is possible to construct ρ∗ in time O(n2) given the entries of ρ.
+Proof. The proof almost exactly follows the proof of Proposition 3.1 in [10], regardless, we present the
+adjusted proof for completeness.
+Our aim is to show that a ζ-precise solution ρ to (5) obtained using
+Algorithm 2 can be used to construct ρ∗ such that nρ∗ is an exactly feasible solution to (3).
+We begin shifting ρ in order to ensure that our solution is positive semidefinite. In particular, choosing
+δ according to (18), we set
+˜ρ =
+1
+1 + nδ (ρ + δI) .
+It then follows from Proposition 10 that ˜ρ satisfies
+˜ρ ⪰ 0,
+∥ρ − ˜ρ∥tr ≤ ζ,
+max
+�
+γ − tr
+�
+C
+∥C∥F
+˜ρ
+�
+,
+��˜ρ − n−1I
+��
+tr
+�
+≤ 2ζ.
+(19)
+Next, we examine the diagonal elements of ˜ρ and check whether modifications need to be made to
+ensure that our solution is an exactly feasible point to the renormalized SDO problem (5).
+Namely, if
+|⟨i|˜ρ|i⟩ − 1
+n| >
+√2ζ
+n
+for i ∈ [n], we replace ˜ρii with 1
+n and set all elements in the i-th row and the i-th column
+to 0, and denote the resulting matrix by ρ′. From here we introduce another matrix W which we obtain
+by replacing each diagonal entry of ρ′ with 1
+n. In general we may not have W ⪰ 0, so the authors in [10]
+suggest using the convex combination:
+ρ∗ =
+1
+1 + √2ζ
+�
+W +
+√2ζ
+n I
+�
+.
+Then, ρ∗ ⪰ 0 and by construction ⟨i|ρ∗|i⟩ =
+1
+n for all i ∈ [n].
+Hence, ρ∗ is a feasible solution to the
+renormalized SDO problem (5).
+What remains is to show that the above reformulations yield the desired approximation. Denote by
+B = {i : |n⟨i|˜ρ|i⟩ − 1| > √2ζ} ⊂ [n] the set of diagonal entries that deviate substantially from 1
+n. Without
+loss of generality, it suffices to assume that such elements are found in the first |B| rows of ˜ρ, in which case
+∥ρ′ − ˜ρ∥tr =
+����
+�
+n−1IB
+0
+0
+˜ρ22
+�
+−
+�
+˜ρ11
+˜ρ12
+˜ρ21
+˜ρ22
+�����
+tr
+=
+����
+�
+n−1IB − ˜ρ11
+−˜ρ12
+−˜ρ21
+0
+�����
+tr
+≤ ∥˜ρ11∥tr + 2∥˜ρ12∥tr + ∥n−1IB∥tr.
+(20)
+Since ˜ρ is a 2ζ-precise solution to (5), ˜ρ obeys
+n
+�
+i=1
+����⟨i|˜ρ|i⟩ − 1
+n
+���� ≤ 2ζ.
+Therefore, we must have
+|B|
+√2ζ
+n
+≤ 2ζ,
+which equates to |B| ≤ n√2ζ. Now, by the definition of B, it follows
+∥˜ρ22∥tr ≥ (n − |B|)1 − √2ζ
+n
+≥ (n − n
+�
+2ζ)1 − √2ζ
+n
+= (1 −
+�
+2ζ)2.
+31
+
+Following [10], we invoke a result from [40], which states
+����
+�∥˜ρ11∥tr
+∥˜ρ12∥tr
+∥˜ρ⊤
+12∥tr
+∥˜ρ22∥tr
+����� ≤
+����
+�˜ρ11
+˜ρ12
+˜ρ⊤
+12
+˜ρ22
+�����
+tr
+= ∥˜ρ∥tr = tr (˜ρ) = 1.
+Using the fact that ∥ · ∥tr ≥ ∥ · ∥2, where ∥ · ∥2 is the Frobenius, or Schatten-2 norm, the above implies
+∥˜ρ11∥2
+tr + 2∥˜ρ12∥2
+tr + ∥˜ρ22∥2
+tr ≤ 1.
+As ∥˜ρ22∥tr ≥ (1 − √2ζ)2, it can be seen trivially that ∥˜ρ22∥2
+tr ≥ (1 − √2ζ)4, and thus
+∥˜ρ11∥2
+tr + 2∥˜ρ12∥2
+tr ≤ 1 − (1 −
+�
+2ζ)4 = O(
+�
+ζ).
+Consequently ∥˜ρ11∥tr + 2∥˜ρ12∥tr = O
+�
+ζ
+1
+4
+�
+, and plugging this into equation (20) asserts
+∥ρ′ − ˜ρ∥tr = O
+�
+ζ
+1
+4
+�
+.
+(21)
+Let R be a diagonal matrix whose elements are Rii ∈
+�
+−
+√2ζ
+n ,
+√2ζ
+n
+�
+for i ∈ [n], such that
+W = ρ′ + R,
+and note that R + √2ζn−1I ⪰ 0. Upon normalizing the trace, one can observe
+ρ∗ =
+1
+1 + √2ζ
+�
+ρ′ + R +
+�
+2ζn−1I
+�
+⪰ 0,
+with ρ∗
+ii = 1
+nn for all i ∈ [n]. Thus, nρ∗ is a feasible solution to the SDO problem (3). Further, by a triangle
+inequality we have
+∥ρ′ − ρ∗∥tr =
+1
+1 + √2ζ
+���
+�
+2ζρ′ + R +
+�
+2ζn−1I
+���
+tr = O(
+�
+ζ).
+(22)
+Combining equations (21) and (22) and noting ζ =
+�
+ǫ
+n∥C∥F
+�4
+, applying another triangle inequality yields
+∥˜ρ − ρ∗∥tr = O
+�
+ζ
+1
+4
+�
+= O
+
+
+��
+ǫ
+n∥C∥F
+�4� 1
+4 
+ = O
+�
+ǫ
+n∥C∥F
+�
+.
+Then, the result follows from a matrix H¨older inequality:
+|tr (nCρ) − tr (nCρ∗)| ≤ n∥C∥∥ρ − ρ∗∥tr ≤ n∥C∥F (∥ρ − ˜ρ∥tr + ∥˜ρ − ρ∗∥tr)
+= O
+�
+n∥C∥F
+�
+ζ + ζ
+1
+4
+��
+= O
+�
+n∥C∥F
+��
+ǫ
+n∥C∥F
+�4
++
+ǫ
+n∥C∥F
+��
+= O
+�
+ǫ4
+(n∥C∥F)3 + ǫ
+�
+= O (ǫ) .
+5
+Complexity
+We now analyze the worst case overall running time of our Iterative Refinement Method given in Algorithm
+2 in both the classical and quantum settings.
+32
+
+5.1
+Classical running time
+As we saw in Section 3, the complexity of using Algorithm 1 to solve the SDO problem (3) scales poorly in
+the inverse precision, with the classical algorithm exhibiting an O(ǫ−12) dependence. In both the classical
+and quantum cases, our iterative refinement scheme reconciles the poor scaling in ǫ because it possesses
+the following two properties. First, we can obtain an arbitrarily precise solution to (5) in at most �
+O 1
+ζ (1)
+iterations. Second, it suffices to treat ξ as fixed for the oracle calls that occur in each iteration, as the
+precision of the final solution is a byproduct of how we use these solution of the refining problems to produce
+a solution to (5).
+The next result formalizes the above argument, and establishes the complexity of Algorithm 2 for the
+classical case.
+Theorem 5. Let C ∈ Sn with row sparsity s and ǫ ∈ (0, 1).
+Then, fixing ξ = 10−2, and setting
+ζ =
+�
+ǫ
+n∥C∥F
+�4
+, a classical implementation of Algorithm 2 solves (3) up to additive error O(ǫ) in time
+O
+�
+min{n2s, nω} · polylog
+�
+n, ∥C∥F, 1
+ǫ
+��
+.
+The output of the algorithm is a classical description of a matrix ˆρ ∈ Sn such that
+˜ρ =
+1
+1 + nδ (ˆρ + δI) ,
+is a 2ζ-precise solution to (5), where δ is defined according to (18). The entries of ˜ρ can be modified to
+construct a matrix ρ∗ at trace distance O
+�
+ǫ
+n∥C∥F
+�
+of ˜ρ in time O(n2), such that nρ∗ is a feasible point of
+the SDO problem (3).
+Proof. Given that C is an s-sparse matrix, we can load C in O(ns) time, and from here we must compute
+∥C∥F , which requires O(ns) arithmetic operations. In every iteration of Algorithm 2, we make a call to our
+subroutine in Algorithm 1, before updating the solution and preparing the next refining problem. Updating
+the solution involves matrix addition between two n × n matrices and requires O(n2) arithmetic opera-
+tions, whereas updating Q and ε for the next refining problem can be accomplished using O(n) arithmetic
+operations, as only the diagonal entries of Q need to be stored and maintained.
+In view of Proposition 6, the dominant operation at each iteration is the use of Algorithm 1 to solve the
+SDO problem at hand. By Proposition 6, Algorithm 1 can be used to solve (14) to additive error ξ in time
+T classical
+HU
+= O
+�
+min{n2s, nω} log2(n)ξ−3�
+.
+If every call to Algorithm 1 is made using precision ξ, then by Corollary 5, Algorithm 2 converges in at most
+O
+�
+log(ζ−1)
+�
+iterations, and we can thus express the overall running time of Algorithm 2 as
+O
+��
+min{n2s, nω} log2(n)ξ−3�
+log(ζ−1)
+�
+.
+In the context of Algorithm 2, it suffices to carry out each of the calls to the SDO subroutine (i.e., calls to
+Algorithm 1) using fixed precision ξ to obtain a 2ζ-precise solution to (5) (see, e.g., Proposition 10). The
+above complexity thus reduces to
+O
+�
+min{n2s, nω} log2(n) log(ζ−1)
+�
+.
+For our choice of ζ =
+�
+ǫ
+n∥C∥F
+�4
+, one can observe
+O
+�
+min{n2s, nω} log2(n) log(ζ−1)
+�
+= O
+�
+min{n2s, nω} · polylog
+�
+n, ∥C∥F, 1
+ǫ
+��
+.
+33
+
+Proposition 11 certifies that the above running time suffices to obtain a ρ from which we can construct ρ∗
+in time O(n2), such that nρ∗ is a feasible point of the SDO problem (3) satisfying
+|γζn∥C∥F − tr (nρ∗C)| = O(ǫ),
+and the proof is complete.
+5.2
+Quantum running time
+Just as in the classical case, we show that a quantum implementation of Algorithm 2 mitigates the poor
+scaling in the running time with respect to the inverse precision.
+Our quantum implementation of Algorithm 2 is provided in Algorithm 3. The relevant error parameters
+are the same as those appearing in Algorithm 2: (i) ξ, the fixed precision used to test closeness to the sets
+Cη(γ−ˆγ) and Dηε in every iteration, (ii) ζ, the precision to which the final solution satisfies the functional
+constraints in (5), and (iii) ǫ, the additive error to which we seek to solve (3). In our initialization steps
+we set the values of Q, ε and η such that the first iteration corresponds to solving the feasibility problem
+(10). We also create a vector p = 0n×1 that will be used to maintain a classical description of the diagonal
+elements of our solution over the course of the algorithm.
+At every iteration k, a call is made to Algorithm 1 with separation oracles OCη(γ−ˆγ) and ODηε to solve
+(14) using fixed precision ξ. If the oracles accept the candidate state, then Algorithm 1 returns a real-valued
+vector y(k) ∈ R2 along with a diagonal matrix D(k) such that the Hamiltonian associated with the Gibbs
+state that solves the refining problem is
+H(k) = y(k)
+1 Q(k) ◦
+C
+∥C∥F
++ y(k)
+2 D(k),
+with ∥y(k)∥1 ≤ 4 log(n)ξ−1 and ∥D(k)∥ ≤ 1 for every k ≥ 0. This allows us to efficiently describe the solution
+to each refining problem, and once the algorithm has terminated, it facilitates an efficient way to describe
+the final solution as well.5 First, observe that the matrices Q(k) and D(k) can be completely described by
+their diagonal elements; letting q(k) ∈ Rn and d(k) ∈ Rn be the vectors that store the diagonal elements of
+Q(k) and D(k), respectively, we have
+Q(k) = (ee⊤ − I) + diag
+�
+q(k)�
+,
+D(k) = diag
+�
+d(k)�
+.
+Therefore, we store the solution to the refining problem at iteration k as the tuple
+(η(k), y(k), q(k), d(k)),
+and the final solution to (5) is defined as
+˜ρ =
+1
+1 + nδ
+
+
+
+
+K
+�
+k=0
+1
+η(k) Q(k) ◦
+exp
+�
+−
+�
+y(k)
+1 Q(k) ◦
+C
+∥C∥F + y(k)
+2
+diag
+�
+d(k)���
+tr
+�
+exp
+�
+−
+�
+y(k)
+1 Q(k) ◦
+C
+∥C∥F + y(k)
+2
+diag
+�
+d(k)����
+
+ + δI
+
+ ,
+(23)
+where δ is defined according to (18) (see, e.g., Proposition 10). We point out that this marks a key difference
+between the output of our algorithm and other quantum SDO solvers based on Gibbs sampling [9, 10, 11, 60,
+61], which need only return a single state preparation pair. This however does not increase the cost of the
+method; the iteration bound in Corollary 5 ensures that there are only at most �O 1
+ζ (1) (i.e., a poly-logarithmic
+number) of these tuples to be stored over the course of the algorithm. Using the QRAM input model, one can
+5Requiring an explicit classical description of the solution would in fact lead to a worse running time overall when compared
+to the classical implementation we studied in Section 5.1.
+34
+
+use the stored tuples to construct a block-encoding of the final solution up to error θ using �On,∥C∥F , 1
+ǫ , 1
+θ (√n)
+queries to the QRAM and �On,∥C∥F , 1
+ǫ (n) classical operations. This construction, and the associated time
+complexity are analyzed later in Proposition 12. We further demonstrate that provided classical access to an
+s-sparse matrix A ∈ Rn×n (with subnormalization factor 1) and access to QRAM, one can estimate tr(A˜ρ)
+to additive error θ using �On,∥C∥F , 1
+ǫ
+� √n
+θ
+�
+queries to the QRAM and �
+On,∥C∥F , 1
+ǫ (ns) classical operations. If
+A has a subnormalization factor αA > 1, then θ must be scaled down by αA, increasing the cost.
+Additionally, we require Algorithm 1 to return the estimates ˜p(k) ∈ Rn (a classical estimate of the diagonal
+elements of the solution to the refining problem) and ˜c(k) ∈ R (a classical estimate of the objective value
+attained by the solution of the refining problem) that are used to test ξ-closeness for the accepted state. In this
+fashion, we can (classically) prepare the refining problem data for the next iteration without increasing the
+cost of the algorithm with respect to n; the objective value can be updated using O(1) arithmetic operations
+using ˜c(k), while updating the residuals along the diagonal of ρ requires O(n) arithmetic operations provided
+classical access to ˜p(k).
+If the current solution is indistinguishable up to precision ζ from the maximally mixed state n−1I, and
+provides an objective value of at least γ − ζ, the algorithm terminates and reports the current solution.
+Otherwise, we construct the refining problem associated with our current solution and proceed to the next
+iteration.
+The next result gives the overall running time required to solve (3) to additive error O(ǫ) using the
+QRAM input model.
+Theorem 6. Let C ∈ Sn, ǫ ∈ (0, 1), and set ζ =
+�
+ǫ
+n∥C∥F
+�4
+. Assume we have classical access to C. Then,
+in the QRAM input model, Algorithm 3 solves (3) up to additive error O(ǫ) using
+O
+�
+n1.5 · polylog
+�
+n, ∥C∥F , 1
+ǫ
+��
+accesses to the QRAM and O(ns) classical arithmetic operations.
+The output of the algorithm is a collection of tuples {(η(k), y(k), q(k), d(k))}K
+k=0 such that
+˜ρ =
+1
+1 + nδ
+
+
+
+
+K
+�
+k=0
+1
+η(k) Q(k) ◦
+exp
+�
+−
+�
+y(k)
+1 Q(k) ◦
+C
+∥C∥F + y(k)
+2
+diag
+�
+d(k)���
+tr
+�
+exp
+�
+−
+�
+y(k)
+1 Q(k) ◦
+C
+∥C∥F + y(k)
+2
+diag
+�
+d(k)����
+
+ + δI
+
+ ,
+is a 2ζ-precise solution to (5), where δ is defined according to (18). The entries of ˜ρ can be modified to
+construct a matrix ρ∗ at trace distance O
+�
+ǫ
+n∥C∥F
+�
+of ˜ρ in time O(n2), such that nρ∗ is a feasible point of
+the SDO problem (3).
+Proof. Given that C is an s-sparse matrix, we can classically load C in O(ns) time. Similarly, for normaliza-
+tion purposes we classically compute ∥C∥F, which requires O(ns) arithmetic operations. In each iteration
+we use Algorithm 1 to solve (14), and use classical estimates of the diagonal elements of the refining solution,
+and a classical estimate of the objective value attained by the refining solution to update the solution and
+data for the refining problem we need to solve in the next iteration.
+Letting T quantum
+HU
+be the cost of using Algorithm 1 as an approximate SDO subroutine, by Proposition 8,
+Algorithm 1 solves (14) to additive error ξ using at most
+T quantum
+HU
+= �
+O n
+ξ
+�
+n1.5ξ−5�
+accesses to the QRAM. Classically updating the objective value requires O(1) arithmetic operations while
+updating the vector p which stores a classical description of the diagonal elements of our solution as
+pi ← pi + Qii
+η(k) ˜p(k)
+i
+35
+
+Algorithm 3 Iterative Refinement for SDO Approximations of QUBOs using a quantum computer
+Input: Error tolerances ǫ ∈ (0, 1) and ζ =
+�
+ǫ
+n∥C∥F
+�4
+, upper bound on objective value γ ∈ [−1, 1]
+Output: Tuples (η(k), y(k), q(k), d(k)) that define 2ζ-precise solution to (5) using Equation (23)
+Initialize: p ← 0n, Q ← ee⊤, εi = 1
+n for i ∈ [n], ˆγ ← 0, η(0) ← 1, k ← 0
+while max {γ − ˆγ, ∥ε∥1} > ζ do
+1. (y(k), D(k), ˜p(k), ˜c(k)) ← Solve (14) to precision ξ using Algorithm 1
+2. Store diagonal elements of Q and D(k)
+q(k)
+i
+← Qii,
+d(k)
+i
+← D(k)
+ii
+for i ∈ [n]
+3. Store description of solution to the refining problem (η(k), y(k), q(k), d(k))
+4. Update the objective value of solution:
+ˆγ ← ˆγ +
+1
+η(k) ˜c(k)
+5. Update estimate of diagonal entries:
+pi ← pi + Qii
+η(k) ˜p(k)
+i
+for i ∈ [n]
+6. Compute element-wise deviations from the maximally mixed state:
+εi ← pi − 1
+n
+for i ∈ [n]
+7. Classically update:
+Qii ← sign(−εi) for i ∈ [n],
+η(k+1) ←
+1
+max {γ − ˆγ, ∥ε∥1}
+8. k ← k + 1
+end
+36
+
+requires O(n) classical arithmetic operations. Likewise, ε and Q can each be updated using O(n) classical
+arithmetic operations, as we only need to store the diagonal elements of Q. This also implies that we can
+update Q◦
+C
+∥C∥F using �On(n) operations, for only the diagonal elements need to be updated. When compared
+to loading and normalizing the coefficient matrix C, or our use of Algorithm 1 as a subroutine for solving
+(14), these intermediate computation steps are negligible and do not factor into the overall running time
+using O notation.
+By Corollary 5, Algorithm 3 terminates in at most �
+O 1
+ζ (1) iterations. Therefore, the worst case complexity
+of Algorithm 3 can be bounded by
+O
+�
+n1.5ξ−5 · polylog
+�
+n, ∥C∥F, 1
+ǫ
+��
+accesses to the QRAM, and O (ns) classical arithmetic operations to load and normalize C. Further, it
+suffices to use fixed precision ξ for the every call to Algorithm 1 to reach a final solution that solves (5) to
+additive error ζ, as the final solution can always be arbitrarily precise using a �O 1
+ζ (1) calls to Algorithm 1.
+Since ξ is a fixed constant in the context of Algorithm 3, the overall running time of Algorithm 3 simplifies
+to
+O
+�
+n1.5 · polylog
+�
+n, ∥C∥F, 1
+ǫ
+��
+.
+accesses to the QRAM, and O (ns) classical arithmetic operations.
+Just as in the proof of Theorem 5, applying Proposition 11 with our choice of ζ =
+�
+ǫ
+n∥C∥F
+�4
+implies that
+the above running time is sufficient to obtain a solution that can be used to solve (3) up to additive error
+O(ǫ), and the proof is complete.
+We analyze the cost of Algorithm 3 without access to QRAM in Appendix A. Using the sparse-access
+input model, one can show that the resulting scheme exhibits an oracle complexity of
+O
+�
+n1.5s0.5+o(1) · polylog
+�
+n, ∥C∥F, 1
+ǫ
+��
+,
+and requires O
+�
+n2.5s0.5+o(1) · polylog
+�
+n, ∥C∥F , 1
+ǫ
+��
+additional gates.
+To summarize, in the absence of
+QRAM, the number of oracle accesses is a factor √s larger due to the Hamiltonian simulation, and the
+gate complexity increases by a factor n due to the cost of constructing OD without QRAM.
+We conclude this section by establishing the costs of preparing a block-encoding of the final solution, and
+estimating trace inner products of the form tr(A˜ρ) for a given matrix A.
+Proposition 12. Suppose that Algorithm 3 is run with ζ =
+�
+ǫ
+n∥C∥F
+�4
+for some ǫ ∈ (0, 1), and terminates
+after K iterations, classically outputting the tuples {(η(k), y(k), q(k), d(k))}K
+k=0. Then, letting C∥C∥−1
+F
+be stored
+in QRAM, and denoting the refining problem at iteration k by ρ(k), one can use {(η(k), y(k), q(k), d(k))}K
+k=0
+to implement an (n, O(log(n), θ)-block-encoding of
+˜ρ =
+1
+1 + nδ
+
+
+
+
+K
+�
+k=0
+1
+η(k) Q(k) ◦
+exp
+�
+−
+�
+y(k)
+1 Q(k) ◦
+C
+∥C∥F + y(k)
+2
+diag
+�
+d(k)���
+tr
+�
+exp
+�
+−
+�
+y(k)
+1 Q(k) ◦
+C
+∥C∥F + y(k)
+2
+diag
+�
+d(k)����
+
+ + δI
+
+ ,
+with at most �On,∥C∥F , 1
+ǫ , 1
+θ (√n) queries to the QRAM and �
+On,∥C∥F , 1
+ǫ , 1
+θ (n) classical operations.
+Proof. First, note that
+exp
+�
+−
+�
+y(k)
+1 Q(k) ◦
+C
+∥C∥F + y(k)
+2
+diag
+�
+d(k)���
+tr
+�
+exp
+�
+−
+�
+y(k)
+1 Q(k) ◦
+C
+∥C∥F + y(k)
+2
+diag
+�
+d(k)���� = n−1I
+37
+
+whenever y = (0, 0)⊤. Thus, by choosing y(K+1) = (0, 0)⊤, η(K+1) =
+1
+nδ , and Q(K+1) = ee⊤ we can simplify
+the expression of the final solution to
+1
+1 + nδ
+
+
+K+1
+�
+k=0
+1
+η(k) Q(k) ◦
+exp
+�
+−
+�
+y(k)
+1 Q(k) ◦
+C
+∥C∥F + y(k)
+2
+diag
+�
+d(k)���
+tr
+�
+exp
+�
+−
+�
+y(k)
+1 Q(k) ◦
+C
+∥C∥F + y(k)
+2
+diag
+�
+d(k)����
+
+ .
+To ensure that the stated complexity holds, for each k ∈ [K + 1], we block-encode
+A(k) = Q(k) ◦ C
+∥C∥F
++ D(k).
+First, note that with classical access to C and q(k), one can store Q(k)◦C
+∥C∥F
+in the QRAM by properly updating
+C∥C∥−1
+F
+in the QRAM. This step requires O(n) classical operations, as the only non-trivial computation that
+is performed is limited to the diagonal elements of the involved matrices. Then, with Q(k)◦C
+∥C∥F
+stored in QRAM,
+noting that
+��� Q(k)◦C
+∥C∥F
+���
+F ≤ 1 holds for every k ∈ [K + 1], we apply Lemma 3 to construct a (1, log(n) + 2, θ1)-
+block-encoding of
+Q(k)◦C
+∥C∥F
+in time O
+�
+polylog
+�
+n
+θ1
+��
+. Similarly, as we saw in the proof of Proposition 7,
+classical access to d(k) and access to QRAM implies one can implement a (1, log(n) + 3, θ1)-block-encoding
+of D(k) can be constructed in time �O n
+θ1 (1).
+Again following the proof of Proposition 7, applying Corollary 3 with y(k) satisfying ∥y(k)∥1 = �
+On(ξ−1)
+implies that we can construct a unitary which prepares a copy of the Gibbs state ρ(k) encoding the solution
+to the refining problem at iteration k with at most
+�
+O n
+ξ
+�√nαξ−1�
+= �
+On
+�√n
+�
+,
+accesses to the QRAM, as α = 1 and ξ is a fixed constant. Therefore, by Lemma 7, preparing a (1, log(n) +
+a, θ1) block-encoding of a purification of ρ(k) thus requires �
+O n
+θ1 (√n) queries to the QRAM.
+Next, provided classical access to the vector q(k) that store the diagonal elements of Q(k), access to
+QRAM implies that we can efficiently implement an oracle OQ(k) that returns the entries of Q(k) in a binary
+description:
+OQ(k) : |i⟩ |j⟩ |0⟩⊗p �→ |i⟩ |j⟩
+���q(k)
+ij
+�
+,
+∀i, j ∈ [2log n] − 1,
+where q(k)
+ij
+is a p-bit binary description of the ij-matrix element of Q(k) for k = 0, . . . , K +1. By construction
+each matrix Q(k) may be fully dense, and hence an application of Lemma 4 with sr = sc = n asserts that in
+the presence of QRAM, one can construct a (n, log(n) + 3, θ2)-block-encoding of Q(k) in time �O n
+θ2 (1).
+From here, we can utilize Proposition 4 with θ1 = θ2 =
+˜θ
+10 to construct an (n, a+ 4 log(n2)+ 12, ˜θ)-block-
+encoding of Q(k) ◦ ρ(k) in time �O n
+˜θ (1). Repeating the above steps for k = 0, . . . , K + 1, it follows that we
+can block-encode each of the terms Q(k) ◦ ρ(k) using at most
+�On, 1
+˜
+θ
+�
+K√n
+�
+= �On,∥C∥F , 1
+ǫ , 1
+˜
+θ
+�√n
+�
+queries to the QRAM and �On,∥C∥F , 1
+ǫ , 1
+˜
+θ (n) classical operations, as K = O
+�
+polylog
+�
+n, ∥C∥F, 1
+ǫ
+��
+by Corollary
+5.
+Finally, what remains is to take the linear combination of these terms. To do so, we choose our weights
+to be wk =
+1
+2(1+nδ)η(k) , which indeed satisfies ∥w∥1 ≤ 1. Then, we can construct a (K + 2, log(K + 2), 0)-
+state-preparation pair PL, PR for w, which can be constructed by taking a log(K + 2)-fold tensor product
+of the Hadamard gate, i.e.,
+PL = PR =
+1
+√
+2
+�1
+1
+1
+−1
+�⊗ log(K+2)
+.
+38
+
+We are now in a position to apply Proposition 1, and choosing ˜θ =
+θ
+n, we can obtain W upon adding a
+control qubit to the circuits used to construct the block-encoding of each Q(k) ◦ ρ(k). As a result, we obtain
+an (n, O(log(n), θ)-block-encoding of ˜ρ with a single use of W, PR and P †
+L. Summing the cost of each step in
+the construction we arrive at total cost of
+�
+On,∥C∥F , 1
+ǫ , 1
+θ
+�√n
+�
+queries to the QRAM and �On,∥C∥F , 1
+ǫ , 1
+θ (n) classical operations, and proof is complete.
+Proposition 13. Suppose that Algorithm 3 is run with ζ =
+�
+ǫ
+n∥C∥F
+�4
+for some ǫ ∈ (0, 1), and terminates
+after K iterations, classically outputting the tuples {(η(k), y(k), q(k), d(k))}K
+k=0. Let A ∈ Rn×n be a matrix with
+∥A∥F ≤ 1 and assume classical access to A and C/∥C∥F. Then, with access to QRAM, one can compute a
+θ-precise estimate of tr(A˜ρ) using at most �On,∥C∥F , 1
+ǫ
+� √n
+θ
+�
+queries to the QRAM and �On,∥C∥F , 1
+ǫ (n) classical
+operations.
+Proof. See the proof of Theorem 8 in Appendix B.
+A QRAM-free version of Proposition 13 is also analyzed in Appendix B, and the cost is summarized
+in Corollary 7.
+Without access to QRAM, the cost increases with respect to n because computing the
+Hadamard product of block-encodings introduces n as a subnormalization factor. This is compounded in
+the running time, upon noting that we then have to scale down the error for the amplitude estimation steps
+by n, and constructing sparse-access oracles for the intermediate block-encodings of Q and D that arise in
+the trace estimation procedure requires �
+On(n) gates.
+5.3
+Comparison to existing SDO algorithms
+Table 1 presents a comparison of the running time results for the algorithms we have proposed with the
+running times of the best performing methods from both the classical and quantum literature when applied
+to solving (3).
+Note that when directly solving (3), m = n, and any feasible solution X to (3) satisfies tr (X) = n,
+implying R = n for the algorithms based on the (Q)MMWU framework. We also point out that the running
+times in Table 1 take into account the role of sparsity in context of the algorithms, which is measured as the
+maximum number of nonzero entries per row of the constraint matrices A1, . . . , An. When using either an
+IPM or CPM to solve (3), the n constraint matrices are Ai = eie⊤
+i (with row sparsity one) enforcing Xii = 1
+for each diagonal element. On the other hand, algorithms based on the (Q)MMWU or HU frameworks solve
+(3) by reducing the problem to a feasibility problem; C enters into the resulting formulation as another
+constraint matrix, and as a result, the relevant sparsity parameter is the maximum number of non-zeroes
+per row of C, which we denote by s in Table 1.
+There are additional considerations that need to be taken into account when making comparisons across
+methodologies listed in Table 1.
+Broadly speaking, both (Q)MMWUs and HU require normalizing the
+problem by an upper bound on the trace of a primal solution, and in the case of (3), we have the natural
+bound tr(X) = n. Moreover, (Q)MMWUs and HU additionally normalize the cost matrix so that it exhibits
+unit norm with respect to some norm. While these modifications amount to scaling the optimal objective
+value of (3) by a fixed quantity, without employing any safeguards such as IR, these modifications impact
+the scaling of the error as reflected in the fourth column of Table 1. On the contrary, (Q)IPMs do not
+require the SDO problem to be normalized in any way. Finally there is a distinction with regard to output;
+(Q)IPMs explicitly report a classical description of the solution X, whereas only the classical HU algorithm
+of [10] and our own classical IR-HU method do so; the primal QMMWU of [60] reports a state-preparation
+pair y, and the MMWU algorithm found in [41] reports a “gradient” G ∈ Sn such that X = W exp(G)W
+for a diagonal matrix W. As we noted earlier, (Q)IPMs and (Q)MMWUs also utilize different definitions of
+optimality.
+39
+
+References
+Method
+Runtime
+Error Scaling
+[34]
+IPM
+�
+On, 1
+ǫ
+�
+nω+0.5�
+ǫ
+[6]
+QIPM
+�
+On,κ, 1
+ǫ
+�√n(n3κǫ−1 + n4)
+�
+ǫ
+[41]
+MMWU
+�
+On, 1
+ǫ
+�
+nsǫ−3.5�
+∥C∥ℓ1ǫ
+[60]
+QMMWU
+�
+On, 1
+ǫ
+�
+n5.5sǫ−4�
+n∥C∥ǫ
+[10] (Classical)
+HU
+�
+On,∥C∥
+�
+min{n2s, nω}ǫ−12�
+n∥C∥ǫ
+[10] (Quantum)
+HU
+�
+On,∥C∥, 1
+ǫ
+�
+n2.5s0.5+o(1)ǫ−28+o(1) exp
+�
+1.6
+�
+log(ǫ−1)
+��
+n∥C∥ǫ
+[10] (Quantum)
+HU-QRAM
+�
+On,∥C∥, 1
+ǫ
+�
+n1.5s0.5+o(1)ǫ−28+o(1) exp
+�
+1.6
+�
+log(ǫ−1)
+��
+n∥C∥ǫ
+This work (Classical)
+IR-HU
+�
+On,∥C∥F , 1
+ǫ
+�
+min{n2s, nω}
+�
+ǫ
+This work (Quantum)
+IR-HU
+�
+On,∥C∥F , 1
+ǫ
+�
+n2.5s0.5+o(1)�
+ǫ
+This work (Quantum)
+IR-HU-QRAM
+�
+On,∥C∥F , 1
+ǫ (n1.5) + ns
+ǫ
+Table 1: Total running times for classical and quantum algorithms to solve (3).
+It can be easily seen that both the classical and quantum implementations of our proposed methodology
+outperform all existing algorithms that exhibit poly-logarithmic dependence on the precision ǫ. Our classical
+algorithm is only outperformed with respect to dimension by our own quantum algorithms, and the algorithm
+from [41], which has an exponentially worse dependence on the inverse prevision. Moreover, to achieve the
+same error scaling as our algorithms, the algorithm from [41] would require time �On, 1
+ǫ
+�
+n4.5sǫ−3.5�
+(as it is
+assumed ∥C∥ℓ1 = n in [41]). Up to poly-logarithmic factors, our quantum algorithms outperform each of
+the classical and quantum solvers in every parameter, suggesting the first evidence of quantum advantage
+for solving a special class of SDO problems. Moreover, our implementation with access to QRAM dominates
+all other algorithms. We therefore conclude that our proposed algorithms are respectively, the fastest both
+in the classical and quantum regimes.
+6
+Conclusion
+In this work we devised an iterative refinement scheme for a particular class of semidefinite optimization
+problems. The key to our idea behind our speedup is to solve a sequence of related SDO problems in fixed
+low precision, rather than solve one SDO problem using high accuracy requirements. Moreover, our solutions
+satisfy a far stronger approximation guarantee over previous quantum solution methodologies for this class of
+problem. We show that, provided access to QRAM, a quantum implementation of our algorithm can produce
+accurate solutions to SDO approximations of QUBO problems in time O
+�
+ns + n1.5 · polylog
+�
+n, ∥C∥F, 1
+ǫ
+��
+in the worst case. In the absence of QRAM, one can bound the running time of the quantum algorithm
+using using the sparse-access input model, in which case the algorithm exhibits an oracle complexity of
+O
+�
+n2.5s0.5+o(1) · polylog
+�
+n, ∥C∥F, 1
+ǫ
+��
+.
+A classical implementation of the algorithm exhibits worst case
+running time of O
+�
+min{n2s, nω} · polylog
+�
+n, ∥C∥F, 1
+ǫ
+��
+, which is at least a √n factor better than classical
+IPMs.
+When compared to the best performing algorithms in the literature, our algorithms are the fastest in both
+the quantum and classical regimes, respectively. We therefore conclude that this work serves as evidence of a
+genuine quantum advantage for a specific class of SDO problems. We believe one can improve the theoretical
+performance of our classical algorithm by not explicitly computing the density operator in our subroutines.
+In particular, it may be possible to construct the separation oracles as we do in the quantum setting using
+techniques to classically estimate trace inner products of the form tr(Aρ) (see, e.g., Appendix A in [61]), and
+applying ideas developed in [3, 41] to estimate the diagonal elements of matrix exponentials via randomized
+projection [36]. It remains an open question as to whether our techniques can be applied to general SDO
+problems using the matrix-multiplicative weights update framework as a subroutine.
+40
+
+Acknowledgements
+This project has been carried out thanks to funding by the Defense Advanced Research Projects Agency
+(DARPA), ONISQ grant W911NF2010022, titled The Quantum Computing Revolution and Optimization:
+Challenges and Opportunities.
+Giacomo Nannicini is partially supported by the Army Research Office under grant number W911NF-20-1-
+0014.
+A
+Running time of Algorithm 3 without QRAM
+The following result from [10] gives the sample complexity of implementing the oracles in the sparse-access
+model.
+Lemma 16 (see, proof of Lemma 3.3 in [10]). We can implement the oracle OCγ on a quantum computer
+given access to O(ǫ−2) copies of a state that is an
+ǫ
+8-approximation of the input state ρ in trace distance.
+The oracle ODn can be implemented using O(nǫ−2)
+ǫ
+8-approximate copies of the input, and the classical
+post-processing time needed to implement the oracle is O(nǫ−2).
+Next, we bound the overall complexity of Algorithm 1 without access to QRAM.
+Proposition 14. Suppose that C ∈ Sn has row sparsity s and ξ ∈ (0, 1). Then, in the sparse-access input
+model, the complexity of solving (5) up to additive error ξ using Algorithm 1 on a quantum computer requires
+�
+On
+�
+n1.5√s
+1+o(1)ξ−7+o(1) exp
+�
+1.6
+�
+log(ξ−1)
+��
+queries to the input oracle OC and �On
+�
+n2.5√s1+o(1)ξ−7+o(1) exp
+�
+1.6
+�
+log(ξ−1)
+��
+additional gates.
+Proof. Our proof can be viewed as the QRAM-free analogue of the discussion found in [10, Section 3.4], and
+we repeat it here for completeness. In order to derive an appropriate bound on the per-iteration cost, we
+need to evaluate the cost of constructing our separation oracles. By Lemma 16, we can conclude that the
+time to construct the oracle ODn for the diagonal elements dominates that of constructing the oracle OCγ to
+test the objective value.
+We now turn our attention to the cost of simulating our Hamiltonian H.
+From the results in [53,
+Appendix] it follows that we can produce a state that is
+ξ
+8 close to ρ using �O(√nξ−3) invocations of a
+controlled U which satisfies
+��U − eit0H�� ≤ O
+�
+ξ3�
+,
+with t0 =
+π
+4∥H∥. Further, the authors in [10] note that each of the Hamiltonians we seek to simulate are
+of the form H = y1C∥C∥−1
+F + y2D where y1, y2 = O(log(n)ξ−1) and D is a diagonal matrix which satisfies
+∥D∥ ≤ 1. Invoking [15, Theorem 1], we can simulate H for time t up to error ξ3 using
+�
+O
+�
+t(a + b) exp
+�
+1.6
+�
+log (log(n)tξ−3)
+��
+separate simulations of y1C∥C∥F and y2D.
+As noted in [10], access to the oracles Osparse and OC we described in Section 2.1.1 allows us to simulate
+exp(itC∥C∥−1
+F ) in time O
+�
+(t√s)1+o(1)ξo(1)�
+if we utilize the algorithm in [44]. Similarly, we follow [10] in
+constructing an oracle OD acting on C ⊗ (C2)⊗a, where a is a sufficiently large constant such that we can
+represent the diagonal elements of D as
+OD |i, z⟩ �→ |i, z ⊕ Dii⟩
+41
+
+to the desired level of precision in binary. Accordingly, we can simulate eiDt for t = �O(ξ−1) using �On(1)
+queries to OD and �
+On(1) elementary operations [7], and we can implement OD using �On(n) gates.
+To summarize, the Gibbs sampler from [53] requires �O(√nξ−3) Hamiltonian simulation steps, each of
+which requires time
+�
+O
+�√s
+1+o(1)ξo(1) exp
+�
+1.6
+�
+log(ξ−1)
+��
+.
+Hence, each iteration of Algorithm 1 requires a total of
+�
+On
+�
+n1.5√s
+1+o(1)ξ−5+o(1) exp
+�
+1.6
+�
+log(ξ−1)
+��
+sparse-access oracle queries. Combining the above per-iteration cost with the iteration bound O(log(n)ξ−2)
+provided in Theorem 3, it follows that Algorithm 1 solves (5) up to additive error ξ with at most
+�
+On
+�
+n1.5√s
+1+o(1)ξ−7+o(1) exp
+�
+1.6
+�
+log(ξ−1)
+��
+queries to the input oracle OC and �
+On
+�
+n2.5√s1+o(1)ξ−7+o(1) exp
+�
+1.6
+�
+log(ξ−1)
+��
+additional gates.
+Theorem 7 formalizes the complexity of of Algorithm 3 in the quantum setting without access to QRAM.
+In our analysis, we employ the same Hamiltonian simulation subroutines and Gibbs sampler used in [10] to
+construct our separation oracles.
+Theorem 7. Let C ∈ Sn with row sparsity s and ǫ ∈ (0, 1).
+Then, setting ζ =
+�
+ǫ
+n∥C∥F
+�4
+and fixing
+ξ = 10−2, a quantum implementation of Algorithm 3 using the sparse-access input model solves (3) up to
+additive error O(ǫ) using
+O
+�
+n1.5s0.5+o(1) · polylog
+�
+n, ∥C∥F , 1
+ǫ
+��
+queries to the input oracle OC and O
+�
+n2.5s0.5+o(1) · polylog
+�
+n, ∥C∥F, 1
+ǫ
+��
+additional gates.
+The output of the algorithm is a collection of tuples {(η(k), y(k), q(k), d(k))}K
+k=0 such that
+˜ρ =
+1
+1 + nδ
+
+
+
+
+K
+�
+k=0
+1
+η(k) Q(k) ◦
+exp
+�
+−
+�
+y(k)
+1 Q(k) ◦
+C
+∥C∥F + y(k)
+2
+diag
+�
+d(k)���
+tr
+�
+exp
+�
+−
+�
+y(k)
+1 Q(k) ◦
+C
+∥C∥F + y(k)
+2
+diag
+�
+d(k)����
+
+ + δI
+
+ ,
+is a 2ζ-precise solution to (5), where δ is defined according to (18). The entries of ˜ρ can be modified to
+construct a matrix ρ∗ at trace distance O
+�
+ǫ
+n∥C∥F
+�
+of ˜ρ in time O(n2), such that nρ∗ is a feasible point of
+the SDO problem (3).
+Proof. Given that C is an s-sparse matrix, we can load C in O(ns) time.
+Similarly, for normalization
+purposes we classically compute ∥C∥F, which requires O(ns) arithmetic operations. In each iteration we use
+Algorithm 1 to solve (14), and use classical estimates of the diagonal elements of the refining solution, and
+a classical estimate of the objective value attained by the refining solution to update the solution and data
+for the refining problem we need to solve in the next iteration.
+Letting T sparse
+HU
+be the cost of using Algorithm 1 as an approximate SDO subroutine, we saw in Proposition
+14, Algorithm 1 solves (14) to additive error ξ using
+T sparse
+HU
+= �On
+�
+n1.5√s
+1+o(1)ξ−7+o(1) exp
+�
+1.6
+�
+log(ξ−1)
+��
+queries to the oracle describing the problem data and �On
+�
+n2.5√s1+o(1)ξ−7+o(1) exp
+�
+1.6
+�
+log(ξ−1)
+��
+addi-
+tional gates. In the context of Algorithm 3, ξ is a fixed constant, so the cost of our oracle call to Algorithm
+1 simplifies to
+T sparse
+HU
+= �On
+�
+n1.5√s
+1+o(1)�
+42
+
+queries to the oracle describing the problem data and �On
+�
+n2.5√s1+o(1)�
+additional gates.
+Classically updating the objective value requires O(1) arithmetic operations while updating the vector p
+which stores a classical description of the diagonal elements of our solution as
+pi ← pi + Qii
+η(k) ˜p(k)
+i
+requires O(n) arithmetic operations. Again, ε and Q can each be updated using O(n) arithmetic operations,
+as we only need to store the diagonal elements of Q. This also implies that we can also calculate Q◦
+C
+∥C∥F in
+time O(n), for only the element-wise products along the diagonal are non-trivial. When compared to loading
+and normalizing the data or our use of Algorithm 1 as a subroutine for solving (14), these intermediate
+computation steps are negligible and do not factor into the overall running time using O notation.
+Factoring in the O
+�
+polylog
+�
+1
+ζ
+��
+= O
+�
+polylog
+�
+n, ∥C∥F, 1
+ǫ
+��
+from Corollary 5, it follows that a quantum
+implementation of Algorithm 3 requires at most
+O
+�
+n1.5s0.5+o(1) · polylog
+�
+n, ∥C∥F , 1
+ǫ
+��
+queries to the input oracle OC and O
+�
+n2.5s0.5+o(1) · polylog
+�
+n, ∥C∥F , 1
+ǫ
+��
+additional gates. Just as in the
+proof of Theorem 5, applying Proposition 11 with our choice of ζ =
+�
+ǫ
+n∥C∥F
+�4
+implies that the above running
+time is sufficient to obtain a solution that can be used to solve (3) up to additive error O(ǫ), and the proof
+is complete.
+B
+Estimating trace inner products with the final solution
+Given that we do not explicitly report a classical description of the final solution ˜ρ defined in equation (23),
+it may be of interest to understand how, for a user specified matrix A, one can compute the trace inner
+product tr(A˜ρ). We outline a procedure for doing so using the state preparation pair description of solution
+{(η(k), y(k), q(k), d(k))}K
+k=0 in Algorithm 4, and subsequently analyze the complexity of doing so.
+Theorem 8. Let A ∈ Rn×n, and
+C
+∥C∥F ∈ Sn be stored in QRAM, θ ∈ (0, 1), and {(η(k), y(k), q(k), d(k))}K
+k=0
+be a state preparation pair description of the solution obtained from running Algorithm 3 to final precision
+ζ =
+�
+ǫ
+n∥C∥F
+�4
+. Suppose A is an s-sparse matrix with ∥A∥F ≤ 1, and assume classical access to A and
+C
+∥C∥F ∈ Sn. Then, Algorithm 4 outputs a θ-precise estimate of
+tr(A˜ρ) =
+1
+1 + nδ tr
+
+A
+
+
+
+
+K
+�
+k=0
+1
+η(k) Q(k) ◦
+exp
+�
+−
+�
+y(k)
+1 Q(k) ◦
+C
+∥C∥F + y(k)
+2
+diag
+�
+d(k)���
+tr
+�
+exp
+�
+−
+�
+y(k)
+1 Q(k) ◦
+C
+∥C∥F + y(k)
+2
+diag
+�
+d(k)����
+
+ + δI
+
+
+
+
+using at most
+�On,∥C∥F , 1
+ǫ
+�√n
+θ
+�
+queries to the QRAM and �
+On,∥C∥F , 1
+ǫ (ns) classical operations.
+Proof. We begin by establishing the correctness of Algorithm 4.
+First, note that following the proof of
+Proposition 12, we can simplify the expression of the final solution to
+˜ρ =
+1
+1 + nδ
+
+
+K+1
+�
+k=0
+1
+η(k) Q(k) ◦
+exp
+�
+−
+�
+y(k)
+1 Q(k) ◦
+C
+∥C∥F + y(k)
+2
+diag
+�
+d(k)���
+tr
+�
+exp
+�
+−
+�
+y(k)
+1 Q(k) ◦
+C
+∥C∥F + y(k)
+2
+diag
+�
+d(k)����
+
+ .
+43
+
+Algorithm 4 Trace estimation procedure for the final solution
+Input: Access to an s-sparse matrix A ∈ Rn×n with ∥A∥F ≤ 1, state preparation pair description of solution
+{(η(k), y(k), q(k), d(k))}K
+k=0, precision θ ∈ (0, 1), ζ =
+�
+ǫ
+n∥C∥F
+�4
+Output: A θ-precise classical estimate of tr(A˜ρ)
+Initialize: a ← 0, k ← 0, y(K+1) ← (0, 0)⊤, η(K+1) ←
+1
+nδ, Q(K+1) ← ee⊤
+for k = 0, . . . , K + 1 do
+1. Implement an (α, a, ζ/2(K + 2))-block-encoding of Q(k) ◦ A
+2. Use block-encoding of Q(k) ◦ A to implement a trace estimator for
+a(k) = tr
+
+
+�
+Q(k) ◦ A
+�
+
+
+exp
+�
+−
+�
+y(k)
+1 Q(k) ◦
+C
+∥C∥F + y(k)
+2
+diag
+�
+d(k)���
+tr
+�
+exp
+�
+−
+�
+y(k)
+1 Q(k) ◦
+C
+∥C∥F + y(k)
+2
+diag
+�
+d(k)����
+
+
+
+
+3. Use O
+� K
+θ
+�
+samples from the trace estimator to produce
+θ
+K+2-precise estimate ˜a(k) of a(k)
+4. Update solution:
+a ← a +
+1
+η(k) ˜a(k)
+5. k ← k + 1
+end
+Scale down estimate to account for spectrum shift:
+a ←
+1
+1 + nδ a
+44
+
+by setting y(K+1) = (0, 0)⊤, η(K+1) =
+1
+nδ, and Q(K+1) = ee⊤. Then, by linearity of the trace and Lemma 1,
+one has:
+tr(A˜ρ) = tr
+
+A
+1
+1 + nδ
+
+
+K+1
+�
+k=0
+1
+η(k) Q(k) ◦
+exp
+�
+−
+�
+y(k)
+1 Q(k) ◦
+C
+∥C∥F + y(k)
+2
+diag
+�
+d(k)���
+tr
+�
+exp
+�
+−
+�
+y(k)
+1 Q(k) ◦
+C
+∥C∥F + y(k)
+2
+diag
+�
+d(k)����
+
+
+
+
+=
+1
+1 + nδ
+K+1
+�
+k=0
+1
+η(k) tr
+
+A
+
+Q(k) ◦
+exp
+�
+−
+�
+y(k)
+1 Q(k) ◦
+C
+∥C∥F + y(k)
+2
+diag
+�
+d(k)���
+tr
+�
+exp
+�
+−
+�
+y(k)
+1 Q(k) ◦
+C
+∥C∥F + y(k)
+2
+diag
+�
+d(k)����
+
+
+
+
+=
+1
+1 + nδ
+K+1
+�
+k=0
+1
+η(k) tr
+
+
+�
+Q(k) ◦ A
+�
+exp
+�
+−
+�
+y(k)
+1 Q(k) ◦
+C
+∥C∥F + y(k)
+2
+diag
+�
+d(k)���
+tr
+�
+exp
+�
+−
+�
+y(k)
+1 Q(k) ◦
+C
+∥C∥F + y(k)
+2
+diag
+�
+d(k)����
+
+ .
+In other words, the output of Algorithm 4 is indeed an estimate of tr(A˜ρ).
+Next, we analyze the complexity of the procedure. If A is classically known, one can store Q(k) ◦ A in the
+QRAM using O(ns) classical operations, as A is s-sparse. With Q ◦ A stored in a QRAM data structure,
+one can apply Lemma 3 to implement an (1, log(n) + 2, ζ/2(K + 2))-block-encoding of Q ◦ A in time �
+O nK
+ζ (1)
+(as ∥Q ◦ A∥F ≤ ∥A∥F ≤ 1 for any Q defined according to (13)). As we saw in the proof of Proposition 12,
+with
+C
+∥C∥F stored in QRAM, one can implement the state
+ρ(k) =
+exp
+�
+−
+�
+y(k)
+1 Q(k) ◦
+C
+∥C∥F + y(k)
+2
+diag
+�
+d(k)���
+tr
+�
+exp
+�
+−
+�
+y(k)
+1 Q(k) ◦
+C
+∥C∥F + y(k)
+2
+diag
+�
+d(k)����
+using at most
+�On
+�√n
+�
+,
+accesses to the QRAM and O(n) classical operations.
+Having prepared the state ρ(k) and a (1, log(n) + 2, ζ/2(K + 2))-block-encoding Uk of Q(k) ◦ A, Lemma
+8 asserts that one can implement a trace estimator for
+tr
+��
+Q(k) ◦ A
+�
+ρ(k)�
+with bias at most
+ζ
+K+2 using �O(1) applications of Uk and U †
+k. Applying amplitude estimation using O
+� K
+θ
+�
+=
+�
+On,∥C∥F , 1
+ǫ
+� 1
+θ
+�
+samples from the estimator, we obtain a
+θ
+K+2-precise classical estimate ˜a(k) of a(k), as K =
+O
+�
+polylog
+�
+n, ∥C∥F , 1
+ǫ
+��
+.
+From here, we classically update a using O(1) arithmetic operations. Therefore, each iteration of Algo-
+rithm 4 requires at most
+�
+On, K
+ζ
+�√n
+θ
+�
+accesses to the QRAM and O(ns) classical operations. Summing over K + 2 iterations implies a total of
+�
+On, K
+ζ
+�
+K
+�√n
+θ
+��
+= �
+On,∥C∥F , 1
+ǫ
+�√n
+θ
+�
+accesses to the QRAM and
+O (Kns) = �On,∥C∥F , 1
+ǫ (ns)
+classical operations. The proof is complete.
+Note that if ∥A∥F > 1, because of the subnormalization to block-encode A we need to increase precision
+of the estimation procedure: the cost increases by a factor proportional to ∥A∥F.
+45
+
+Corollary 7. Let A ∈ Rn×n, θ ∈ (0, 1), and {(η(k), y(k), q(k), d(k))}K
+k=0 be a state preparation pair description
+of the solution obtained from running Algorithm 3 to final precision ζ =
+�
+ǫ
+n∥C∥F
+�4
+. Suppose A is an s-sparse
+matrix with ∥A∥F ≤ 1, and assume sparse oracle access to A and
+C
+∥C∥F ∈ Sn. Then, Algorithm 4 outputs a
+θ-precise estimate of
+tr(A˜ρ) =
+1
+1 + nδ tr
+
+A
+
+
+
+
+K
+�
+k=0
+1
+η(k) Q(k) ◦
+exp
+�
+−
+�
+y(k)
+1 Q(k) ◦
+C
+∥C∥F + y(k)
+2
+diag
+�
+d(k)���
+tr
+�
+exp
+�
+−
+�
+y(k)
+1 Q(k) ◦
+C
+∥C∥F + y(k)
+2
+diag
+�
+d(k)����
+
+ + δI
+
+
+
+
+using at most
+�
+On,∥C∥F , 1
+ǫ
+�n2.5s2
+θ
+�
+queries to OA, OC, and �
+On,∥C∥F , 1
+ǫ
+�
+n3.5s2
+θ
+�
+additional gates.
+Proof. Provided classical access to A, we use Lemma 4 with sr = sc to construct an (s, log(n) + 3, θ/n)-
+block-encoding of A with two uses of OA (an oracle describing the elements of A in binary), and additionally
+using �On (1) one and two qubit gates.
+Likewise, with access to the oracle OC describing the elements of C∥C∥−1
+F , one can construct an (s, log(n)+
+3, θ/n)-block-encoding of C∥C∥−1
+F with two uses of OC, and additionally using �
+On (1) one and two qubit gates.
+Note that without access to QRAM, we must compute the Hadamard products by taking the Hadamard prod-
+ucts of block-encodings, which causes the subnormalization factor for the Hadamard product Q(k) ◦ C∥C∥−1
+F
+to be ns, as Q(k) may be fully dense and C is s-sparse. It follows that preparing one copy of each Gibbs
+state requires
+�
+On
+�√n(ns)
+�
+= �
+On
+�
+n1.5s
+�
+accesses to block-encodings of Q(k) ◦ C∥C∥−1
+F
+and D, which each require an additional �
+On(n) gates (to
+construct sparse-access oracles for Q(k) and D).
+Similarly, the subnormalization factor for a block-encoding Uk of Q(k) ◦ A will be ns. Having prepared
+the state ρ(k) and a block-encoding Q(k) ◦ A, Lemma 8 asserts that one can implement a trace estimator for
+tr
+��
+Q(k) ◦ A
+�
+ρ(k)�
+with bias at most
+ζ
+K+2 using �O(ns) applications of Uk and U †
+k.
+Applying amplitude estimation using
+O
+� K
+θ
+�
+= �On,∥C∥F , 1
+ǫ
+� 1
+θ
+�
+samples from the estimator to obtain a
+θ
+K+2-precise classical estimate ˜a(k) of a(k),
+as K = O
+�
+polylog
+�
+n, ∥C∥F , 1
+ǫ
+��
+.
+Just as in the QRAM setting, classically updating a requires O(1) arithmetic operations. Therefore,
+without access to QRAM, each iteration of Algorithm 4 requires at most
+�
+On, K
+ζ
+�n2.5s2
+θ
+�
+= �
+On,∥C∥F , 1
+ǫ
+�n2.5s2
+θ
+�
+applications of block-encodings for Q(k)◦C∥C∥−1
+F , D(k) and Q(k)◦A and �On,∥C∥F , 1
+ǫ
+�
+n3.5s2
+θ
+�
+additional gates.
+This corresponds to �On,∥C∥F , 1
+ǫ
+�
+n2.5s2
+θ
+�
+queries to OA and OC in each iteration, and �On,∥C∥F , 1
+ǫ
+�
+n3.5s2
+θ
+�
+additional gates. Summing over the K + 2 = �
+On,∥C∥F , 1
+ǫ (1) iterations yields the stated complexity.
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diff --git a/FtE2T4oBgHgl3EQf-Qmd/content/tmp_files/load_file.txt b/FtE2T4oBgHgl3EQf-Qmd/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..7a2a1c759efa010db31016f35988598831e67435
--- /dev/null
+++ b/FtE2T4oBgHgl3EQf-Qmd/content/tmp_files/load_file.txt
@@ -0,0 +1,1644 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf,len=1643
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='04237v1  [quant-ph]  10 Jan 2023  Solving the semidefinite relaxation of QUBOs in matrix  multiplication time, and faster with a quantum computer  Brandon Augustino ∗†, Giacomo Nannicini ‡, Tam´as Terlaky†, and Luis F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Zuluaga†  January 12, 2023  Abstract  Recent works on quantum algorithms for solving semidefinite optimization (SDO) problems have  leveraged a quantum-mechanical interpretation of positive semidefinite matrices to develop methods  that obtain quantum speedups with respect to the dimension n and number of constraints m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' While  their dependence on other parameters suggests no overall speedup over classical methodologies, some  quantum SDO solvers provide speedups in the low-precision regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' We exploit this fact to our advan-  tage, and present an iterative refinement scheme for the Hamiltonian Updates algorithm of Brand˜ao et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' (Quantum 6, 625 (2022)) to exponentially improve the dependence of their algorithm on precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' As  a result, we obtain a classical algorithm to solve the semidefinite relaxation of Quadratic Unconstrained  Binary Optimization problems (QUBOs) in matrix multiplication time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Provided access to a quan-  tum read/classical write random access memory (QRAM), a quantum implementation of our algorithm  exhibits a worst case running time of O  �  ns + n1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='5 · polylog  �  n, ∥C∥F , 1  ǫ  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  1  Introduction  We consider optimization problems of the form:  max x⊤Cx  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' x ∈ {−1, 1}n,  (1)  where C ∈ Sn is the problem data and Sn is the space of symmetric matrices in Rn×n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Solving (1) can  be viewed as computing the ∞ → 1 norm of the coefficient matrix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' This particular norm is intrinsically  related the cut norm of a matrix, which plays a crucial role in developing efficient approximation algorithms  for dense graph and matrix problems [1, 21], with perhaps the most well-known application being the task  of finding the largest cut in a graph (MaxCut).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' These problems also play an important role in quantum  information sciences;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' the Ising model belongs to this class of problems [52], and quantum algorithms such as  the Quantum Approximate Optimization Algorithm (QAOA) [18] and quantum annealing [19] can address  its solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Computing the cut norm corresponds to replacing x ∈ {−1, 1}n with z ∈ {0, 1}n in (1), giving rise to  quadratic unconstrained binary optimization (QUBO) problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' A standard QUBO is of the form  max z⊤Cz  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' z ∈ {0, 1}n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  (2)  ∗Corresponding Author: bra216@lehigh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='edu  †Department of Industrial and Systems Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Quantum Computing and Optimization Lab,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Lehigh University  ‡Department of Industrial and Systems Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' University of Southern California  1  Provided that we allow for linear terms (in both formulations),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' it is well known that solutions to (1) can  be used to compute a solution to (2) which differs only by a constant factor,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' and vice-versa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' due to the  equivalence z = x+e  2  if z ∈ {0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' 1}n and x ∈ {−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' 1}n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' where e ∈ Rn is the all ones vector of dimension n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Although (1) and (2) cover many applications of interest, they are intrinsically difficult to solve;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' computing  optimal solutions to either (1) or (2) is NP-Hard in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Following the seminal work of Lov´asz [43] and the  theoretical and practical development of Interior Point Methods (IPMs) for solving semidefinite optimization  (SDO) problems [46, 47, 48, 49, 50, 55, 56], a prevailing approach has been to obtain approximate solutions  to (1) and (2) by relaxing integrality and lifting the problem from a vector space of dimension n, to the  space of n × n symmetric matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' The quadratic form x⊤Cx can be equivalently expressed by tr (Cxx⊤),  where tr (U) denotes the sum of the diagonal elements (or, trace) of a matrix U ∈ Rn×n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' To deal with the  bilinear term xx⊤, we introduce a matrix variable X ∈ Rn×n, and require that X satisfies the following:  diag(X) = e,  X ⪰ 0,  rank(X) = 1,  where the notation U ⪰ V means that the matrix U − V is a symmetric positive semidefinite matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Under  these requirements, X is guaranteed to be of the form X = xx⊤ for x ∈ {−1, 1}n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' The rank constraint,  however, is not convex, and thus dropping it yields the following (convex) SDO relaxation of (1):  max  tr (CX)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  diag (X) = e,  X ⪰ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  (3)  Although the optimal solution X∗ to (3) is no longer guaranteed to satisfy X∗ = x∗x∗⊤ and may not be  integral in general, the approximation of x∗ provided by X∗ is of sufficient quality to justify its use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' In  fact, SDO approximations cover some of the most celebrated results in optimization, such as the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='878-  approximation guarantee of Goemans and Williamson for MaxCut [28] and the Lov´asz-ϑ number [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='1  Literature Review  More generally, a (primal) SDO problem involving n × n matrices and m constraints is of the form  max  X  tr (CX)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  tr (AiX) = bi  for i ∈ [m],  X ⪰ 0,  where [m] = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' , m} and A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' , Am, C ∈ Sn, and b ∈ Rm are the (given) problem data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' The dual SDO  problem associated with the primal is given by  min  (u,S)  b⊤u  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  S =  m  �  i=1  uiAi − C ⪰ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  where S is the dual slack matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='1 The classical literature on algorithms for solving SDO problems is rich  and can be categorized into two classes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' algorithms that depend poly-logarithimically on the inverse precision  to which we solve the problem and the size of the minimally inscribed ellipsoid, and algorithms that depend  polynomially on these quantities but exhibit an advantage with respect to n and m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' For instances with  m ≤ √n, the cutting plane methods (CPMs) of [35, 42] are the best performing classical algorithms,2 and  can solve SDO problems in time  O  �  m(mns + m2 + nω) · polylog  �  m, n, R, 1  ǫ  ��  ,  1While the dual variable is typically denoted by y rather than u, it is also customary in the literature to use y to denote a  certain state preparation pair, and we do so later in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  2We remark that the running time in [35] does however exhibit improved dependence with respect to poly-logarithmic factors  compared to the running time of [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  2  where ω ∈ [2, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='38] is the matrix multiplication exponent, R is an upper bound on the trace of a primal  optimal solution X (which can be exponentially large), ǫ is the precision parameter, s denotes the maximum  number of nonzeros per row of the input matrices and hence, O(mns) is the total number of nonzeros in the  constraints of SDO problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' However, we typically have m ∈ [Ω(n), O(n2)], in which case the CPMs given  in [35, 42] are outperformed by the IPM for SDO from Jiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Their IPM exhibits a worst case  running time of  O  �√n(mns + mω + nω) · polylog  �  m, n, 1  ǫ  ��  ,  where the term mω + nω represents the per-iteration cost of inverting the Hessian and matrices of the  variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  While quantum SDO solvers could also be categorized in a somewhat similar fashion, it is perhaps more  natural to do so according to how they attempt to obtain quantum speedups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' In this case we also have two  classes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' at a high level, all proposed quantum SDO solution methodologies quantize a classical algorithm by  either using quantum linear system algorithms (QLSAs) [12, 14, 31], or a quantum mechanical interpretation  of normalized positive semidefinite matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' We now review these works in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  The former class is comprised of algorithms that quantize IPMs, giving rise to quantum IPMs (QIPMs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  QIPMs attempt to speedup the bottleneck of the classical IPM by substituting the classical solution of the  Newton linear system with the combined use of QLSA and quantum state tomography (with some classical  computation between iterates).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Augustino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' [6] present a convergent QIPM for SDO, avoiding the  shortcomings prevalent in early works on QIPMs (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=', [39]), by properly symmetrizing the Newton linear  system, and utilizing an orthogonal subspace representation of the search directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' This representation  guarantees that primal and dual feasibility are satisfied exactly by all the iterates generated by inexact  solutions of the Newton linear system obtained via quantum subroutines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' The worst case complexity of their  algorithm is  �On,κ, 1  ǫ  �√n  �n3κ2  ǫ  + n4  ��  ,  where κ is an upper bound on the condition numbers of the intermediate Newton linear system coefficient ma-  trices that arise over the course of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Here, the notation �Oa,b(f(x)) suppresses poly-logarithmic  factors in f(x), a and b that appear in the overall running time, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=', �  Oa,b(f(x)) ≡ O(f(x)·polylog(a, b, f(x))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  While this QIPM achieves a speedup in n over the IPM from [34] when m = O(n2), its dependence on κ and  ǫ suggest no quantum advantage overall: the complexity of the classical IPM does not depend on κ and its  dependence on ǫ−1 is logarithmic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' As the authors in [6] note, dependence on the condition number bound κ  is particularly problematic in the context of IPMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  The second class of quantum SDO solvers are those that quantize algorithms based on matrix exponentials  and Gibbs states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' The most prominent example is the Matrix Multiplicative Weights Update (MMWU)  Method of Arora and Kale [3], which can solve SDO problems in time  �On,R, 1  ǫ  �  nms  �Rr  ǫ  �4  + ns  �Rr  ǫ  �7�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  where r is a known ℓ1-norm upper bound3 on a dual optimal solution u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Unlike IPMs, the MMWU framework  does not involve the solution of linear systems;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' rather, these algorithms alternate between candidate solutions  to the primal and dual SDO problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' IPMs and MMWUs also employ different definitions of optimality;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  for IPMs, ǫ-optimality implies that the primal and dual feasible solutions exhibit a normalized duality gap  bounded by ǫ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' :  tr (XS)  n  ≤ ǫ,  whereas an ǫ-optimal solution obtained using an MMWU approximates the optimal objective value to addi-  tive error ǫ (via binary search).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Finally, we point out a distinction between these algorithms with respect to  3It is also assumed that R, r ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  3  output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' While primal-dual IPMs return the primal-dual optimal solution (X, u, S), MMWUs report u, but  may avoid explicitly reporting X and S to maintain the speedups they offer with respect to n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Reporting X  or S under the MMWU framework necessitates the computation of matrix exponentials, which may impose  a considerable overhead because it generally resorts to matrix multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  The MMWU framework has been specialized to solve SDO problems of the form in (3) (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=', [4]),  and the current state of the art is attributed to Lee and Padmanabhan [41], who give an algorithm that can  solve (3) to additive error ∥C∥ℓ1ǫ with overall complexity  �On, 1  ǫ  �  nsǫ−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='5�  ,  where ∥C∥ℓ1 = �  i,j |Cij|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' It is important to note however, that to achieve the stated complexity their  methodology does not explicitly report the solution X4 and the authors assume �  i,j |Cij| = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Thus, to  achieve the same error scaling as the algorithms we present in this work, the algorithm found in [41] would  incur overall cost �  On, 1  ǫ  �  n4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='5sǫ−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='5�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Brand˜ao and Svore [11] and van Apeldoorn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' [61] were the first to quantize the MMWU framework,  utilizing a clever interpretation of the primal variables: Gibbs states, which can be efficiently prepared on a  quantum computer, naturally correspond to trace-normalized positive definite matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' The running time of  these MMWU-based algorithms was subsequently improved [30, 60], and the current state of the art running  time of the quantum MMWU (QMMWU) algorithm for SDO problems is:  �  On,s,R, 1  ǫ  ��√m + √nRr  ǫ  �  s  �Rr  ǫ  �4�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Similar to the complexity of QIPMs, QMMWU algorithms are faster with respect to m and n when compared  to their classical counterparts, but these algorithms still exhibit a non-polynomial running time, due to their  polynomial dependence on the scale invariant parameter Rr  ǫ , whereas the natural input size depends on the  logarithm of this quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Seeking to improve the performance of quantum SDO solvers, Brand˜ao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' [10] present an algorithm,  which they call Hamiltonian Updates (HU), for solving the SDO approximation (3) of (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' The HU method  is a primal-only algorithm closely related to the QMMWU framework, in that it leverages a Gibbs state  representation of the primal variable and progression towards the optimal solution is made via matrix-  exponentiated gradient updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Specifically, the authors in [10] are interested in solving an SDO feasibility  problem that arises upon renormalizing and relaxing (3):  find  X  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  tr  � C  ∥C∥X  �  ≥ γ − ǫ  �  i∈[n]  ����⟨i|X|i⟩ − 1  n  ���� ≤ ǫ  tr (X) = 1,  X ⪰ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  (4)  Here, γ is an upper bound on the absolute value of the optimal objective value of (3) when the cost matrix  C is normalized, obtained via binary search over [−1, 1], and |i⟩ for i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' , n} are the computational  basis states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Since any log(n)-qubit Gibbs state is an element of the set {X ∈ Rn×n : tr(X) = 1, X ⪰ 0} by  definition, solutions to (4) can be naturally be expressed as a Gibbs state  ρ =  exp(−H)  tr(exp(−H)),  where H is the Hamiltonian associated with ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' The key observation in [10] is that upon using the Gibbs  state change of variables in (4), one can model the n constraints on the diagonal elements as single constraint  4Alternatively, they report a “gradient” G ∈ Sn such that X = W exp(G)W for a diagonal matrix W .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  4  which requires that the distribution on the diagonal elements of a feasible solution ρ to (4) be at most ǫ in  total variation distance to the uniform distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' In other words, the task of solving (4) reduces to finding  a log(n)-qubit mixed quantum state that upon measurement in the computational basis is approximately  indistinguishable from the maximally-mixed state, and whose trace inner product with the normalized cost  matrix C∥C∥−1 is at least γ − ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Using a quantum computer, the HU method of [10] solves (3) to additive error O (n∥C∥ǫ) in time  �On, 1  ǫ  �  n1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='5√s  1+o(1)ǫ−28+o(1) exp  �  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='6  �  log(ǫ−1)  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  The authors in [10] also provide an analysis of essentially the same algorithm when using a classical computer,  and show that the classical algorithm has a complexity of  �  On  �  min{n2s, nω}ǫ−12�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  The quantum algorithm yields a speedup in n over classical algorithms, for a specific class of SDO problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  However, as we have already seen with QIPMs and QMMWU algorithms, its dependence on other parameters  (in this case the inverse precision) is prohibitive unless a very low precision solution is acceptable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' This  raises the question as to whether the poor scaling in the inverse precision can be mitigated without incurring  additional cost in n and s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' We answer this question in the affirmative using iterative refinement techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Iterative Refinement (IR) is a methodology for computing high-precision solutions to linear system of  equations [29], as well as linear [25, 26, 27] and mixed integer optimization problems [2, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' We summarize  the methodology at a high level as follows, and present a detailed discussion for the case of convex feasibility  problems later in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Given an initial solution x(0) ∈ Rd, at each iteration k IR produces a refined  solution x(k+1) ← x(k) + u(k), where u(k) acts as a correction of the error r(k) associated with x(k), and  is determined by solving a refining problem induced by the current solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' These operations can all be  carried out using the same level of accuracy, called the fixed-precision approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Alternatively, one may  increase the accuracy with which the residuals r(k) are computed as compared to u(k), and this approach is  called a mixed precision approach [29, 62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' In this paper, we utilize the fixed precision approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='2  Contributions  In this paper we develop an iterative refinement scheme for SDO approximations of QUBO problems that uses  the HU algorithm of [10] as a subroutine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' We show that proceeding in this way allows one to exponentially  improve the dependence on the inverse precision for both the quantum and classical algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  With the proposed IR scheme, the classical algorithm solves the SDO problem (3) up to additive error  O(ǫ) with a worst-case overall complexity of  O  �  min{n2s, nω} · polylog  �  n, ∥C∥F, 1  ǫ  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  This is a significant speedup compared to general-purpose SDO solvers, such as IPMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' This algorithm can  be quantized following a similar strategy to [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' When provided access to quantum random access memory  (QRAM), the quantum algorithm requires at most  O  �  n1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='5 · polylog  �  n, ∥C∥F , 1  ǫ  ��  accesses to the QRAM data structure in the worst case, and O(ns) classical arithmetic operations (to load  and normalize the cost matrix C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Summarizing, the combination of HU with IR described in this paper provides exponential speedups over  the methodology proposed in [10] with respect to the precision parameter ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' To the best of our knowledge, our  classical and quantum algorithms are the fastest known algorithms in their respective model of computation  for this class of problems, and our quantum algorithm provides a genuine asymptotic speedup over known  5  classical solution methodologies, provided that we have access to QRAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Without access to QRAM, one can  bound the running time of the quantum algorithm using the sparse-access input model, in which case the  algorithm requires O  ��  n1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='5s0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='5+o(1)�  polylog  �  n, ∥C∥F, 1  ǫ  ��  accesses to an oracle describing the coefficient  matrix C and O  ��  n2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='5s0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='5+o(1)�  polylog  �  n, ∥C∥F , 1  ǫ  ��  additional gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  The remainder of this paper is organized in the following manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Section 2 introduces notation, as  well as the relevant input models and quantum subroutines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' In Section 3 we introduce the Hamiltonian  Updates (HU) algorithm from [10], and our Iterative Refinement scheme for SDO approximations of QUBOs  is presented in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' The running time analysis is performed in Section 5, and Section 6 concludes the  manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  2  Preliminaries  We write [n] to represent the set of elements {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' We denote the i-th element of a vector x ∈ Rn by  xi for i ∈ [n], and the ij-th element of a matrix A ∈ Rm×n by Aij for i ∈ [m] and j ∈ [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' To refer to the  i-th row of a matrix A, we write Ai,· and write A·,j when referring to its j-th column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' We distinguish the  quantity a to the k-th power and the value of a at iterate k using round brackets, writing ak and a(k) to  denote these quantities, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  The smallest and largest singular values of a matrix A are denoted σmin(A), σmax(A), and if A ∈ Sn, then  the smallest and largest eigenvalues are denoted λmin(A), λmax(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' We let Sn  + and Sn  ++ represent the cones  of symmetric positive semidefinite, and symmetric positive definite matrices, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' For A, B ∈ Sn,  we write A ⪰ B (A ≻ B) to indicate that the matrix A − B is symmetric positive semidefinite (symmetric  positive definite), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=', A − B ∈ Sn  + (A − B ∈ Sn  ++).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' The matrix exponential exp(A), which is defined by the  power series  exp(A) = I + A + 1  2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='A2 + 1  3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='A3 + · · · ,  maps symmetric matrices to the space of symmetric positive definite matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Given the spectral decompo-  sition A = V ΛV ⊤, then exp(A) = V exp(Λ)V ⊤, where exp(Λ) = diag(exp(Λ11), exp(Λ22) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' , exp(Λnn)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  We let A ◦ B denote the Hadamard (or element-wise) product of two matrices, and A ⊗ B denotes their  tensor product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Later in this work, we make use of the following facts regarding Hadamard products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Lemma 1 (Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='4 in [33]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Let E, F and G be m × n matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Then, the i-th diagonal entry of the  matrix (E ◦ F)G⊤ coincides with the i-th diagonal entry of the matrix (E ◦ G)F ⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' That is,  [(E ◦ F)G⊤)]ii = [(E ◦ G)F ⊤)]ii  ∀i ∈ [m].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Lemma 2 (Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='4 in [33]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Let A and B be n×n Hermitian matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' If A ∈ Sn  +, then any eigenvalue  λ(A ◦ B) of A ◦ B satisfies  λmin(A) · λmin(B) ≤ λ(A ◦ B) ≤ λmax(A) · λmax(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Suppose A and B are n × n Hermitian matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' If A ∈ Sn  + and B ∈ Sn with B having at  least one negative eigenvalue, then  λmax(A) · λmin(B) ≤ λmin(A ◦ B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' First, note that by Lemma 2, we have λmin(A ◦ B) ≥ λmin(A) · λmin(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Moreover, since B has at  least one negative eigenvalue, we have λmin(B) < 0, which combined with the fact that λmin(A) ≥ 0 yields  λmax(A) · λmin(B) ≤ λmin(A) · λmin(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Hence,  λmax(A) · λmin(B) ≤ λmin(A ◦ B),  and the proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  6  We write e to refer to the vector of all ones in Rn, and use the notation ei to refer to the i-th unit  vector in the standard orthonormal basis {e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' , en} for Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Analogously, the computational basis states  are denoted by |i⟩ for i ∈ [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Hence, for x ∈ Rn, we denote its amplitude encoding by |x⟩, defined as  |x⟩ =  1  ∥x∥  �  i∈[n]  xi |i⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Observe that |x⟩ is a log(n)-qubit state;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' for simplicity, we assume that the dimensions of all spaces are powers  of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' All logarithms are base 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Where appropriate, our analysis makes use of the Schatten p-norm, defined for a bounded linear operator  A as  ∥A∥p = [tr (|A|p)]  1  p ,  where |A| = (A†A)  1  2 with A† denoting the conjugate transpose of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Notice that the trace and operator  norms ∥ · ∥tr and ∥ · ∥ are the Schatten-1 and Schatten-∞ norms, respectively, and the Frobenius norm ∥ · ∥F  corresponds to the Schatten-2 norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  For a scalar x ∈ R define the sign function sign(x) as  sign(x) =  \uf8f1  \uf8f4  \uf8f2  \uf8f4  \uf8f3  −1  if x < 0  0  if x = 0  1  if x > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  When x ∈ Rn, sign(x) = (sign(x1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' , sign(xn))⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  For any positive integer q, and binary strings j, k ∈ {0, 1}q, we denote by j ⊕ k the bitwise modulo 2  addition of q-digit strings, defined as  j ⊕ k = h  where h ∈ {0, 1}q is the bitstring whose elements hp are defined for p ∈ [q] as  hp =  �  0  if jp = kp,  1  otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  “Big-O” notation  We define O(·) as  f(x) = O(g(x)) ⇐⇒ ∃ℓ ∈ R, c ∈ R+, such that f(x) ≤ cg(x)  ∀x > ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  We write f(x) = Ω(g(x)) ⇐⇒ g(x) = O(f(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' We also define �  O(f(x)) = O(f(x)·polylog(f(x))) and when  the function depends poly-logarithmically on other variables we write  �  Oa,b (f(x)) = O(f(x) · polylog(a, b, f(x))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='1  Input models and subroutines  For our quantum algorithm, we provide analyses for two distinct models of input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' One model considers  a quantum-read/classical-write RAM (QRAM), and the other is the sparse-access model, which we use to  bound the running time without access to QRAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  7  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='1  Sparse-access model  In the sparse-access model, the input matrix C is assumed to be s-row sparse for some known bound s ∈ [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  In other words, C has at most s nonzero entries per row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' The sparse-access model is closely related to the  classical notion, in that we assume access to an oracle Osparse, which upon being queried with input (i, j)  returns the index of the j-th nonzero entry of the i-th row of C by calculating the index function:  index : [n] × [s] → [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  That is, for i ∈ [n] and j ∈ [s], Osparse computes the position in place:  Osparse |i, j⟩ = |i, index(i, j)⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  We also assume access to an oracle that returns a bitstring representation of the individual entries of the  normalized cost matrix C∥C∥−1  F  for every i, j ∈ [n]:  OC |i, j, z⟩ =  ��i, j, z ⊕ (Cij∥C∥−1  F )  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='2  Quantum random access memory  We consider a quantum-read/classical-write RAM (QRAM), which enables us to store classical data that  our quantum algorithms can make oracle calls to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Note that while the QRAM we consider does not need to  be able to store a quantum state, the data is addressable in a superposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Accessing a QRAM of size n  requires O(n) gates [5, 24], however, one can arrange these gates in parallel in order to ensure that the circuit  depth remains O(polylog(n)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Therefore, when we analyze the complexity of our quantum algorithms, we  make the standard assumption that the cost of accessing a QRAM of size n is O(polylog(n)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  The next result from Chakraborty et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' [12], is adapted from an earlier result of Kerenidis and Prakash  [38] and summarizes the aspects of the data structure we utilize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Theorem 1 (Theorem 1 in [12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' (Implementing quantum operators using an efficient data structure) Let  A ∈ Rm×n be a matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' If w is the number of non-zero entries of A, then there exists a data structure  of size O  �  w log2(mn)  �  that, given the entries (i, j, Aij) in an arbitrary order, stores them such that time  taken to store each entry of A is O(log(mn)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Once this data structure has been initiated with all non-zero  entries of A, there exists a quantum algorithm that can perform the following maps with ξ-precision in time  O  �  polylog  �  mn  ξ  ��  :  �U : |i⟩ |0⟩ �→ |i⟩  1  ∥Ai,·∥  n  �  j=1  Aij |j⟩ = |i, Ai,·⟩ ,  �V : |0⟩ |j⟩ �→  1  ∥A∥F  m  �  i=1  ∥Ai,·∥ |i⟩ |j⟩ =  ��� �A, j  �  ,  where |Ai,·⟩ is the normalized quantum state corresponding to the i-th row of A and  ��� �A  �  is a normalized  quantum state such that ⟨i| �A⟩ = ∥Ai,·∥, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=', the norm of the i-th row of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='3  Working with block-encoded matrices  We now give a formal definition of a block-encoding from [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Definition 1 (Block-encoding).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Let A ∈ C2w×2w be a w-qubit operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Then, a (w + a)-qubit unitary U is  an (α, a, ξ)-block-encoding of A if U =  � �A �  , with the property that  ∥α �A − A∥ ≤ ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  8  It was shown by Kerenidis and Prakash [38] and Chakraborty et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' [12] how to efficiently implement  block-encodings of matrices that are stored in a QRAM data structure, which is formalized in the next result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Lemma 3 (Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='7 in [22]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Let A ∈ C2w×2w and ξ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  (i) Fix q ∈ [0, 2] and define µq(A) =  �  nq(A)n(2−q)(A⊤) where nq(A) = maxi ∥Ai,·∥q  q is the q-th power of  the maximum q-norm of the rows of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Defining A{q} to be the matrix with elements A{q}  ij  =  �  Aq  ij, if  A{q} and (A{2−q})† are both stored in QRAM data structures, then there exist unitaries UR and UL that  can be implemented in time O(poly(w log 1  ξ )) and such that U †  RUL is a (µq(A), w + 2, ξ)-block-encoding  of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  (ii) If A is stored in a QRAM data structure, then there exist unitaries UR and UL that can be implemented  in time O(poly(w log 1  ξ )) and such that U †  RUL is an (∥A∥F , w + 2, ξ)-block-encoding of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Linear combinations of block-encodings can also be constructed at cost that is merely logarithmic in the  dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Definition 2 (Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='8 in [22]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' (State preparation pair) Let y ∈ Cm and ∥y∥1 ≤ β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  The pair  of unitaries (PL, PR) is called a (β, p, ξ)-state-preparation-pair if PL |0⟩⊗p = �2p−1  j=0 cj |j⟩ and PR |0⟩⊗p =  �2p−1  j=1 dj |j⟩ such that �m−1  j=0 |β(c∗  jdj) − yj| ≤ ξ and for all j ∈ m, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' , 2p − 1 we have c∗  jdj = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Proposition 1 (Lemma 52 in [23]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' (Linear combination of block-encoded matrices, with weights given  by a state preparation pair) Let A = �m−1  j=0 yjAj be a w-qubit operator, where Aj are matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Suppose  PL, PR is a (β, p, ξ1)-state-preparation pair for y, W = �m−1  j=0 |j⟩ ⟨j| ⊗ Uj + ((I − �m−1  j=0 |j⟩ ⟨j|) ⊗ Ia ⊗ Is)  is an (w + a + p)-qubit unitary with the property that Uj is an (α, a, ξ2)-block-encoding of Aj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Then we can  implement a (αβ, a + p, αξ1 + αβξ2)-block-encoding of A with a single use of W, PR and P †  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  It turns out that the sparse-access model reduces to the quantum operator model upon choosing α = s (if  row and column sparsity are the same).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' The next result from [23] describes how to implement block-encodings  using the sparse-access input model, and the associated costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Lemma 4 (Lemma 48 in [23]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Let A ∈ C2w×2w be a matrix that is sr-row-sparse and sc-column-sparse, and  each element of A has absolute value at most 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Suppose that we have access to the following sparse-access  oracles acting on two (w + 1) qubit registers:  Or : |i⟩ |k⟩ �→ |i⟩ |rik⟩  ∀i ∈ [2w] − 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' k ∈ [sr],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' and  Oc : |ℓ⟩ |j⟩ �→ |cℓj⟩ |j⟩  ∀ℓ ∈ [sc],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' j ∈ [2w] − 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' where  rij is the index for the j-th non-zero entry of the i-th row of A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' or if there are less than i non-zero entries,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  then it is j + 2w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' and similarly cij is the index for the i-th non-zero entry of the j-th column of A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' or if there  are less than j non-zero entries,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' then it is i + 2w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Additionally, assume that we have access to an oracle OA  that returns the entries of A in a binary description:  OA : |i⟩ |j⟩ |0⟩⊗p �→ |i⟩ |j⟩ |aij⟩ ,  ∀i, j ∈ [2w] − 1,  where aij is a p-bit binary description of the ij-matrix element of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Then, we can implement a (√srsc, w +  3, ξ)-block-encoding of A with a single use of Or, Oc and two uses of OA, and additionally using O  �  w + log2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='5 �  srsc  ξ  ��  one and two qubit gates while using O  �  p + log2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='5 �  srsc  ξ  ��  ancilla qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  The block-encoding framework will be useful in speeding up the overall running time found in [10], as it  allows us to perform matrix computations and Hamiltonian simulation efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  9  Theorem 2 (Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='7 in [22]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' (Optimal block-Hamiltonian simulation) Suppose that U is an (α, a, ξ/|2t|)-  block-encoding of the Hamiltonian H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Then, we can implement a ξ-precise Hamiltonian simulation unitary V  which is an (1, a + 2, ξ)-block-encoding of eitH, with O  �  |αt| +  log(1/ξ)  log log(1/ξ)  �  uses of controlled-U or its inverse  and with O  �  a|αt| + a  log(1/ξ)  log log(1/ξ)  �  two-qubit gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Additionally, one can easily take the product of block-encodings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Proposition 2 (Lemma 4 in [12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' (Product of block-encoded matrices) If UA is an (α1, a1, ξA)-block-encoding  of an s-qubit operator A, and UB is an (α2, a2, ξB)-block-encoding of an s-qubit operator B, then (Ia2 ⊗  UA)(Ia1 ⊗ UB) is an (α1α2, a1 + a2, α1ξB + α2ξA)-block-encoding of AB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Relevant to our work in the quantum operator input model is the idea of block-encoding the Hadamard,  or element-wise product of two matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' We will demonstrate how one can carry out the Hadamard product  of block-encodings of matrices A and B as a reduction of the Kronecker product of block-encodings, which  is straightforward to construct given block encodings of A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' (Kronecker product of block-encoded matrices) Suppose that UA is an (α1, a1, ξA)-block-  encoding of A ∈ Rn×n, and UB is an (α2, a2, ξB)-block-encoding of B ∈ Rn×n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Then, taking the tensor  product of UA and UB, we obtain a (α1α2, a1 + a2, ξA + ξB)-block-encoding of A ⊗ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  We do not give a formal proof here as the result directly follows from the definition of a block-encoding;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  to obtain the tensor product of two block-encoded matrices, it suffices to take the tensor product of their  block-encodings while keeping the ancilla qubits separate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Proposition 4 (Hadamard product of block-encoded matrices).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Suppose that UA is an (α1, a1, ξA)-block-  encoding of A ∈ Rn×n, and UB is a (α2, a2, ξB)-block-encoding B ∈ Rn×n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Then, using UA and UB, we can  implement an (α1α2, a1 + a2 + 8 log(n) + 12, 5(ξA + ξB))-block-encoding of A ◦ B using one application of UA  and UB, and �On(1) additional gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' First, note that  A ◦ B = (A ⊗ B)[ιA, ιB],  where ιA = ιB = {1, n + 2, 2n + 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' , n2} are index sets of cardinality n (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=', Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='1 in [33]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Our goal is to use the index sets ιA and ιB along with a block encoding of A ⊗ B to construct a unitary  which block-encodes M ∈ Rn2×n2, a matrix which contains the elements of A◦B in its upper left-most n×n  block, while all other entries are 0:  Mij =  �  Aij · Bij  for i, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' , n,  0  otherwise,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=',  M =  �  A ◦ B  0n×(n2−n)  0(n2−n)×n  0(n2−n)×(n2−n)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  We will first show how one can use ιA and ιB to construct sparse matrices that map A ⊗ B to M, and then  subsequently analyze the cost of constructing the corresponding unitary block-encoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Consider the matrix Z ∈ Rn2×n2, whose elements are defined as  Zij =  �  1  if i = j = (k − 1)n + k,  k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' , n,  0  otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Multiplying A ⊗ B on the left by Z sets the rows of A ⊗ B which do not contain elements of A ◦ B to zero,  and subsequently multiplying Z(A ⊗ B) on the right by Z will set the columns of Z(A ⊗ B) which do not  10  appear in A ◦ B to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' As a result, a block-encoding of Z(A ⊗ B)Z corresponds to block-encoding A ⊗ B,  and setting all terms not appearing in A ◦ B to zero:  [Z(A ⊗ B)Z]ij =  �  [A ⊗ B]ij  if i = (k − 1)n + k and j = (ℓ − 1)n + ℓ  k, ℓ = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' , n,  0  otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Next, let G ∈ Rn2×n2 be a matrix whose elements are defined as follows:  Gij =  \uf8f1  \uf8f4  \uf8f2  \uf8f4  \uf8f3  1  if i ∈ [n2] and i = j = (k − 1)n + k,  k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' , n,  1  if i ∈ [n2] \\ {1, n + 2, 2n + 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' , n2} and j = (i − 1)n + i,  0  otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  We will now establish that GZ(A⊗B)Z)G⊤ is precisely the matrix we seek to block-encode, by demonstrating  that G(Z(A⊗B)Z)G⊤ = M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' First, observe that G is a (partial) permutation matrix: multiplying Z(A⊗B)Z  on the left by G performs the necessary row-exchanges, as the elements of G(Z(A ⊗ B)Z) are given by  [G (Z(A ⊗ B)Z)]ik =  �  Aij · Bij  for k = (j − 1)n + j,  i, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' , n,  0  otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  On the other hand, multiplying Z(A ⊗ B)Z) on the right by G⊤ performs this transformation with respect  to the columns such that  [(Z(A ⊗ B)Z) G]kj =  �  Aij · Bij  for k = (i − 1)n + i,  i, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' , n,  0  otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Hence, multiplying G (Z(A ⊗ B)Z) on the right by G⊤ conducts the column exchanges to move A◦ B to the  top left n-dimensional block of Z(A ⊗ B)Z, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=',  [G (Z(A ⊗ B)Z) G]ij =  �  Aij · Bij  for i, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' , n,  0  otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Therefore, G(Z(A ⊗ B)Z)G⊤ = M as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  We now analyze the cost associated with block-encoding M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Under the stated hypothesis, we have  access to an (α1, a1, ξA)-block-encoding UA of A, and an (α2, a2, ξB)-block-encoding UB of B, and thus  applying Proposition 3 we can construct an (α1α2, a1 + a2, ξA + ξB)-block-encoding UA⊗B of A ⊗ B using  one application of UA and of UB, and no additional gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Using the description of Z, we can construct the sparse-access oracles Or and Oc as defined in Lemma 4  (which act on two (2 log n + 1) qubit registers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Additionally, from the definition of Z, we can construct an  oracle OZ, which returns the entries of Z in a binary description:  OZ : |i⟩ |j⟩ |0⟩⊗p �→ |i⟩ |j⟩ |zij⟩ ,  ∀i, j ∈ [22 log n] − 1,  where zij is a p-bit binary description of the ij-matrix element of Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Note that the circuit for the position  and value of the nonzero elements of Z using �  On(1) gates because they admit an efficient description: their  value is 1 and we have a compact description of their position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' By construction the matrix Z is 1-row sparse  and 1-column sparse, and hence an application of Lemma 4 with sr = sc = 1 asserts that one can construct a  (1, 2 log(n)+3, ξZ)-block-encoding UZ of Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Given block-encodings UZ and UA⊗B, we can apply Proposition  2 with  ξZ = ξA + ξB  α1α2  ,  ξA⊗B = ξA + ξB,  yielding an (α1α2, a1 + a2 + 2 log(n) + 3, 2(ξA + ξB))-block-encoding of Z(A ⊗ B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Applying Proposition 2  once more with  ξZ = ξA + ξB  α1α2  ,  ξZ(A⊗B) = 2(ξA + ξB),  11  we obtain an (α1α2, a1 + a2 + 4 log(n) + 6, 3(ξA + ξB))-block-encoding of Z(A ⊗ B)Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Just as was the case with Z, we can use the description of G to construct the sparse-access oracles Or  and Oc as defined in Lemma 4 (which again, act on two (2 log n + 1) qubit registers), as well as an oracle  OG using �  On(1) gates, that returns the entries of G in a binary description:  OG : |i⟩ |j⟩ |0⟩⊗p �→ |i⟩ |j⟩ |gij⟩ ,  ∀i, j ∈ [22 log n] − 1,  where gij is a p-bit binary description of Gij (the ij-matrix element of G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Noting that G is 1-row sparse  and 1-column sparse (and hence, so its transpose);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' applying Lemma 4 twice more allows us to construct  a (1, 2 log(n) + 3, ξG)-block-encoding UG of G, as well as a (1, 2 log(n) + 3, ξ⊤  G)-block-encoding UG⊤ of the  transpose G⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' We can then use UG and our (α1α2, a1+a2+4 log(n)+6, 3(ξA+ξB))-block-encoding UZ(A⊗B)Z  of Z(A ⊗ B)Z to construct an (α1α2, a1 + a2 + 6 log(n) + 9, 4(ξA + ξB))-block-encoding of G(Z(A ⊗ B)Z)  by applying Proposition 2 with  ξG = ξA + ξB  α1α2  ,  ξZ(A⊗B)Z = 3(ξA + ξB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Applying Proposition 2 a final time, with  ξG⊤ = ξA + ξB  α1α2  ,  ξG(Z(A⊗B)Z) = 4(ξA + ξB),  produces an (α1α2, a1 + a2 + 8 log(n) + 12, 5(ξA + ξB))-block-encoding UM of M = G(Z(A ⊗ B)Z)G⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  The stated complexity result follows upon noting that the steps required to construct the unitary  UM = UGUZUA⊗BUZUG⊤  requires one application of UA⊗B and one application of each of the other matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' In turn, this amounts  to 1 application of UA and UB each, plus the �On(1) gate cost of the remaining matrices UG, UZ and UG⊤,  and the proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  We remark that a similar result to Proposition 4 was independently derived and discussed in the recent  paper [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='4  Gibbs Samplers and Trace Estimators  For clarity, we begin with a formal definition of a subnormalized density operators and their purifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Definition 3 (Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='1 in [22]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' (Subnormalized density operators & Purification) A subnormalized  density operator ρ is a positive semidefinite matrix of trace at most 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' A purification ̺ of a subnormalized  density operator ρ is a 3-register pure state such that tracing out the third register and projecting on the  subspace where the second register is |0⟩ yields ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  The frameworks introduced later in this paper require that we implement a Gibbs sampler and a trace  estimator, which we define next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Definition 4 (Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='11 in [58]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' (Gibbs Sampler) A θ-precise Gibbs-sampler for the input matrix H,  is a unitary that takes as input a data structure storing a Hamiltonian H and creates as output a purification  of a θ-approximation (in trace distance) of the Gibbs state  ρ =  exp(−H)  tr(exp(−H)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  We will use these approximate Gibbs states in order to check the diagonal entries of our solutions, as well  as compute the trace inner products of matrices (or, expectation values), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=', quantities of the form tr(Aρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  12  Definition 5 (Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='12 in [58]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' (Trace Estimator) A θ-precise trace estimator is a unitary that as  input takes a state ρ and a matrix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' It outputs a sample from a random variable x ∈ R such that x is an  estimator for tr(Aρ) that is at most θ/4 biased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  These implementations require polynomial approximations of the exponential function, which can be  obtained using quantum singular value transformation techniques introduced in [22, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Lemma 5 (Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='14 in [58]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Let ξ ∈ (0, 1/6] and β ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' There exists a polynomial P(x) such that  For all x ∈ [−1, 0], we have |P(x) − exp(2βx)/4| ≤ ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  For all x ∈ [−1, 1], we have |P(x)| ≤ 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  deg(P) = �O 1  ξ (β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Lemma 6 (Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='15 in [58]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Let θ ∈ (0, 1/3], β > 1, and let d be the degree of the polynomial from Lemma  5 when we let ξ =  θ  128n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Let U be a (β, a,  θ2β  10242d2n2 )-block-encoding of a Hermitian operator H ∈ Rn×n, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='e,,  a (β, a, �  O(θ/βn2))-block-encoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Then, we can create a purification of a state ˜ρ such that  ����˜ρ −  exp(H)  tr (exp(H))  ����  tr  ≤ θ  using �O 1  θ (√nβ) applications of U and �O 1  θ (√nβa) elementary operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Provided access to a unitary that prepares a purification of a density operator, we can also construct a  block-encoding of it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' This is formalized in the following lemma from [22], which was based on ideas found  in [45, Corollary 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Lemma 7 (Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='4 in [22]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' (Block-encoding of a (subnormalized) density operator) Let G be a (w +a)  unitary which on the input state |0⟩w |0⟩a prepares a purification |̺⟩ of the subnormalized w-qubit density  operator ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Then we can implement a (1, w + a, 0)-block-encoding of ρ with a single use of G and its inverse  and with w + 1 two-qubit gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  We are now in a position to define a trace estimator using the quantum operator input model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Lemma 8 (Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='18 in [58]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Let ρ be an n-dimensional quantum state and U an (α, a, θ/2)-block-  encoding of a matrix A ∈ Rn×n with ∥A∥ ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  A trace estimator for tr (Aρ) with bias at most θ and  σ = O(1) can be implemented using �  O(α) uses of U and U † and �  O 1  θ (α) elementary operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='5  Computational complexity  When discussing the computational complexity of quantum algorithms we generally express the cost in terms  of the number of calls to some input oracles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Unless otherwise specified, the gate complexity is at most a  poly-logarithmic factor larger than the stated oracle complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' The meaning of “input oracles access”  depends on the input model:  For the sparse-oracle access model, it refers to a query to the oracle describing C/∥C∥F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  For the QRAM model, it refers to the number of accesses to QRAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' A QRAM of size O  �  ns log2(n)  �  is sufficient for our algorithms, and in particular, we only need classical write access to the QRAM,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=', we do not write in superposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  It is straightforward to translate each of these oracle costs into a running time in the gate model, by  considering the cost of implementing each oracle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  13  3  Hamiltonian Updates  In this section, we present the algorithm from [10] and relevant results required to prove its convergence and  analyze its cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='1  Convex Feasibility Problems  In order to avoid any normalization issues for the problems that arise over the course of our IR scheme,  we deviate slightly from [10] and renormalize the problem (3) using the Frobenius norm of the cost matrix  rather than use its operator norm:  find  X  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  tr  �  C  ∥C∥F  X  �  ≥ γ − ǫ  �  i∈[n]  ����⟨i|X|i⟩ − 1  n  ���� ≤ ǫ  tr (X) = 1,  X ⪰ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  (5)  The relaxed renormalized SDO problem (5) is a specific example of the convex optimization problem  max  f(X)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  X ∈ P1 ∩ P2 ∩ · · · ∩ Pm,  tr(X) = 1, X ⪰ 0,  (6)  where P1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' , Pm are convex sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  In this context, the trace constraint enforces normalization, but also allows us to obtain a bound on  the optimal objective value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Letting �C = C∥C∥−1  F  and invoking the tracial matrix H¨older inequality [8], it  follows that any X∗ that solves (6) satisfies the following relation:  ���tr( �CX∗)  ��� ≤ ∥ �C∥∥X∗∥tr = ∥ �C∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  It is well known in the optimization literature that performing binary search over the range of values  γ ∈  �  −∥ �C∥, ∥ �C∥  �  ⊆ [−1, 1]  that the objective can take reduces the task of solving (6) to solving a sequence of feasibility problems of  the form:  find  X ∈ Sn  + ∩ {X : tr(X) = 1}  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  tr( �CX) ≥ γ  X ∈ P1 ∩ P2 ∩ · · · ∩ Pm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  (7)  In particular, log(∥ �C∥ǫ−1) queries to (7) are sufficient to estimate the optimal objective value of of (6) up  to additive error ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='2  Solving Convex Feasibility Problems via Hamiltonian Updates  Hamiltonian Updates (HU) is a meta-algorithm for solving convex feasibility problems of the form (7), which  can be viewed as an adaptation of the work of Tsdua, R¨atsch and Warmuth [57] as well as [4, 9, 32, 42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  At a high level, HU can be viewed as a mirror descent algorithm with von Neumann entropy as the mirror  map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' In each iteration, the method makes use of subroutines that can be used to test ǫ-closeness to convex  sets P1, P2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' , Pm, which we formally define next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  14  Definition 6 (Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='1 in [10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Let P ⊂ {X ∈ Sn  + : tr(X) = 1} be a closed, convex subset of quantum  states, and �P ⊂ {X ∈ Cn×n : X = X†, ∥X∥ ≤ 1} be a closed, convex subset of observables of operator norm  at most 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' For ǫ > 0, an ǫ-separation oracle with respect to �P is a subroutine that either accepts a state ρ  (in the sense that observables from �P cannot distinguish ρ from the elements of P), or provides a normal  vector (in the matrix space) P of a hyperplane that separates ρ from the set P using a test from �P:  OP,ǫ(ρ) =  �  accept ρ  if minY ∈P maxP ∈ �  P tr(P(ρ − Y )) ≤ ǫ,  output P ∈ �P s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' tr(P(ρ − Y )) ≥ ǫ  2 for all Y ∈ P  otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  The authors in [10] point out that the above oracle construction is well defined, as we can always choose  some hyperplane P ∈ �P such that  tr (P(ρ − Y )) ≥ ǫ  2,  holds for all Y ∈ P whenever  min  Y ∈P max  P ∈ �  P  tr(P(ρ − Y )) > ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  From Sion’s min-max theorem [54], it follows that  max  P ∈ �  P  min  Y ∈P tr(P(ρ − Y )) = min  Y ∈P max  P ∈ �  P  tr(P(ρ − Y )) > ǫ,  and hence there exists a hyperplane which separates ρ from P by ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' By relaxing the requirement to  ǫ  2-  separation, the algorithm is able to reconcile with the errors that result from approximating quantities  computed with ρ, or estimating its entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  The Hamiltonian Updates (HU) algorithm of Brand˜ao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' [10] is provided in full detail in Algorithm  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' The algorithm takes as input the precision parameter ǫ, and m ǫ-separation oracles O1,ǫ, O2,ǫ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' , Om,ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  In the initialization steps, the starting point is defined to be the maximally mixed state ρ ← n−1I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' This is  critical to ensuring the convergence of mirror descent-based approaches such as Algorithm 1 and the works  in [4, 9, 32, 42, 57];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' initialization to the maximally mixed state ensures that the quantum relative entropy  between any feasible state and the initial state is bounded by log(n) (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=', Theorem 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='8 pt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' 2 [51]), and  is reduced at every iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Consequently, Algorithm 1 terminates in a finite number of iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  As noted in [10], how we define �P determines the number of closeness conditions that need to be tested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  By using the Gibbs state change of variables, we do not need to test if our candidate solution is trace  normalized or positive semidefinite;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' any Gibbs state  ρH =  exp(−H)  tr(exp(−H))  is an element of the set {X ∈ Sn  + : tr(X) = 1} by definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Our task therefore reduces to finding a  log(n)-qubit mixed state state ρ which is ǫ-close to the convex sets Pi that arise from any other constraints  included in the feasibility problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' At each iteration, ǫ-closeness is tested by querying ǫ-separation oracles  which are constructed using observables in �Pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' If each of our oracles accepts the candidate state, the algorithm  terminates and reports (ρ, H) as an ǫ-precise solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Otherwise, upon detecting infeasibility the matrix  exponent is updated to penalize the infeasible directions using the rule  H ← H + ǫ  16P,  where P is a normal vector in the matrix space of a hyperplane that witnesses infeasibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  15  Algorithm 1 Hamiltonian Updates for Convex Feasibility Problems  Input: Query access to m ǫ-separation oracles O1,ǫ(·), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' , Om,ǫ(·)  Initialize ρ ← n−1I and H ← 0n×n  for t = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' , T do  for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' , m do  if Oi,ǫ(ρ) = P then  H ← H +  ǫ  16P  ρ ←  exp(−H)  tr(exp(−H))  break  end  end  return (ρ, H) and exit  end  The following result establishes the iteration complexity of Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Theorem 3 (Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='1 in [10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Algorithm 1 requires at most T = ⌈64 log(n)ǫ−2⌉ + 1 iterations to certify  that (7) is infeasible or output a state ρ satisfying  for all 1 ≤ i ≤ m :  max  Pi∈ �  Pi  min  Yi∈Pi tr(Pi(ρ − Yi)) ≤ ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Note that Theorem 3 applies to any convex feasibility problem (on density operators, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=', trace-normalized  positive semidefinite matrices) for which we have separation oracles as outlined in Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' This is crucial  for the development of an iterative refinement scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  There is an important distinction with respect to output across the models of computation we study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  A classical implementation of Algorithm 1 outputs an explicit description of an ǫ-precise solution ρ∗ to (5)  and its associated Hamiltonian H∗, whereas a quantum implementation reports a real valued vector y ∈ R2  along with a diagonal matrix D (with ∥D∥ ≤ 1) such that H∗ = y1  C  ∥C∥F + y2D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' The vector y = (y1, y2)⊤ is  the state preparation pair of ρ∗, in particular:  ρ∗ =  exp  �  −  �  y1  C  ∥C∥F + y2D  ��  tr  �  exp  �  −  �  y1  C  ∥C∥F + y2D  ���,  and we refer to this type of output as a state preparation pair description of ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' This choice of output is  used in all quantum SDO solvers based on Gibbs sampling techniques (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=', [9, 10, 11, 60, 61]), and is  motivated by the fact that it is difficult to develop quantum algorithms that are substantially faster than  classical algorithms if we still have to output each entry of the solution (an n × n matrix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  The Gibbs sampling approaches that we apply later exhibit a cost that depends on a norm bound for y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Observe that we initialize y to the all zeros vector of appropriate dimension, and in every iteration, at most  one entry of y changes by a magnitude of  ǫ  16 (specifically, an entry yi, where the oracle Oi,ǫ has detected  infeasibility).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' As a consequence, the vector y satisfies the inequality  ���y(t+1) − y(t)��� ≤ ǫ  16  (8)  for each iteration t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' In view of the iteration bound for Algorithm 1 provided in Theorem 3, it is easy to see  that for any y obtained from Algorithm 1 we have  ∥y∥1 ≤ ⌈64 log(n)ǫ−2⌉  ���y(t+1) − y(t)��� ≤ ⌈64 log(n)ǫ−2⌉ ǫ  16 ≤ 4 log(n)ǫ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  (9)  To instantiate the algorithm to solve problem (3) we need to choose the sets Pi, and provide separation  oracles for them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' This is what we do in the following section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  16  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='1  Oracle Construction  The goal of Hamiltonian Updates is to solve, for fixed γ ∈ [−1, 1], the following feasibility problem:  find  ρ ∈ {X ∈ Sn  + : tr(X) = 1} ∩ Cγ ∩ Dn  where  Cγ =  �  X : tr  �  C  ∥C∥F  X  �  ≥ γ  �  ,  Dn =  �  X : ⟨i|X|i⟩ = 1  n, i ∈ [n]  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  (10)  One can observe that the set Cγ constitutes a halfspace, while Dn is an affine space of codimension n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' The  sets of observables for Cγ and Dn are given by �Cγ and �Dn respectively, with  �Cγ = {−C∥C∥−1  F }, and �Dn = {D ∈ Rn×n : ∥D∥ ≤ 1, D is diagonal}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  As noted in [10], it follows  max  P ∈ �  Cγ  min  Y ∈Cγ tr(P(ρ − Y )) ≤ ǫ ⇐⇒ − tr  �  C∥C∥−1  F (ρ − Y )  �  ≤ ǫ  for some Y ∈ Cγ,  which in turn implies tr (C∥C∥−1  F ρ) ≥ γ − ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Given the structure of Cγ and Dn, the authors in [10] suggest the following two separation oracles:  OCγ : compute an approximation ˜c of tr  �  C∥C∥−1  F ρ  �  up to additive error ǫ  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Check if ˜c ≥ γ − 3ǫ  4 and  output P = −C∥C∥−1  F  if the inequality is violated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  ODn : compute an approximation ˜p ∈ Rn of pi = ⟨i|ρ|i⟩ satisfying  n  �  i=1  |pi − ˜pi| ≤ ǫ  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Check if  n  �  i=1  ����˜pi − 1  n  ���� ≤ 3ǫ  4 and output P =  n  �  i=1  �  I  �  ˜pi > 1  n  �  − I  �  ˜pi < 1  n  ��  |i⟩ ⟨i|  if the inequality is violated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  For any given  ρH =  exp(−H)  tr(exp(−H)),  the required separation oracles are straightforward to implement on a classical computer that has access to  ρH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Thus, classically we only need to prepare ρH once and store it to build the separation oracles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' The next  result from [10] establishes that computing an O(log(n)ǫ−1)-degree Taylor series suffices to produce accurate  approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Lemma 9 (Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='2 in [10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Fix a Hermitian n×n matrix H, an accuracy ǫ, and let ℓ be the smallest even  number satisfying (ℓ + 1)(log(ℓ + 1) − 1) ≥ 2∥H∥ + log(n) + log  � 1  ǫ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Then, the truncated matrix exponential  Tℓ = �ℓ  k=0  1  k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' (−H)k satisfies  ����  exp(−H)  tr (exp(−H)) −  Tℓ  tr(Tℓ)  ����  tr  ≤ ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  The task of implementing our separation oracles and testing feasibility on a quantum computer reduces  to preparing Gibbs states [10], which are subsequently used to test closeness to the sets Cγ and Dn via  quantum measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' The next result can be viewed as the quantum analogue to Lemma 9;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' contrary to  bounding the number of required Taylor series steps for computing ρ via a matrix exponential, it bounds  the number of copies of ρ required to estimate its diagonal entries and expectation values tr(Aρ) using the  QRAM model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  17  Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Fix ǫ ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Let ρ be a log(n)-qubit quantum state and U a (1, log(n) + 2, ǫ/(2n))-block-  encoding of C∥C∥−1  F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Then, in the QRAM model, we can implement the oracle OCγ on a quantum computer  given access to O(ǫ−1) copies of a state that is an ǫ  8-approximation of the input state ρ in trace distance and  O(ǫ−1) applications of U and U †.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' The oracle ODn can be implemented using O(nǫ−2) ǫ  8-approximate copies  of the input, and the classical post-processing time needed to implement the oracle is O(nǫ−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' First, note that we can obtain an estimate ˜p of the diagonal elements of ρ whose total variation  distance from p is no more than  ǫ  8 using �  On  �  nǫ−2�  copies of ρ to measure ρ in the computational basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Further, provided accesses to ρ and a (1, log(n) + 2, ǫ/(2n))-block-encoding U of C∥C∥−1  F , by Lemma 8, a  trace estimator for tr  �  C∥C∥−1  F ρ  �  with bias at most ǫ  n can be implemented using �  O(1) uses of U and U † and  �  O n  ǫ (1) elementary operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' From here, applying amplitude estimation using O(ǫ−1) samples from the  trace estimator to suffice to compute an approximation tr  �  C∥C∥−1  F ρ  �  up to additive ǫ  8 to implement OCγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  The rest of the proof exactly follows the proof of [10, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  We remark that in the presence of QRAM, utilizing multidimensional phase estimation techniques from  [59] improves the dependence on ǫ−1 for estimating the diagonal elements of ρ to linear, which is a factor  ǫ−1 better than a n¨aive application of computational basis measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' However, in the context of the  iterative refinement scheme we present later, the improvement would only reduce the amount of constant  overhead in the overall running time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' There are also numerous ways to prepare Gibbs states using a quantum  computer [16, 20, 37, 53, 60, 61, 63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Following [10], we utilize the Gibbs sampler from [53] when working  with the sparse-access input model, and for the QRAM input model we consider Gibbs sampling techniques  introduced in [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='3  Complexity  Having understood the cost of constructing the oracles in both the classical and quantum settings, we are  now in a position to analyze the complexity associated with using Algorithm 1 to obtain solutions to (5)  and approximations to (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Relevant to this discussion is the following result, which imposes precision  requirements on solving (3) to an additive error of the order O (n∥C∥F ǫ) using Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Proposition 5 (Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='1 in [10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Let ρ be an ǫ4-accurate solution to the relaxed SDO problem (5)  with input matrix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Let γǫ4 = tr (Cρ) be the value attained by ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Then, there is a quantum state ρ∗ at trace  distance O(ǫ) of ρ such that nρ∗ is a feasible point of SDO problem (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' In particular  |γǫ4n∥C∥F − tr (nρ∗C)| = O (n∥C∥Fǫ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Moreover, it is possible to construct ρ∗ in time O(n2) given the entries of ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  We do not provide a proof of this result here, as later we will provide an improved approximation  guarantee and a proof of the improved statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='1  Classical running time  Using Lemma 9 in combination with Theorem 3, we can bound the running time required to solve (5) to  additive error ǫ using a classical implementation of Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Suppose that C has row sparsity s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Then, the classical cost of solving (5) up to additive  error ǫ using Algorithm 1 is O  �  min{n2s, nω} log2(n)ǫ−3�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' The result follows directly from the proof of Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='1 in [10], but we repeat the argument here  for completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  First, observe that over the course of the iterations t = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' , T , the operator norms ∥H(t)∥ do not  become prohibitively large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' This follows from initializing H(0) = 0n×n, and that by (8), the inequality  ���H(t+1) − H(t)��� ≤ ǫ  16  ���P (t)��� ≤ ǫ  16  18  holds for all t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' By Theorem 3, Algorithm 1 requires at most T = ⌈64 log(n)ǫ−2⌉ iterations, which implies  ∥H(t)∥ ≤ 4 log(n)ǫ−1 for all t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  By Lemma 9, it suffices to compute O(log(n)ǫ−1) steps of the Taylor series corresponding to exp(−H(t))  in order to obtain a matrix ˜ρ(t) that is at most a trace distance of ǫ  4 from ρ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Moreover, given that H(t)  is defined as a linear combination of C∥C∥−1  F  with a diagonal matrix, matrix multiplication involving H(t)  can be carried out in O(min{n2s, nω}) arithmetic operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Given classical access to ˜ρ(t), the diagonal  constraints comprising Dn can be checked in time O(n), whereas computing tr  �  C  ∥C∥F ˜ρ(t)�  requires O(ns)  arithmetic operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Thus, the dominant operation at each iteration is computing the matrix exponential  and the classical per-iteration cost of Algorithm 1 is given by  O  �  min{n2s, nω} log(n)ǫ−1�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Taking into account the iteration bound O(log(n)ǫ−2) provided in Theorem 3, we arrive at an overall running  time of  O  �  min{n2s, nω} log2(n)ǫ−3�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  The proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  The next corollary from [10] follows from Proposition 5 in the context of the previous result, and provides  the overall running time of Algorithm 1 to solve (3) to additive error O (n∥C∥Fǫ) in the classical setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Suppose that C has row-sparsity s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Then, the classical cost of solving (3) up to an additive  error O (n∥C∥Fǫ) using Algorithm 1 is O  �  min{n2s, nω} log2(n)ǫ−12�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' By Proposition 6, Algorithm 1 requires time  O  �  min{n2s, nω} log2(n)˜ǫ−3�  ,  to solve (5) up to additive error ˜ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' In order to satisfy the approximation guarantee for (3) given in Proposition  5, it suffices to solve (5) to error ˜ǫ = ǫ4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Plugging in this value for the precision parameter, the total cost  required to solve (3) up to an additive error O (n∥C∥Fǫ) using Algorithm 1 is  O  �  min{n2s, nω} log2(n)˜ǫ−3�  = O  �  min{n2s, nω} log2(n)(ǫ4)−3�  = O  �  min{n2s, nω} log2(n)ǫ−12�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='2  Quantum running time  Combining the sampling requirements provided in Lemma 10 with the cost of preparing a single Gibbs state  and the iteration bound from Theorem 3 gives the complexity of Algorithm 1 when run on a quantum com-  puter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' However, Gibbs samplers based on the block-encoding framework depend only poly-logarithmically  on the inverse precision, therefore they are exponentially faster (in the parameter ǫ−1) compared to the  Gibbs sampling algorithm from [53] utilized in [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' It thus makes sense to analyze the running time in the  more efficient model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' This will require an efficient data structure for storing y so that we can efficiently  prepare linear combinations of block-encodings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Lemma 11 (Lemma 15 in [60]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' There is a data structure that can store an m-dimensional χ-sparse vector  y with θ-precision using a QRAM of size �O m  θ (χ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Furthermore:  Given a classical O(1)-sparse vector, adding it to the stored vector has classical cost �  O m  θ (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Given that β ≥ ∥y∥1, we can implement a (symmetric) (β, �  O m  θ (1), θ)-state preparation pair for y with  �O m  θ (1) queries to the QRAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  19  Corollary 3 (Corollary 16 in [60]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Suppose A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' , Am are Hermitian matrices with operator norm at most  1, and that y ∈ Rm satisfies ∥y∥1 ≤ β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Having access to the above data structure for y, we can prepare one  copy of the Gibbs state  ρ =  exp (− �m  i=1 yiAi)  tr (exp (− �m  i=1 yiAi))  using �Oθ(√nαβ) accesses to the data structure for y and block-encodings of A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' , Am.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  We can now use Corollary 3 in combination with results from Sections 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='3 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='4 to establish the  running time of Algorithm 1 in the QRAM input model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Let  C  ∥C∥F ∈ Sn be stored in QRAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Then, the complexity of solving (5) up to additive error  ǫ with Algorithm 1 using the QRAM input model is  �  O n  ǫ  �  n1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='5ǫ−5�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Here, the complexity corresponds to the number of accesses to the QRAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Given that  C  ∥C∥F is stored in QRAM, Lemma 3(ii) asserts that when constructing a block-encoding  of  C  ∥C∥F , one can set the subnormalization factor to be αC =  ���  C  ∥C∥F  ���  F = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Hence, one can construct a  (1, log(n) + 2, ǫ/(2n))-block-encoding of C∥C∥−1  F  in time �O n  ǫ (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Next, recall that in iteration t ∈ [T ] of Algorithm 1, our Hamiltonian is defined as  H(t) = y(t)  1  C  ∥C∥F  + y(t)  2 D(t),  where D(t) is a diagonal matrix with the diagonal entries taking value −1, 0 or 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Now, the diagonal elements  of D change in each iteration, and therefore, a new D must be block-encoded in each iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' For this,  we can utilize the QRAM model described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='2, which allows for insertions to be made in time  �  On(1) to keep the cost of this step negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' In this case, provided a classical description of D, we can store  D in the QRAM in time O(n log(n)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Thus, applying Lemma 4, a (1, log(n) + 3, ǫ)-block-encoding of D(t)  can be constructed in time �O n  ǫ (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  In an earlier discussion we saw that any y obtained from a call to Algorithm 1 will satisfy ∥y∥1 = �On(ǫ−1)  if we call Algorithm 1 using precision ǫ (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=', equation (9)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Hence, an application of Corollary 3 with  β = �  On(ǫ−1) implies that we can prepare one copy of our Gibbs state using  �  O n  ǫ  �√nαǫ−1�  accesses to the data structure for y and the block-encodings of C∥C∥−1  F  and D, where α is defined as the  maximum over the subnormalization factors used to block-encode  C  ∥C∥F and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Since α = max{αC, αD} = 1,  it follows  �  O n  ǫ  �√nαǫ−1�  = �O n  ǫ  �√nǫ−1�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Now, one can see from Lemma 10 that the cost of constructing ODn dominates that of constructing OCγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Noting that ODn can be implemented using O(nǫ−2) copies of a state that is an  ǫ  8-approximation of the  input state ρ in trace distance and its inverse, the per-iteration cost of Algorithm 1 in the QRAM input  model is given by  �  O n  ǫ  �  n1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='5ǫ−3�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Factoring in the iteration bound of �  On(ǫ−2) from Theorem 3, it follows that when provided access to QRAM,  Algorithm 1 solves (5) up to additive error ǫ using  T quantum  HU  = �O n  ǫ  �  n1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='5ǫ−5�  accesses to the QRAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' The proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  20  Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Let  C  ∥C∥F ∈ Sn be stored in QRAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Then, the complexity of solving (3) up to additive error  O(n∥C∥F ǫ) with Algorithm 1 using the QRAM input model is  �O n  ǫ  �  n1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='5ǫ−20�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Here, the complexity corresponds to the number of accesses to the QRAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' By Proposition 7, Algorithm 1 requires  �  O n  ˜ǫ  �  n1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='5˜ǫ−5�  ,  accesses to the QRAM to solve (5) up to additive error ˜ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' In order to satisfy the approximation guarantee  for (3) given in Proposition 5, it suffices to solve (5) to error ˜ǫ = ǫ4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Plugging in this value for the precision  parameter, the total cost required to solve (3) up to an additive error O (n∥C∥F ǫ) using Algorithm 1 is  �O n  ˜ǫ  �  n1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='5˜ǫ−5�  = �O n  ǫ  �  n1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='5(ǫ4)−5�  = �O n  ǫ  �  n1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='5ǫ−20�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  The proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Corollary 4 establishes that utilizing Gibbs samplers and trace estimators based on the block-encoding  framework for our oracle construction in Algorithm 1 leads to an  O  �√s  1+o(1)ǫ−8+o(1) exp  �  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='6  �  log(ǫ−4)  ��  speedup over the running time result provided in [10, Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='2] when applied to solving (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Yet, the  costly accuracy requirements for the rounding procedure (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=', Proposition 5) lead to a prohibitive scaling  in the inverse precision for the overall running time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Given the advantageous dependence on the dimension,  as compared to classical algorithms, we study how to improve the dependence on the precision parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  This is discussed next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  4  Iterative Refinement for SDO approximations of QUBOs  In this section, we introduce an iterative refinement method for obtaining accurate solutions to the renor-  malized relaxed SDO problem (5), that at a high level can be viewed as solving a series of problems related  to the feasibility problem (10) associated with (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' We then discuss how to test ǫ-closeness to the convex sets  which comprise the feasible regions of the intermediate refining problems before presenting our algorithm in  full detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' We conclude the section by proving our algorithm’s correctness and iteration complexity, and use  these results to provide an improved approximation guarantee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='1  The refining problem  To develop an iterative refinement scheme for (5), we need to design a problem whose solution can be used  to improve the quality of solutions to (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Suppose we run Algorithm 1 and obtain an ǫ-precise solution ˆρ  to (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Letting ˆγ = tr  �  C∥C∥−1  F ˆρ  �  , ˆρ must satisfy  tr  �  C∥C∥−1  F ˆρ  �  = ˆγ ≥ γ − ǫ,  n  �  i=1  ����⟨i|ˆρ|i⟩ − 1  n  ���� ≤ ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  In refining our solution to (5), we should aim to reduce the trace distance to the maximally mixed state  n−1I, while also improving the precision to which the optimal objective value is approximated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Thus, an  21  improved solution ρ′ should obey  tr  �  C∥C∥−1  F ρ′�  ≥ γ − ǫ′,  n  �  i=1  ����⟨i|ρ′|i⟩ − 1  n  ���� ≤ ǫ′,  with ǫ′ < ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' The basic idea behind constructing the refining problem is to use our current solution ˆρ to first  shift the renormalized relaxed SDO problem (5) to the origin, and then scale the shifted problem back to  the domain of the original problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' In particular, we solve a series of problems related to the feasibility  problem (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Let ε ∈ Rn be a vector whose elements are the residuals along the diagonal εi = ˆρii − 1  n for i ∈ [n], and  η ≥ 1 to be a scalar defined as  η =  1  max  �  γ − tr  �  C∥C∥−1  F ˆρ  �  , �n  i=1 |εi|  � =  1  max  �  γ − tr  �  C∥C∥−1  F ˆρ  �  , ∥ˆρ − n−1I∥tr  �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Using these quantities, the refining problem is given by:  find  ρr ∈ {X ∈ Sn  + : tr(X) = 1} ∩ Cη(γ−ˆγ) ∩ Dηε  where  Cη(γ−ˆγ) =  �  X : tr  �  C  ∥C∥F  (Q ◦ X)  �  ≥ η(γ − ˆγ)  �  ,  Dηε = {X : ⟨i|X|i⟩ = η|εi|, ∀i ∈ [n]} ,  (11)  where Q ∈ Sn is a matrix whose diagonal elements are chosen such that for any X ∈ Dηε, we have  (Q ◦ X)ii = ηεi  for i ∈ [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Further details and requirements on the structure of Q are specified later in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' We  refer to solutions ρr to (11) as refining solutions, which we use to update our current solution ˆρ to (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  The set Dηε is comprised of the diagonal constraints  ⟨i|X|i⟩ = η|εi|,  ∀i ∈ [n],  and similar to Dn, is an affine space with codimension n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Our use of the absolute value function of the  residuals and scaling by η ensures the viability of applying Gibbs sampling techniques to solve the refining  problem (11);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' the diagonal terms of any density matrix must be nonnegative and sum to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Whenever  n  �  i=1  |εi| > γ − tr  �  C∥C∥−1  F ˆρ  �  ,  then η∥ε∥1 = 1, and the parameter η therefore scales the shifted problem back to the space of the log(n)-qubit  mixed states, ensuring that any solution ρr to (11) is indeed a (trace normalized) Gibbs state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  On the other hand, should it be the case that  n  �  i=1  |εi| ≤ γ − tr  �  C∥C∥−1  F ˆρ  �  ,  then for any X ∈ Dηε we have tr(X) ≤ 1, rather than tr(X) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Our primal SDO oracle in Algorithm 1  solves feasibility problems in which the trace upper bound is tight, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=', tr(X) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' The authors in [60] note  that this can be dealt with adding one extra variable w such that  ¯ρr :=  �ρr  0  0  w  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  22  Then, tr (¯ρr) = 1 and ¯ρr ⪰ 0 imply that tr(ρr) ≤ 1, and as a result we obtain an SDO problem that is  equivalent to (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Since we know exactly the amount of subnormalization, we can also get rid of the extra  variable in subsequent calculations and rescale the trace back to 1 when necessary (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=', when combining  solutions from multiple iterative refinement iterations for trace estimations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Crucially, using the input  models described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='1, these modifications do not introduce more than constant overhead in the  overall complexity, as the problem data in this case is simply given by  C =  �  C  ∥C∥F  0  0  0  �  ,  Q =  �  Q  0  0  0  �  ,  with (C, Q) ∈ Sn+1 × Sn+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  The Hadamard product Q ◦ ρr that appears in the definition of Cη(γ−ˆγ) is required for similar reasons;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  properly setting Q allows us to drive the trace distance to the maximally mixed state to zero using the  solutions to the refining problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Later, in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='3 we demonstrate that this can be achieved by updating  the current solution ˆρ using the rule  ˆρ = ˆρ + 1  η Q ◦ ρr,  (12)  with a suitable choice for Q being  Q = (ee⊤ − I) + diag (sign(−ε)) =  \uf8eb  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ed  sign(−ε1)  1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  1  1  sign(−ε2)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  1  1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  1  sign(−εn)  \uf8f6  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f8  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  (13)  Choosing Q in this manner also implies that the Hadamard product Q ◦ A can be carried out classically  using O(n) arithmetic operations for any A ∈ Rn×n, as the element-wise products QijAij = Aij for i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  We also point out that updating Q at each iterate only requires updating its diagonal elements, which again  is an O(n) operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  It is important to note that the update we propose in (12) does not preserve positive semidefiniteness  in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' However, later in our analysis, we demonstrate that the eigenvalues of the final solution ˆρ are  only slightly negative in the worst case, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=', λmin(ˆρ) ≥ −δ for a small constant δ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' one can restore positive  semidefiniteness by adding δ to the diagonal elements of the final solution, and we renormalize by (1 + nδ)  to obtain unit trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' It turns out that the trace distance from resulting matrix to ˆρ is bounded by the final  precision tolerance parameter of our refining scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' We show that these modifications required to restore  positive semidefiniteness have only a mild (in fact, constant) impact on feasibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' In order to accomplish  this, we will need to utilize bounds on the eigenvalues of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' This will require use of the next result, which  is a special instance of Weyl’s inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Lemma 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Suppose that A ∈ Rn×n and B ∈ Rn×n are Hermitian matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Then  λmin(A + B) ≥ λmin(A) + λmin(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Using the preceding lemma, the following result bounds the minimum eigenvalue of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Lemma 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Suppose that Q ∈ Sn is defined according to Equation (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Then, λmin(Q) ≥ −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Let A = (ee⊤ − I) and B = diag (sign(−ε)), such that Q = A + B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Now, it can be easily seen  from the definition of A that A + I is an all-ones matrix of dimension n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Upon performing row-reduction  (via, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=', Guassian elimination) on A, it is trivial to observe that the resulting row-echelon form will have  n − 1 zero rows, and as a consequence, A has the eigenvalue −1, repeated (at least) n − 1 times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Further,  since tr (A) = 0, the other eigenvalue is n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Therefore, we have λmin(A) ≥ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' On the other hand, B  is a diagonal matrix whose diagonal elements can take value −1, 0, or 1, from which λmin(B) ≥ −1 readily  follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  23  Applying Lemma 12, we obtain  λmin(Q) = λmin(A + B) ≥ λmin(A) + λmin(B) ≥ −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  The proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='2  Oracle construction for the refining problem  In order to construct separation oracles for testing closeness to Cη(γ−ˆγ), we rely on the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Lemma 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Let E, F and G ∈ Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' We have  tr (G(E ◦ F)) = tr ((E ◦ G)F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Applying Lemma 1 with m = n, we have  [(E ◦ F)G]ii = [(E ◦ G)F]ii  ∀i ∈ [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Note that we have dropped the transpose terms, as E, F and G are symmetric matrices, and hence, so are  E ◦ F and E ◦ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' It follows  tr (G(E ◦ F)) = tr ((E ◦ F)G) =  �  i∈[n]  [(E ◦ F)G]ii =  �  i∈[n]  [(E ◦ G)F]ii = tr ((E ◦ G)F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  In addition to Q ∈ Sn, we also require maxi,j∈[n]{|Qij|} ≤ 1 to avoid any normalization issues with  respect to Q ◦  C  ∥C∥F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Note that defining of Q according to equation (13) satisfies both of these properties  trivially, as each of the diagonal elements are 1, 0, or −1, while the off-diagonal elements are all set to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  This idea is formalized next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Lemma 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Let A, Q ∈ Sn be matrices satisfying maxi,j∈[n]{|Qij|} ≤ 1 and ∥A∥F ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Then,  ∥Q ◦ A∥ ≤ ∥Q ◦ A∥F ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Under the stated conditions for Q, it follows  ∥Q ◦ A∥2  F =  �  i∈[n]  �  j∈[n]  �  [Q ◦ A]ij  �2  =  �  i∈[n]  �  j∈[n]  (Qij · Aij)2 =  �  i∈[n]  �  j∈[n]  (Qij)2 (Aij)2  ≤  �  i∈[n]  �  j∈[n]  (Aij)2 = ∥A∥2  F ,  and applying the square root throughout the above we obtain ∥Q ◦ A∥F ≤ ∥A∥F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' From here, the result  follows upon noting ∥A∥F ≤ 1 and ∥A∥ ≤ ∥A∥F is true for any A ∈ Rn×n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Although the sets Cγ and Dn differ from their refining counterparts Cη(γ−ˆγ) and Dηε, their dissimilarity  merely affects the right hand side of the inequality defining the sets, and are thus no more difficult to  construct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Just as in the case of (10), the task of obtaining separation oracles for the refining problem (11)  in the quantum regime reduces to preparing many copies of Gibbs states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Likewise, these oracles can also  be implemented on a classical computer, given access to ρr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  The similarities between (10) and (11) become transparent when we demonstrate that they are specific  instances of the same problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' In particular, it is easy to see that solving (10) corresponds to solving  find  ρ ∈ {X ∈ Sn  + : tr(X) = 1} ∩ Cη(γ−ˆγ) ∩ Dηε  where  Cη(ˆγ−γ) =  �  X : tr  �  C  ∥C∥F  Q ◦ X  �  ≥ η(γ − ˆγ)  �  ,  Dηε = {X : ⟨i|X|i⟩ = η|εi|, ∀i ∈ [n]} ,  (14)  24  with εi = 1  n, η = 1, Q = ee⊤, and ˆγ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' In view of this relationship, we can unify the oracle construction  for (10) and (11) as follows:  OCη(γ−ˆγ) : Compute an approximation ˜c of tr  �  Q ◦ C∥C∥−1  F ρ  �  up to additive error ǫ  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Check if ˜c ≥ η(γ − ˆγ) + 3ǫ  4 and output P = −Q ◦ C∥C∥−1  F  if the inequality is violated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  ODηε : Compute an approximation ˜p ∈ Rn of pi = ⟨i|ρ|i⟩ satisfying  �  i∈[n]  |pi − ˜pi| ≤ ǫ  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Check if  �  i∈[n]  |˜pi − η|εi|| ≤ 3ǫ  4 and output P =  �  i∈[n]  (I{˜pi > η|εi|} − I{˜pi < η|εi|}) |i⟩ ⟨i|  if the inequality is violated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Again, the sets of observables for Cη(γ−ˆγ) and Dηε are given by  �Cη(γ−ˆγ) = {−Q ◦ C∥C∥−1  F }, and �Dηε = {D ∈ Rn×n : ∥D∥ ≤ 1, D is diagonal}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Although these observations are straightforward, they justify our use of Algorithm 1 as a semidefinite opti-  mization oracle that solves a convex feasibility problem at hand in every iteration for different values of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  In particular, these facts, along with Lemmas 14 and 15 ensure that the complexity results in Propositions  6 and 7 hold when applying Algorithm 1 to solve (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Let Q ◦  C  ∥C∥F ∈ Sn be stored in QRAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Then, the complexity of solving (14) up to additive  error ǫ with Algorithm 1 using the QRAM input model is  �  O n  ǫ  �  n1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='5ǫ−5�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Here, the complexity corresponds to the number of accesses to the QRAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Given that Q◦  C  ∥C∥F is stored in QRAM, Lemma 3(ii) asserts that when constructing a block-encoding  of Q◦  C  ∥C∥F , one can set the subnormalization factor to be αC =  ���Q ◦  C  ∥C∥F  ���  F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' In particular, one can always  choose αC = 1, as it can be seen from the proof of Lemma 15 that the inequality  ����Q ◦  C  ∥C∥F  ����  F  ≤  ����  C  ∥C∥F  ����  F  = 1  always holds for any Q defined according to equation (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Collecting these facts, one can construct a  (1, O(log(n)), ǫ/(2n))-block-encoding of Q◦C∥C∥−1  F  in time �  O n  ǫ (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Note that the quantity Q◦  C  ∥C∥F remains  unchanged for the duration of Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' From here, the rest of the proof follows exactly that of Proposition  7 upon replacing  C  ∥C∥F , OCγ and ODn with Q ◦  C  ∥C∥F , OCη(γ−ˆγ) and ODηε, respectively, in what remains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='3  Iterative Refinement using Hamiltonian Updates  We are now in a position to provide our iterative refinement method for SDO approximations of QUBOs  presented in full detail in Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  The algorithm takes three parameters as input;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' (i) ξ, the fixed precision used to test closeness to the sets  Cη(γ−ˆγ) and Dηε in every iteration, (ii) ζ, the precision to which the final solution satisfies the functional  constraints of (5), and (iii) ǫ, the additive error to which we seek to solve (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' In our initialization steps we  set the values of Q, ε and η such that the first iteration corresponds to solving the feasibility problem (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  In each iteration k, Algorithm 2 calls Algorithm 1 with separation oracles OCη(γ−ˆγ) and ODηε using  fixed precision ξ such that every call to Algorithm 1 produces a ξ-precise classical solution ρ(k) to (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' If  ˆρ is indistinguishable up to precision ζ from the maximally mixed state n−1I upon measurement in the  computational basis, and satisfies tr  �  C  ∥C∥F ˆρ  �  ≥ γ − ζ, the algorithm terminates and reports ˆρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Otherwise,  we construct the refining problem associated with our current solution, and proceed to the next iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  25  Algorithm 2 Iterative Refinement for SDO Approximations of QUBOs  Input: Error tolerances ǫ ∈ (0, 1) and ζ =  �  ǫ  n∥C∥F  �4  , upper bound on objective value γ ∈ [−1, 1]  Output: A matrix ˆρ ∈ Sn satisfying max  �  γ − tr  �  C  ∥C∥F ˆρ  �  , ∥ˆρ − n−1I∥tr  �  ≤ ζ  Initialize: ˆρ ← 0n×n, Q ← ee⊤, εi = 1  n for i ∈ [n], ˆγ ← 0, η(0) ← 1, k ← 0  while max  �  γ − tr  �  C  ∥C∥F ρ  �  , ∥ε∥1  �  > ζ do  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Solve (14) to precision ξ  4 for ρ(k) using Algorithm 1 with oracles OCη(γ−ˆγ) and ODηε  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Update Solution:  ˆρ ← ˆρ +  1  η(k) Q ◦ ρ(k),  ˆγ ← ˆγ +  1  η(k) tr  �  C  ∥C∥F  Q ◦ ρ(k)  �  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Compute element-wise deviations from the maximally mixed state:  εi ← ˆρii − 1  n for i ∈ [n]  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Update refining problem parameters:  Qii ← sign(−εi) for i ∈ [n],  η(k+1) ←  1  max  �  γ − tr  �  C  ∥C∥ρ  �  , ∥ε∥1  �  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' k ← k + 1  end  26  To define the parameters for the next refining problem, we first calculate the deviation of the diagonal  elements from 1  n, and the violation with respect to satisfying our objective value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Then, we define our scaling  factor to be the maximum over the ℓ1-norm of the diagonal deviations, and the objective violation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' We stress  that ξ is a (chosen) constant, and does not change throughout the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  We now state a series of results in order to bound the iteration complexity of Algorithm 2, and use our  findings to improve the approximation guarantee given in Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' We begin by proving that the  iterates generated by Algorithm 2 satisfy the constraints in (5) with increasing accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Let ˆρ be the current overall solution, and let ρ(k) be a solution to (14) obtained from running  Algorithm 1 using fixed precision ξ ∈ (0, 1) in iteration k of Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Then, the following hold:  (a) For k ≥ 0, η(k) ≥  1  ξk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  (b) For k ≥ 0, ρ = ˆρ +  1  η(k) Q ◦ ρ(k) satisfies  max  �  γ − tr  �  C  ∥C∥F  ρ  �  , ∥ρ − n−1I∥tr  �  ≤ ξk+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' We begin by establishing that for k ≥ 0, the current solution satisfies  max  �  γ − tr  �  C  ∥C∥F  ρ  �  , ∥ρ − n−1I∥tr  �  ≤  ξ  η(k) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  (15)  First, observe that when k = 0, we have εi = 1  n for i ∈ [n], ˆρ = 0n×n, η(0) = 1 and Q = ee⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Under these  conditions, one can observe that if ρ(0) solves (14) to precision ξ, then  n  �  i=1  ����⟨i|ρ|i⟩ − 1  n  ���� =  n  �  i=1  ����⟨i|ˆρ +  1  η(0) ρ(0)|i⟩ − 1  n  ���� =  n  �  i=1  ����  1  η(0) ρ(0) − εi  ���� =  1  η(0)  n  �  i=1  ���ρ(0) − η(0)εi  ��� ≤  ξ  η(0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  In other words, ρ = ρ(0) satisfies  ∥ρ − n−1I∥tr ≤  ξ  η(0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Next, by the definition of OCη(γ−ˆγ) we have  tr  �  C∥C∥−1  F ρ  �  = tr  �  C∥C∥−1  F  �  ˆρ +  1  η(0) Q ◦ ρ(0)  ��  = tr  �  C∥C∥−1  F ˆρ  �  +  1  η(0) tr  �  C∥C∥−1  F  �  Q ◦ ρ(0)��  = tr  �  C∥C∥−1  F 0n×n�  �  ��  �  =0  + 1  η(0) tr  �  C∥C∥−1  F  �  Q ◦ ρ(0)��  =  1  η(0) tr  �  C∥C∥−1  F  �  Q ◦ ρ(0)��  =  1  η(0) tr  �  C∥C∥−1  F ρ(0)�  ≥  1  η(0)  �  [η(0)(γ − ˆγ(0))] − ξ  �  = γ −  ξ  η(0) ,  where we used the fact that ˆγ(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Next, letting ˆρ be our current solution and ˆγ = tr  �  C  ∥C∥F ˆρ  �  be the objective value attained at this  solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' When k ≥ 1, we have εi = ˆρii − 1  n for i ∈ [n] and Q = (ee⊤ − I) + diag(sign(−ε)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' For this choice  of parameters, the general feasibility problem (14) reduces to the refining problem (11) and the solution  ρ(k) obtained via Algorithm 1 is therefore a ξ-precise solution to (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Accordingly,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' for k ≥ 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' setting  27  ρ = ˆρ +  1  η(k) Q ◦ ρ(k) improves the precision to which the maximally mixed state is approximated:  n  �  i=1  ����⟨i|ρ|i⟩ − 1  n  ���� =  n  �  i=1  ����⟨i|ˆρ +  1  η(k) Q ◦ ρ(k)|i⟩ − 1  n  ����  =  n  �  i=1  ����  �  ˆρii +  1  η(k)  �  sign(−εi) · ρ(k)  ii  ��  − 1  n  ����  =  n  �  i=1  ����  �  ˆρii − 1  n  �  +  1  η(k) sign(−εi)ρ(k)  ii  ����  =  n  �  i=1  ����εi +  1  η(k) sign(−εi)ρ(k)  ii  ���� =  1  η(k)  n  �  i=1  ���η(k)εi + sign(−εi)ρ(k)  ii  ��� ≤  ξ  η(k) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Consequently, we can conclude that at iteration k ≥ 1, the trace distance from our solution to the maximally  mixed state is  ∥ρ − n−1I∥tr ≤  ξ  η(k) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  (16)  Next, letting ˜γ(k) = tr  �  C  ∥C∥F Q ◦ ρ(k)�  , one can observe  tr  �  C  ∥C∥F  ρ  �  = tr  �  C  ∥C∥F  �  ˆρ +  1  η(k) Q ◦ ρ(k)  ��  = tr  �  C  ∥C∥F  ˆρ  �  +  1  η(k) tr  �  C  ∥C∥F  Q ◦ ρ(k)  �  = ˆγ + ˜γ(k)  η(k) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  For any ρ(k) which is ξ-close to the set Cη(γ−ˆγ) we must have  ˜γ(k) ≥ η(k)(γ − ˆγ) − ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  It follows:  tr  �  C  ∥C∥F  ρ  �  = ˆγ + ˜γ(k)  η(k) ≥ ˆγ +  1  η(k)  �  η(k)(γ − ˆγ) − ξ  �  = γ −  ξ  η(k) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  (17)  It therefore follows from (16) and (17) that ρ = ˆρ +  1  η(k) Q ◦ ρ(k) satisfies inequality (15) for all k ≥ 0, and  we can now use this fact to establish the lower bound on η(k), which we prove by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  For k = 0, we have η(0) = 1, for which η(k) ≥  1  ξk trivially holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' By the induction hypothesis, it assumed  that η(ℓ) ≥  1  ξℓ is true for ℓ = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' , k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' From here, applying (15) yields  η(k+1) =  1  max  �  γ − tr  �  C  ∥C∥F ρ  �  , ∥ρ − n−1I∥tr  � ≥  1  ξ  η(k)  ≥  1  ξk+1 ,  which completes the proof of (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Having demonstrated that (a) holds, to prove (b), we can simply combine inequality (15) with the lower  bound η(k) ≥  1  ξk , which together imply  max  �  γ − tr  �  C  ∥C∥F  ρ  �  , ∥ρ − n−1I∥tr  �  ≤  ξ  η(k) ≤ ξk+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  That is, (b) holds, and the proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  The next result establishes polynomial convergence of Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Let 0 < ζ ≪ ξ < 1, and η(0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Then, Algorithm 2 terminates in at most  K = O  �  log  �1  ζ  ��  iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  28  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' The result follows from Theorem 4(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Observe that in Theorem 4, we do not consider whether the updated solution remains positive semidef-  inite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' It turns out that, by the nature of our update scheme, the minimum eigenvalue of the final solution  obtained via Algorithm 2 will never fall significantly below zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Moreover, we employ a rounding procedure  to modify the entries of the solution we obtain from Algorithm 2 so that we arrive at an exactly feasible  solution to (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Before proceeding further, we formalize this notion, and derive a lower bound on the small-  est eigenvalue of the final solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' We utilize Lemma 13 to bound the minimum eigenvalue of the terms  1  η(k) Q(k) ◦ ρ(k) that are used to update the overall solution in each iteration, from which a lower bound  minimum eigenvalue bound for the final solution obtained by Algorithm 2 readily follows upon applying  Lemma 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Let ρ(k) be a solution to (14) obtained from running Algorithm 1 using fixed precision  ξ ∈ (0, 1) in iteration k of Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Then, the following hold:  (a) For k ≥ 1,  1  η(k) Q(k) ◦ ρ(k) ⪰ −2 · ξkI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  (b) Suppose Algorithm 2 is run with final precision ζ, and terminates after K iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Then, the solution  ρ output by Algorithm 2 satisfies  λmin(ρ) ≥ λmin  �  ρ(0)�  − 2 ·  K  �  k=1  ξk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' We begin with a proof of (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' In what follows, we assume without loss of generality that Q(k) has at  least one negative eigenvalue (otherwise, Q(k) ◦ ρ(k) ⪰ 0 trivially holds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Hence, a combined application of  Corollary 1 and Lemma 13 yields  λmin  �  Q(k) ◦ ρ(k)�  ≥ λmax  �  ρ(k)�  λmin  �  Q(k)�  ≥ −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Therefore, recalling that by Theorem 4(a) we have η(k) ≥  1  ξk , it follows  λmin  � 1  η(k) Q ◦ ρ(k)  �  =  1  η(k) λmin  �  Q ◦ ρ(k)�  ≥ −2  η(k) ≥ −2 · ξk,  which completes the proof of (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Noting that the final solution can be expressed as  ρ =  K  �  k=0  1  η(k) Q(k) ◦ ρ(k) = ρ(0) +  K  �  k=1  1  η(k) Q(k) ◦ ρ(k),  the result in (b) follows from (a) and repeated application of Lemma 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  The next corollary computes bounds for the geometric series that appears in Proposition 9(b) for different  values of the fixed precision parameter ξ, by computing the value of the series to ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Let ξ =  ˜ξ  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Then, for any positive integer K, the following hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  (a) If ˜ξ = 10−2, then −2 · �K  k=1 ξk ≥ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  (b) If ˜ξ = 10−4, then −2 · �K  k=1 ξk ≥ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='00005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  The goal of Corollary 6 is simply to show that with fixed precision we obtain matrices with eigenvalues that  are, in the worst case, slightly negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' A shift of the spectrum suffices to restore positive semidefiniteness,  and it does not change the constraint violation or the objective function value by a large amount, as we show  next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  29  Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Suppose Algorithm 2 is run with final precision ζ, and terminates after K iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Let  ρ be the solution output by Algorithm 2, and let ξ be the fixed precision used in every iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Then, letting  δ = 2 ·  K  �  k=1  ξk,  (18)  it follows that  ˜ρ =  1  1 + nδ (ρ + δI)  is a positive semidefinite matrix at trace distance ζ from ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Moreover, ˜ρ satisfies:  max  �  γ − tr  �  C  ∥C∥F  ˜ρ  �  ,  ��˜ρ − n−1I  ��  tr  �  ≤ 2ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  In other words, ˜ρ is a 2ζ-precise solution to (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' First, observe that ˜ρ ⪰ 0 by the definition of ˜ρ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' applying Proposition 9(b), we have λmin(ρ) ≥ −δ,  which implies that ρ + δI ⪰ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' From the definition of ˜ρ, we also have:  ∥ρ − ˜ρ∥tr =  ����ρ −  �  1  1 + nδ (ρ + δI)  �����  tr  =  ����  1 + nδ − 1  1 + nδ  ρ −  δ  1 + nδ I  ����  tr  =  δ  1 + nδ ∥nρ − I∥tr  =  nδ  1 + nδ  ��ρ − n−1I  ��  tr  ≤  nδ  1 + nδζ  < ζ,  where the second to last inequality follows from the fact that ρ is obtained from running Algorithm 2 with  ζ as the final precision parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Next, we leverage our bound on the trace distance from ˜ρ to ρ to establish that ˜ρ is indeed an accurate  solution to the renormalized SDO problem (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' First, note that  ��˜ρ − n−1I  ��  tr =  ��˜ρ − n−1I + (ρ − ρ)  ��  tr  =  ��(˜ρ − ρ) +  �  ρ − n−1I  ���  tr ≤ ∥˜ρ − ρ∥tr +  ��ρ − n−1I  ��  tr ≤ 2ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Further, applying a matrix H¨older inequality, one can observe:  ����tr  �  C  ∥C∥F  ρ  �  − tr  �  C  ∥C∥F  ˜ρ  ����� ≤  ����  C  ∥C∥F  ���� ∥ρ − ˜ρ∥tr ≤ ∥ρ − ˜ρ∥tr < ζ,  from which we can conclude  γ − tr  �  C  ∥C∥F  ˜ρ  �  = γ − tr  �  C  ∥C∥F  ˜ρ  �  +  �  tr  �  C  ∥C∥F  ρ  �  − tr  �  C  ∥C∥F  ρ  ��  =  �  γ − tr  �  C  ∥C∥F  ρ  ��  +  �  tr  �  C  ∥C∥F  ρ  �  − tr  �  C  ∥C∥F  ˜ρ  ��  ≤ 2ζ,  as γ − tr  �  C  ∥C∥F ρ  �  ≤ ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  It is important at this point for us to remark that fixing ξ ∈ (0, 1) does not limit us with respect to how  accurately we can solve (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' We can always make the final precision parameter arbitrarily small using only  �  O 1  ζ (1) iterations, as the overall running time depends only poly-logarithmically on ζ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Accordingly, we  take advantage of this fact and revisit the approximation guarantee provided in Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  30  Proposition 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Let ρ be a ζ-accurate solution to the renormalized and relaxed SDO problem (5) with input  matrix C and ζ =  �  ǫ  n∥C∥F  �4  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Let γζ = tr (Cρ) be the value attained by ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Then, there is a quantum state  ρ∗ at trace distance O  �  ǫ  n∥C∥F  �  of ρ such that nρ∗ is a feasible point of SDO problem (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' In particular  |γζn∥C∥F − tr (nρ∗C)| = O (ǫ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Moreover, it is possible to construct ρ∗ in time O(n2) given the entries of ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' The proof almost exactly follows the proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='1 in [10], regardless, we present the  adjusted proof for completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Our aim is to show that a ζ-precise solution ρ to (5) obtained using  Algorithm 2 can be used to construct ρ∗ such that nρ∗ is an exactly feasible solution to (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  We begin shifting ρ in order to ensure that our solution is positive semidefinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' In particular, choosing  δ according to (18), we set  ˜ρ =  1  1 + nδ (ρ + δI) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  It then follows from Proposition 10 that ˜ρ satisfies  ˜ρ ⪰ 0,  ∥ρ − ˜ρ∥tr ≤ ζ,  max  �  γ − tr  �  C  ∥C∥F  ˜ρ  �  ,  ��˜ρ − n−1I  ��  tr  �  ≤ 2ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  (19)  Next, we examine the diagonal elements of ˜ρ and check whether modifications need to be made to  ensure that our solution is an exactly feasible point to the renormalized SDO problem (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Namely, if  |⟨i|˜ρ|i⟩ − 1  n| >  √2ζ  n  for i ∈ [n], we replace ˜ρii with 1  n and set all elements in the i-th row and the i-th column  to 0, and denote the resulting matrix by ρ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' From here we introduce another matrix W which we obtain  by replacing each diagonal entry of ρ′ with 1  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' In general we may not have W ⪰ 0, so the authors in [10]  suggest using the convex combination:  ρ∗ =  1  1 + √2ζ  �  W +  √2ζ  n I  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Then, ρ∗ ⪰ 0 and by construction ⟨i|ρ∗|i⟩ =  1  n for all i ∈ [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Hence, ρ∗ is a feasible solution to the  renormalized SDO problem (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  What remains is to show that the above reformulations yield the desired approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Denote by  B = {i : |n⟨i|˜ρ|i⟩ − 1| > √2ζ} ⊂ [n] the set of diagonal entries that deviate substantially from 1  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Without  loss of generality, it suffices to assume that such elements are found in the first |B| rows of ˜ρ, in which case  ∥ρ′ − ˜ρ∥tr =  ����  �  n−1IB  0  0  ˜ρ22  �  −  �  ˜ρ11  ˜ρ12  ˜ρ21  ˜ρ22  �����  tr  =  ����  �  n−1IB − ˜ρ11  −˜ρ12  −˜ρ21  0  �����  tr  ≤ ∥˜ρ11∥tr + 2∥˜ρ12∥tr + ∥n−1IB∥tr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  (20)  Since ˜ρ is a 2ζ-precise solution to (5), ˜ρ obeys  n  �  i=1  ����⟨i|˜ρ|i⟩ − 1  n  ���� ≤ 2ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Therefore, we must have  |B|  √2ζ  n  ≤ 2ζ,  which equates to |B| ≤ n√2ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Now, by the definition of B, it follows  ∥˜ρ22∥tr ≥ (n − |B|)1 − √2ζ  n  ≥ (n − n  �  2ζ)1 − √2ζ  n  = (1 −  �  2ζ)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  31  Following [10], we invoke a result from [40], which states  ����  �∥˜ρ11∥tr  ∥˜ρ12∥tr  ∥˜ρ⊤  12∥tr  ∥˜ρ22∥tr  ����� ≤  ����  �˜ρ11  ˜ρ12  ˜ρ⊤  12  ˜ρ22  �����  tr  = ∥˜ρ∥tr = tr (˜ρ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Using the fact that ∥ · ∥tr ≥ ∥ · ∥2, where ∥ · ∥2 is the Frobenius, or Schatten-2 norm, the above implies  ∥˜ρ11∥2  tr + 2∥˜ρ12∥2  tr + ∥˜ρ22∥2  tr ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  As ∥˜ρ22∥tr ≥ (1 − √2ζ)2, it can be seen trivially that ∥˜ρ22∥2  tr ≥ (1 − √2ζ)4, and thus  ∥˜ρ11∥2  tr + 2∥˜ρ12∥2  tr ≤ 1 − (1 −  �  2ζ)4 = O(  �  ζ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Consequently ∥˜ρ11∥tr + 2∥˜ρ12∥tr = O  �  ζ  1  4  �  , and plugging this into equation (20) asserts  ∥ρ′ − ˜ρ∥tr = O  �  ζ  1  4  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  (21)  Let R be a diagonal matrix whose elements are Rii ∈  �  −  √2ζ  n ,  √2ζ  n  �  for i ∈ [n], such that  W = ρ′ + R,  and note that R + √2ζn−1I ⪰ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Upon normalizing the trace, one can observe  ρ∗ =  1  1 + √2ζ  �  ρ′ + R +  �  2ζn−1I  �  ⪰ 0,  with ρ∗  ii = 1  nn for all i ∈ [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Thus, nρ∗ is a feasible solution to the SDO problem (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Further, by a triangle  inequality we have  ∥ρ′ − ρ∗∥tr =  1  1 + √2ζ  ���  �  2ζρ′ + R +  �  2ζn−1I  ���  tr = O(  �  ζ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  (22)  Combining equations (21) and (22) and noting ζ =  �  ǫ  n∥C∥F  �4  , applying another triangle inequality yields  ∥˜ρ − ρ∗∥tr = O  �  ζ  1  4  �  = O  \uf8eb  \uf8ed  ��  ǫ  n∥C∥F  �4� 1  4 \uf8f6  \uf8f8 = O  �  ǫ  n∥C∥F  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Then, the result follows from a matrix H¨older inequality:  |tr (nCρ) − tr (nCρ∗)| ≤ n∥C∥∥ρ − ρ∗∥tr ≤ n∥C∥F (∥ρ − ˜ρ∥tr + ∥˜ρ − ρ∗∥tr)  = O  �  n∥C∥F  �  ζ + ζ  1  4  ��  = O  �  n∥C∥F  ��  ǫ  n∥C∥F  �4  +  ǫ  n∥C∥F  ��  = O  �  ǫ4  (n∥C∥F)3 + ǫ  �  = O (ǫ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  5  Complexity  We now analyze the worst case overall running time of our Iterative Refinement Method given in Algorithm  2 in both the classical and quantum settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  32  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='1  Classical running time  As we saw in Section 3, the complexity of using Algorithm 1 to solve the SDO problem (3) scales poorly in  the inverse precision, with the classical algorithm exhibiting an O(ǫ−12) dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' In both the classical  and quantum cases, our iterative refinement scheme reconciles the poor scaling in ǫ because it possesses  the following two properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' First, we can obtain an arbitrarily precise solution to (5) in at most �  O 1  ζ (1)  iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Second, it suffices to treat ξ as fixed for the oracle calls that occur in each iteration, as the  precision of the final solution is a byproduct of how we use these solution of the refining problems to produce  a solution to (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  The next result formalizes the above argument, and establishes the complexity of Algorithm 2 for the  classical case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Let C ∈ Sn with row sparsity s and ǫ ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Then, fixing ξ = 10−2, and setting  ζ =  �  ǫ  n∥C∥F  �4  , a classical implementation of Algorithm 2 solves (3) up to additive error O(ǫ) in time  O  �  min{n2s, nω} · polylog  �  n, ∥C∥F, 1  ǫ  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  The output of the algorithm is a classical description of a matrix ˆρ ∈ Sn such that  ˜ρ =  1  1 + nδ (ˆρ + δI) ,  is a 2ζ-precise solution to (5), where δ is defined according to (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' The entries of ˜ρ can be modified to  construct a matrix ρ∗ at trace distance O  �  ǫ  n∥C∥F  �  of ˜ρ in time O(n2), such that nρ∗ is a feasible point of  the SDO problem (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Given that C is an s-sparse matrix, we can load C in O(ns) time, and from here we must compute  ∥C∥F , which requires O(ns) arithmetic operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' In every iteration of Algorithm 2, we make a call to our  subroutine in Algorithm 1, before updating the solution and preparing the next refining problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Updating  the solution involves matrix addition between two n × n matrices and requires O(n2) arithmetic opera-  tions, whereas updating Q and ε for the next refining problem can be accomplished using O(n) arithmetic  operations, as only the diagonal entries of Q need to be stored and maintained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  In view of Proposition 6, the dominant operation at each iteration is the use of Algorithm 1 to solve the  SDO problem at hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' By Proposition 6, Algorithm 1 can be used to solve (14) to additive error ξ in time  T classical  HU  = O  �  min{n2s, nω} log2(n)ξ−3�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  If every call to Algorithm 1 is made using precision ξ, then by Corollary 5, Algorithm 2 converges in at most  O  �  log(ζ−1)  �  iterations, and we can thus express the overall running time of Algorithm 2 as  O  ��  min{n2s, nω} log2(n)ξ−3�  log(ζ−1)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  In the context of Algorithm 2, it suffices to carry out each of the calls to the SDO subroutine (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=', calls to  Algorithm 1) using fixed precision ξ to obtain a 2ζ-precise solution to (5) (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=', Proposition 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' The  above complexity thus reduces to  O  �  min{n2s, nω} log2(n) log(ζ−1)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  For our choice of ζ =  �  ǫ  n∥C∥F  �4  , one can observe  O  �  min{n2s, nω} log2(n) log(ζ−1)  �  = O  �  min{n2s, nω} · polylog  �  n, ∥C∥F, 1  ǫ  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  33  Proposition 11 certifies that the above running time suffices to obtain a ρ from which we can construct ρ∗  in time O(n2), such that nρ∗ is a feasible point of the SDO problem (3) satisfying  |γζn∥C∥F − tr (nρ∗C)| = O(ǫ),  and the proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='2  Quantum running time  Just as in the classical case, we show that a quantum implementation of Algorithm 2 mitigates the poor  scaling in the running time with respect to the inverse precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Our quantum implementation of Algorithm 2 is provided in Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' The relevant error parameters  are the same as those appearing in Algorithm 2: (i) ξ, the fixed precision used to test closeness to the sets  Cη(γ−ˆγ) and Dηε in every iteration, (ii) ζ, the precision to which the final solution satisfies the functional  constraints in (5), and (iii) ǫ, the additive error to which we seek to solve (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' In our initialization steps  we set the values of Q, ε and η such that the first iteration corresponds to solving the feasibility problem  (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' We also create a vector p = 0n×1 that will be used to maintain a classical description of the diagonal  elements of our solution over the course of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  At every iteration k, a call is made to Algorithm 1 with separation oracles OCη(γ−ˆγ) and ODηε to solve  (14) using fixed precision ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' If the oracles accept the candidate state, then Algorithm 1 returns a real-valued  vector y(k) ∈ R2 along with a diagonal matrix D(k) such that the Hamiltonian associated with the Gibbs  state that solves the refining problem is  H(k) = y(k)  1 Q(k) ◦  C  ∥C∥F  + y(k)  2 D(k),  with ∥y(k)∥1 ≤ 4 log(n)ξ−1 and ∥D(k)∥ ≤ 1 for every k ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' This allows us to efficiently describe the solution  to each refining problem, and once the algorithm has terminated, it facilitates an efficient way to describe  the final solution as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='5 First, observe that the matrices Q(k) and D(k) can be completely described by  their diagonal elements;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' letting q(k) ∈ Rn and d(k) ∈ Rn be the vectors that store the diagonal elements of  Q(k) and D(k), respectively, we have  Q(k) = (ee⊤ − I) + diag  �  q(k)�  ,  D(k) = diag  �  d(k)�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Therefore, we store the solution to the refining problem at iteration k as the tuple  (η(k), y(k), q(k), d(k)),  and the final solution to (5) is defined as  ˜ρ =  1  1 + nδ  \uf8ee  \uf8f0  \uf8eb  \uf8ed  K  �  k=0  1  η(k) Q(k) ◦  exp  �  −  �  y(k)  1 Q(k) ◦  C  ∥C∥F + y(k)  2  diag  �  d(k)���  tr  �  exp  �  −  �  y(k)  1 Q(k) ◦  C  ∥C∥F + y(k)  2  diag  �  d(k)����  \uf8f6  \uf8f8 + δI  \uf8f9  \uf8fb ,  (23)  where δ is defined according to (18) (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=', Proposition 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' We point out that this marks a key difference  between the output of our algorithm and other quantum SDO solvers based on Gibbs sampling [9, 10, 11, 60,  61], which need only return a single state preparation pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' This however does not increase the cost of the  method;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' the iteration bound in Corollary 5 ensures that there are only at most �O 1  ζ (1) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=', a poly-logarithmic  number) of these tuples to be stored over the course of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Using the QRAM input model, one can  5Requiring an explicit classical description of the solution would in fact lead to a worse running time overall when compared  to the classical implementation we studied in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  34  use the stored tuples to construct a block-encoding of the final solution up to error θ using �On,∥C∥F , 1  ǫ , 1  θ (√n)  queries to the QRAM and �On,∥C∥F , 1  ǫ (n) classical operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' This construction, and the associated time  complexity are analyzed later in Proposition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' We further demonstrate that provided classical access to an  s-sparse matrix A ∈ Rn×n (with subnormalization factor 1) and access to QRAM, one can estimate tr(A˜ρ)  to additive error θ using �On,∥C∥F , 1  ǫ  � √n  θ  �  queries to the QRAM and �  On,∥C∥F , 1  ǫ (ns) classical operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' If  A has a subnormalization factor αA > 1, then θ must be scaled down by αA, increasing the cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Additionally, we require Algorithm 1 to return the estimates ˜p(k) ∈ Rn (a classical estimate of the diagonal  elements of the solution to the refining problem) and ˜c(k) ∈ R (a classical estimate of the objective value  attained by the solution of the refining problem) that are used to test ξ-closeness for the accepted state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' In this  fashion, we can (classically) prepare the refining problem data for the next iteration without increasing the  cost of the algorithm with respect to n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' the objective value can be updated using O(1) arithmetic operations  using ˜c(k), while updating the residuals along the diagonal of ρ requires O(n) arithmetic operations provided  classical access to ˜p(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  If the current solution is indistinguishable up to precision ζ from the maximally mixed state n−1I, and  provides an objective value of at least γ − ζ, the algorithm terminates and reports the current solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Otherwise, we construct the refining problem associated with our current solution and proceed to the next  iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  The next result gives the overall running time required to solve (3) to additive error O(ǫ) using the  QRAM input model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Let C ∈ Sn, ǫ ∈ (0, 1), and set ζ =  �  ǫ  n∥C∥F  �4  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Assume we have classical access to C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Then,  in the QRAM input model, Algorithm 3 solves (3) up to additive error O(ǫ) using  O  �  n1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='5 · polylog  �  n, ∥C∥F , 1  ǫ  ��  accesses to the QRAM and O(ns) classical arithmetic operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  The output of the algorithm is a collection of tuples {(η(k), y(k), q(k), d(k))}K  k=0 such that  ˜ρ =  1  1 + nδ  \uf8ee  \uf8f0  \uf8eb  \uf8ed  K  �  k=0  1  η(k) Q(k) ◦  exp  �  −  �  y(k)  1 Q(k) ◦  C  ∥C∥F + y(k)  2  diag  �  d(k)���  tr  �  exp  �  −  �  y(k)  1 Q(k) ◦  C  ∥C∥F + y(k)  2  diag  �  d(k)����  \uf8f6  \uf8f8 + δI  \uf8f9  \uf8fb ,  is a 2ζ-precise solution to (5), where δ is defined according to (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' The entries of ˜ρ can be modified to  construct a matrix ρ∗ at trace distance O  �  ǫ  n∥C∥F  �  of ˜ρ in time O(n2), such that nρ∗ is a feasible point of  the SDO problem (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Given that C is an s-sparse matrix, we can classically load C in O(ns) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Similarly, for normaliza-  tion purposes we classically compute ∥C∥F, which requires O(ns) arithmetic operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' In each iteration  we use Algorithm 1 to solve (14), and use classical estimates of the diagonal elements of the refining solution,  and a classical estimate of the objective value attained by the refining solution to update the solution and  data for the refining problem we need to solve in the next iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Letting T quantum  HU  be the cost of using Algorithm 1 as an approximate SDO subroutine, by Proposition 8,  Algorithm 1 solves (14) to additive error ξ using at most  T quantum  HU  = �  O n  ξ  �  n1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='5ξ−5�  accesses to the QRAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Classically updating the objective value requires O(1) arithmetic operations while  updating the vector p which stores a classical description of the diagonal elements of our solution as  pi ← pi + Qii  η(k) ˜p(k)  i  35  Algorithm 3 Iterative Refinement for SDO Approximations of QUBOs using a quantum computer  Input: Error tolerances ǫ ∈ (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' 1) and ζ =  �  ǫ  n∥C∥F  �4  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' upper bound on objective value γ ∈ [−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' 1]  Output: Tuples (η(k),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' y(k),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' q(k),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' d(k)) that define 2ζ-precise solution to (5) using Equation (23)  Initialize: p ← 0n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Q ← ee⊤,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' εi = 1  n for i ∈ [n],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' ˆγ ← 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' η(0) ← 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' k ← 0  while max {γ − ˆγ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' ∥ε∥1} > ζ do  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' (y(k), D(k), ˜p(k), ˜c(k)) ← Solve (14) to precision ξ using Algorithm 1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Store diagonal elements of Q and D(k)  q(k)  i  ← Qii,  d(k)  i  ← D(k)  ii  for i ∈ [n]  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Store description of solution to the refining problem (η(k), y(k), q(k), d(k))  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Update the objective value of solution:  ˆγ ← ˆγ +  1  η(k) ˜c(k)  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Update estimate of diagonal entries:  pi ← pi + Qii  η(k) ˜p(k)  i  for i ∈ [n]  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Compute element-wise deviations from the maximally mixed state:  εi ← pi − 1  n  for i ∈ [n]  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Classically update:  Qii ← sign(−εi) for i ∈ [n],  η(k+1) ←  1  max {γ − ˆγ, ∥ε∥1}  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' k ← k + 1  end  36  requires O(n) classical arithmetic operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Likewise, ε and Q can each be updated using O(n) classical  arithmetic operations, as we only need to store the diagonal elements of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' This also implies that we can  update Q◦  C  ∥C∥F using �On(n) operations, for only the diagonal elements need to be updated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' When compared  to loading and normalizing the coefficient matrix C, or our use of Algorithm 1 as a subroutine for solving  (14), these intermediate computation steps are negligible and do not factor into the overall running time  using O notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  By Corollary 5, Algorithm 3 terminates in at most �  O 1  ζ (1) iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Therefore, the worst case complexity  of Algorithm 3 can be bounded by  O  �  n1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='5ξ−5 · polylog  �  n, ∥C∥F, 1  ǫ  ��  accesses to the QRAM, and O (ns) classical arithmetic operations to load and normalize C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Further, it  suffices to use fixed precision ξ for the every call to Algorithm 1 to reach a final solution that solves (5) to  additive error ζ, as the final solution can always be arbitrarily precise using a �O 1  ζ (1) calls to Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Since ξ is a fixed constant in the context of Algorithm 3, the overall running time of Algorithm 3 simplifies  to  O  �  n1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='5 · polylog  �  n, ∥C∥F, 1  ǫ  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  accesses to the QRAM, and O (ns) classical arithmetic operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Just as in the proof of Theorem 5, applying Proposition 11 with our choice of ζ =  �  ǫ  n∥C∥F  �4  implies that  the above running time is sufficient to obtain a solution that can be used to solve (3) up to additive error  O(ǫ), and the proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  We analyze the cost of Algorithm 3 without access to QRAM in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Using the sparse-access  input model, one can show that the resulting scheme exhibits an oracle complexity of  O  �  n1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='5s0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='5+o(1) · polylog  �  n, ∥C∥F, 1  ǫ  ��  ,  and requires O  �  n2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='5s0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='5+o(1) · polylog  �  n, ∥C∥F , 1  ǫ  ��  additional gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  To summarize, in the absence of  QRAM, the number of oracle accesses is a factor √s larger due to the Hamiltonian simulation, and the  gate complexity increases by a factor n due to the cost of constructing OD without QRAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  We conclude this section by establishing the costs of preparing a block-encoding of the final solution, and  estimating trace inner products of the form tr(A˜ρ) for a given matrix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Proposition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Suppose that Algorithm 3 is run with ζ =  �  ǫ  n∥C∥F  �4  for some ǫ ∈ (0, 1), and terminates  after K iterations, classically outputting the tuples {(η(k), y(k), q(k), d(k))}K  k=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Then,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' letting C∥C∥−1  F  be stored  in QRAM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' and denoting the refining problem at iteration k by ρ(k),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' one can use {(η(k),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' y(k),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' q(k),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' d(k))}K  k=0  to implement an (n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' O(log(n),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' θ)-block-encoding of  ˜ρ =  1  1 + nδ  \uf8ee  \uf8f0  \uf8eb  \uf8ed  K  �  k=0  1  η(k) Q(k) ◦  exp  �  −  �  y(k)  1 Q(k) ◦  C  ∥C∥F + y(k)  2  diag  �  d(k)���  tr  �  exp  �  −  �  y(k)  1 Q(k) ◦  C  ∥C∥F + y(k)  2  diag  �  d(k)����  \uf8f6  \uf8f8 + δI  \uf8f9  \uf8fb ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  with at most �On,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='∥C∥F ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' 1  ǫ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' 1  θ (√n) queries to the QRAM and �  On,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='∥C∥F ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' 1  ǫ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' 1  θ (n) classical operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' First, note that  exp  �  −  �  y(k)  1 Q(k) ◦  C  ∥C∥F + y(k)  2  diag  �  d(k)���  tr  �  exp  �  −  �  y(k)  1 Q(k) ◦  C  ∥C∥F + y(k)  2  diag  �  d(k)���� = n−1I  37  whenever y = (0, 0)⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Thus, by choosing y(K+1) = (0, 0)⊤, η(K+1) =  1  nδ , and Q(K+1) = ee⊤ we can simplify  the expression of the final solution to  1  1 + nδ  \uf8ee  \uf8f0  K+1  �  k=0  1  η(k) Q(k) ◦  exp  �  −  �  y(k)  1 Q(k) ◦  C  ∥C∥F + y(k)  2  diag  �  d(k)���  tr  �  exp  �  −  �  y(k)  1 Q(k) ◦  C  ∥C∥F + y(k)  2  diag  �  d(k)����  \uf8f9  \uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  To ensure that the stated complexity holds, for each k ∈ [K + 1], we block-encode  A(k) = Q(k) ◦ C  ∥C∥F  + D(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  First, note that with classical access to C and q(k), one can store Q(k)◦C  ∥C∥F  in the QRAM by properly updating  C∥C∥−1  F  in the QRAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' This step requires O(n) classical operations, as the only non-trivial computation that  is performed is limited to the diagonal elements of the involved matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Then, with Q(k)◦C  ∥C∥F  stored in QRAM,  noting that  ��� Q(k)◦C  ∥C∥F  ���  F ≤ 1 holds for every k ∈ [K + 1], we apply Lemma 3 to construct a (1, log(n) + 2, θ1)-  block-encoding of  Q(k)◦C  ∥C∥F  in time O  �  polylog  �  n  θ1  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Similarly, as we saw in the proof of Proposition 7,  classical access to d(k) and access to QRAM implies one can implement a (1, log(n) + 3, θ1)-block-encoding  of D(k) can be constructed in time �O n  θ1 (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Again following the proof of Proposition 7, applying Corollary 3 with y(k) satisfying ∥y(k)∥1 = �  On(ξ−1)  implies that we can construct a unitary which prepares a copy of the Gibbs state ρ(k) encoding the solution  to the refining problem at iteration k with at most  �  O n  ξ  �√nαξ−1�  = �  On  �√n  �  ,  accesses to the QRAM, as α = 1 and ξ is a fixed constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Therefore, by Lemma 7, preparing a (1, log(n) +  a, θ1) block-encoding of a purification of ρ(k) thus requires �  O n  θ1 (√n) queries to the QRAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Next, provided classical access to the vector q(k) that store the diagonal elements of Q(k), access to  QRAM implies that we can efficiently implement an oracle OQ(k) that returns the entries of Q(k) in a binary  description:  OQ(k) : |i⟩ |j⟩ |0⟩⊗p �→ |i⟩ |j⟩  ���q(k)  ij  �  ,  ∀i, j ∈ [2log n] − 1,  where q(k)  ij  is a p-bit binary description of the ij-matrix element of Q(k) for k = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' , K +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' By construction  each matrix Q(k) may be fully dense, and hence an application of Lemma 4 with sr = sc = n asserts that in  the presence of QRAM, one can construct a (n, log(n) + 3, θ2)-block-encoding of Q(k) in time �O n  θ2 (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  From here, we can utilize Proposition 4 with θ1 = θ2 =  ˜θ  10 to construct an (n, a+ 4 log(n2)+ 12, ˜θ)-block-  encoding of Q(k) ◦ ρ(k) in time �O n  ˜θ (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Repeating the above steps for k = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' , K + 1, it follows that we  can block-encode each of the terms Q(k) ◦ ρ(k) using at most  �On, 1  ˜  θ  �  K√n  �  = �On,∥C∥F , 1  ǫ , 1  ˜  θ  �√n  �  queries to the QRAM and �On,∥C∥F , 1  ǫ , 1  ˜  θ (n) classical operations, as K = O  �  polylog  �  n, ∥C∥F, 1  ǫ  ��  by Corollary  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Finally, what remains is to take the linear combination of these terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' To do so, we choose our weights  to be wk =  1  2(1+nδ)η(k) , which indeed satisfies ∥w∥1 ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Then, we can construct a (K + 2, log(K + 2), 0)-  state-preparation pair PL, PR for w, which can be constructed by taking a log(K + 2)-fold tensor product  of the Hadamard gate, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=',  PL = PR =  1  √  2  �1  1  1  −1  �⊗ log(K+2)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  38  We are now in a position to apply Proposition 1, and choosing ˜θ =  θ  n, we can obtain W upon adding a  control qubit to the circuits used to construct the block-encoding of each Q(k) ◦ ρ(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' As a result, we obtain  an (n, O(log(n), θ)-block-encoding of ˜ρ with a single use of W, PR and P †  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Summing the cost of each step in  the construction we arrive at total cost of  �  On,∥C∥F , 1  ǫ , 1  θ  �√n  �  queries to the QRAM and �On,∥C∥F , 1  ǫ , 1  θ (n) classical operations, and proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Proposition 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Suppose that Algorithm 3 is run with ζ =  �  ǫ  n∥C∥F  �4  for some ǫ ∈ (0, 1), and terminates  after K iterations, classically outputting the tuples {(η(k), y(k), q(k), d(k))}K  k=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Let A ∈ Rn×n be a matrix with  ∥A∥F ≤ 1 and assume classical access to A and C/∥C∥F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Then, with access to QRAM, one can compute a  θ-precise estimate of tr(A˜ρ) using at most �On,∥C∥F , 1  ǫ  � √n  θ  �  queries to the QRAM and �On,∥C∥F , 1  ǫ (n) classical  operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' See the proof of Theorem 8 in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  A QRAM-free version of Proposition 13 is also analyzed in Appendix B, and the cost is summarized  in Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Without access to QRAM, the cost increases with respect to n because computing the  Hadamard product of block-encodings introduces n as a subnormalization factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' This is compounded in  the running time, upon noting that we then have to scale down the error for the amplitude estimation steps  by n, and constructing sparse-access oracles for the intermediate block-encodings of Q and D that arise in  the trace estimation procedure requires �  On(n) gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='3  Comparison to existing SDO algorithms  Table 1 presents a comparison of the running time results for the algorithms we have proposed with the  running times of the best performing methods from both the classical and quantum literature when applied  to solving (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Note that when directly solving (3), m = n, and any feasible solution X to (3) satisfies tr (X) = n,  implying R = n for the algorithms based on the (Q)MMWU framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' We also point out that the running  times in Table 1 take into account the role of sparsity in context of the algorithms, which is measured as the  maximum number of nonzero entries per row of the constraint matrices A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' , An.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' When using either an  IPM or CPM to solve (3), the n constraint matrices are Ai = eie⊤  i (with row sparsity one) enforcing Xii = 1  for each diagonal element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' On the other hand, algorithms based on the (Q)MMWU or HU frameworks solve  (3) by reducing the problem to a feasibility problem;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' C enters into the resulting formulation as another  constraint matrix, and as a result, the relevant sparsity parameter is the maximum number of non-zeroes  per row of C, which we denote by s in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  There are additional considerations that need to be taken into account when making comparisons across  methodologies listed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Broadly speaking, both (Q)MMWUs and HU require normalizing the  problem by an upper bound on the trace of a primal solution, and in the case of (3), we have the natural  bound tr(X) = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Moreover, (Q)MMWUs and HU additionally normalize the cost matrix so that it exhibits  unit norm with respect to some norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' While these modifications amount to scaling the optimal objective  value of (3) by a fixed quantity, without employing any safeguards such as IR, these modifications impact  the scaling of the error as reflected in the fourth column of Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' On the contrary, (Q)IPMs do not  require the SDO problem to be normalized in any way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Finally there is a distinction with regard to output;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  (Q)IPMs explicitly report a classical description of the solution X, whereas only the classical HU algorithm  of [10] and our own classical IR-HU method do so;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' the primal QMMWU of [60] reports a state-preparation  pair y, and the MMWU algorithm found in [41] reports a “gradient” G ∈ Sn such that X = W exp(G)W  for a diagonal matrix W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' As we noted earlier, (Q)IPMs and (Q)MMWUs also utilize different definitions of  optimality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  39  References  Method  Runtime  Error Scaling  [34]  IPM  �  On, 1  ǫ  �  nω+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='5�  ǫ  [6]  QIPM  �  On,κ, 1  ǫ  �√n(n3κǫ−1 + n4)  �  ǫ  [41]  MMWU  �  On, 1  ǫ  �  nsǫ−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='5�  ∥C∥ℓ1ǫ  [60]  QMMWU  �  On, 1  ǫ  �  n5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='5sǫ−4�  n∥C∥ǫ  [10] (Classical)  HU  �  On,∥C∥  �  min{n2s, nω}ǫ−12�  n∥C∥ǫ  [10] (Quantum)  HU  �  On,∥C∥, 1  ǫ  �  n2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='5s0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='5+o(1)ǫ−28+o(1) exp  �  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='6  �  log(ǫ−1)  ��  n∥C∥ǫ  [10] (Quantum)  HU-QRAM  �  On,∥C∥, 1  ǫ  �  n1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='5s0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='5+o(1)ǫ−28+o(1) exp  �  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='6  �  log(ǫ−1)  ��  n∥C∥ǫ  This work (Classical)  IR-HU  �  On,∥C∥F , 1  ǫ  �  min{n2s, nω}  �  ǫ  This work (Quantum)  IR-HU  �  On,∥C∥F , 1  ǫ  �  n2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='5s0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='5+o(1)�  ǫ  This work (Quantum)  IR-HU-QRAM  �  On,∥C∥F , 1  ǫ (n1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='5) + ns  ǫ  Table 1: Total running times for classical and quantum algorithms to solve (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  It can be easily seen that both the classical and quantum implementations of our proposed methodology  outperform all existing algorithms that exhibit poly-logarithmic dependence on the precision ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Our classical  algorithm is only outperformed with respect to dimension by our own quantum algorithms, and the algorithm  from [41], which has an exponentially worse dependence on the inverse prevision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Moreover, to achieve the  same error scaling as our algorithms, the algorithm from [41] would require time �On, 1  ǫ  �  n4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='5sǫ−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='5�  (as it is  assumed ∥C∥ℓ1 = n in [41]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Up to poly-logarithmic factors, our quantum algorithms outperform each of  the classical and quantum solvers in every parameter, suggesting the first evidence of quantum advantage  for solving a special class of SDO problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Moreover, our implementation with access to QRAM dominates  all other algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' We therefore conclude that our proposed algorithms are respectively, the fastest both  in the classical and quantum regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  6  Conclusion  In this work we devised an iterative refinement scheme for a particular class of semidefinite optimization  problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' The key to our idea behind our speedup is to solve a sequence of related SDO problems in fixed  low precision, rather than solve one SDO problem using high accuracy requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Moreover, our solutions  satisfy a far stronger approximation guarantee over previous quantum solution methodologies for this class of  problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' We show that, provided access to QRAM, a quantum implementation of our algorithm can produce  accurate solutions to SDO approximations of QUBO problems in time O  �  ns + n1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='5 · polylog  �  n, ∥C∥F, 1  ǫ  ��  in the worst case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' In the absence of QRAM, one can bound the running time of the quantum algorithm  using using the sparse-access input model, in which case the algorithm exhibits an oracle complexity of  O  �  n2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='5s0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='5+o(1) · polylog  �  n, ∥C∥F, 1  ǫ  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  A classical implementation of the algorithm exhibits worst case  running time of O  �  min{n2s, nω} · polylog  �  n, ∥C∥F, 1  ǫ  ��  , which is at least a √n factor better than classical  IPMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  When compared to the best performing algorithms in the literature, our algorithms are the fastest in both  the quantum and classical regimes, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' We therefore conclude that this work serves as evidence of a  genuine quantum advantage for a specific class of SDO problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' We believe one can improve the theoretical  performance of our classical algorithm by not explicitly computing the density operator in our subroutines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  In particular, it may be possible to construct the separation oracles as we do in the quantum setting using  techniques to classically estimate trace inner products of the form tr(Aρ) (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=', Appendix A in [61]), and  applying ideas developed in [3, 41] to estimate the diagonal elements of matrix exponentials via randomized  projection [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' It remains an open question as to whether our techniques can be applied to general SDO  problems using the matrix-multiplicative weights update framework as a subroutine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  40  Acknowledgements  This project has been carried out thanks to funding by the Defense Advanced Research Projects Agency  (DARPA), ONISQ grant W911NF2010022, titled The Quantum Computing Revolution and Optimization:  Challenges and Opportunities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Giacomo Nannicini is partially supported by the Army Research Office under grant number W911NF-20-1-  0014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  A  Running time of Algorithm 3 without QRAM  The following result from [10] gives the sample complexity of implementing the oracles in the sparse-access  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Lemma 16 (see, proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='3 in [10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' We can implement the oracle OCγ on a quantum computer  given access to O(ǫ−2) copies of a state that is an  ǫ  8-approximation of the input state ρ in trace distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  The oracle ODn can be implemented using O(nǫ−2)  ǫ  8-approximate copies of the input, and the classical  post-processing time needed to implement the oracle is O(nǫ−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Next, we bound the overall complexity of Algorithm 1 without access to QRAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Proposition 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Suppose that C ∈ Sn has row sparsity s and ξ ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Then, in the sparse-access input  model, the complexity of solving (5) up to additive error ξ using Algorithm 1 on a quantum computer requires  �  On  �  n1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='5√s  1+o(1)ξ−7+o(1) exp  �  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='6  �  log(ξ−1)  ��  queries to the input oracle OC and �On  �  n2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='5√s1+o(1)ξ−7+o(1) exp  �  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='6  �  log(ξ−1)  ��  additional gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Our proof can be viewed as the QRAM-free analogue of the discussion found in [10, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='4], and  we repeat it here for completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' In order to derive an appropriate bound on the per-iteration cost, we  need to evaluate the cost of constructing our separation oracles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' By Lemma 16, we can conclude that the  time to construct the oracle ODn for the diagonal elements dominates that of constructing the oracle OCγ to  test the objective value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  We now turn our attention to the cost of simulating our Hamiltonian H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  From the results in [53,  Appendix] it follows that we can produce a state that is  ξ  8 close to ρ using �O(√nξ−3) invocations of a  controlled U which satisfies  ��U − eit0H�� ≤ O  �  ξ3�  ,  with t0 =  π  4∥H∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Further, the authors in [10] note that each of the Hamiltonians we seek to simulate are  of the form H = y1C∥C∥−1  F + y2D where y1, y2 = O(log(n)ξ−1) and D is a diagonal matrix which satisfies  ∥D∥ ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Invoking [15, Theorem 1], we can simulate H for time t up to error ξ3 using  �  O  �  t(a + b) exp  �  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='6  �  log (log(n)tξ−3)  ��  separate simulations of y1C∥C∥F and y2D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  As noted in [10], access to the oracles Osparse and OC we described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='1 allows us to simulate  exp(itC∥C∥−1  F ) in time O  �  (t√s)1+o(1)ξo(1)�  if we utilize the algorithm in [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Similarly, we follow [10] in  constructing an oracle OD acting on C ⊗ (C2)⊗a, where a is a sufficiently large constant such that we can  represent the diagonal elements of D as  OD |i, z⟩ �→ |i, z ⊕ Dii⟩  41  to the desired level of precision in binary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Accordingly, we can simulate eiDt for t = �O(ξ−1) using �On(1)  queries to OD and �  On(1) elementary operations [7], and we can implement OD using �On(n) gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  To summarize, the Gibbs sampler from [53] requires �O(√nξ−3) Hamiltonian simulation steps, each of  which requires time  �  O  �√s  1+o(1)ξo(1) exp  �  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='6  �  log(ξ−1)  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Hence, each iteration of Algorithm 1 requires a total of  �  On  �  n1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='5√s  1+o(1)ξ−5+o(1) exp  �  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='6  �  log(ξ−1)  ��  sparse-access oracle queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Combining the above per-iteration cost with the iteration bound O(log(n)ξ−2)  provided in Theorem 3, it follows that Algorithm 1 solves (5) up to additive error ξ with at most  �  On  �  n1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='5√s  1+o(1)ξ−7+o(1) exp  �  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='6  �  log(ξ−1)  ��  queries to the input oracle OC and �  On  �  n2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='5√s1+o(1)ξ−7+o(1) exp  �  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='6  �  log(ξ−1)  ��  additional gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Theorem 7 formalizes the complexity of of Algorithm 3 in the quantum setting without access to QRAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  In our analysis, we employ the same Hamiltonian simulation subroutines and Gibbs sampler used in [10] to  construct our separation oracles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Let C ∈ Sn with row sparsity s and ǫ ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Then, setting ζ =  �  ǫ  n∥C∥F  �4  and fixing  ξ = 10−2, a quantum implementation of Algorithm 3 using the sparse-access input model solves (3) up to  additive error O(ǫ) using  O  �  n1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='5s0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='5+o(1) · polylog  �  n, ∥C∥F , 1  ǫ  ��  queries to the input oracle OC and O  �  n2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='5s0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='5+o(1) · polylog  �  n, ∥C∥F, 1  ǫ  ��  additional gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  The output of the algorithm is a collection of tuples {(η(k), y(k), q(k), d(k))}K  k=0 such that  ˜ρ =  1  1 + nδ  \uf8ee  \uf8f0  \uf8eb  \uf8ed  K  �  k=0  1  η(k) Q(k) ◦  exp  �  −  �  y(k)  1 Q(k) ◦  C  ∥C∥F + y(k)  2  diag  �  d(k)���  tr  �  exp  �  −  �  y(k)  1 Q(k) ◦  C  ∥C∥F + y(k)  2  diag  �  d(k)����  \uf8f6  \uf8f8 + δI  \uf8f9  \uf8fb ,  is a 2ζ-precise solution to (5), where δ is defined according to (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' The entries of ˜ρ can be modified to  construct a matrix ρ∗ at trace distance O  �  ǫ  n∥C∥F  �  of ˜ρ in time O(n2), such that nρ∗ is a feasible point of  the SDO problem (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Given that C is an s-sparse matrix, we can load C in O(ns) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Similarly, for normalization  purposes we classically compute ∥C∥F, which requires O(ns) arithmetic operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' In each iteration we use  Algorithm 1 to solve (14), and use classical estimates of the diagonal elements of the refining solution, and  a classical estimate of the objective value attained by the refining solution to update the solution and data  for the refining problem we need to solve in the next iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Letting T sparse  HU  be the cost of using Algorithm 1 as an approximate SDO subroutine, we saw in Proposition  14, Algorithm 1 solves (14) to additive error ξ using  T sparse  HU  = �On  �  n1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='5√s  1+o(1)ξ−7+o(1) exp  �  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='6  �  log(ξ−1)  ��  queries to the oracle describing the problem data and �On  �  n2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='5√s1+o(1)ξ−7+o(1) exp  �  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='6  �  log(ξ−1)  ��  addi-  tional gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' In the context of Algorithm 3, ξ is a fixed constant, so the cost of our oracle call to Algorithm  1 simplifies to  T sparse  HU  = �On  �  n1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='5√s  1+o(1)�  42  queries to the oracle describing the problem data and �On  �  n2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='5√s1+o(1)�  additional gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Classically updating the objective value requires O(1) arithmetic operations while updating the vector p  which stores a classical description of the diagonal elements of our solution as  pi ← pi + Qii  η(k) ˜p(k)  i  requires O(n) arithmetic operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Again, ε and Q can each be updated using O(n) arithmetic operations,  as we only need to store the diagonal elements of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' This also implies that we can also calculate Q◦  C  ∥C∥F in  time O(n), for only the element-wise products along the diagonal are non-trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' When compared to loading  and normalizing the data or our use of Algorithm 1 as a subroutine for solving (14), these intermediate  computation steps are negligible and do not factor into the overall running time using O notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Factoring in the O  �  polylog  �  1  ζ  ��  = O  �  polylog  �  n, ∥C∥F, 1  ǫ  ��  from Corollary 5, it follows that a quantum  implementation of Algorithm 3 requires at most  O  �  n1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='5s0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='5+o(1) · polylog  �  n, ∥C∥F , 1  ǫ  ��  queries to the input oracle OC and O  �  n2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='5s0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='5+o(1) · polylog  �  n, ∥C∥F , 1  ǫ  ��  additional gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Just as in the  proof of Theorem 5, applying Proposition 11 with our choice of ζ =  �  ǫ  n∥C∥F  �4  implies that the above running  time is sufficient to obtain a solution that can be used to solve (3) up to additive error O(ǫ), and the proof  is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  B  Estimating trace inner products with the final solution  Given that we do not explicitly report a classical description of the final solution ˜ρ defined in equation (23),  it may be of interest to understand how, for a user specified matrix A, one can compute the trace inner  product tr(A˜ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' We outline a procedure for doing so using the state preparation pair description of solution  {(η(k), y(k), q(k), d(k))}K  k=0 in Algorithm 4, and subsequently analyze the complexity of doing so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Let A ∈ Rn×n, and  C  ∥C∥F ∈ Sn be stored in QRAM, θ ∈ (0, 1), and {(η(k), y(k), q(k), d(k))}K  k=0  be a state preparation pair description of the solution obtained from running Algorithm 3 to final precision  ζ =  �  ǫ  n∥C∥F  �4  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Suppose A is an s-sparse matrix with ∥A∥F ≤ 1, and assume classical access to A and  C  ∥C∥F ∈ Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Then, Algorithm 4 outputs a θ-precise estimate of  tr(A˜ρ) =  1  1 + nδ tr  \uf8eb  \uf8edA  \uf8ee  \uf8f0  \uf8eb  \uf8ed  K  �  k=0  1  η(k) Q(k) ◦  exp  �  −  �  y(k)  1 Q(k) ◦  C  ∥C∥F + y(k)  2  diag  �  d(k)���  tr  �  exp  �  −  �  y(k)  1 Q(k) ◦  C  ∥C∥F + y(k)  2  diag  �  d(k)����  \uf8f6  \uf8f8 + δI  \uf8f9  \uf8fb  \uf8f6  \uf8f8  using at most  �On,∥C∥F , 1  ǫ  �√n  θ  �  queries to the QRAM and �  On,∥C∥F , 1  ǫ (ns) classical operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' We begin by establishing the correctness of Algorithm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  First, note that following the proof of  Proposition 12, we can simplify the expression of the final solution to  ˜ρ =  1  1 + nδ  \uf8ee  \uf8f0  K+1  �  k=0  1  η(k) Q(k) ◦  exp  �  −  �  y(k)  1 Q(k) ◦  C  ∥C∥F + y(k)  2  diag  �  d(k)���  tr  �  exp  �  −  �  y(k)  1 Q(k) ◦  C  ∥C∥F + y(k)  2  diag  �  d(k)����  \uf8f9  \uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  43  Algorithm 4 Trace estimation procedure for the final solution  Input: Access to an s-sparse matrix A ∈ Rn×n with ∥A∥F ≤ 1, state preparation pair description of solution  {(η(k), y(k), q(k), d(k))}K  k=0, precision θ ∈ (0, 1), ζ =  �  ǫ  n∥C∥F  �4  Output: A θ-precise classical estimate of tr(A˜ρ)  Initialize: a ← 0, k ← 0, y(K+1) ← (0, 0)⊤, η(K+1) ←  1  nδ, Q(K+1) ← ee⊤  for k = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' , K + 1 do  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Implement an (α, a, ζ/2(K + 2))-block-encoding of Q(k) ◦ A  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Use block-encoding of Q(k) ◦ A to implement a trace estimator for  a(k) = tr  \uf8ee  \uf8f0  �  Q(k) ◦ A  �  \uf8eb  \uf8ed  exp  �  −  �  y(k)  1 Q(k) ◦  C  ∥C∥F + y(k)  2  diag  �  d(k)���  tr  �  exp  �  −  �  y(k)  1 Q(k) ◦  C  ∥C∥F + y(k)  2  diag  �  d(k)����  \uf8f6  \uf8f8  \uf8f9  \uf8fb  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Use O  � K  θ  �  samples from the trace estimator to produce  θ  K+2-precise estimate ˜a(k) of a(k)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Update solution:  a ← a +  1  η(k) ˜a(k)  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' k ← k + 1  end  Scale down estimate to account for spectrum shift:  a ←  1  1 + nδ a  44  by setting y(K+1) = (0, 0)⊤, η(K+1) =  1  nδ, and Q(K+1) = ee⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Then,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' by linearity of the trace and Lemma 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='one has:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='tr(A˜ρ) = tr  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='\uf8eb  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='\uf8edA  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='1 + nδ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='\uf8ee  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='\uf8f0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='K+1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='k=0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='η(k) Q(k) ◦  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='exp  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='−  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='y(k)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='1 Q(k) ◦  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='C  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='∥C∥F + y(k)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='diag  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='d(k)���  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='tr  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='exp  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='−  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='y(k)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='1 Q(k) ◦  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='C  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='∥C∥F + y(k)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='diag  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
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+page_content='  In other words, the output of Algorithm 4 is indeed an estimate of tr(A˜ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Next, we analyze the complexity of the procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' If A is classically known, one can store Q(k) ◦ A in the  QRAM using O(ns) classical operations, as A is s-sparse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' With Q ◦ A stored in a QRAM data structure,  one can apply Lemma 3 to implement an (1, log(n) + 2, ζ/2(K + 2))-block-encoding of Q ◦ A in time �  O nK  ζ (1)  (as ∥Q ◦ A∥F ≤ ∥A∥F ≤ 1 for any Q defined according to (13)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' As we saw in the proof of Proposition 12,  with  C  ∥C∥F stored in QRAM, one can implement the state  ρ(k) =  exp  �  −  �  y(k)  1 Q(k) ◦  C  ∥C∥F + y(k)  2  diag  �  d(k)���  tr  �  exp  �  −  �  y(k)  1 Q(k) ◦  C  ∥C∥F + y(k)  2  diag  �  d(k)����  using at most  �On  �√n  �  ,  accesses to the QRAM and O(n) classical operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Having prepared the state ρ(k) and a (1, log(n) + 2, ζ/2(K + 2))-block-encoding Uk of Q(k) ◦ A, Lemma  8 asserts that one can implement a trace estimator for  tr  ��  Q(k) ◦ A  �  ρ(k)�  with bias at most  ζ  K+2 using �O(1) applications of Uk and U †  k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Applying amplitude estimation using O  � K  θ  �  =  �  On,∥C∥F , 1  ǫ  � 1  θ  �  samples from the estimator, we obtain a  θ  K+2-precise classical estimate ˜a(k) of a(k), as K =  O  �  polylog  �  n, ∥C∥F , 1  ǫ  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  From here, we classically update a using O(1) arithmetic operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Therefore, each iteration of Algo-  rithm 4 requires at most  �  On, K  ζ  �√n  θ  �  accesses to the QRAM and O(ns) classical operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Summing over K + 2 iterations implies a total of  �  On, K  ζ  �  K  �√n  θ  ��  = �  On,∥C∥F , 1  ǫ  �√n  θ  �  accesses to the QRAM and  O (Kns) = �On,∥C∥F , 1  ǫ (ns)  classical operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' The proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Note that if ∥A∥F > 1, because of the subnormalization to block-encode A we need to increase precision  of the estimation procedure: the cost increases by a factor proportional to ∥A∥F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  45  Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Let A ∈ Rn×n, θ ∈ (0, 1), and {(η(k), y(k), q(k), d(k))}K  k=0 be a state preparation pair description  of the solution obtained from running Algorithm 3 to final precision ζ =  �  ǫ  n∥C∥F  �4  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Suppose A is an s-sparse  matrix with ∥A∥F ≤ 1, and assume sparse oracle access to A and  C  ∥C∥F ∈ Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Then, Algorithm 4 outputs a  θ-precise estimate of  tr(A˜ρ) =  1  1 + nδ tr  \uf8eb  \uf8edA  \uf8ee  \uf8f0  \uf8eb  \uf8ed  K  �  k=0  1  η(k) Q(k) ◦  exp  �  −  �  y(k)  1 Q(k) ◦  C  ∥C∥F + y(k)  2  diag  �  d(k)���  tr  �  exp  �  −  �  y(k)  1 Q(k) ◦  C  ∥C∥F + y(k)  2  diag  �  d(k)����  \uf8f6  \uf8f8 + δI  \uf8f9  \uf8fb  \uf8f6  \uf8f8  using at most  �  On,∥C∥F , 1  ǫ  �n2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='5s2  θ  �  queries to OA, OC, and �  On,∥C∥F , 1  ǫ  �  n3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='5s2  θ  �  additional gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Provided classical access to A, we use Lemma 4 with sr = sc to construct an (s, log(n) + 3, θ/n)-  block-encoding of A with two uses of OA (an oracle describing the elements of A in binary), and additionally  using �On (1) one and two qubit gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Likewise, with access to the oracle OC describing the elements of C∥C∥−1  F , one can construct an (s, log(n)+  3, θ/n)-block-encoding of C∥C∥−1  F with two uses of OC, and additionally using �  On (1) one and two qubit gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Note that without access to QRAM, we must compute the Hadamard products by taking the Hadamard prod-  ucts of block-encodings, which causes the subnormalization factor for the Hadamard product Q(k) ◦ C∥C∥−1  F  to be ns, as Q(k) may be fully dense and C is s-sparse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' It follows that preparing one copy of each Gibbs  state requires  �  On  �√n(ns)  �  = �  On  �  n1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='5s  �  accesses to block-encodings of Q(k) ◦ C∥C∥−1  F  and D, which each require an additional �  On(n) gates (to  construct sparse-access oracles for Q(k) and D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Similarly, the subnormalization factor for a block-encoding Uk of Q(k) ◦ A will be ns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Having prepared  the state ρ(k) and a block-encoding Q(k) ◦ A, Lemma 8 asserts that one can implement a trace estimator for  tr  ��  Q(k) ◦ A  �  ρ(k)�  with bias at most  ζ  K+2 using �O(ns) applications of Uk and U †  k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Applying amplitude estimation using  O  � K  θ  �  = �On,∥C∥F , 1  ǫ  � 1  θ  �  samples from the estimator to obtain a  θ  K+2-precise classical estimate ˜a(k) of a(k),  as K = O  �  polylog  �  n, ∥C∥F , 1  ǫ  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  Just as in the QRAM setting, classically updating a requires O(1) arithmetic operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content=' Therefore,  without access to QRAM, each iteration of Algorithm 4 requires at most  �  On, K  ζ  �n2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='5s2  θ  �  = �  On,∥C∥F , 1  ǫ  �n2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='5s2  θ  �  applications of block-encodings for Q(k)◦C∥C∥−1  F , D(k) and Q(k)◦A and �On,∥C∥F , 1  ǫ  �  n3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='5s2  θ  �  additional gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='  This corresponds to �On,∥C∥F , 1  ǫ  �  n2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='5s2  θ  �  queries to OA and OC in each iteration, and �On,∥C∥F , 1  ǫ  �  n3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
+page_content='5s2  θ  �  additional gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQf-Qmd/content/2301.04237v1.pdf'}
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diff --git a/G9AyT4oBgHgl3EQf5fpe/content/tmp_files/2301.00805v1.pdf.txt b/G9AyT4oBgHgl3EQf5fpe/content/tmp_files/2301.00805v1.pdf.txt
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@@ -0,0 +1,1915 @@
+Betrayed by Captions: Joint Caption Grounding and Generation for
+Open Vocabulary Instance Segmentation
+Jianzong Wu1*
+Xiangtai Li2∗ †
+Henghui Ding2
+Xia Li3
+Guangliang Cheng4
+Yunhai Tong1
+Chen Change Loy2
+1 Key Laboratory of Machine Perception, MOE, School of Artificial Intelligence, Peking University
+2 S-Lab, Nanyang Technological University
+3 ETH Zurich
+4 SenseTime Research
+jzwu@stu.pku.edu.cn
+{xiangtai.li, henghui.ding, ccloy}@ntu.edu.sg
+Abstract
+In this work, we focus on instance-level open vocab-
+ulary segmentation, intending to expand a segmenter for
+instance-wise novel categories without mask annotations.
+We investigate a simple yet effective framework with the
+help of image captions, focusing on exploiting thousands
+of object nouns in captions to discover instances of novel
+classes. Rather than adopting pretrained caption models or
+using massive caption datasets with complex pipelines, we
+propose an end-to-end solution from two aspects: caption
+grounding and caption generation. In particular, we devise
+a joint Caption Grounding and Generation (CGG) frame-
+work based on a Mask Transformer baseline. The frame-
+work has a novel grounding loss that performs explicit and
+implicit multi-modal feature alignments. We further design
+a lightweight caption generation head to allow for addi-
+tional caption supervision. We find that grounding and gen-
+eration complement each other, significantly enhancing the
+segmentation performance for novel categories. We con-
+duct extensive experiments on the COCO dataset with two
+settings: Open Vocabulary Instance Segmentation (OVIS)
+and Open Set Panoptic Segmentation (OSPS). The results
+demonstrate the superiority of our CGG framework over
+previous OVIS methods, achieving a large improvement of
+6.8% mAP on novel classes without extra caption data. Our
+method also achieves over 15% PQ improvements for novel
+classes on the OSPS benchmark under various settings.
+1. Introduction
+Instance-Level Segmentation [17, 40] is a core vision
+task that goes beyond object detection [38, 39, 50] via seg-
+*The first two authors contribute equally to this work. Parts of the work
+are done when Xiangtai was an intern at SenseTime Research. † Project
+Leader. Code and model will be made available at https://github.
+com/jzwu48033552/betrayed-by-captions.
+menting and classifying each object. Although it continues
+to attract significant research effort [2,4–6,8–10,18,26,34–
+37,44,54–57,60,61,64,72,74], current solutions mainly fo-
+cus on a closed-set problem that assumes a pre-defined set
+of object categories [24,31,40]. In practice, many applica-
+tions need to detect and segment new categories. To save
+the need of annotating new object categories, zero-shot ob-
+ject detection/segmentation [3,48] is proposed, where mod-
+els are trained on base classes and equipped with the ability
+to segment new classes. However, zero-shot setting suffers
+from low novel-class performance, as high-level word em-
+beddings cannot effectively encode fine-grained visual in-
+formation.
+To address this issue, recent work [69] takes an open vo-
+cabulary setting by pretraining a visual backbone on cap-
+tioned images for learning rich visual features. With the
+success of pretrained Vision Language Models (VLMs) [30,
+46], several approaches, e.g., ViLD [23], propose effective
+methods to distill knowledge from VLMs into detectors or
+segmentation methods. Meanwhile, several works decou-
+ple the learning of open vocabulary classification and de-
+tection/segmentation into a two-stage pipeline [15,21]. Re-
+cently, state-of-the-art solutions [19,28,33,71,79] for open
+vocabulary detection/segmentation try to adopt larger-scale
+dataset pre-training with the help of VLMs. For example,
+Detic [79] adopts the ImageNet-21k [51] dataset to enlarge
+the detector in a weakly supervised manner, while Prompt-
+Det [19] augments the detection dataset with image-caption
+pairs scraped from the Internet.
+Recent XPM [28] also
+pretrains their model on caption datasets [52]. These ap-
+proaches typically require a complex architecture design to
+leverage extra datasets [31, 51]. Despite the performance
+improvement, these methods are not cost-effective in terms
+of data utilization. In this paper, we explore the use of cap-
+tion data with more effective designs.
+Caption-related vision tasks can be broadly divided into
+grounding and generation. The former [13, 14, 22, 41, 67]
+1
+arXiv:2301.00805v1  [cs.CV]  2 Jan 2023
+
+Text 
+Encoder
+“A teddy bear, alarm clock
+and key, and flask on a bed”
+Vision 
+Encoder
+Text 
+Encoder
+Vision 
+Encoder
+“A teddy bear, alarm clock
+and key, and flask on a bed”
+Word 
+Encoder
+Vision 
+Encoder
+Sentence 
+Encoder
+grounding loss
+grounding loss
+grounding loss
+generation 
+loss
+（c) Ours: Object Nouns + Caption Generation
+word features
+word features
+sentence features
+object nouns features
+“A teddy bear sleeping 
+with alarm clock on a bed”
+（a) OVR-CNN: All Words
+（b) OpenSeg: Nouns & Adjectives
+Nouns & Adjectives
+extract object nouns
+“A teddy bear, alarm clock
+and key, and flask on a bed”
+Stage 2: Training detection
+Stage 1: Pretraining Using 
+Caption dataset.
+Vision 
+Encoder
+agnostic 
+seg/det loss
+det/seg loss
+det/seg loss
+Figure 1. (a) OVR-CNN [69] uses all words for caption grounding, then finetunes, in a two-stage pipeline. (b) OpenSeg [21] uses extra
+agnostic head for segmentation and Nouns for grounding. (c) Our method encodes only object nouns in captions for caption grounding,
+and all words for caption generation in one unified framework.
+A woman walking down the street holding an umbrella
+*car
+*umbrella
+*bus
+*car
+There is a red double decker bus parked on the street
+Figure 2. Example of Instance Segmentation and Panoptic Segmentation results of CGG. Categories marked by ’*’ are novel categories.
+Sentences are generated by the Instance Segmentation model. To the best of our knowledge, we are the first to unify OVIS, OSPS and
+Caption Generation in one framework.
+requires a model to align the text and corresponding region
+features, e.g., OVR-CNN [69] and OpenSeg [21] in Fig. 1
+(a) and (b), while the latter [59, 66, 73] learns a model that
+output a caption for a given imagery input. The relationship
+between the two tasks and open vocabulary instance seg-
+mentation is not well explored. We argue that caption data
+encode rich structural and semantic information, which may
+help the process of novel class detection. Different from the
+OVR-CNN [69] (see Fig. 1(a)) that adopts a caption model
+in the pre-training process where the caption data and detec-
+tion results are not well aligned, we propose a unified frame-
+work to jointly perform caption grounding, generation, and
+instance segmentation.
+Our framework presents a novel caption grounding loss
+and an extra caption decoder for the generation loss, as
+shown in Fig. 1(c). The caption data is thus well exploited
+in both the input and output stages. In particular, we use
+object queries as inputs following Mask2Former [10]. At
+the input stage, we adopt separated object nouns to ground
+each object query, providing us with the grounding loss. At
+the output stage, with a lightweight Transformer decoder,
+we add supervision to the generated caption, resulting in
+the generation loss. Both losses are well coupled and have
+a mutual effect for novel class segmentation, adding only
+0.8% GFlops during the training. For inference, our method
+drops the caption generation module for OVIS and OSPS
+with no extra computation cost.
+We carry out experiments in two different settings, in-
+cluding Open Vocabulary Instance Segmentation (OVIS)
+and Open Set Panoptic Segmentation (OSPS) [29].
+Ex-
+perimental results demonstrate that our proposed method
+achieves significant improvements for novel classes de-
+spite using a strong baseline [10] as the encoder.
+The
+proposed method achieves new state-of-the-art results on
+COCO OVIS and COCO OSPS without any data pre-
+training and complex pipelines. Figure 2 shows that our
+method predicts instance segmentation, panoptic segmenta-
+tion, and the corresponding caption in one unified frame-
+work while predicting novel classes.
+In particular, our
+method achieves a large improvement of 6.8% mAP over
+previous XPM [28] on OVIS and 15% PQ improvements
+over previous method [63].
+2. Related Work
+Zero-Shot Detection and Segmentation. Scaling up data
+collection and annotation is laborious and expensive for
+large vocabulary detection and segmentation.
+Zero-Shot
+Detection [48] and Segmentation [3] tries to detect/segment
+novel categories that the annotations are not accessible dur-
+ing the training process. Many studies address this prob-
+lem by aligning region features to the fixed text embed-
+dings [1, 20, 47, 70, 80].
+Due to the limited capacity of
+word embeddings and the advent of large Vision-Language-
+Models (VLMs), recent studies [23, 68, 69] have moved to
+the open vocabulary setting.
+Open Vocabulary Object Detection (OVOD). Recent
+studies [16,23,68,69,79] focus on the open vocabulary set-
+2
+
+Grand Canvon*bus/0.97
+carj0.97
+carl0.98sky-other-merged
+tree-merged
+trafficlight
+bus
+gravel
+road
+pavement-mergedtree-merged
+building-other-merged
+SOI
+personperson
+truck
+umbrella
+*car
+*car
+person
+person
+pavement-merged
+roadtruck|0.98
+tario.95
+personj0.97
+car10.93
+carj0.94
+carlo.960.95arjo.96
+*umbrellal0.89
+cari0.93
+personj0.98
+personj0.85
+handbag/0.97ting, in which models are trained by leveraging pre-trained
+language-text pairs including captions and text prompts.
+For instance, OVR-CNN [69] is first pretrained on image-
+caption data to recognize novel objects, then fine-tunes the
+model for zero-shot detection. Recently, many works on
+image classification successfully expand their vocabulary
+sizes by pretraining on large-scale image-text pairs datasets.
+ViLD [23] proposes to distill the rich representation of pre-
+trained CLIP [46] into the detector, while DetPro [16] adds
+a fine-grained automatic prompt learning. Meanwhile, sev-
+eral works extract pseudo region annotations from the pre-
+trained VLMs and employ them as the additional training
+data for detectors. Detic [79] improves the performance
+on the novel classes with image classification datasets by
+supervising the max-size proposal with all image labels.
+Methods above share the same idea of trying to enlarge
+the capacity of training data to find the rare classes, thus
+they need more computation/annotation costs and complex
+pipelines. On the contrary, we focus on designing a way
+to discover novel classes from the caption data in one uni-
+fied framework without pre-training on extra datasets nor
+distilling knowledge from pretrained VLMs.
+Open Vocabulary Segmentation (OVS). Beyond OVOD,
+OVS further requires the model to segment the novel
+classes. Current solutions for OVS usually decouple mask
+generation and mask classification as two different steps.
+The former generates mask regions, while the latter per-
+forms classification with pre-trained VLMs [21,32]. Dense-
+CLIP [78] proposes a similar pipeline to that in OVD by
+distilling CLIP knowledge through generating pseudo mask
+labels. Our method proposes an end-to-end pipeline to per-
+form caption learning (grounding/generation) and segmen-
+tation learning jointly. The differences with OpenSeg [21]
+are: 1. We extract object nouns from captions, rather than
+nouns and adjectives as in OpenSeg. 2. For text encoders,
+we use BERT embeddings that are purely trained on text
+corpus, while OpenSeg employs a state-of-the-art VLM
+(ALIGN [30]). 3. We mainly focus on instance-level open
+vocabulary segmentation task rather than semantic segmen-
+tation.
+Image Captioning. This task requires the model to gen-
+erate captions to describe the content of images [59].
+State-of-the-art methods follow multi-modal attention de-
+signs, treating the task as a multi-modal translation prob-
+lem [66, 73, 75]. Our focus in this work is not to design a
+new captioning model, but to explore image captioning as a
+sub-task for open vocabulary learning to enhance the novel
+class discovery ability. To our best knowledge, this study is
+the first attempt that explores caption generation on OVS.
+3. Methodology
+In this section, we first review the background of open
+vocabulary instance segmentation. Then, we present our
+Caption Grounding and Generation framework, which aims
+to fully exploit caption data through joint caption grounding
+and generation.
+3.1. Background
+Problem Setting. We first describe the open-vocabulary
+problem setting. Let DB = {(Im, Ym)}NB
+m=1 be the set of
+training images and instance annotations for a limited set of
+base classes VB. Among these images, there are also novel
+classes VN, whose annotations cannot be accessed during
+the training. For OSPS, novel classes come from the thing
+classes, while the stuff classes are treated as base classes.
+Each image Im is associated with a set of ground-truth (GT)
+annotations Ym, which comprises instance masks and their
+corresponding object classes. In order to detect and segment
+novel classes, following previous works [69], we leverage
+additional image-level annotations, i.e., image captions. Let
+DC = {(Ic, Yc)}NC
+c=1 be another set of training images with
+image caption annotations. Each image Ic is annotated with
+a caption Yc. Compared to pixel-level annotations, captions
+are easier to collect, and its vocabulary VC is much larger
+than base classes, i.e., |VC| ≫ |VB|. Therefore, exploiting
+the additional information from the image caption dataset
+would be beneficial.
+Open-vocabulary instance segmentation aims to train a
+model to segment both base classes VB and novel classes
+VN. Following previous methods [21, 28, 69], our model
+uses high-level semantic embeddings from a pretrained text
+Transformer (BERT [12]) as the weights of the linear clas-
+sifier. We focus on distilling knowledge in the captions to
+the target classes via representation similarities.
+Baseline Method. We adopt the recent Mask2Former [10]
+model as our baseline since the mask-based Transformer ar-
+chitecture can be readily extended into multi-modal training
+with captions. Mask2Former takes a Transformer encoder-
+decoder architecture with a set of object queries, where the
+object queries interact with encoder features via masked
+cross-attention. Given an image I, during the inference,
+Mask2Former directly outputs a set of object queries Q =
+{qi}, i = 1, ..Nq, where each object query qi represents
+one entity. Then, two different Multiple Layer Perceptrons
+(MLPs) project the queries into two embeddings for mask
+classification and mask prediction, respectively. During the
+training, each object query is matched to the ground truth
+mask via masked-based bipartite matching. The loss func-
+tion is Lmask = λclsLcls + λceLce + λdiceLdice, where
+Lcls is the Cross-Entropy (CE) loss for mask classification,
+and Lce and Ldice are the Cross-Entropy (CE) loss and Dice
+loss [43] for segmentation, respectively. In particular, fol-
+lowing [69], we use pretrained embeddings to replace the
+learnable classifier for training and inference, as shown in
+Fig. 3.
+The original Mask2Former can only detect and segment
+3
+
+People sitting around a table and talking
+A group of people sitting 
+around a table
+𝑁𝑞 mask
+predictions
+caption grounding loss
+caption generation loss
+classification  
+loss
+mask loss
+backbone
+pixel
+decoder
+image features ℱ
+Transformer 
+decoder
+𝑁𝑞 queries
+MLP
+base categories
+novel categories
+person
+train
+car
+pizza
+pretrained
+embeddings
+𝑁𝑞 multi-modal
+embeddings 
+caption
+generator
+sentence
+encoder
+word
+encoder
+sentence features
+𝑁𝑠 × 𝐶𝑠
+…
+…
+word features
+𝑁𝑤 × 𝐶𝑤
+𝑁𝑞 classification
+predictions
+𝑁𝑐 × 𝐶𝑤
+loss
+legend
+trainable
+fixed
+output
+categories
+object queries
+output
+𝑁𝑞 × 𝐶𝑤
+Input Image
+remaining after training
+𝑓𝑔
+people
+table
+𝑒𝑖
+𝑀
+𝑒𝑗
+𝑊
+𝑒𝑘
+𝑆
+Figure 3. The illustration of CGG framework. The input image I is first provided to Mask2Former. The output of the Transformer decoder
+is then fed into an MLP, which generates Nq mask predictions together with the output of the pixel decoder. Then the query is transferred
+into Nq multi-modal embeddings {eM
+i }, of which the similarity with class embeddings is computed to produce classification predictions.
+{eM
+i } also outputs grounding loss and generation loss with text features extracted by word encoder and sentence encoder.
+closed-set objects and cannot handle the novel classes. Our
+method extends it to perform open-vocabulary segmenta-
+tion in a new framework.
+3.2. CGG Framework for OVS
+Overview.
+Figure 3 presents the overall pipeline of our
+CGG framework.
+Based on Mask2Former [10], follow-
+ing [69], we set the pretrained text embeddings as the
+weights of the final linear classifier. We add two losses:
+the caption grounding loss and the caption generation loss.
+A caption generator is appended at the end of the output
+queries, directly producing the image caption. During the
+training, we adopt a pre-trained sentence encoder and word
+encoder to encode both captions and object nouns extracted
+from captions into sentence features and word features. The
+former is used for caption generation loss, while the latter is
+used for caption grounding loss. During the inference, we
+discard all the newly-introduced modules, and perform the
+same inference procedure as Mask2Former.
+Class-Agnostic Pretraining.
+Following previous works
+[21,28,69], we first pretrain our framework using only base
+data annotations in a class-agnostic manner. Such a process
+is similar to training a Region Proposal Network (RPN) at
+the first stage. The goal of pretraining is to encode instance-
+wised information into object queries. Then we load the
+pretrained model for joint training with caption data.
+End-to-End Caption Grounding.
+Previous works like
+OVR-CNN [69] pre-train their models with caption data.
+The core idea is to learn a Vision to Language (V2L) pro-
+jection layer where the language data from novel classes are
+transformed into vision features via multiple multi-modal
+losses including grounding loss and a set of auxiliary self-
+supervision loss.
+There are two potential issues with the previous design.
+Firstly, training caption and segmentation separately can-
+not fully explore caption data and detection/segmentation
+annotations. The training of segmenter is isolated and the
+connection between the two models is broken. Secondly,
+there is a weakened region-word alignment in the tradi-
+tional grounding process by calculating similarities between
+multi-modal embeddings and all words in caption data,
+because object-unrelated words may encounter the vision-
+language implicit matching. We argue that object nouns in
+caption data should be well aligned with region features in
+a more fine-grained manner since the class categories are
+always nouns in captions.
+Therefore, rather than sending the entire sentence as in-
+puts, we extract only object nouns from the sentence and
+feed it to a word encoder.
+Such extraction finds more
+precise semantic embeddings, verified effectively for novel
+class grounding. For multi-modal embeddings, we adopt an
+MLP as the V2L projection and take the outputs of Trans-
+former decoder in Mask2Former as the inputs, since the ob-
+ject queries group region features naturally, which is well
+proved in many previous works [5, 10]. Given an image-
+caption pair (I, C), we first calculate similarities between
+Nq multi-modal embeddings {eM
+i } and Nw word features
+4
+
+{eW
+j } extracted from the caption.
+SC(I, C) =
+1
+Nw
+Nw
+�
+j=1
+Nq
+�
+i=1
+aC
+i,j⟨eM
+i , eW
+j ⟩,
+SI(I, C) = 1
+Nq
+Nq
+�
+i=1
+Nw
+�
+j=1
+aI
+i,j⟨eM
+i , eW
+j ⟩,
+(1)
+where ⟨·, ·⟩ is the dot product between two vectors.
+SC(I, C) and SI(I, C) are similarities between image I
+and caption C normalized for text or image separately. The
+normalization term is formulated as
+aC
+i,j =
+exp⟨eM
+i , eM
+j ⟩
+�Nq
+i′=1 exp⟨eM
+i′ , eW
+j ⟩
+,
+aI
+i,j =
+exp⟨eM
+i , eM
+j ⟩
+�Nw
+j′=1 exp⟨eM
+i , eW
+j′ ⟩
+.
+(2)
+The core idea of grounding is that: the similarity between
+the matched image-caption pairs should be high, while for
+the unmatched pairs, it should be low. During the training,
+given a batch of image-caption pairs (BI, BC), for each
+similarity S ∈ {SC, SI}, the grounding loss is composed
+of two aspects. Take the similarity normalized along text
+dimension SC as an example, from the image perspective,
+the grounding loss is as follows:
+LIC
+gro(I) = − log
+exp SC(I, C)
+�
+C′∈BC exp SC(I, C′),
+(3)
+and from the caption perspective, the grounding loss is for-
+mulated as:
+LCC
+gro(C) = − log
+exp SC(I, C)
+�
+I′∈BI exp SC(I′, C).
+(4)
+The final grounding loss is formulated as the sum of four
+losses:
+Lgro = 1
+|BI|
+BI
+�
+i=1
+(LII
+Gro(Ii) + LIC
+Gro(Ii)) +
+1
+|BC|
+BC
+�
+j=1
+(LCI
+Gro(Cj) + LCC
+Gro(Cj)).
+(5)
+Optimizing the grounding loss aligns the multi-modal em-
+beddings and language embeddings in a large noun vocab-
+ulary.
+End-to-End Caption Generation. Besides using caption
+data for grounding loss to align regions and words, we ar-
+gue that caption data can also be employed as a generative
+supervision signal for a more fine-grained multi-modal un-
+derstanding. The key insight is that we force the model
+a couple of dogs
+standing next 
+to each other.
+CGG
+w/o generation
+Input Image
+Figure 4. The effectiveness of caption generation. The generated
+caption depicts rich information beyond object nouns.
+to predict the occurring instances and their relationships
+in the image to identify novel classes. Unlike grounding
+loss that aims to push nouns and query embeddings as close
+as possible, generative loss decodes the visual features into
+the semantic embeddings, which are complementary to the
+grounding loss. As shown in Fig. 4, the caption generation
+module can help the model learn specific status and rela-
+tionships of objects in the scene.
+Specifically, since the multi-modal embeddings encode
+the region-wise information, we directly take these embed-
+dings {eM
+i } as the input of a lightweight caption generator,
+which includes a stack of Transformer decoder layers. To
+supervise the caption generator, we simply adopt a Cross
+Entropy Loss on the predicted distribution of text vocabu-
+laries, which is the commonly used objective function in the
+research field of caption generation.
+Lgen = −
+Ns
+�
+t=1
+log(pθ(wt|w1, ..., wt−1)),
+(6)
+where pθ( ˆwt| ˆw1, . . . , ˆwt−1) is the probability of predicting
+a particular word from the caption, θ denotes the parame-
+ters of the generation network. Hence, this loss function
+enforces the predicted sentence to be consistent with the in-
+put caption C, making the multi-modal embeddings {eM
+i }
+capable of representing various objects and their potential
+relations in the image.
+Overall Loss Design. The overall training loss contains
+four items, i.e., the classification loss Lcls, the segmen-
+tation loss Lmask, the caption grounding loss Lgro, and
+the caption generation loss Lgen. Following the previous
+method [69], the classification loss is selected as the Cross-
+Entropy Loss that takes the dot product of multi-modal em-
+beddings eM
+i
+and base class embeddings as its logit inputs.
+The final loss function L is the weighted summation of the
+four losses: L = λclsLcls + λmaskLmask + λgroLgro +
+λgenLgen. We follow the default setting in the MMDetec-
+tion framework, where the weights are set to 2.0, 5.0, 2.0
+and 2.0 in all our experiments.
+Inference. Compared to the baseline model, CGG only in-
+troduces extra losses and a caption generation head during
+the training. During the inference, following [69], we use
+5
+
+*dog0.94
+*dogl0.95*doglo.9
+dog0.96the pretrained embeddings of all classes to perform open
+vocabulary segmentation via dot product, including base
+classes and novel classes. The remaining inference proce-
+dure is the same as the Mask2Former [10].
+4. Experiments
+4.1. Experimental Setup
+Dataset Settings. For Open Vocabulary Instance Segmen-
+tation, we mainly conduct experiments on COCO dataset
+[40]. Following previous works [28, 69], we split 48 base
+classes with mask annotations and 17 target classes with-
+out mask annotations. There are 107,761 training images
+with 665,387 mask annotations from base classes and 4,836
+testing images consisting of 28,538 and 4,614 mask in-
+stances for base and target classes, respectively. For cap-
+tioned images, we use the entire MS-COCO training set
+with 118,287 images. Each image is annotated with five
+captions describing the visually-grounded objects in the im-
+age. Unlike previous works [19, 49, 79] that adopt extra
+caption datasets, like Conceptual Captions [53] with 3M
+image-caption pairs for pre-training, we do not use extra
+caption datasets or detection datasets. We follow the ori-
+gin OVR-CNN [69] setting by only exploring a limited cap-
+tion dataset within COCO. For Open Set Panoptic Segmen-
+tation [29], we mainly adopt the COCO-panoptic dataset.
+We follow the previous works [29, 63] by splitting part of
+thing classes into unknown classes. We obtain three differ-
+ent splits by varying the numbers of unknown classes (K%
+ratios, 5%, 10%, 20%). Different from previous works, we
+use extra caption data for training.
+Metric. For OVIS setting, we report the mask-based mean
+Average Precision (mAP) at intersection-over-union (IoU)
+of 0.5.
+To analyze the performances on base and target
+classes, we carry out experiments in two settings: con-
+strained setting where the model is only evaluated on test
+image inputs, which belong to either base classes or target
+classes; generalized setting in which a model is tested on
+both base and target class images. The latter is more chal-
+lenging as it requires the model to segment target classes
+and avoid class bias from base classes, mostly with very
+high scores. We further report open vocabulary detection
+with box-based mAP. For OSPS setting, we employ the
+panoptic segmentation metrics, including Panoptic Qual-
+ity (PQ) and Segmentation Quality (SQ), where we report
+known classes and unknown classes separately for refer-
+ence. More details about the data preparation can be found
+in the appendix.
+Implementation Details. We implement our models in Py-
+Torch [45] with MMDetection framework [7].
+For both
+settings, we use the distributed training framework with 8
+GPUs. Each mini-batch has one image per GPU. The opti-
+mizer is AdamW [42] with a weight decay of 0.0001. We
+Table 1. Results on Open Vocabulary Instance Segmentation.
+Method
+Constrained
+Generalized
+Base
+Novel
+Base
+Novel
+All
+OVR [69]
+42.0
+20.9
+41.6
+17.1
+35.2
+SB [1]
+41.6
+20.8
+41.0
+16.0
+34.5
+BA-RPN [76]
+41.8
+20.1
+41.3
+15.4
+34.5
+XPM [28]
+42.4
+24.0
+41.5
+21.6
+36.3
+CGG (Ours)
+46.8
+29.5
+46.0
+28.4
+41.4
+Table 2.
+Results on Open Set Panoptic Segmentation (OSPS).
+Note that previous methods EOPSN [29] and Dual [63] treat all
+unknown things as one class while not classifying them. In con-
+trast, CGG performs complete Open Vocabulary Panoptic Seg-
+mentation, classifying each unknown thing into its specific cate-
+gory. We report the mean PQ and SQ for all unknown categories.
+* indicates that the scores are averaged from each unknown class.
+Method
+K(%)
+Known
+Unknown
+PQTh
+SQTh
+PQSt
+SQSt
+PQTh
+SQTh
+EOPSN [29]
+5
+44.8
+80.5
+28.3
+73.1
+23.1
+74.7
+Dual [63]
+45.1
+80.9
+28.1
+73.1
+30.2
+80.0
+CGG (Ours)
+50.2
+83.1
+34.3
+81.5
+45.0*
+85.2*
+EOPSN
+10
+44.5
+80.6
+28.4
+71.8
+17.9
+76.8
+Dual
+45.0
+80.7
+27.8
+72.2
+24.5
+79.9
+CGG (Ours)
+49.2
+82.8
+34.6
+81.2
+41.6*
+82.6*
+EOPSN
+20
+45.0
+80.3
+28.2
+71.2
+11.3
+73.8
+Dual
+45.0
+80.6
+27.6
+70.1
+21.4
+79.1
+CGG (Ours)
+48.4
+82.3
+34.4
+81.1
+36.5*
+78.0*
+adopt full image size for a random crop in both the pre-
+training and training process following Mask2Former [10].
+For classification head, word encoder, and sentence en-
+coder, we all adopt the BERT embeddings (pre-trained with
+fixed input embeddings, not the output of transformer lay-
+ers). We use a LVIS class name parser to extract object
+nouns from captions to ensure that the extracted nouns rep-
+resent objects in the image [24]. For OVIS, we keep the
+top-100 queries as the model outputs. For OSPS, following
+previous work [29, 63], rather than Mask2Former baseline,
+we put thing mask predictions first and fill the remaining
+background with stuff mask predictions. Note that all ex-
+periments use ResNet-50 backbone for fair comparison.
+4.2. Main Results
+Results on OVIS. We evaluate the performance of CGG
+and baselines on the Open Vocabulary Instance Segmenta-
+tion task on MSCOCO. As shown in Tab. 1, our model
+outperforms the best baseline XPM by 5.5% mAP in the
+constrained setting where only novel categories input and
+6.8% mAP in the generalized setting where both base and
+novel categories are employed as input. The generalized
+setting is more challenging because the model also needs
+to distinguish novel categories from given base categories,
+which have a data distribution bias from training data. We
+observed that CGG has a greater improvement in general-
+6
+
+Table 3. Results on COCO Open Vocabulary Object Detection
+(OVOD). IN-21K indicates ImageNet-21K [11].
+CC indicates
+Conceptual Captions [53]
+Method
+Epochs
+Extra Data
+AP50box
+novel
+AP50box
+all
+DLWL [49]
+96
+YFCC100M
+19.6
+42.9
+Cap2Det [65]
+8.5
+None
+20.3
+20.1
+OVR-CNN [69]
+12
+None
+22.8
+39.9
+Detic [79]
+96
+IN-21K & CC
+24.1
+44.7
+PromptDet [19]
+24
+LAION-novel
+26.6
+50.6
+CGG (Ours)
+12
+None
+29.3
+42.8
+ized settings compared to constrained settings, which fur-
+ther shows the effectiveness of CGG in identifying the lan-
+guage embeddings of novel classes and distinguishing them
+from base classes.
+Results on OSPS. To show the scalability of CGG, we also
+perform experiments on the Open Set Panoptic Segmen-
+tation task by expanding the base classes from base thing
+classes to including stuff classes, maintaining the whole
+training pipeline of CGG unchanged. The results are shown
+in Tab.
+2.
+Compared with traditional Open Set Panop-
+tic Segmentation, CGG actually performs a more difficult
+Open Vocabulary Panoptic Segmentation and still outper-
+forms previous methods EOPSN and Dual by a large margin
+of 14.9% PQ on unknown things in 20% unknown things
+setting [29], and 16.9%, 14.8% in 10% and 5% settings, re-
+spectively. Due to the scalability of Mask2Former and our
+simple yet effective training pipeline, which can fully uti-
+lize the open vocabulary knowledge from caption data, our
+model can perform well on open vocabulary panoptic seg-
+mentation.
+Results on OVOD. Besides the segmentation task, we also
+evaluate Open Vocabulary Object Detection task by match-
+ing Ground Truth with bounding boxes in the testing stage.
+As shown in Tab. 3, CGG achieves better AP50 score in
+novel classes compared to several previous works [19, 79]
+in a shorter training schedule with no extra data used (only
+COCO-Caption). Previous methods like PromptDet [19]
+and Detic [79] tend to use large-scale image-text datasets,
+thus causing a longer training schedule and higher compu-
+tational cost. We observe that CGG has inferior results in
+AP50box
+all . It may cause by the shorter training schedule and
+exposure to base classes compared with other methods.
+4.3. Ablation Study and Analysis
+We do ablation studies of our model to validate the ef-
+fectiveness of each component. We do all the ablations on
+the MSCOCO 48/17 split [69] with the metric mAP.
+Effectiveness of the CGG framework. First, we evalu-
+ate the significance of each proposed module. As shown in
+Tab. 4a, the baseline Class Emb., which transfers class la-
+bels to their corresponding text embeddings, only achieves
+a meager AP score of 0.2 for the novel class. As a com-
+parison, adding Caption Grounding increases the Novel AP
+to 22.2, verifying the significance of Caption Grounding,
+which helps align multi-modal embeddings explicitly. Fur-
+ther, with the Caption Generation module, the final score
+reaches 28.4. The increase arises from a more strict regu-
+larization of Caption Generation, which supervises beyond
+nouns.
+For example, the caption “a woman holding an
+umbrella” requires the network to capture the relationship
+“holding” as well. If the Caption Generation module plays
+independently, the performance decreases to 0.3.
+Training Pipeline.
+Previous methods like OVR-CNN
+[69] train their embeddings before fine-tuning the seg-
+mentor/detector, called ’emb-segm.’ In this paper, we in-
+stead pre-train a class-agnostic segmentor and then train the
+multi-modal embeddings eM
+i
+on image-text data. We name
+it ’segm-emb.’ These pipelines are compared over CGG in
+Tab. 4b. Note that “segm-emb-segm” is also included as
+a candidate. Results show that though “segm-emb” is in-
+ferior to others for base classes, it achieves significantly
+higher scores for novel classes. This phenomenon arises
+because training the segmentor in the last stage overfits the
+base classes, thus leading to a worse recall for novel classes.
+Grounding Nouns Extraction. In CGG, we extract only
+object nouns from the sentences, leaving other words un-
+touched. To validate the effectiveness of different word se-
+lection strategies, we also try to extract all words, and all
+nouns, except for only object nouns. The results are shown
+in Tab. 4c. By extracting all words, the novel AP decreases
+by 20.8, and by extracting all words without distinguishing
+whether they refer to objects or not, the novel AP decreases
+by 13.3. In conclusion, different extracting strategies sig-
+nificantly impact the model’s performance.
+Layers of Caption Generator. We evaluate the influence
+of layers of the transformer decoder in the caption genera-
+tor. The results are in Tab. 4d. We test 2, 4, and 6 layers
+transformer decoders and observe that the middle number of
+4 is a better choice, while fewer layers of 2 and more layers
+of 6 both harm the performance. Moreover, heavier caption
+generators may improve mAP for base classes a little, but
+also increase the computational cost.
+Ablation on Class-Agnostic Pretraining.
+As a class-
+agnostic segmentor is trained to segment base and potential
+novel objects before training the multi-modal embeddings
+and caption generator, we do ablations on the effectiveness
+of class-agnostic pretraining and its alternatives. As shown
+in Tab. 4e, without any class-agnostic pretraining, the mAP
+on novel classes decreases by 5.7%. Moreover, if we freeze
+the Mask2Former, and only trains multi-modal embeddings
+and caption generator, the mAP on novel classes decreases
+by 2.0%, compared to the CGG model.
+GFLOPs and Parameter Analysis.
+CGG adds a
+lightweight Transformer decoder as the caption generator in
+7
+
+Table 4. Ablation studies and comparison analysis on COCO OVIS.
+(a) The Effectiveness of Each Components.
+baseline
+Gro.
+Gen.
+Base
+Novel
+Class Emb.
+48.6
+0.2
+w. Gro.
+✓
+49.1
+22.2
+w. Gen.
+✓
+49.4
+0.3
+Both (CGG)
+✓
+✓
+48.0
+28.4
+(b) Training Pipeline Comparison
+Settings
+Base
+Novel
+All
+emb-segm
+49.2
+20.3
+41.6
+segm-emb-segm
+50.2
+24.3
+43.4
+segm-emb (CGG)
+46.0
+28.4
+41.4
+(c) Nouns Extraction in Caption Grounding
+Method
+Base
+Novel
+All
+All Words
+44.7
+7.6
+35.0
+All Nouns
+48.8
+15.0
+40.0
+Object Nouns
+46.0
+28.4
+41.4
+(d) Caption Generator Design
+#layers
+Base
+Novel
+All
+2
+46.7
+23.4
+40.6
+4 (CGG)
+46.0
+28.4
+41.4
+6
+48.2
+26.9
+42.6
+(e) Effect of Class-Agnostic Pretraining
+Settings
+Base
+Novel
+All
+No class-agnostic
+46.2
+22.7
+40.0
+Freeze class-agnostic
+47.6
+26.4
+42.1
+CGG
+46.0
+28.4
+41.4
+(f) GFlops and Parameters
+Schedule
+Parameters
+GFLOPs
+baseline
+35.65M
+227.48
+Ours: Inference
+35.65M
+227.48
+Ours: Training
+81.19M
+229.33
+a person standing next to a large group of cows
+a couple of people that are standing next to a table
+a herd of zebras grazing in the grass
+a group of cows standing next to each other on the grass
+there is a lot of food on this table
+a black and white photo of a large brown dog
+a couple of elephants that are standing in the dirt
+a man riding a snowboarding down the side of a hill
+Figure 5. Visualization of CGG results. Instance Segmentation (Top) and Panoptic Segmentation (Bottom). Categories marked by ’*’ are
+novel categories. Captions generated are depicted upon each image-prediction pair. Novel categories are colored in the caption if it has.
+CGG
+class-agnostic pretrain
+Figure 6.
+The embedding space of multi-modal embeddings
+{eM
+i }. The dimension of the embeddings is reduced to 2 dimen-
+sions using t-SNE [58]. Each color represents a class label in the
+17 novel COCO classes. Each dot represents the embedding with
+the corresponding mask matched with ground truth annotations.
+training. As shown in Tab. 4f, the #Parameters increases by
+127.7% in training, while the total GFLOPs only increase
+by a small margin of 0.8%. Since text data is much smaller
+than images under the same batch size, the increased com-
+putational cost by the caption generator can be ignored. The
+GFLOPs and Parameters during inference are the same as
+the Mask2Former baseline.
+Segmentation Results Visualization.
+Fig. 5 shows the
+qualitative results of CGG. The row shows panoptic results
+and the second row shows instance results. Novel classes
+detected in the image are marked with ’*’ and highlighted
+in the caption. The result shows that our framework can
+identify and segment base and novel classes. We show the
+generated comprehensive captions above the images.
+Embeddings Space Visualization. Fig. 6 shows the t-SNE
+visualization result of the trained multi-modal embeddings
+(right) and the embeddings from a class-agnostic pretrain-
+ing model (left). Specifically, we evaluate our model on
+the COCO validation set and get all the embeddings for
+each image. We change the bipartite matching strategy of
+Mask2Former to calculating mask loss only, thus getting
+the matching result between ground truth labels and multi-
+modal embeddings. The original Mask2Former [10] model
+cannot distinguish different novel classes in the embedding
+space, with only class-agnostic pretraining. After training
+using caption grounding and generation, the embeddings
+can formulate groups consistent with their categories.
+5. Conclusion
+In this paper, we present a joint Caption Grounding and
+Generation (CGG) framework for instance-level open vo-
+cabulary segmentation.
+Our core insights are two folds:
+8
+
+ceiling-merged
+sky-other-merged
+wall-other-mer
+fence-merged
+*cow
+FCOW
+*COW
+person
+*cow
+graveldoor-stuff
+wall-other-merge
+person
+*bicycle
+*cake
+table-mergedzebra
+*zebra
+grass-mergedsky-other-merged
+tree-merged
+*cow
+*CoW
+*CoW
+*cow
+*cow
+AM
+grass-merged*dog0.97chairj0.90onj0.89
+bottlelo.
+bird0.93cakej0.90
+broccolil0.97
+knifej0.89
+bowl/0.41
+bowlj0.94
+forkj0.43
+spoonj0.70
+*cake0.71
+fork/0.98
+*cup0.77personj0.96
+backpackj0.93
+*snowboard|0.94
+person|personj0.98nj0.98
+personjo
+skis10.80
+backpack0.9*elephantl0.94
+*elephantj0.93Firstly, the caption contains fine-grained nouns, which leads
+to better fine-grained grounding with object queries. Sec-
+ondly, the caption can be a supervision signal that forces
+the model to predict novel objects. To our knowledge, we
+are the first to unify segmentation and caption generation
+for open vocabulary learning. We obtain significant per-
+formance improvement on both OVIS and OSPS and com-
+parable results on OVOD without extra large-scale datasets
+pre-training.
+Limitation and Future Work. Due to the limited com-
+putation resources, we do not pre-train our framework on
+extra caption datasets.
+Moreover, we do not use VLMs
+such as CLIP for distillation or supervision, and we do not
+experiment on larger scale datasets, like LVIS and Open-
+Image [25,31]. We will put these as future work.
+Acknowledgement.
+This work is supported by the Na-
+tional Key Research and Development Program of China
+(No.2020YFB2103402). We also gratefully acknowledge
+the support of SenseTime Research for providing the com-
+puting resources for this work.
+6. More Implementation Details
+Baseline Details. All the table results in main paper use the
+same ResNet50 [27] backbone for a fair comparison. The
+number of object queries is 100 by default. Our method is
+trained by only 12 epochs on the COCO train set and eval-
+uated on the COCO validation set. All the experiments are
+carried out on 8 V100 GPUs. Following previous meth-
+ods [28,69], the metric we use for OVIS is mAP (mean AP
+on the IoU threshold of 0.5).
+Training and Inference Details.
+We adopt the default
+training of Mask2Former [7,10,62]. A learning rate multi-
+plier of 0.1 is applied to the backbone. For data augmenta-
+tion, we use the default large-scale jittering (LSJ) augmen-
+tation with a random scale sampled from the range 0.1 to
+2.0 with the crop size of 1024 × 1024. We use the default
+Mask R-CNN inference setting [26], where we resize an
+image with shorter side to 800 and longer side to 1333. For
+the Inference of OSPS, we do not use the default joint merge
+for things and stuff. We put the thing mask first and fill the
+remaining area with stuff mask prediction because the thing
+predictions for unknown are usually in a low score, and they
+may be covered by high score stuff mask prediction.
+Training Splits For OVIS and OSPS. For OVIS, we fol-
+low the 48/17 split in COCO proposed by [48], in which 48
+classes are base classes, and 17 are novel classes. For OSPS,
+we follow the unknown things split proposed by [29]. The
+unknown percentages are 5%, 10%, and 20% separately.
+Concretely, for 48/17 split of OVIS, the base classes
+are:
+“person”, “bicycle”, “car”, “motorcycle”, “truck”,
+“boat”, “bench”, “bird”, “horse”, “sheep”, “zebra”, “gi-
+raffe”, “backpack”, “handbag”, “skis”, “kite”, “surfboard”,
+“bottle”, “spoon”, “bowl”, “banana”, “apple”, “orange”,
+Table 5. Ablation on fully supervised instance segmentation, ob-
+ject detection, and panoptic segmentation. AP-novel indicates the
+mean AP on the 17 novel classes (trained in the fully supervised
+setting). AP-bbox indicates object detection.
+Method
+Instance
+Panoptic
+AP
+AP-novel
+AP-bbox
+PQ
+PQ-th
+PQ-st
+class-label
+59.3
+66.6
+58.9
+46.4
+51.9
+38.2
+class-emb.
+50.6
+57.8
+50.2
+44.4
+50.5
+35.1
+w/ gro.
+50.8
+57.4
+50.3
+44.1
+50.3
+35.0
+w/ gen.
+50.9
+57.6
+50.7
+44.2
+50.5
+34.8
+w/ both.
+51.3
+57.5
+50.7
+44.3
+50.6
+34.9
+“broccoli”, “carrot”, “pizza”, “donut”, “chair”, “bed”,
+“tv”, “laptop”, “remote”, “microwave”, “oven”, “refrig-
+erator”, “book”, “clock”, “vase”, “toothbrush”, “train”,
+“bear”, “suitcase”, “frisbee”, “fork”, “sandwich”, “toilet”,
+“mouse”, “toaster”.
+The novel classes are:
+’bus’, ’dog’, ’cow’, ’ele-
+phant’, ’umbrella’, ’tie’, ’skateboard’, ’cup’, ’knife’, ’cake’,
+’couch’, ’keyboard’, ’sink’, ’scissors’, ’airplane’, ’cat’,
+’snowboard’.
+For OSPS, the unknown things are: 5%: “car”, “cow”,
+“pizza”, “toilet”. 10%: “boat”, “tie”, “zebra”, “stop sign”.
+20%: “dining table”, “banana”, “bicycle”, “cake”, “sink”,
+“cat”, “keyboard”, “bear”.
+7. More Experiments Results
+Will Joint Grounding and Caption Help the Fully Su-
+pervised Baseline? To answer this question, we perform
+ablation on fully supervised settings in Tab. 5.
+For the
+proposed CGG, we verify two main components, includ-
+ing caption grounding and caption generation. Class-emb
+means only using pre-trained text embeddings for mask
+classification. Class-label is a traditional learnable, fully
+connected layer that converts the classes into contiguous la-
+bels. In Tab. 5, we observe that the fully supervised method
+achieves better results than using class embeddings in all
+three tasks. As shown in the last three rows of Tab. 5, for
+within class embedding settings, the added caption ground-
+ing and generation modules help to improve the perfor-
+mance on OVIS, but no performance gain on OSPS. We
+conclude that joint grounding and caption have limited ben-
+efits (0.5% improvements) in supervised settings.
+Will Better Caption Generator Help Open Vocabulary
+Instance Segmentation? We further explore the influence
+of the caption generation module to open vocabulary in-
+stance segmentation.
+Fig. 6 shows the results.
+As the
+caption generator becomes larger, the overall segmentation
+quality (AP all) increases. On the contrary, the quality of
+the caption (including BLUE and CIDEr) generation drops.
+This means a better caption generator may not be a better
+open vocabulary instance segmenter. The role of the cap-
+9
+
+airplane
+bus
+cat
+dog
+cow
+elephant
+umbrella
+tie
+snowboard
+skateboard
+cup
+knife
+cake
+couch
+keyboard
+sink
+scissors
+airplane
+bus
+cat
+dog
+cow
+elephant
+umbrella
+tie
+snowboard
+skateboard
+cup
+knife
+cake
+couch
+keyboard
+sink
+scissors
+Ground Truth
+Predict
+Figure 7. The correlation map between Ground Truth and model
+predictions on novel classes. The noun embeddings and object
+queries for novel classes are highly correlated.
+tion generator is to force the model to know the existence
+of novel objects, and pursuing a better caption generation
+model is not our goal of OVIS and OSPS.
+8. Visual Analysis and Comparison
+Visualization Analysis both Nouns and Object Queries.
+We calculate the correlation map between the predicted
+multi-modal embeddings eM
+i
+and the Ground Truth class
+embeddings. As shown in Fig. 7, our model can correctly
+distinguish novel classes based on the segmentation masks.
+More Visual Examples from Caption Generation. We
+observe that in some cases, the caption generated by CGG
+can predict objects that are not in the category list. Cate-
+gories beyond the given list cannot be correctly classified
+using the similarity between multi-modal embeddings and
+class embeddings since the class embeddings are not acces-
+sible during inference, like in Fig. 8, images top. There is
+a couple of luggage on the floor, but ’luggage’ is not a class
+in the validation dataset. Without a caption generator, the
+model classifies the luggage as ’suitcase.’ However, with
+the caption generation module, the generated caption suc-
+cessfully depicts the word ’luggage’. In the bottom images,
+’tennis’ is also described by captions. Fig. 9 shows more
+visualization results with captions.
+More Visualization Results on OVIS and OSPS. In
+Fig. 10, we present more visual results of OVIS and OSPS
+tasks. The CGG model can well segment and classify novel
+categories well.
+Zero Shot Visualization on ADE20K dataset. In Fig. 11,
+we show the visualization results on ADE20K dataset [77].
+CGG can detect and segment novel classes in a zero shot
+a couple pieces of luggage on top of the floor
+CGG
+CGG w/o generator
+a couple of people that are playing tennis
+Figure 8. Examples of captions predicting objects that are not in
+the category list.
+manner on ADE20K. At the same time, CGG generates
+comprehensive captions that well depict the content of the
+images.
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+0.8
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+0.4
+0.2
+0.0personj0.97
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+42.6
+0.387
+0.226
+0.138
+0.088
+0.171
+0.289
+a group of people riding skateboards down a street
+there is a bathroom sink with a large mirror above it
+Figure 9. Visualization results of generated captions and the related segmentations of CGG. Input Image (Left), CGG (Middle), CGG w/o
+caption generation (Right). ’mirror’ is not in the category list but is depicted by the generated caption.
+a red double decker bus driving down the street
+there is an airplane that can be seen on the ground
+there is a lot of stuff on the table
+a close up of two plates of food
+a zebra looking down
+there is a lot of cars on the road
+a person sitting on the ground under an umbrella
+a couple of elephants standing next to each other
+a bunch of bananas hanging from the metal pole
+a plate of pizza on a table
+a bunch of bikes parked next to each other
+two different sized bears
+Figure 10. More visualization results of OVIS (Top two rows) and OSPS (Bottom two rows). Novel classes are marked by ’*’.
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+oatj0.98
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf,len=1155
+page_content='Betrayed by Captions: Joint Caption Grounding and Generation for  Open Vocabulary Instance Segmentation  Jianzong Wu1*  Xiangtai Li2∗ †  Henghui Ding2  Xia Li3  Guangliang Cheng4  Yunhai Tong1  Chen Change Loy2  1 Key Laboratory of Machine Perception, MOE, School of Artificial Intelligence, Peking University  2 S-Lab, Nanyang Technological University  3 ETH Zurich  4 SenseTime Research  jzwu@stu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='pku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='cn  {xiangtai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='li, henghui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='ding, ccloy}@ntu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='sg  Abstract  In this work, we focus on instance-level open vocab-  ulary segmentation, intending to expand a segmenter for  instance-wise novel categories without mask annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  We investigate a simple yet effective framework with the  help of image captions, focusing on exploiting thousands  of object nouns in captions to discover instances of novel  classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' Rather than adopting pretrained caption models or  using massive caption datasets with complex pipelines, we  propose an end-to-end solution from two aspects: caption  grounding and caption generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' In particular, we devise  a joint Caption Grounding and Generation (CGG) frame-  work based on a Mask Transformer baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' The frame-  work has a novel grounding loss that performs explicit and  implicit multi-modal feature alignments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' We further design  a lightweight caption generation head to allow for addi-  tional caption supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' We find that grounding and gen-  eration complement each other, significantly enhancing the  segmentation performance for novel categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' We con-  duct extensive experiments on the COCO dataset with two  settings: Open Vocabulary Instance Segmentation (OVIS)  and Open Set Panoptic Segmentation (OSPS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' The results  demonstrate the superiority of our CGG framework over  previous OVIS methods, achieving a large improvement of  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='8% mAP on novel classes without extra caption data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' Our  method also achieves over 15% PQ improvements for novel  classes on the OSPS benchmark under various settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' Introduction  Instance-Level Segmentation [17, 40] is a core vision  task that goes beyond object detection [38, 39, 50] via seg-  The first two authors contribute equally to this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' Parts of the work  are done when Xiangtai was an intern at SenseTime Research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' † Project  Leader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' Code and model will be made available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  com/jzwu48033552/betrayed-by-captions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  menting and classifying each object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' Although it continues  to attract significant research effort [2,4–6,8–10,18,26,34–  37,44,54–57,60,61,64,72,74], current solutions mainly fo-  cus on a closed-set problem that assumes a pre-defined set  of object categories [24,31,40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' In practice, many applica-  tions need to detect and segment new categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' To save  the need of annotating new object categories, zero-shot ob-  ject detection/segmentation [3,48] is proposed, where mod-  els are trained on base classes and equipped with the ability  to segment new classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' However, zero-shot setting suffers  from low novel-class performance, as high-level word em-  beddings cannot effectively encode fine-grained visual in-  formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  To address this issue, recent work [69] takes an open vo-  cabulary setting by pretraining a visual backbone on cap-  tioned images for learning rich visual features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' With the  success of pretrained Vision Language Models (VLMs) [30,  46], several approaches, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=', ViLD [23], propose effective  methods to distill knowledge from VLMs into detectors or  segmentation methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' Meanwhile, several works decou-  ple the learning of open vocabulary classification and de-  tection/segmentation into a two-stage pipeline [15,21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' Re-  cently, state-of-the-art solutions [19,28,33,71,79] for open  vocabulary detection/segmentation try to adopt larger-scale  dataset pre-training with the help of VLMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' For example,  Detic [79] adopts the ImageNet-21k [51] dataset to enlarge  the detector in a weakly supervised manner, while Prompt-  Det [19] augments the detection dataset with image-caption  pairs scraped from the Internet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  Recent XPM [28] also  pretrains their model on caption datasets [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' These ap-  proaches typically require a complex architecture design to  leverage extra datasets [31, 51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' Despite the performance  improvement, these methods are not cost-effective in terms  of data utilization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' In this paper, we explore the use of cap-  tion data with more effective designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  Caption-related vision tasks can be broadly divided into  grounding and generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' The former [13, 14, 22, 41, 67]  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='00805v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='CV]  2 Jan 2023  Text  Encoder  “A teddy bear,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' alarm clock  and key,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' and flask on a bed”  Vision  Encoder  Text  Encoder  Vision  Encoder  “A teddy bear,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' alarm clock  and key,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' and flask on a bed”  Word  Encoder  Vision  Encoder  Sentence  Encoder  grounding loss  grounding loss  grounding loss  generation  loss  （c) Ours: Object Nouns + Caption Generation  word features  word features  sentence features  object nouns features  “A teddy bear sleeping  with alarm clock on a bed”  （a) OVR-CNN: All Words  （b) OpenSeg: Nouns & Adjectives  Nouns & Adjectives  extract object nouns  “A teddy bear,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' alarm clock  and key,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' and flask on a bed”  Stage 2: Training detection  Stage 1: Pretraining Using  Caption dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  Vision  Encoder  agnostic  seg/det loss  det/seg loss  det/seg loss  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' (a) OVR-CNN [69] uses all words for caption grounding, then finetunes, in a two-stage pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' (b) OpenSeg [21] uses extra  agnostic head for segmentation and Nouns for grounding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' (c) Our method encodes only object nouns in captions for caption grounding,  and all words for caption generation in one unified framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  A woman walking down the street holding an umbrella  car  umbrella  bus  car  There is a red double decker bus parked on the street  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' Example of Instance Segmentation and Panoptic Segmentation results of CGG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' Categories marked by ’*’ are novel categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  Sentences are generated by the Instance Segmentation model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' To the best of our knowledge, we are the first to unify OVIS, OSPS and  Caption Generation in one framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  requires a model to align the text and corresponding region  features, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=', OVR-CNN [69] and OpenSeg [21] in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' 1  (a) and (b), while the latter [59, 66, 73] learns a model that  output a caption for a given imagery input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' The relationship  between the two tasks and open vocabulary instance seg-  mentation is not well explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' We argue that caption data  encode rich structural and semantic information, which may  help the process of novel class detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' Different from the  OVR-CNN [69] (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' 1(a)) that adopts a caption model  in the pre-training process where the caption data and detec-  tion results are not well aligned, we propose a unified frame-  work to jointly perform caption grounding, generation, and  instance segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  Our framework presents a novel caption grounding loss  and an extra caption decoder for the generation loss, as  shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' 1(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' The caption data is thus well exploited  in both the input and output stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' In particular, we use  object queries as inputs following Mask2Former [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' At  the input stage, we adopt separated object nouns to ground  each object query, providing us with the grounding loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' At  the output stage, with a lightweight Transformer decoder,  we add supervision to the generated caption, resulting in  the generation loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' Both losses are well coupled and have  a mutual effect for novel class segmentation, adding only  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='8% GFlops during the training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' For inference, our method  drops the caption generation module for OVIS and OSPS  with no extra computation cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  We carry out experiments in two different settings, in-  cluding Open Vocabulary Instance Segmentation (OVIS)  and Open Set Panoptic Segmentation (OSPS) [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  Ex-  perimental results demonstrate that our proposed method  achieves significant improvements for novel classes de-  spite using a strong baseline [10] as the encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  The  proposed method achieves new state-of-the-art results on  COCO OVIS and COCO OSPS without any data pre-  training and complex pipelines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' Figure 2 shows that our  method predicts instance segmentation, panoptic segmenta-  tion, and the corresponding caption in one unified frame-  work while predicting novel classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  In particular, our  method achieves a large improvement of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='8% mAP over  previous XPM [28] on OVIS and 15% PQ improvements  over previous method [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' Related Work  Zero-Shot Detection and Segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' Scaling up data  collection and annotation is laborious and expensive for  large vocabulary detection and segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  Zero-Shot  Detection [48] and Segmentation [3] tries to detect/segment  novel categories that the annotations are not accessible dur-  ing the training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' Many studies address this prob-  lem by aligning region features to the fixed text embed-  dings [1, 20, 47, 70, 80].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  Due to the limited capacity of  word embeddings and the advent of large Vision-Language-  Models (VLMs), recent studies [23, 68, 69] have moved to  the open vocabulary setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  Open Vocabulary Object Detection (OVOD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' Recent  studies [16,23,68,69,79] focus on the open vocabulary set-  2  Grand Canvon*bus/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='97  carj0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='97  carl0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='98sky-other-merged  tree-merged  trafficlight  bus  gravel  road  pavement-mergedtree-merged  building-other-merged  SOI  personperson  truck  umbrella  car  car  person  person  pavement-merged  roadtruck|0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='98  tario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='95  personj0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='97  car10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='93  carj0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='94  carlo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='960.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='95arjo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='96  umbrellal0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='89  cari0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='93  personj0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='98  personj0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='85  handbag/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='97ting, in which models are trained by leveraging pre-trained  language-text pairs including captions and text prompts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  For instance, OVR-CNN [69] is first pretrained on image-  caption data to recognize novel objects, then fine-tunes the  model for zero-shot detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' Recently, many works on  image classification successfully expand their vocabulary  sizes by pretraining on large-scale image-text pairs datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  ViLD [23] proposes to distill the rich representation of pre-  trained CLIP [46] into the detector, while DetPro [16] adds  a fine-grained automatic prompt learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' Meanwhile, sev-  eral works extract pseudo region annotations from the pre-  trained VLMs and employ them as the additional training  data for detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' Detic [79] improves the performance  on the novel classes with image classification datasets by  supervising the max-size proposal with all image labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  Methods above share the same idea of trying to enlarge  the capacity of training data to find the rare classes, thus  they need more computation/annotation costs and complex  pipelines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' On the contrary, we focus on designing a way  to discover novel classes from the caption data in one uni-  fied framework without pre-training on extra datasets nor  distilling knowledge from pretrained VLMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  Open Vocabulary Segmentation (OVS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' Beyond OVOD,  OVS further requires the model to segment the novel  classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' Current solutions for OVS usually decouple mask  generation and mask classification as two different steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  The former generates mask regions, while the latter per-  forms classification with pre-trained VLMs [21,32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' Dense-  CLIP [78] proposes a similar pipeline to that in OVD by  distilling CLIP knowledge through generating pseudo mask  labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' Our method proposes an end-to-end pipeline to per-  form caption learning (grounding/generation) and segmen-  tation learning jointly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' The differences with OpenSeg [21]  are: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' We extract object nouns from captions, rather than  nouns and adjectives as in OpenSeg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' For text encoders,  we use BERT embeddings that are purely trained on text  corpus, while OpenSeg employs a state-of-the-art VLM  (ALIGN [30]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' We mainly focus on instance-level open  vocabulary segmentation task rather than semantic segmen-  tation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  Image Captioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' This task requires the model to gen-  erate captions to describe the content of images [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  State-of-the-art methods follow multi-modal attention de-  signs, treating the task as a multi-modal translation prob-  lem [66, 73, 75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' Our focus in this work is not to design a  new captioning model, but to explore image captioning as a  sub-task for open vocabulary learning to enhance the novel  class discovery ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' To our best knowledge, this study is  the first attempt that explores caption generation on OVS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' Methodology  In this section, we first review the background of open  vocabulary instance segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' Then, we present our  Caption Grounding and Generation framework, which aims  to fully exploit caption data through joint caption grounding  and generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' Background  Problem Setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' We first describe the open-vocabulary  problem setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' Let DB = {(Im, Ym)}NB  m=1 be the set of  training images and instance annotations for a limited set of  base classes VB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' Among these images, there are also novel  classes VN, whose annotations cannot be accessed during  the training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' For OSPS, novel classes come from the thing  classes, while the stuff classes are treated as base classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  Each image Im is associated with a set of ground-truth (GT)  annotations Ym, which comprises instance masks and their  corresponding object classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' In order to detect and segment  novel classes, following previous works [69], we leverage  additional image-level annotations, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=', image captions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' Let  DC = {(Ic, Yc)}NC  c=1 be another set of training images with  image caption annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' Each image Ic is annotated with  a caption Yc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' Compared to pixel-level annotations, captions  are easier to collect, and its vocabulary VC is much larger  than base classes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=', |VC| ≫ |VB|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' Therefore, exploiting  the additional information from the image caption dataset  would be beneficial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  Open-vocabulary instance segmentation aims to train a  model to segment both base classes VB and novel classes  VN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' Following previous methods [21, 28, 69], our model  uses high-level semantic embeddings from a pretrained text  Transformer (BERT [12]) as the weights of the linear clas-  sifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' We focus on distilling knowledge in the captions to  the target classes via representation similarities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  Baseline Method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' We adopt the recent Mask2Former [10]  model as our baseline since the mask-based Transformer ar-  chitecture can be readily extended into multi-modal training  with captions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' Mask2Former takes a Transformer encoder-  decoder architecture with a set of object queries, where the  object queries interact with encoder features via masked  cross-attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' Given an image I, during the inference,  Mask2Former directly outputs a set of object queries Q =  {qi}, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='.Nq, where each object query qi represents  one entity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' Then, two different Multiple Layer Perceptrons  (MLPs) project the queries into two embeddings for mask  classification and mask prediction, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' During the  training, each object query is matched to the ground truth  mask via masked-based bipartite matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' The loss func-  tion is Lmask = λclsLcls + λceLce + λdiceLdice, where  Lcls is the Cross-Entropy (CE) loss for mask classification,  and Lce and Ldice are the Cross-Entropy (CE) loss and Dice  loss [43] for segmentation, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' In particular, fol-  lowing [69], we use pretrained embeddings to replace the  learnable classifier for training and inference, as shown in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='The original Mask2Former can only detect and segment  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='People sitting around a table and talking  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='A group of people sitting  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='around a table  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='𝑁𝑞 mask  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='predictions  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='caption grounding loss  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='caption generation loss  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='classification  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='loss  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='mask loss  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='backbone  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='pixel  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='decoder  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='image features ℱ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='Transformer  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='decoder  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='𝑁𝑞 queries  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='MLP  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='base categories  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='novel categories  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='person  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='train  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='car  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='pizza  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='pretrained  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='embeddings  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='𝑁𝑞 multi-modal  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='embeddings  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='caption  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='generator  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='sentence  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='encoder  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='word  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='encoder  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='sentence features  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='𝑁𝑠 × 𝐶𝑠  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='…  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='…  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='word features  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='𝑁𝑤 × 𝐶𝑤  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='𝑁𝑞 classification  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='predictions  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='𝑁𝑐 × 𝐶𝑤  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='loss  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='legend  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='trainable  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='fixed  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='output  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='categories  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='object queries  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='output  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='𝑁𝑞 × 𝐶𝑤  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='Input Image  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='remaining after training  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='𝑓𝑔  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='people  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='table  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='𝑒𝑖  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='𝑀  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='𝑒𝑗  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='𝑊  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='𝑒𝑘  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='𝑆  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' The illustration of CGG framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' The input image I is first provided to Mask2Former.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' The output of the Transformer decoder  is then fed into an MLP, which generates Nq mask predictions together with the output of the pixel decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' Then the query is transferred  into Nq multi-modal embeddings {eM  i }, of which the similarity with class embeddings is computed to produce classification predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  {eM  i } also outputs grounding loss and generation loss with text features extracted by word encoder and sentence encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  closed-set objects and cannot handle the novel classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' Our  method extends it to perform open-vocabulary segmenta-  tion in a new framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' CGG Framework for OVS  Overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  Figure 3 presents the overall pipeline of our  CGG framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  Based on Mask2Former [10], follow-  ing [69], we set the pretrained text embeddings as the  weights of the final linear classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' We add two losses:  the caption grounding loss and the caption generation loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  A caption generator is appended at the end of the output  queries, directly producing the image caption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' During the  training, we adopt a pre-trained sentence encoder and word  encoder to encode both captions and object nouns extracted  from captions into sentence features and word features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' The  former is used for caption generation loss, while the latter is  used for caption grounding loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' During the inference, we  discard all the newly-introduced modules, and perform the  same inference procedure as Mask2Former.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  Class-Agnostic Pretraining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  Following previous works  [21,28,69], we first pretrain our framework using only base  data annotations in a class-agnostic manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' Such a process  is similar to training a Region Proposal Network (RPN) at  the first stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' The goal of pretraining is to encode instance-  wised information into object queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' Then we load the  pretrained model for joint training with caption data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  End-to-End Caption Grounding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  Previous works like  OVR-CNN [69] pre-train their models with caption data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  The core idea is to learn a Vision to Language (V2L) pro-  jection layer where the language data from novel classes are  transformed into vision features via multiple multi-modal  losses including grounding loss and a set of auxiliary self-  supervision loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  There are two potential issues with the previous design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  Firstly, training caption and segmentation separately can-  not fully explore caption data and detection/segmentation  annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' The training of segmenter is isolated and the  connection between the two models is broken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' Secondly,  there is a weakened region-word alignment in the tradi-  tional grounding process by calculating similarities between  multi-modal embeddings and all words in caption data,  because object-unrelated words may encounter the vision-  language implicit matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' We argue that object nouns in  caption data should be well aligned with region features in  a more fine-grained manner since the class categories are  always nouns in captions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  Therefore, rather than sending the entire sentence as in-  puts, we extract only object nouns from the sentence and  feed it to a word encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  Such extraction finds more  precise semantic embeddings, verified effectively for novel  class grounding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' For multi-modal embeddings, we adopt an  MLP as the V2L projection and take the outputs of Trans-  former decoder in Mask2Former as the inputs, since the ob-  ject queries group region features naturally, which is well  proved in many previous works [5, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' Given an image-  caption pair (I, C), we first calculate similarities between  Nq multi-modal embeddings {eM  i } and Nw word features  4  {eW  j } extracted from the caption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  SC(I, C) =  1  Nw  Nw  �  j=1  Nq  �  i=1  aC  i,j⟨eM  i , eW  j ⟩,  SI(I, C) = 1  Nq  Nq  �  i=1  Nw  �  j=1  aI  i,j⟨eM  i , eW  j ⟩,  (1)  where ⟨·, ·⟩ is the dot product between two vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  SC(I, C) and SI(I, C) are similarities between image I  and caption C normalized for text or image separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' The  normalization term is formulated as  aC  i,j =  exp⟨eM  i , eM  j ⟩  �Nq  i′=1 exp⟨eM  i′ , eW  j ⟩  ,  aI  i,j =  exp⟨eM  i , eM  j ⟩  �Nw  j′=1 exp⟨eM  i , eW  j′ ⟩  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  (2)  The core idea of grounding is that: the similarity between  the matched image-caption pairs should be high, while for  the unmatched pairs, it should be low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' During the training,  given a batch of image-caption pairs (BI, BC), for each  similarity S ∈ {SC, SI}, the grounding loss is composed  of two aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' Take the similarity normalized along text  dimension SC as an example, from the image perspective,  the grounding loss is as follows:  LIC  gro(I) = − log  exp SC(I, C)  �  C′∈BC exp SC(I, C′),  (3)  and from the caption perspective, the grounding loss is for-  mulated as:  LCC  gro(C) = − log  exp SC(I, C)  �  I′∈BI exp SC(I′, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  (4)  The final grounding loss is formulated as the sum of four  losses:  Lgro = 1  |BI|  BI  �  i=1  (LII  Gro(Ii) + LIC  Gro(Ii)) +  1  |BC|  BC  �  j=1  (LCI  Gro(Cj) + LCC  Gro(Cj)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  (5)  Optimizing the grounding loss aligns the multi-modal em-  beddings and language embeddings in a large noun vocab-  ulary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  End-to-End Caption Generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' Besides using caption  data for grounding loss to align regions and words, we ar-  gue that caption data can also be employed as a generative  supervision signal for a more fine-grained multi-modal un-  derstanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' The key insight is that we force the model  a couple of dogs  standing next  to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  CGG  w/o generation  Input Image  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' The effectiveness of caption generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' The generated  caption depicts rich information beyond object nouns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  to predict the occurring instances and their relationships  in the image to identify novel classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' Unlike grounding  loss that aims to push nouns and query embeddings as close  as possible, generative loss decodes the visual features into  the semantic embeddings, which are complementary to the  grounding loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' 4, the caption generation  module can help the model learn specific status and rela-  tionships of objects in the scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  Specifically, since the multi-modal embeddings encode  the region-wise information, we directly take these embed-  dings {eM  i } as the input of a lightweight caption generator,  which includes a stack of Transformer decoder layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' To  supervise the caption generator, we simply adopt a Cross  Entropy Loss on the predicted distribution of text vocabu-  laries, which is the commonly used objective function in the  research field of caption generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  Lgen = −  Ns  �  t=1  log(pθ(wt|w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=', wt−1)),  (6)  where pθ( ˆwt| ˆw1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' , ˆwt−1) is the probability of predicting  a particular word from the caption, θ denotes the parame-  ters of the generation network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' Hence, this loss function  enforces the predicted sentence to be consistent with the in-  put caption C, making the multi-modal embeddings {eM  i }  capable of representing various objects and their potential  relations in the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  Overall Loss Design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' The overall training loss contains  four items, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=', the classification loss Lcls, the segmen-  tation loss Lmask, the caption grounding loss Lgro, and  the caption generation loss Lgen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' Following the previous  method [69], the classification loss is selected as the Cross-  Entropy Loss that takes the dot product of multi-modal em-  beddings eM  i  and base class embeddings as its logit inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  The final loss function L is the weighted summation of the  four losses: L = λclsLcls + λmaskLmask + λgroLgro +  λgenLgen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' We follow the default setting in the MMDetec-  tion framework, where the weights are set to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='0, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='0, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='0  and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='0 in all our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  Inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' Compared to the baseline model, CGG only in-  troduces extra losses and a caption generation head during  the training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' During the inference, following [69], we use  5  dog0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='94  dogl0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='95*doglo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='9  dog0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='96the pretrained embeddings of all classes to perform open  vocabulary segmentation via dot product, including base  classes and novel classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' The remaining inference proce-  dure is the same as the Mask2Former [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' Experiments  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' Experimental Setup  Dataset Settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' For Open Vocabulary Instance Segmen-  tation, we mainly conduct experiments on COCO dataset  [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' Following previous works [28, 69], we split 48 base  classes with mask annotations and 17 target classes with-  out mask annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' There are 107,761 training images  with 665,387 mask annotations from base classes and 4,836  testing images consisting of 28,538 and 4,614 mask in-  stances for base and target classes, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' For cap-  tioned images, we use the entire MS-COCO training set  with 118,287 images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' Each image is annotated with five  captions describing the visually-grounded objects in the im-  age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' Unlike previous works [19, 49, 79] that adopt extra  caption datasets, like Conceptual Captions [53] with 3M  image-caption pairs for pre-training, we do not use extra  caption datasets or detection datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' We follow the ori-  gin OVR-CNN [69] setting by only exploring a limited cap-  tion dataset within COCO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' For Open Set Panoptic Segmen-  tation [29], we mainly adopt the COCO-panoptic dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  We follow the previous works [29, 63] by splitting part of  thing classes into unknown classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' We obtain three differ-  ent splits by varying the numbers of unknown classes (K%  ratios, 5%, 10%, 20%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' Different from previous works, we  use extra caption data for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  Metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' For OVIS setting, we report the mask-based mean  Average Precision (mAP) at intersection-over-union (IoU)  of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  To analyze the performances on base and target  classes, we carry out experiments in two settings: con-  strained setting where the model is only evaluated on test  image inputs, which belong to either base classes or target  classes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' generalized setting in which a model is tested on  both base and target class images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' The latter is more chal-  lenging as it requires the model to segment target classes  and avoid class bias from base classes, mostly with very  high scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' We further report open vocabulary detection  with box-based mAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' For OSPS setting, we employ the  panoptic segmentation metrics, including Panoptic Qual-  ity (PQ) and Segmentation Quality (SQ), where we report  known classes and unknown classes separately for refer-  ence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' More details about the data preparation can be found  in the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  Implementation Details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' We implement our models in Py-  Torch [45] with MMDetection framework [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  For both  settings, we use the distributed training framework with 8  GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' Each mini-batch has one image per GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' The opti-  mizer is AdamW [42] with a weight decay of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='0001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' We  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' Results on Open Vocabulary Instance Segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  Method  Constrained  Generalized  Base  Novel  Base  Novel  All  OVR [69]  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='0  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='9  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='6  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='1  35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='2  SB [1]  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='6  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='8  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='0  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='0  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='5  BA-RPN [76]  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='8  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='1  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='3  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='4  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='5  XPM [28]  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='4  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='0  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='5  21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='6  36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='3  CGG (Ours)  46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='8  29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='5  46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='0  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='4  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='4  Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  Results on Open Set Panoptic Segmentation (OSPS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  Note that previous methods EOPSN [29] and Dual [63] treat all  unknown things as one class while not classifying them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' In con-  trast, CGG performs complete Open Vocabulary Panoptic Seg-  mentation, classifying each unknown thing into its specific cate-  gory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' We report the mean PQ and SQ for all unknown categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  indicates that the scores are averaged from each unknown class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  Method  K(%)  Known  Unknown  PQTh  SQTh  PQSt  SQSt  PQTh  SQTh  EOPSN [29]  5  44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='8  80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='5  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='3  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='1  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='1  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='7  Dual [63]  45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='1  80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='9  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='1  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='1  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='2  80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='0  CGG (Ours)  50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='2  83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='1  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='3  81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='5  45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='0*  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='2*  EOPSN  10  44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='5  80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='6  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='4  71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='8  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='9  76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='8  Dual  45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='0  80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='7  27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='8  72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='2  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='5  79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='9  CGG (Ours)  49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='2  82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='8  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='6  81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='2  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='6*  82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='6*  EOPSN  20  45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='0  80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='3  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='2  71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='2  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='3  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='8  Dual  45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='0  80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='6  27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='6  70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='1  21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='4  79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='1  CGG (Ours)  48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='4  82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='3  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='4  81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='1  36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='5*  78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='0*  adopt full image size for a random crop in both the pre-  training and training process following Mask2Former [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  For classification head, word encoder, and sentence en-  coder, we all adopt the BERT embeddings (pre-trained with  fixed input embeddings, not the output of transformer lay-  ers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' We use a LVIS class name parser to extract object  nouns from captions to ensure that the extracted nouns rep-  resent objects in the image [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' For OVIS, we keep the  top-100 queries as the model outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' For OSPS, following  previous work [29, 63], rather than Mask2Former baseline,  we put thing mask predictions first and fill the remaining  background with stuff mask predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' Note that all ex-  periments use ResNet-50 backbone for fair comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' Main Results  Results on OVIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' We evaluate the performance of CGG  and baselines on the Open Vocabulary Instance Segmenta-  tion task on MSCOCO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' As shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' 1, our model  outperforms the best baseline XPM by 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='5% mAP in the  constrained setting where only novel categories input and  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='8% mAP in the generalized setting where both base and  novel categories are employed as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' The generalized  setting is more challenging because the model also needs  to distinguish novel categories from given base categories,  which have a data distribution bias from training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' We  observed that CGG has a greater improvement in general-  6  Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' Results on COCO Open Vocabulary Object Detection  (OVOD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' IN-21K indicates ImageNet-21K [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  CC indicates  Conceptual Captions [53]  Method  Epochs  Extra Data  AP50box  novel  AP50box  all  DLWL [49]  96  YFCC100M  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='6  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='9  Cap2Det [65]  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='5  None  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='3  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='1  OVR-CNN [69]  12  None  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='8  39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='9  Detic [79]  96  IN-21K & CC  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='1  44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='7  PromptDet [19]  24  LAION-novel  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='6  50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='6  CGG (Ours)  12  None  29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='3  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='8  ized settings compared to constrained settings, which fur-  ther shows the effectiveness of CGG in identifying the lan-  guage embeddings of novel classes and distinguishing them  from base classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  Results on OSPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' To show the scalability of CGG, we also  perform experiments on the Open Set Panoptic Segmen-  tation task by expanding the base classes from base thing  classes to including stuff classes, maintaining the whole  training pipeline of CGG unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' The results are shown  in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  Compared with traditional Open Set Panop-  tic Segmentation, CGG actually performs a more difficult  Open Vocabulary Panoptic Segmentation and still outper-  forms previous methods EOPSN and Dual by a large margin  of 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='9% PQ on unknown things in 20% unknown things  setting [29], and 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='9%, 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='8% in 10% and 5% settings, re-  spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' Due to the scalability of Mask2Former and our  simple yet effective training pipeline, which can fully uti-  lize the open vocabulary knowledge from caption data, our  model can perform well on open vocabulary panoptic seg-  mentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  Results on OVOD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' Besides the segmentation task, we also  evaluate Open Vocabulary Object Detection task by match-  ing Ground Truth with bounding boxes in the testing stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  As shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' 3, CGG achieves better AP50 score in  novel classes compared to several previous works [19, 79]  in a shorter training schedule with no extra data used (only  COCO-Caption).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' Previous methods like PromptDet [19]  and Detic [79] tend to use large-scale image-text datasets,  thus causing a longer training schedule and higher compu-  tational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' We observe that CGG has inferior results in  AP50box  all .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' It may cause by the shorter training schedule and  exposure to base classes compared with other methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' Ablation Study and Analysis  We do ablation studies of our model to validate the ef-  fectiveness of each component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' We do all the ablations on  the MSCOCO 48/17 split [69] with the metric mAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  Effectiveness of the CGG framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' First, we evalu-  ate the significance of each proposed module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' As shown in  Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' 4a, the baseline Class Emb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=', which transfers class la-  bels to their corresponding text embeddings, only achieves  a meager AP score of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='2 for the novel class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' As a com-  parison, adding Caption Grounding increases the Novel AP  to 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='2, verifying the significance of Caption Grounding,  which helps align multi-modal embeddings explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' Fur-  ther, with the Caption Generation module, the final score  reaches 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' The increase arises from a more strict regu-  larization of Caption Generation, which supervises beyond  nouns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  For example, the caption “a woman holding an  umbrella” requires the network to capture the relationship  “holding” as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' If the Caption Generation module plays  independently, the performance decreases to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  Training Pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  Previous methods like OVR-CNN  [69] train their embeddings before fine-tuning the seg-  mentor/detector, called ’emb-segm.’ In this paper, we in-  stead pre-train a class-agnostic segmentor and then train the  multi-modal embeddings eM  i  on image-text data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' We name  it ’segm-emb.’ These pipelines are compared over CGG in  Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' 4b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' Note that “segm-emb-segm” is also included as  a candidate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' Results show that though “segm-emb” is in-  ferior to others for base classes, it achieves significantly  higher scores for novel classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' This phenomenon arises  because training the segmentor in the last stage overfits the  base classes, thus leading to a worse recall for novel classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  Grounding Nouns Extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' In CGG, we extract only  object nouns from the sentences, leaving other words un-  touched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' To validate the effectiveness of different word se-  lection strategies, we also try to extract all words, and all  nouns, except for only object nouns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' The results are shown  in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' 4c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' By extracting all words, the novel AP decreases  by 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='8, and by extracting all words without distinguishing  whether they refer to objects or not, the novel AP decreases  by 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' In conclusion, different extracting strategies sig-  nificantly impact the model’s performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  Layers of Caption Generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' We evaluate the influence  of layers of the transformer decoder in the caption genera-  tor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' The results are in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' 4d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' We test 2, 4, and 6 layers  transformer decoders and observe that the middle number of  4 is a better choice, while fewer layers of 2 and more layers  of 6 both harm the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' Moreover, heavier caption  generators may improve mAP for base classes a little, but  also increase the computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  Ablation on Class-Agnostic Pretraining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  As a class-  agnostic segmentor is trained to segment base and potential  novel objects before training the multi-modal embeddings  and caption generator, we do ablations on the effectiveness  of class-agnostic pretraining and its alternatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' As shown  in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' 4e, without any class-agnostic pretraining, the mAP  on novel classes decreases by 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='7%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' Moreover, if we freeze  the Mask2Former, and only trains multi-modal embeddings  and caption generator, the mAP on novel classes decreases  by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='0%, compared to the CGG model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  GFLOPs and Parameter Analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  CGG adds a  lightweight Transformer decoder as the caption generator in  7  Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' Ablation studies and comparison analysis on COCO OVIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  (a) The Effectiveness of Each Components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  baseline  Gro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  Gen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  Base  Novel  Class Emb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='2  w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' Gro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  ✓  49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='1  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='2  w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' Gen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  ✓  49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='3  Both (CGG)  ✓  ✓  48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='0  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='4  (b) Training Pipeline Comparison  Settings  Base  Novel  All  emb-segm  49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='2  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='3  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='6  segm-emb-segm  50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='2  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='3  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='4  segm-emb (CGG)  46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='0  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='4  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='4  (c) Nouns Extraction in Caption Grounding  Method  Base  Novel  All  All Words  44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='7  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='6  35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='0  All Nouns  48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='8  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='0  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='0  Object Nouns  46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='0  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='4  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='4  (d) Caption Generator Design  #layers  Base  Novel  All  2  46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='7  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='4  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='6  4 (CGG)  46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='0  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='4  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='4  6  48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='2  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='9  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='6  (e) Effect of Class-Agnostic Pretraining  Settings  Base  Novel  All  No class-agnostic  46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='2  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='7  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='0  Freeze class-agnostic  47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='6  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='4  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='1  CGG  46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='0  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='4  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='4  (f) GFlops and Parameters  Schedule  Parameters  GFLOPs  baseline  35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='65M  227.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='48  Ours: Inference  35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='65M  227.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='48  Ours: Training  81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='19M  229.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='33  a person standing next to a large group of cows  a couple of people that are standing next to a table  a herd of zebras grazing in the grass  a group of cows standing next to each other on the grass  there is a lot of food on this table  a black and white photo of a large brown dog  a couple of elephants that are standing in the dirt  a man riding a snowboarding down the side of a hill  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' Visualization of CGG results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' Instance Segmentation (Top) and Panoptic Segmentation (Bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' Categories marked by ’*’ are  novel categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' Captions generated are depicted upon each image-prediction pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' Novel categories are colored in the caption if it has.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  CGG  class-agnostic pretrain  Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  The embedding space of multi-modal embeddings  {eM  i }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' The dimension of the embeddings is reduced to 2 dimen-  sions using t-SNE [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' Each color represents a class label in the  17 novel COCO classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' Each dot represents the embedding with  the corresponding mask matched with ground truth annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' As shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' 4f, the #Parameters increases by  127.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='7% in training, while the total GFLOPs only increase  by a small margin of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='8%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' Since text data is much smaller  than images under the same batch size, the increased com-  putational cost by the caption generator can be ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' The  GFLOPs and Parameters during inference are the same as  the Mask2Former baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  Segmentation Results Visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' 5 shows the  qualitative results of CGG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' The row shows panoptic results  and the second row shows instance results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' Novel classes  detected in the image are marked with ’*’ and highlighted  in the caption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' The result shows that our framework can  identify and segment base and novel classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' We show the  generated comprehensive captions above the images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  Embeddings Space Visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' 6 shows the t-SNE  visualization result of the trained multi-modal embeddings  (right) and the embeddings from a class-agnostic pretrain-  ing model (left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' Specifically, we evaluate our model on  the COCO validation set and get all the embeddings for  each image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' We change the bipartite matching strategy of  Mask2Former to calculating mask loss only, thus getting  the matching result between ground truth labels and multi-  modal embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' The original Mask2Former [10] model  cannot distinguish different novel classes in the embedding  space, with only class-agnostic pretraining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' After training  using caption grounding and generation, the embeddings  can formulate groups consistent with their categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' Conclusion  In this paper, we present a joint Caption Grounding and  Generation (CGG) framework for instance-level open vo-  cabulary segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  Our core insights are two folds:  8  ceiling-merged  sky-other-merged  wall-other-mer  fence-merged  cow  FCOW  COW  person  cow  graveldoor-stuff  wall-other-merge  person  bicycle  cake  table-mergedzebra  zebra  grass-mergedsky-other-merged  tree-merged  cow  CoW  CoW  cow  cow  AM  grass-merged*dog0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='97chairj0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='90onj0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='89  bottlelo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  bird0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='93cakej0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='90  broccolil0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='97  knifej0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='89  bowl/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='41  bowlj0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='94  forkj0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='43  spoonj0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='70  cake0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='71  fork/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='98  cup0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='77personj0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='96  backpackj0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='93  snowboard|0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='94  person|personj0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='98nj0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='98  personjo  skis10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='80  backpack0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='9*elephantl0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='94  elephantj0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='93Firstly, the caption contains fine-grained nouns, which leads  to better fine-grained grounding with object queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' Sec-  ondly, the caption can be a supervision signal that forces  the model to predict novel objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' To our knowledge, we  are the first to unify segmentation and caption generation  for open vocabulary learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' We obtain significant per-  formance improvement on both OVIS and OSPS and com-  parable results on OVOD without extra large-scale datasets  pre-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  Limitation and Future Work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' Due to the limited com-  putation resources, we do not pre-train our framework on  extra caption datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  Moreover, we do not use VLMs  such as CLIP for distillation or supervision, and we do not  experiment on larger scale datasets, like LVIS and Open-  Image [25,31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' We will put these as future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  Acknowledgement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  This work is supported by the Na-  tional Key Research and Development Program of China  (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='2020YFB2103402).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' We also gratefully acknowledge  the support of SenseTime Research for providing the com-  puting resources for this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' More Implementation Details  Baseline Details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' All the table results in main paper use the  same ResNet50 [27] backbone for a fair comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' The  number of object queries is 100 by default.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' Our method is  trained by only 12 epochs on the COCO train set and eval-  uated on the COCO validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' All the experiments are  carried out on 8 V100 GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' Following previous meth-  ods [28,69], the metric we use for OVIS is mAP (mean AP  on the IoU threshold of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  Training and Inference Details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  We adopt the default  training of Mask2Former [7,10,62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' A learning rate multi-  plier of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='1 is applied to the backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' For data augmenta-  tion, we use the default large-scale jittering (LSJ) augmen-  tation with a random scale sampled from the range 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='1 to  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='0 with the crop size of 1024 × 1024.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' We use the default  Mask R-CNN inference setting [26], where we resize an  image with shorter side to 800 and longer side to 1333.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' For  the Inference of OSPS, we do not use the default joint merge  for things and stuff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' We put the thing mask first and fill the  remaining area with stuff mask prediction because the thing  predictions for unknown are usually in a low score, and they  may be covered by high score stuff mask prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  Training Splits For OVIS and OSPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' For OVIS, we fol-  low the 48/17 split in COCO proposed by [48], in which 48  classes are base classes, and 17 are novel classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' For OSPS,  we follow the unknown things split proposed by [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' The  unknown percentages are 5%, 10%, and 20% separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  Concretely, for 48/17 split of OVIS, the base classes  are:  “person”, “bicycle”, “car”, “motorcycle”, “truck”,  “boat”, “bench”, “bird”, “horse”, “sheep”, “zebra”, “gi-  raffe”, “backpack”, “handbag”, “skis”, “kite”, “surfboard”,  “bottle”, “spoon”, “bowl”, “banana”, “apple”, “orange”,  Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' Ablation on fully supervised instance segmentation, ob-  ject detection, and panoptic segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' AP-novel indicates the  mean AP on the 17 novel classes (trained in the fully supervised  setting).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' AP-bbox indicates object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  Method  Instance  Panoptic  AP  AP-novel  AP-bbox  PQ  PQ-th  PQ-st  class-label  59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='3  66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='6  58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='9  46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='4  51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='9  38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='2  class-emb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='6  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='8  50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='2  44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='4  50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='5  35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='1  w/ gro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='8  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='4  50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='3  44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='1  50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='3  35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='0  w/ gen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='9  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='6  50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='7  44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='2  50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='5  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='8  w/ both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='3  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='5  50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='7  44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='3  50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='6  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='9  “broccoli”, “carrot”, “pizza”, “donut”, “chair”, “bed”,  “tv”, “laptop”, “remote”, “microwave”, “oven”, “refrig-  erator”, “book”, “clock”, “vase”, “toothbrush”, “train”,  “bear”, “suitcase”, “frisbee”, “fork”, “sandwich”, “toilet”,  “mouse”, “toaster”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  The novel classes are:  ’bus’, ’dog’, ’cow’, ’ele-  phant’, ’umbrella’, ’tie’, ’skateboard’, ’cup’, ’knife’, ’cake’,  ’couch’, ’keyboard’, ’sink’, ’scissors’, ’airplane’, ’cat’,  ’snowboard’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  For OSPS, the unknown things are: 5%: “car”, “cow”,  “pizza”, “toilet”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' 10%: “boat”, “tie”, “zebra”, “stop sign”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  20%: “dining table”, “banana”, “bicycle”, “cake”, “sink”,  “cat”, “keyboard”, “bear”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' More Experiments Results  Will Joint Grounding and Caption Help the Fully Su-  pervised Baseline?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' To answer this question, we perform  ablation on fully supervised settings in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  For the  proposed CGG, we verify two main components, includ-  ing caption grounding and caption generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' Class-emb  means only using pre-trained text embeddings for mask  classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' Class-label is a traditional learnable, fully  connected layer that converts the classes into contiguous la-  bels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' In Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' 5, we observe that the fully supervised method  achieves better results than using class embeddings in all  three tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' As shown in the last three rows of Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' 5, for  within class embedding settings, the added caption ground-  ing and generation modules help to improve the perfor-  mance on OVIS, but no performance gain on OSPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' We  conclude that joint grounding and caption have limited ben-  efits (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='5% improvements) in supervised settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  Will Better Caption Generator Help Open Vocabulary  Instance Segmentation?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' We further explore the influence  of the caption generation module to open vocabulary in-  stance segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' 6 shows the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  As the  caption generator becomes larger, the overall segmentation  quality (AP all) increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' On the contrary, the quality of  the caption (including BLUE and CIDEr) generation drops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  This means a better caption generator may not be a better  open vocabulary instance segmenter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' The role of the cap-  9  airplane  bus  cat  dog  cow  elephant  umbrella  tie  snowboard  skateboard  cup  knife  cake  couch  keyboard  sink  scissors  airplane  bus  cat  dog  cow  elephant  umbrella  tie  snowboard  skateboard  cup  knife  cake  couch  keyboard  sink  scissors  Ground Truth  Predict  Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' The correlation map between Ground Truth and model  predictions on novel classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' The noun embeddings and object  queries for novel classes are highly correlated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  tion generator is to force the model to know the existence  of novel objects, and pursuing a better caption generation  model is not our goal of OVIS and OSPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' Visual Analysis and Comparison  Visualization Analysis both Nouns and Object Queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  We calculate the correlation map between the predicted  multi-modal embeddings eM  i  and the Ground Truth class  embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' 7, our model can correctly  distinguish novel classes based on the segmentation masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  More Visual Examples from Caption Generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' We  observe that in some cases, the caption generated by CGG  can predict objects that are not in the category list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' Cate-  gories beyond the given list cannot be correctly classified  using the similarity between multi-modal embeddings and  class embeddings since the class embeddings are not acces-  sible during inference, like in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' 8, images top.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' There is  a couple of luggage on the floor, but ’luggage’ is not a class  in the validation dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' Without a caption generator, the  model classifies the luggage as ’suitcase.’ However, with  the caption generation module, the generated caption suc-  cessfully depicts the word ’luggage’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' In the bottom images,  ’tennis’ is also described by captions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' 9 shows more  visualization results with captions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  More Visualization Results on OVIS and OSPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' In  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' 10, we present more visual results of OVIS and OSPS  tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' The CGG model can well segment and classify novel  categories well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  Zero Shot Visualization on ADE20K dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' 11,  we show the visualization results on ADE20K dataset [77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  CGG can detect and segment novel classes in a zero shot  a couple pieces of luggage on top of the floor  CGG  CGG w/o generator  a couple of people that are playing tennis  Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' Examples of captions predicting objects that are not in  the category list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  manner on ADE20K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' At the same time, CGG generates  comprehensive captions that well depict the content of the  images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  References  [1] Ankan Bansal, Karan Sikka, Gaurav Sharma, Rama Chel-  lappa, and Ajay Divakaran.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' Zero-shot object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' In  ECCV, pages 384–400, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' 2, 6  [2] Daniel Bolya, Chong Zhou, Fanyi Xiao, and Yong Jae Lee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  Yolact: Real-time instance segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' In ICCV, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' 1  [3] Maxime Bucher, Tuan-Hung Vu, Matthieu Cord, and Patrick  P´erez.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' Zero-shot semantic segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' NIPS, 32, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' 1,  2  [4] Jiale Cao, Rao Muhammad Anwer, Hisham Cholakkal, Fa-  had Shahbaz Khan, Yanwei Pang, and Ling Shao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' Sipmask:  Spatial information preservation for fast image and video in-  stance segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' ECCV, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' 1  [5] Nicolas Carion, Francisco Massa, Gabriel Synnaeve, Nicolas  Usunier, Alexander Kirillov, and Sergey Zagoruyko.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' End-to-  end object detection with transformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' In ECCV, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' 1,  4  [6] Hao Chen, Kunyang Sun, Zhi Tian, Chunhua Shen, Yong-  ming Huang, and Youliang Yan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' Blendmask top-down meets  bottom-up for instance segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' In CVPR, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' 1  [7] Kai Chen, Jiaqi Wang, Jiangmiao Pang, Yuhang Cao, Yu  Xiong, Xiaoxiao Li, Shuyang Sun, Wansen Feng, Ziwei Liu,  Jiarui Xu, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' Mmdetection: Open mmlab detection tool-  box and benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' arXiv preprint arXiv:1906.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='07155, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  6, 9  [8] Xinlei Chen, Ross Girshick, Kaiming He, and Piotr Doll´ar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  Tensormask: A foundation for dense object segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' In  ICCV, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' 1  [9] Bowen Cheng, Maxwell D Collins, Yukun Zhu, Ting Liu,  Thomas S Huang, Hartwig Adam, and Liang-Chieh Chen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  Panoptic-deeplab: A simple, strong, and fast baseline for  bottom-up panoptic segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' In CVPR, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' 1  10  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
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+page_content='98Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' Ablation on layers of Caption Generator and quality of Open Vocabulary Instance Segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' We adopt BLUE, CIDEr, and  ROUGE as the metrics to evaluate the quality of generated captions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  # layers  Segmentation  Caption Generation  Base  Novel  All  BLUE-1  BLUE-2  BLUE-3  BLUE-4  CIDEr  ROUGE  2  46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
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+page_content='2  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
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+page_content='289  a group of people riding skateboards down a street  there is a bathroom sink with a large mirror above it  Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' Visualization results of generated captions and the related segmentations of CGG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' Input Image (Left), CGG (Middle), CGG w/o  caption generation (Right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' ’mirror’ is not in the category list but is depicted by the generated caption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  a red double decker bus driving down the street  there is an airplane that can be seen on the ground  there is a lot of stuff on the table  a close up of two plates of food  a zebra looking down  there is a lot of cars on the road  a person sitting on the ground under an umbrella  a couple of elephants standing next to each other  a bunch of bananas hanging from the metal pole  a plate of pizza on a table  a bunch of bikes parked next to each other  two different sized bears  Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' More visualization results of OVIS (Top two rows) and OSPS (Bottom two rows).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' Novel classes are marked by ’*’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  [10] Bowen Cheng, Ishan Misra, Alexander G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' Schwing, Alexan-  der Kirillov, and Rohit Girdhar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
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+page_content=' CVPR, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
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+page_content='95a group of people standing next to each other on display  a bedroom scene with focus on a small table  a couple of cars parked next to a fire hydrant  a room filled with furniture next to each other  Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' Visualization on ADE20k [77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' Following [10], we apply instance segmentation on 100 instance classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' Classes not in COCO  are marked by ’*’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  [11] Jia Deng, Wei Dong, Richard Socher, Li-Jia Li, Kai Li,  and Li Fei-Fei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' Imagenet: A large-scale hierarchical image  database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' In CVPR, pages 248–255.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' Ieee, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' 7  [12] Jacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina  Toutanova.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  Bert:  Pre-training of deep bidirectional  transformers for language understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  arXiv preprint  arXiv:1810.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='04805, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' 3  [13] Henghui Ding, Chang Liu, Suchen Wang, and Xudong Jiang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  Vision-language transformer and query generation for refer-  ring segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' In ICCV, pages 16321–16330, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
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+page_content='  Vlt: Vision-language transformer and query generation for  referring segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' TPAMI, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
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+page_content=' De-  coupling zero-shot semantic segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' In CVPR, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
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+page_content=' Learning to prompt for open-vocabulary ob-  ject detection with vision-language model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' In CVPR, pages  14084–14093, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
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+page_content=' The pascal visual object  classes (voc) challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' IJCV, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
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+page_content=' Instances as  queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' arXiv preprint arXiv:2105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
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+page_content=' Prompt-  det: Towards open-vocabulary detection using uncurated im-  ages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
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+page_content=' Mask r-cnn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' In ICCV, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
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+page_content='  Deep residual learning for image recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content='  In CVPR,  2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
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+page_content=' Open-vocabulary instance segmentation via ro-  bust cross-modal pseudo-labeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' In CVPR, pages 7020–  7031, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
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+page_content=' Exemplar-based open-set panoptic segmentation  network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' In CVPR, pages 1175–1184, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' 2, 6, 7, 9  [30] Chao Jia, Yinfei Yang, Ye Xia, Yi-Ting Chen, Zarana Parekh,  Hieu Pham, Quoc Le, Yun-Hsuan Sung, Zhen Li, and Tom  Duerig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' Scaling up visual and vision-language representation  learning with noisy text supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' In ICML, pages 4904–  4916.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' PMLR, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
+page_content=' 1, 3  [31] Alina Kuznetsova, Hassan Rom, Neil Alldrin, Jasper Ui-  jlings, Ivan Krasin, Jordi Pont-Tuset, Shahab Kamali, Stefan  12  monitor0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AyT4oBgHgl3EQf5fpe/content/2301.00805v1.pdf'}
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+The Evolutionary Computation Methods No One Should Use
+Jakub K˚udela
+Institute of Automation and Computer Science, Brno University of Technology
+Jakub.Kudela@vutbr.cz
+Abstract
+The center-bias (or zero-bias) operator has recently been identified as one of the problems plaguing the benchmarking
+of evolutionary computation methods. This operator lets the methods that utilize it easily optimize functions that have
+their respective optima in the center of the feasible set. In this paper, we describe a simple procedure that can be
+used to identify methods that incorporate a center-bias operator and use it to investigate 90 evolutionary computation
+methods that were published between 1987 and 2022. We show that more than half (47 out of the 90) of the considered
+methods have the center-bias problem. We also show that the center-bias is a relatively new phenomenon (with the first
+identified method being from 2012), but its inclusion has become extremely prevalent in the last few years. Lastly, we
+briefly discuss the possible root causes of this issue.
+Keywords: Evolutionary Computation, Benchmarking, Metaheuristics, Center-bias, Zero-bias
+1. Introduction
+Imagine the following situation. Encountering a challenging optimization task, you decide to find the most recently
+developed algorithm for optimization published in some of the most prestigious journals. The analysis of the method
+performed on several standard benchmarks clearly shows that it is superior to all the other old methods. The paper also
+contains a link to a repository with the code. So, you give it a try. And it fails. The best results it provides are hardly
+better (or worse) than the ones you got from a simple implementation of a method that is more than two decades old.
+Maybe the problem you tried to solve is too challenging? Perhaps a bit of hyperparameter optimization could help the
+method perform as advertised? Or, maybe the method is not as good as it presented itself.
+Through inspiration from natural behaviors, the field of evolutionary computation (EC) produced over its long
+history a great number of important metaheuristic algorithms, such as Evolutionary Strategy, Genetic Algorithms,
+Particle Swarm Optimization, or Differential Evolution. Such methods found applications in complex systems where
+the use of exact algorithms was either inadequate or computationally too prohibitive. However, over the past few years
+we have witnessed an explosion of ”novel” methods that are based on natural/evolutionary principles. The bestiary
+of EC1, which tries to catalog of a portion of these nature-based methods, now contains over 250 methods that claim
+their inspiration in natural processes. And new methods are emerging at an ever-increasing rate. It is also becoming
+clearer that there is more creativity being spent at naming these ”novel” methods, than in making sure they contain
+anything new computation-wise. After many of these methods have been found to conceal their lack of novelty behind
+a methaphor-rich jargon [1, 2, 3, 4, 5], a call was made from within the EC community [6]. In the letter, the collective
+of authors and signatories identified four main issues with the high-volume inflow of new methods: useless metaphors,
+lack of novelty, poor experimental validation and comparison, and publishing these methods in off-topic journals.
+In this text, we will focus on the poor experimental validation of some of the EC methods. Most of the reasoning
+about the viability of metaheuristics is done through benchmarking [7]. If a new method performs well on a universally
+accepted set of benchmark problems, it is likely to be seen as valid. There have been several benchmark functions/sets
+proposed over the years, but the most widely recognized ones came from special sessions (competitions) on black-box
+optimization at two conferences: the IEEE Congress on Evolutionary Computation (CEC), and the Genetic and Evo-
+lutionary Computation Conference (GECCO), where the Black-Box Optimization Benchmarking (BBOB) workshop
+was held.
+There is, however, another quite widely used benchmark set that contains some of the most well-known functions
+such as Griewank, Ackley, Rastrigin, Rosenbrock, and Schwefel. It was recently uncovered [8] that this set contains a
+serious design flaw, as a large portion of the functions in the set have their respective optimum at a zero vector (or in the
+1Campelo, F., Aranha, C. Evolutionary computation bestiary. https://github.com/fcampelo/EC-Bestiary
+Preprint submitted to ...
+January 6, 2023
+arXiv:2301.01984v1  [cs.NE]  5 Jan 2023
+
+Table 1: The 13 benchmark functions, dimension 30. U -– unimodal, M — multimodal, S — separable, N — non-separable, f ∗ – the optimal
+function value, f(0) – function value at the zero vector, x∗ – optimal solution.
+ID
+name
+type
+range
+f ∗
+f(0)
+x∗
+F01
+Sphere
+U, S
+[-100,100]
+0
+0
+[0,0,...]
+F02
+Schwefel 2.22
+U, N
+[-100,100]
+0
+0
+[0,0,...]
+F03
+Schwefel 1.2
+U, N
+[-100,100]
+0
+0
+[0,0,...]
+F04
+Schwefel 2.21
+U, S
+[-100,100]
+0
+0
+[0,0,...]
+F05
+Rosenbrock
+U, N
+[-30,30]
+0
+2.90E+01
+[1,1,...]
+F06
+Step
+U, S
+[-100,100]
+0
+7.50E+00
+[-0.5,-0.5,...]
+F07
+Quartic with noise
+U, S
+[-1.28,1.28]
+0
+0
+[0,0,...]
+F08
+Schwefel 2.26
+M, S
+[-500,500]
+-1.25E+04
+0
+[420.9, 420.9,...]
+F09
+Rastrigin
+M, S
+[-5.12,5.12]
+0
+0
+[0,0,...]
+F10
+Ackley
+M, N
+[-32,32]
+0
+0
+[0,0,...]
+F11
+Griewank
+M, N
+[-600,600]
+0
+0
+[0,0,...]
+F12
+Penalized1
+M, N
+[-50,50]
+0
+1.67E+00
+[-1,-1,...]
+F13
+Penalized2
+M, S
+[-50,50]
+0
+3.00E+00
+[1,1,...]
+centre of the feasible set). This would be fine, if it were not for the methods that utilize this flaw to appear competitive.
+These methods incorporate a “check-the-middle” routine or have a centre-bias (or zero-bias) operator that draws them
+towards the center of the feasible set. One would expect that such methods do not get published very often, are easily
+spotted, or at the very least do not appear in high-profile journals.
+In this paper, we describe a simple methodology that we use to uncover whether or not a given evolutionary
+computation method utilizes a center-bias operator. We then investigate 90 evolutionary computation methods from
+the mealpy library2 and Mathworks code repositories3 for the inclusion of the center-bias.
+2. Methodology
+We utilize the same methodology that was used to uncover the center-bias problem in [9] and [8]. The 13 benchmark
+function used for our test (and optimization ranges) are shown in Table 1. One can easily see that many of these
+functions, apart from F08, have the corresponding optima either at the zero vector or very close to it. The problem F08
+is quite different from the rest, as its optimum is far away from the center.
+For the evaluation we set the dimension of the problems to 30 and allow for at most 50,000 function evaluations.
+We also chose a simple performance measure - the mean error (as the difference between the optimal function value
+and best function value found) over 20 independent runs. Here, we also treat any value smaller than 1e–08 as identical
+to 1e–08, as the problem is essentially solved and additional precision is not needed (we could treat it as a 0 as well,
+but we will shortly use fractions of these numbers, which would bring unwanted hassle). We refer to the results of the
+computation as the “unshifted” ones. Afterwards, we introduce a shift operation, that “moves” the benchmark function
+by a predetermined vector s, meaning that function f(x) becomes f(x+ s). One expects that a “small” value of s should
+not result in a large deviation in the behaviour of the optimization method, as the two problems are very similar. We
+chose the shift vector as 10% of the range - e.g., for F01, s = [20, 20, . . . ]. We use the same computational framework
+(i.e., dimension 30, at maximum 50,000 function evaluations, and 20 independent runs) and refer to the results of these
+computations as the “shifted” ones.
+What we are interested in is the “ratio” between the “shifted” and “unshifted” results for the individual benchmark
+functions, i.e., how many times are the results on the shifted problem worse than on the unshifted one. For the methods
+that do not incorporate a center-bias, one expects this number to be close to 1 (as the ushifted and shifted problems
+are similar), while for the methods that include a center-bias, this ratio should be much bigger than 1. Naturally, the
+value of this ratio will fluctuate depending on the given benchmark function, as well as on the number of independent
+runs of the algorithms. As a simple indicator of the center-bias, we look at the geometric mean of the ratios for the
+different benchmarks – if this value is bigger than 1E+01 (meaning that the method performs roughly at least on order
+of magnitude better on unshifted problems), we take it as a confirmation of the presence of the center-bias operator.
+A small example of this computation is shown in Table 2, where we investigate five EC methods – Artificial
+Bee Colony (ABC) [10], Differential Evolution (DE) [11], LSHADE [12], Satin Bowerbird Optimizer (SBO) [13],
+and Runge Kutta Optimizer (RKO) [14]. The first two methods (ABC and DE) can be thought of as the “standard”
+2N. V. Thieu, “A collection of the state-of-the-art meta-heuristics algorithms in python: Mealpy,” Available: https://doi.org/10.5281/
+zenodo.3711948
+3https://www.mathworks.com/
+2
+
+Table 2: The results of proposed methodology demonstrated on five methods.
+ABC
+DE
+LSHADE
+SBO
+RKO
+unshifted
+shifted
+ratio
+unshifted
+shifted
+ratio
+unshifted
+shifted
+ratio
+unshifted
+shifted
+ratio
+unshifted
+shifted
+ratio
+F1
+5.19E+03
+4.19E+03
+8.08E−01
+3.82E−02
+2.61E−02
+6.83E−01
+1.00E−08
+1.00E−08
+1.00E+00
+5.85E+00
+1.20E+01
+2.05E+00
+1.00E−08
+1.30E−06
+1.30E+02
+F2
+3.08E+03
+5.95E+02
+1.93E−01
+2.79E+00
+3.06E+00
+1.10E+00
+6.37E−06
+9.55E−06
+1.50E+00
+1.25E+15
+6.19E+19
+4.95E+04
+1.00E−08
+2.05E+00
+2.05E+08
+F3
+8.16E+04
+8.04E+04
+9.86E−01
+1.89E+04
+1.64E+04
+8.68E−01
+4.13E−04
+4.16E−04
+1.01E+00
+1.48E+04
+9.08E+04
+6.14E+00
+1.00E−08
+5.03E−02
+5.03E+06
+F4
+5.81E+01
+5.82E+01
+1.00E+00
+1.35E+01
+1.31E+01
+9.70E−01
+5.08E−03
+3.75E−03
+7.38E−01
+1.28E+01
+1.98E+01
+1.55E+00
+1.00E−08
+2.54E+00
+2.54E+08
+F5
+1.40E+03
+1.38E+04
+9.86E+00
+6.09E+02
+5.88E+02
+9.66E−01
+2.07E+01
+2.12E+01
+1.02E+00
+5.63E+02
+1.17E+03
+2.07E+00
+2.53E+01
+3.99E+01
+1.58E+00
+F6
+5.36E+03
+6.12E+03
+1.14E+00
+6.68E+00
+7.10E+00
+1.06E+00
+1.00E−08
+1.00E−08
+1.00E+00
+1.35E+01
+2.42E+01
+1.79E+00
+1.00E−08
+1.74E−06
+1.74E+02
+F7
+1.45E+00
+2.26E+00
+1.55E+00
+8.56E−02
+8.18E−02
+9.56E−01
+3.19E−03
+3.33E−03
+1.04E+00
+4.37E−01
+8.21E−01
+1.88E+00
+9.26E−05
+1.64E−02
+1.76E+02
+F8
+5.07E+03
+5.79E+03
+1.14E+00
+1.43E+04
+1.50E+04
+1.05E+00
+1.64E+02
+1.80E+02
+1.10E+00
+6.99E+03
+5.99E+03
+8.58E−01
+4.19E+03
+4.90E+03
+1.17E+00
+F9
+2.49E+01
+3.03E+01
+1.22E+00
+3.98E+02
+3.98E+02
+1.00E+00
+1.08E+01
+1.10E+01
+1.02E+00
+3.66E+01
+5.16E+01
+1.41E+00
+1.00E−08
+2.99E+01
+2.99E+09
+F10
+8.69E+00
+7.57E+00
+8.70E−01
+1.53E+00
+1.44E+00
+9.41E−01
+2.38E−07
+2.57E−07
+1.08E+00
+4.72E+00
+5.48E+00
+1.16E+00
+1.00E−08
+2.86E+00
+2.86E+08
+F11
+3.16E+02
+3.35E+02
+1.06E+00
+1.07E+00
+1.06E+00
+9.91E−01
+1.00E−08
+1.00E−08
+1.00E+00
+1.06E+00
+1.10E+00
+1.04E+00
+1.00E−08
+1.25E−02
+1.25E+06
+F12
+7.71E+05
+1.09E+06
+1.41E+00
+1.13E+00
+1.27E+00
+1.12E+00
+1.00E−08
+1.00E−08
+1.00E+00
+6.81E+00
+1.29E+01
+1.89E+00
+1.00E−08
+3.37E+00
+3.37E+08
+F13
+9.98E+05
+5.45E+06
+5.46E+00
+6.36E+00
+5.94E+00
+9.34E−01
+1.00E−08
+1.00E−08
+1.00E+00
+1.51E+00
+4.63E+00
+3.07E+00
+4.40E−03
+1.18E−02
+2.68E+00
+geomean
+-
+-
+1.29E+00
+-
+-
+9.66E−01
+-
+-
+1.03E+00
+-
+-
+3.95E+00
+-
+-
+7.36E+04
+ones, LSADE is among the state-of-the ones (as it served as a basis of many of the best methods for recent CEC
+competitions), and the last two (SBO and RKO) are the “new” ones. One can quite easily see that for the first three
+methods (ABC, DE, and LSHADE) the geometric mean of the ratios is roughly 1, meaning that no center-bias was
+detected. For SBO, the situation is a bit more complicated, as on many benchmark functions the ratio is relatively low
+(roughly between 1 and 2), but is very large (almost 5E+04) on F2. This could be a fluke. Fortunately, the nature of the
+geometric mean will suppress some of the individual flukes - the value for SBO is 3.95E+00 (i.e., <1E+01), so we do
+not label it as a method with a center-bias. The same cannot be said about RKO. Here, many of the ratios are extremely
+big (>1E+06), and the value of the geometric mean is 7.36E+04. We can confidently say that RKO incorporates a
+center bias operator.
+An interesting observation can be made regarding the benchmark function F08. For all five methods, the ratio
+between the shifted and unshifted results on F08 is very close to 1. Recall that F08 is the only function in the benchmark
+set that has the optimum quite far away from the center of the feasible set, and its function value at the zero-vector is
+also quite far away from the optimal value. Although it is arguably not surprising that the methods have a ratio around
+1 on this function, it is still valuable to have it confirmed – the function F08 serves as a sanity check in the benchmark
+set.
+3. Results and Discussion
+In this section we report the results of using the methodology described in the previous section on 90 selected
+EC methods. The selected methods, the year of the publication that describes them, and the geometric mean of the
+ratios are shown (in alphabetical order) in Table 3, with the ones with a confirmed center-bias (i.e., values >1E+01)
+highlighted in red. These results are extremely worrying, as more than a half (47 out of the 90) methods have a
+confirmed center-bias. And they become even worse when we take a look at the number of methods with center-bias
+that were proposed recently, as shown in Figure 1.
+Figure 1: Number of papers proposing methods with/without center-bias in time.
+3
+
+14
+without center-bias
+with center-bias
+12
+10
+Number of papers
+8
+6
+4
+2
+9
+7
+6
+30
+YearTable 3: Considered algorithms and results.
+.
+Abbreviation
+Method name
+Year
+Geomean
+Abbreviation
+Method name
+Year
+geomean
+ABC [10]
+Artificial Bee Colony
+2008
+1.29E+00
+HC [15]
+Hill Climbing
+1993
+1.13E+00
+ACOR [16]
+Ant Colony Optimization Continuous
+2008
+7.40E-01
+HGS [17]
+Hunger Games Search
+2021
+3.69E+06
+AEO [18]
+Artificial Ecosystem-based Optimization
+2020
+1.01E+07
+HGSO [19]
+Henry Gas Solubility Optimization
+2019
+8.07E+03
+ALO [20]
+Ant Lion Optimizer
+2015
+1.44E+00
+HHO [21]
+Harris Hawks Optimization
+2019
+1.62E+05
+AO [22]
+Aquila Optimization
+2021
+2.26E+05
+HS [23]
+Harmony Search
+2001
+9.97E-01
+AOA [24]
+Arithmetic Optimization Algorithm
+2021
+1.01E+10
+IWO [25]
+Invasive Weed Optimization
+2006
+1.88E+00
+ArchOA [26]
+Archimedes Optimization Algorithm
+2021
+3.75E+07
+JA [27]
+Jaya Algorithm
+2016
+1.19E+01
+ASO [28]
+Atom Search Optimization
+2019
+8.71E-01
+KMA [29]
+Komodo Mlipir Algorithm
+2022
+1.84E+05
+BA [30]
+Bat-inspired Algorithm
+2010
+1.44E+00
+LCO [31]
+Life Choice-based Optimization
+2020
+8.31E+07
+BBO [32]
+Biogeography-Based Optimization
+2008
+6.43E-01
+MA [33]
+Memetic Algorithm
+1989
+1.68E-03
+BeesA [34]
+Bees Algorithm
+2006
+1.16E+00
+MFO [35]
+Moth-Flame Optimization
+2015
+1.73E-01
+BES [36]
+Bald Eagle Search
+2020
+2.62E+08
+MGO [37]
+Mountain Gazelle Optimizer
+2022
+1.28E+01
+BFO [38]
+Bacterial Foraging Optimization
+2002
+9.66E-01
+MPA [39]
+Marine Predators Algorithm
+2020
+1.06E+02
+BOA [40]
+Butterfly Optimization Algorithm
+2019
+9.57E+05
+MRFO [41]
+Manta Ray Foraging Optimization
+2020
+6.40E+07
+BRO [42]
+Battle Royale Optimization
+2021
+2.59E+09
+MSA [43]
+Moth Search Algorithm
+2018
+8.37E+00
+BSA [44]
+Bird Swarm Algorithm
+2016
+1.09E+01
+MVO [45]
+Multi-Verse Optimizer
+2016
+1.75E+00
+BSO [46]
+Brain Storm Optimization
+2011
+7.85E+00
+NMRA [47]
+Naked Mole-Rat Algorithm
+2019
+5.65E+08
+CA [48]
+Culture Algorithm
+2009
+7.18E-01
+NRO [49]
+Nuclear Reaction Optimization
+2019
+2.30E+06
+CEM [50]
+Cross-Entropy Method
+2005
+1.33E+00
+PFA [51]
+Pathfinder Algorithm
+2019
+3.11E+08
+CGO [52]
+Chaos Game Optimization
+2021
+2.14E+07
+PSO [53]
+Particle Swarm Optimization
+1995
+9.70E-01
+ChOA [54]
+Chimp optimization algorithm
+2020
+3.89E+03
+PSS [55]
+Pareto-like Sequential Sampling
+2021
+1.77E+03
+COA [56]
+Coyote Optimization Algorithm
+2018
+4.00E+06
+QSA [57]
+Queuing Search Algorithm
+2021
+7.91E-01
+CRO [58]
+Coral Reefs Optimization
+2014
+9.69E-01
+RKO [14]
+Runge Kutta Optimizer
+2021
+7.36E+04
+CSA [59]
+Cuckoo Search Algorithm
+2009
+1.10E+00
+SA [60]
+Simulated Annealing
+1987
+8.95E-01
+CSO [61]
+Cat Swarm Optimization
+2006
+9.58E-01
+SARO [62]
+Search And Rescue Optimization
+2019
+2.27E+00
+DE [11]
+Differential Evolution
+1997
+9.66E-01
+SBO [13]
+Satin Bowerbird Optimizer
+2017
+3.95E+00
+DandO [63]
+Dandelion Optimizer
+2022
+3.59E+02
+SCA [64]
+Sine Cosine Algorithm
+2016
+1.18E+04
+DO [65]
+Dragonfly Optimization
+2016
+6.62E+02
+SFO [66]
+SailFish Optimizer
+2019
+2.57E+07
+EFO [67]
+Electromagnetic Field Optimization
+2016
+6.78E-01
+SHO [68]
+Spotted Hyena Optimizer
+2017
+1.31E+00
+EHO [69]
+Elephant Herding Optimization
+2015
+3.99E+03
+SLO [70]
+Sea Lion Optimization Algorithm
+2019
+2.83E+00
+EO [71]
+Equilibrium Optimizer
+2020
+4.65E+03
+SMA [72]
+Slime Mould Algorithm
+2020
+4.54E+06
+EOA [73]
+Earthworm Optimisation Algorithm
+2018
+2.55E+05
+SRSR [74]
+Swarm Robotics Search And Rescue
+2017
+2.03E+00
+EP [75]
+Evolutionary Programming
+1999
+1.43E+00
+SSA [76]
+Sparrow Search Algorithm
+2020
+2.61E+06
+ES [77]
+Evolution Strategies
+2002
+1.14E+00
+SSDO [78]
+Social Ski-Driver Optimization
+2020
+5.40E+08
+FA [79]
+Fireworks Algorithm
+2010
+1.34E+00
+SSO [80]
+Salp Swarm Optimization
+2017
+2.28E+01
+FBIO [81]
+Forensic-Based Investigation Optimization
+2020
+5.07E+06
+SSpiderA [82]
+Social Spider Algorithm
+2015
+1.12E+00
+FFA [83]
+Firefly Algorithm
+2011
+1.18E+00
+STOA [84]
+Sooty Tern Optimization Algorithm
+2019
+6.78E+04
+FOA [85]
+Fruit-fly Optimization Algorithm
+2012
+4.01E+00
+TLO [86]
+Teaching Learning-based Optimization
+2012
+3.19E+04
+FPA [87]
+Flower Pollination Algorithm
+2012
+9.74E-01
+TPO [88]
+Tree Physiology Optimization
+2019
+2.20E+01
+GA [89]
+Genetic Algorithm
+1994
+1.02E+00
+TSA [90]
+Tunicate Swarm Algorithm
+2020
+6.25E+06
+GBO [91]
+Gradient-Based Optimizer
+2020
+7.17E+07
+TWO [92]
+Tug of War Optimization
+2017
+9.69E-01
+GCO [93]
+Germinal Center Optimization
+2018
+1.02E+00
+VCS [94]
+Virus Colony Search
+2016
+2.90E+04
+GOA [95]
+Grasshopper Optimization Algorithm
+2017
+3.39E+00
+WDO [96]
+Wind Driven Optimization
+2013
+4.86E+01
+GSKA [97]
+Gaining Sharing Knowledge-based Algorithm
+2020
+4.51E-01
+WHO [98]
+Wildebeest Herd Optimization
+2019
+8.63E+02
+GWO [99]
+Grey Wolf Optimizer
+2014
+8.89E+05
+WOA [100]
+Whale Optimization Algorithm
+2016
+1.87E+03
+We can find that while the number newly proposed methods that do not have the center-bias problem increased only
+slightly over the last three decades, the number of methods that we have identified as having a center-bias problem
+is growing extremely fast, especially in the last five years. It has gotten so bad that an overwhelming majority of
+newly proposed methods have the center-bias problem. An important thing to remark is that we only considered the
+“baseline” (or original) versions of the methods, and not any of the “improved” or “enhanced” variants that are also
+being published at an ever-increasing rate. If these were considered as well, we suspect that the graph would look even
+worse.
+We can also see that the first method that we have found to incorporate the center-bias was Teaching Learning-
+based Optimization (TLO) in 2012, followed by Wind Driven Optimization (WDO) in 2013, and Grey Wolf Optimizer
+(GWO) in 2014. From these three, TLO and GWO have become extremely influential (gathering thousand of citations)
+and spawned a large number of variants and modifications. Our failure to quickly identify that they are defective is
+one of the root causes of the mess we have to deal with now. Although the defect of the GWO was uncovered in 2019
+[101], GWO is still used in numerical comparisons (even on problems that are susceptible to the center-bias operator).
+Similar defects have been also found for the Salp Swarm Optimization (SSO), Sooty Tern Optimization Algorithm
+(STOA), Tunicate Swarm Algorithm (TSA), Harris Hawks Optimization (HHO), Butterfly Optimization Algorithm
+(BOA), Slime Mould Algorithm (SMA), Gradient-Based Optimizer (GBO), Marine Predators Algorithm (MPA), and
+Komodo Mlipir Algorithm (KMA), all in 2022 [102, 9, 8].
+For the most part, the methods that incorporate a center-bias procedure have been developed by a diverse groups
+of authors (i.e., most authors have only one or two such methods). There is, however, one very notable exception.
+The group of S. Mirjalili, A. H. Gandomi, and A. A. Heidari is collectively responsible for 20 of the 47 methods that
+contain center-bias (and S. Mirjalili is also one of the authors of GWO).
+4
+
+Another interesing point to make is that some of methods that display the worst center-bias properties (i.e., the
+largest values of the geometric mean of the ratios) are the ones which were supposedly based on “mathematical”
+processes – Arithmetic Optimization Algorithm (AOA), Gradient-Based Optimizer (GBO), Runge Kutta Optimizer
+(RKO), and Sine Cosine Algorithm (SCA). The following are the first few sentences from the abstract of the paper
+describing RKO [14]:
+“The optimization field suffers from the metaphor-based “pseudo-novel” or “fancy” optimizers. Most of these clich´e
+methods mimic animals’ searching trends and possess a small contribution to the optimization process itself. Most of
+these clich´e methods suffer from the locally efficient performance, biased verification methods on easy problems, and
+high similarity between their components’ interactions. This study attempts to go beyond the traps of metaphors and
+introduce a novel metaphor-free population-based optimization method based on the mathematical foundations and
+ideas of the Runge Kutta (RK) method widely well-known in mathematics.”
+The irony is rich.
+4. Conclusion
+The center-bias problem is right now one of the major issues plaguing the field of evolutionary computation. In
+this paper, we have described a simple procedure for identifying methods with center-bias and used it to investigate 90
+methods that were proposed in the last three decades. We have found that 47 of the 90 methods utilize center-bias. We
+have also shown that the utilization of center-bias is a relatively new phenomenon, with first instances from 2012-2014.
+However, the number of methods that use it grew extremely fast in the last five years.
+We should note that there is an additional problem that plagues the field right now, which is the equivalence of
+some of the methods that is hidden under a metaphore-rich jargon. Some of the methods that we have identified as not
+having a center bias, such as Harmony Search (HS), Cockoo Search Algorith (CSA), Firefly Algorithm (FA), Moth-
+Flame Optimization (MFO), Ant Lion Optimizer (ALO) should also not be used, as they have been found to be either
+extremely similar (or identical) to other methods [1, 4, 103].
+Further utilization, development and improvement of the methods that contain a center-bias is an exercise in fu-
+tility, as by their very nature they cannot be considered as efficient algorithms. Enough computational and human
+resources were already wasted in writing, testing, comparing, and reviewing these methods. The field of evolutionary
+computation needs a spring cleaning. The sooner the better.
+Acknowledgment
+This work was supported by IGA BUT: FSI-S-20-6538.
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf,len=997
+page_content='The Evolutionary Computation Methods No One Should Use  Jakub K˚udela  Institute of Automation and Computer Science, Brno University of Technology  Jakub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='Kudela@vutbr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='cz  Abstract  The center-bias (or zero-bias) operator has recently been identified as one of the problems plaguing the benchmarking  of evolutionary computation methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content=' This operator lets the methods that utilize it easily optimize functions that have  their respective optima in the center of the feasible set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content=' In this paper, we describe a simple procedure that can be  used to identify methods that incorporate a center-bias operator and use it to investigate 90 evolutionary computation  methods that were published between 1987 and 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content=' We show that more than half (47 out of the 90) of the considered  methods have the center-bias problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content=' We also show that the center-bias is a relatively new phenomenon (with the first  identified method being from 2012), but its inclusion has become extremely prevalent in the last few years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content=' Lastly, we  briefly discuss the possible root causes of this issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='  Keywords: Evolutionary Computation, Benchmarking, Metaheuristics, Center-bias, Zero-bias  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content=' Introduction  Imagine the following situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content=' Encountering a challenging optimization task, you decide to find the most recently  developed algorithm for optimization published in some of the most prestigious journals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content=' The analysis of the method  performed on several standard benchmarks clearly shows that it is superior to all the other old methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content=' The paper also  contains a link to a repository with the code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content=' So, you give it a try.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content=' And it fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content=' The best results it provides are hardly  better (or worse) than the ones you got from a simple implementation of a method that is more than two decades old.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='  Maybe the problem you tried to solve is too challenging?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content=' Perhaps a bit of hyperparameter optimization could help the  method perform as advertised?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content=' Or, maybe the method is not as good as it presented itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='  Through inspiration from natural behaviors, the field of evolutionary computation (EC) produced over its long  history a great number of important metaheuristic algorithms, such as Evolutionary Strategy, Genetic Algorithms,  Particle Swarm Optimization, or Differential Evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content=' Such methods found applications in complex systems where  the use of exact algorithms was either inadequate or computationally too prohibitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content=' However, over the past few years  we have witnessed an explosion of ”novel” methods that are based on natural/evolutionary principles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content=' The bestiary  of EC1, which tries to catalog of a portion of these nature-based methods, now contains over 250 methods that claim  their inspiration in natural processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content=' And new methods are emerging at an ever-increasing rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content=' It is also becoming  clearer that there is more creativity being spent at naming these ”novel” methods, than in making sure they contain  anything new computation-wise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content=' After many of these methods have been found to conceal their lack of novelty behind  a methaphor-rich jargon [1, 2, 3, 4, 5], a call was made from within the EC community [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content=' In the letter, the collective  of authors and signatories identified four main issues with the high-volume inflow of new methods: useless metaphors,  lack of novelty, poor experimental validation and comparison, and publishing these methods in off-topic journals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='  In this text, we will focus on the poor experimental validation of some of the EC methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content=' Most of the reasoning  about the viability of metaheuristics is done through benchmarking [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content=' If a new method performs well on a universally  accepted set of benchmark problems, it is likely to be seen as valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content=' There have been several benchmark functions/sets  proposed over the years, but the most widely recognized ones came from special sessions (competitions) on black-box  optimization at two conferences: the IEEE Congress on Evolutionary Computation (CEC), and the Genetic and Evo-  lutionary Computation Conference (GECCO), where the Black-Box Optimization Benchmarking (BBOB) workshop  was held.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='  There is, however, another quite widely used benchmark set that contains some of the most well-known functions  such as Griewank, Ackley, Rastrigin, Rosenbrock, and Schwefel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content=' It was recently uncovered [8] that this set contains a  serious design flaw, as a large portion of the functions in the set have their respective optimum at a zero vector (or in the  1Campelo, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content=', Aranha, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content=' Evolutionary computation bestiary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content=' https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='com/fcampelo/EC-Bestiary  Preprint submitted to .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='  January 6, 2023  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='01984v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='NE]  5 Jan 2023  Table 1: The 13 benchmark functions, dimension 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content=' U -– unimodal, M — multimodal, S — separable, N — non-separable, f ∗ – the optimal  function value, f(0) – function value at the zero vector, x∗ – optimal solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='  ID  name  type  range  f ∗  f(0)  x∗  F01  Sphere  U, S  [-100,100]  0  0  [0,0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
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+page_content=']  F04  Schwefel 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
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+page_content='26  M, S  [-500,500]  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='25E+04  0  [420.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='9, 420.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='9,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
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+page_content=']  F09  Rastrigin  M, S  [-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
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+page_content=']  F13  Penalized2  M, S  [-50,50]  0  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='00E+00  [1,1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
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+page_content=']  centre of the feasible set).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content=' This would be fine, if it were not for the methods that utilize this flaw to appear competitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='  These methods incorporate a “check-the-middle” routine or have a centre-bias (or zero-bias) operator that draws them  towards the center of the feasible set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content=' One would expect that such methods do not get published very often, are easily  spotted, or at the very least do not appear in high-profile journals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='  In this paper, we describe a simple methodology that we use to uncover whether or not a given evolutionary  computation method utilizes a center-bias operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content=' We then investigate 90 evolutionary computation methods from  the mealpy library2 and Mathworks code repositories3 for the inclusion of the center-bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content=' Methodology  We utilize the same methodology that was used to uncover the center-bias problem in [9] and [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content=' The 13 benchmark  function used for our test (and optimization ranges) are shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content=' One can easily see that many of these  functions, apart from F08, have the corresponding optima either at the zero vector or very close to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content=' The problem F08  is quite different from the rest, as its optimum is far away from the center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='  For the evaluation we set the dimension of the problems to 30 and allow for at most 50,000 function evaluations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='  We also chose a simple performance measure - the mean error (as the difference between the optimal function value  and best function value found) over 20 independent runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content=' Here, we also treat any value smaller than 1e–08 as identical  to 1e–08, as the problem is essentially solved and additional precision is not needed (we could treat it as a 0 as well,  but we will shortly use fractions of these numbers, which would bring unwanted hassle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content=' We refer to the results of the  computation as the “unshifted” ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content=' Afterwards, we introduce a shift operation, that “moves” the benchmark function  by a predetermined vector s, meaning that function f(x) becomes f(x+ s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content=' One expects that a “small” value of s should  not result in a large deviation in the behaviour of the optimization method, as the two problems are very similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content=' We  chose the shift vector as 10% of the range - e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content=', for F01, s = [20, 20, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content=' ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content=' We use the same computational framework  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content=', dimension 30, at maximum 50,000 function evaluations, and 20 independent runs) and refer to the results of these  computations as the “shifted” ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='  What we are interested in is the “ratio” between the “shifted” and “unshifted” results for the individual benchmark  functions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content=', how many times are the results on the shifted problem worse than on the unshifted one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content=' For the methods  that do not incorporate a center-bias, one expects this number to be close to 1 (as the ushifted and shifted problems  are similar), while for the methods that include a center-bias, this ratio should be much bigger than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content=' Naturally, the  value of this ratio will fluctuate depending on the given benchmark function, as well as on the number of independent  runs of the algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content=' As a simple indicator of the center-bias, we look at the geometric mean of the ratios for the  different benchmarks – if this value is bigger than 1E+01 (meaning that the method performs roughly at least on order  of magnitude better on unshifted problems), we take it as a confirmation of the presence of the center-bias operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='  A small example of this computation is shown in Table 2, where we investigate five EC methods – Artificial  Bee Colony (ABC) [10], Differential Evolution (DE) [11], LSHADE [12], Satin Bowerbird Optimizer (SBO) [13],  and Runge Kutta Optimizer (RKO) [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content=' The first two methods (ABC and DE) can be thought of as the “standard”  2N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content=' Thieu, “A collection of the state-of-the-art meta-heuristics algorithms in python: Mealpy,” Available: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='5281/  zenodo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='3711948  3https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='mathworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='com/  2  Table 2: The results of proposed methodology demonstrated on five methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='  ABC  DE  LSHADE  SBO  RKO  unshifted  shifted  ratio  unshifted  shifted  ratio  unshifted  shifted  ratio  unshifted  shifted  ratio  unshifted  shifted  ratio  F1  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='19E+03  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
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+page_content='00E+00  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
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+page_content='30E−06  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='30E+02  F2  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='08E+03  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
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+page_content='68E+00  geomean 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='29E+00 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='66E−01 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='03E+00 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='95E+00 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='36E+04  ones, LSADE is among the state-of-the ones (as it served as a basis of many of the best methods for recent CEC  competitions), and the last two (SBO and RKO) are the “new” ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content=' One can quite easily see that for the first three  methods (ABC, DE, and LSHADE) the geometric mean of the ratios is roughly 1, meaning that no center-bias was  detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content=' For SBO, the situation is a bit more complicated, as on many benchmark functions the ratio is relatively low  (roughly between 1 and 2), but is very large (almost 5E+04) on F2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content=' This could be a fluke.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content=' Fortunately, the nature of the  geometric mean will suppress some of the individual flukes - the value for SBO is 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='95E+00 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content=', <1E+01), so we do  not label it as a method with a center-bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content=' The same cannot be said about RKO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content=' Here, many of the ratios are extremely  big (>1E+06), and the value of the geometric mean is 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='36E+04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content=' We can confidently say that RKO incorporates a  center bias operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='  An interesting observation can be made regarding the benchmark function F08.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content=' For all five methods, the ratio  between the shifted and unshifted results on F08 is very close to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content=' Recall that F08 is the only function in the benchmark  set that has the optimum quite far away from the center of the feasible set, and its function value at the zero-vector is  also quite far away from the optimal value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content=' Although it is arguably not surprising that the methods have a ratio around  1 on this function, it is still valuable to have it confirmed – the function F08 serves as a sanity check in the benchmark  set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content=' Results and Discussion  In this section we report the results of using the methodology described in the previous section on 90 selected  EC methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content=' The selected methods, the year of the publication that describes them, and the geometric mean of the  ratios are shown (in alphabetical order) in Table 3, with the ones with a confirmed center-bias (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content=', values >1E+01)  highlighted in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content=' These results are extremely worrying, as more than a half (47 out of the 90) methods have a  confirmed center-bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content=' And they become even worse when we take a look at the number of methods with center-bias  that were proposed recently, as shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='  Figure 1: Number of papers proposing methods with/without center-bias in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='  3  14  without center-bias  with center-bias  12  10  Number of papers  8  6  4  2  9  7  6  30  YearTable 3: Considered algorithms and results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='  Abbreviation  Method name  Year  Geomean  Abbreviation  Method name  Year  geomean  ABC [10]  Artificial Bee Colony  2008  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='29E+00  HC [15]  Hill Climbing  1993  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='13E+00  ACOR [16]  Ant Colony Optimization Continuous  2008  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='40E-01  HGS [17]  Hunger Games Search  2021  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='69E+06  AEO [18]  Artificial Ecosystem-based Optimization  2020  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='01E+07  HGSO [19]  Henry Gas Solubility Optimization  2019  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='07E+03  ALO [20]  Ant Lion Optimizer  2015  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='44E+00  HHO [21]  Harris Hawks Optimization  2019  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='62E+05  AO [22]  Aquila Optimization  2021  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='26E+05  HS [23]  Harmony Search  2001  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='97E-01  AOA [24]  Arithmetic Optimization Algorithm  2021  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='01E+10  IWO [25]  Invasive Weed Optimization  2006  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='88E+00  ArchOA [26]  Archimedes Optimization Algorithm  2021  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='75E+07  JA [27]  Jaya Algorithm  2016  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='19E+01  ASO [28]  Atom Search Optimization  2019  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='71E-01  KMA [29]  Komodo Mlipir Algorithm  2022  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='84E+05  BA [30]  Bat-inspired Algorithm  2010  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='44E+00  LCO [31]  Life Choice-based Optimization  2020  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='31E+07  BBO [32]  Biogeography-Based Optimization  2008  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='43E-01  MA [33]  Memetic Algorithm  1989  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='68E-03  BeesA [34]  Bees Algorithm  2006  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='16E+00  MFO [35]  Moth-Flame Optimization  2015  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
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+page_content='27E+00  DE [11]  Differential Evolution  1997  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='66E-01  SBO [13]  Satin Bowerbird Optimizer  2017  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='95E+00  DandO [63]  Dandelion Optimizer  2022  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
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+page_content='18E+04  DO [65]  Dragonfly Optimization  2016  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='62E+02  SFO [66]  SailFish Optimizer  2019  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='57E+07  EFO [67]  Electromagnetic Field Optimization  2016  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='78E-01  SHO [68]  Spotted Hyena Optimizer  2017  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='31E+00  EHO [69]  Elephant Herding Optimization  2015  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
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+page_content='83E+00  EO [71]  Equilibrium Optimizer  2020  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='65E+03  SMA [72]  Slime Mould Algorithm  2020  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='54E+06  EOA [73]  Earthworm Optimisation Algorithm  2018  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='55E+05  SRSR [74]  Swarm Robotics Search And Rescue  2017  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='03E+00  EP [75]  Evolutionary Programming  1999  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='43E+00  SSA [76]  Sparrow Search Algorithm  2020  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='61E+06  ES [77]  Evolution Strategies  2002  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='14E+00  SSDO [78]  Social Ski-Driver Optimization  2020  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='40E+08  FA [79]  Fireworks Algorithm  2010  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='34E+00  SSO [80]  Salp Swarm Optimization  2017  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='28E+01  FBIO [81]  Forensic-Based Investigation Optimization  2020  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='07E+06  SSpiderA [82]  Social Spider Algorithm  2015  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='12E+00  FFA [83]  Firefly Algorithm  2011  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='18E+00  STOA [84]  Sooty Tern Optimization Algorithm  2019  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='78E+04  FOA [85]  Fruit-fly Optimization Algorithm  2012  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='01E+00  TLO [86]  Teaching Learning-based Optimization  2012  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='19E+04  FPA [87]  Flower Pollination Algorithm  2012  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='74E-01  TPO [88]  Tree Physiology Optimization  2019  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='20E+01  GA [89]  Genetic Algorithm  1994  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='02E+00  TSA [90]  Tunicate Swarm Algorithm  2020  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='25E+06  GBO [91]  Gradient-Based Optimizer  2020  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='17E+07  TWO [92]  Tug of War Optimization  2017  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='69E-01  GCO [93]  Germinal Center Optimization  2018  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='02E+00  VCS [94]  Virus Colony Search  2016  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='90E+04  GOA [95]  Grasshopper Optimization Algorithm  2017  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='39E+00  WDO [96]  Wind Driven Optimization  2013  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='86E+01  GSKA [97]  Gaining Sharing Knowledge-based Algorithm  2020  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='51E-01  WHO [98]  Wildebeest Herd Optimization  2019  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='63E+02  GWO [99]  Grey Wolf Optimizer  2014  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='89E+05  WOA [100]  Whale Optimization Algorithm  2016  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='87E+03  We can find that while the number newly proposed methods that do not have the center-bias problem increased only  slightly over the last three decades, the number of methods that we have identified as having a center-bias problem  is growing extremely fast, especially in the last five years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content=' It has gotten so bad that an overwhelming majority of  newly proposed methods have the center-bias problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content=' An important thing to remark is that we only considered the  “baseline” (or original) versions of the methods, and not any of the “improved” or “enhanced” variants that are also  being published at an ever-increasing rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content=' If these were considered as well, we suspect that the graph would look even  worse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='  We can also see that the first method that we have found to incorporate the center-bias was Teaching Learning-  based Optimization (TLO) in 2012, followed by Wind Driven Optimization (WDO) in 2013, and Grey Wolf Optimizer  (GWO) in 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content=' From these three, TLO and GWO have become extremely influential (gathering thousand of citations)  and spawned a large number of variants and modifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content=' Our failure to quickly identify that they are defective is  one of the root causes of the mess we have to deal with now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content=' Although the defect of the GWO was uncovered in 2019  [101], GWO is still used in numerical comparisons (even on problems that are susceptible to the center-bias operator).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='  Similar defects have been also found for the Salp Swarm Optimization (SSO), Sooty Tern Optimization Algorithm  (STOA), Tunicate Swarm Algorithm (TSA), Harris Hawks Optimization (HHO), Butterfly Optimization Algorithm  (BOA), Slime Mould Algorithm (SMA), Gradient-Based Optimizer (GBO), Marine Predators Algorithm (MPA), and  Komodo Mlipir Algorithm (KMA), all in 2022 [102, 9, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='  For the most part, the methods that incorporate a center-bias procedure have been developed by a diverse groups  of authors (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content=', most authors have only one or two such methods).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content=' There is, however, one very notable exception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='  The group of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content=' Mirjalili, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content=' Gandomi, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content=' Heidari is collectively responsible for 20 of the 47 methods that  contain center-bias (and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content=' Mirjalili is also one of the authors of GWO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='  4  Another interesing point to make is that some of methods that display the worst center-bias properties (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content=', the  largest values of the geometric mean of the ratios) are the ones which were supposedly based on “mathematical”  processes – Arithmetic Optimization Algorithm (AOA), Gradient-Based Optimizer (GBO), Runge Kutta Optimizer  (RKO), and Sine Cosine Algorithm (SCA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content=' The following are the first few sentences from the abstract of the paper  describing RKO [14]:  “The optimization field suffers from the metaphor-based “pseudo-novel” or “fancy” optimizers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content=' Most of these clich´e  methods mimic animals’ searching trends and possess a small contribution to the optimization process itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content=' Most of  these clich´e methods suffer from the locally efficient performance, biased verification methods on easy problems, and  high similarity between their components’ interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content=' This study attempts to go beyond the traps of metaphors and  introduce a novel metaphor-free population-based optimization method based on the mathematical foundations and  ideas of the Runge Kutta (RK) method widely well-known in mathematics.”  The irony is rich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content=' Conclusion  The center-bias problem is right now one of the major issues plaguing the field of evolutionary computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content=' In  this paper, we have described a simple procedure for identifying methods with center-bias and used it to investigate 90  methods that were proposed in the last three decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content=' We have found that 47 of the 90 methods utilize center-bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content=' We  have also shown that the utilization of center-bias is a relatively new phenomenon, with first instances from 2012-2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='  However, the number of methods that use it grew extremely fast in the last five years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='  We should note that there is an additional problem that plagues the field right now, which is the equivalence of  some of the methods that is hidden under a metaphore-rich jargon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content=' Some of the methods that we have identified as not  having a center bias, such as Harmony Search (HS), Cockoo Search Algorith (CSA), Firefly Algorithm (FA), Moth-  Flame Optimization (MFO), Ant Lion Optimizer (ALO) should also not be used, as they have been found to be either  extremely similar (or identical) to other methods [1, 4, 103].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='  Further utilization, development and improvement of the methods that contain a center-bias is an exercise in fu-  tility, as by their very nature they cannot be considered as efficient algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content=' Enough computational and human  resources were already wasted in writing, testing, comparing, and reviewing these methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content=' The field of evolutionary  computation needs a spring cleaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content=' The sooner the better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
+page_content='  Acknowledgment  This work was supported by IGA BUT: FSI-S-20-6538.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9A0T4oBgHgl3EQfB_-i/content/2301.01984v1.pdf'}
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diff --git a/HtFKT4oBgHgl3EQfdC4X/content/tmp_files/2301.11818v1.pdf.txt b/HtFKT4oBgHgl3EQfdC4X/content/tmp_files/2301.11818v1.pdf.txt
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+arXiv:2301.11818v1  [physics.flu-dyn]  27 Jan 2023
+Predicting extreme events in a data-driven model of turbulent shear flow using an
+atlas of charts
+Andrew J. Fox and Michael D. Graham∗
+Department of Chemical and Biological Engineering,
+University of Wisconsin-Madison,
+Madison, WI 53706, USA
+(Dated: January 30, 2023)
+Dynamical systems with extreme events are difficult to capture with data-driven modeling, due to
+the relative scarcity of data within extreme events compared to the typical dynamics of the system,
+and the strong dependence of the long-time occurrence of extreme events on short-time conditions.
+A recently developed technique [Floryan, D. & Graham, M. D. Data-driven discovery of intrinsic
+dynamics. Nat Mach Intell 4, 1113–1120 (2022)], here denoted as Charts and Atlases for Nonlinear
+Data-Driven Dynamics on Manifolds, or CANDyMan, overcomes these difficulties by decomposing
+the time series into separate charts based on data similarity, learning dynamical models on each
+chart via individual time-mapping neural networks, then stitching the charts together to create a
+single atlas to yield a global dynamical model. We apply CANDyMan to a nine-dimensional model
+of turbulent shear flow between infinite parallel free-slip walls under a sinusoidal body force [Moehlis,
+J., Faisst, H. & Eckhardt, B. A low-dimensional model for turbulent shear flows. New J Phys 6,
+56 (2004)], which undergoes extreme events in the form of intermittent quasi-laminarization and
+long-time full laminarization.
+We demonstrate that the CANDyMan method allows the trained
+dynamical models to more accurately forecast the evolution of the model coefficients, reducing the
+error in the predictions as the model evolves forward in time. The technique exhibits more accurate
+predictions of extreme events, capturing the frequency of quasi-laminarization events and predicting
+the time until full laminarization more accurately than a single neural network.
+I.
+INTRODUCTION
+Real world dynamical systems often produce unusual
+behaviors in the form extreme events.
+These extreme
+events are characterized by a dissimilarity to the typical
+dynamics of the system, usually greater in scope or scale,
+that occur relatively infrequently compared to the typ-
+ical dynamics. Common examples include rogue waves
+in the ocean [1], extreme weather patterns such as hur-
+ricanes and tornadoes [2, 3], and intermittency in turbu-
+lent flows [4]. While extreme events are a consequence of
+the same dynamical system that governs the non-extreme
+state, they are often difficult to forecast using data-driven
+modeling. The relative scarcity of data within extreme
+events both limits the overall observations of the extreme
+events on which to train the model and reduces the rel-
+ative influence of extreme event behavior on data-driven
+model training. Thus, creating a data-driven model that
+can accurately capture extreme events remains a active
+challenge.
+Recent studies have proposed various techniques for
+analyzing and forecasting the occurrence of extreme
+events.
+Guth and Sapsis [5] developed a probabilistic
+framework for the use of indicator observables as predic-
+tors of the extreme events. Ragone and Bouchet [6] sup-
+plemented climate model simulations with a rare event
+algorithm to examine and more accurately capture the
+increasing frequency of extreme heatwaves in Europe.
+Blanchard et al. [7] built a machine learning framework to
+∗ mdgraham@wisc.edu
+correct a biased climate model to produce better forecasts
+of extreme events. Mendez and Farazmand [8] applied
+probabilistic models toward predicting indirect spread-
+ing of wildfires by wind to improve forecasts of new wild-
+fire locations. Gom´e et al. [9] applied a rare-event algo-
+rithm to analyze the transition between states in turbu-
+lent pressure-driven flow and more efficiently predict pas-
+sage time between states. While these studies improved
+predictions of extreme events, they primarily corrected
+and supplemented the forecasts of existing models; we
+will instead aim to develop an improved model.
+One attractive test case of a dynamical system with ex-
+treme events is the nine-dimensional model for turbulent
+flow developed by Moehlist, Faisst, and Eckhardt (MFE)
+[10]. The MFE model, an extension of a model by Waleffe
+[11], governs the evolution of nine amplitudes of spatial
+Fourier modes describing a turbulent shear flow between
+walls. These nine modes provide a minimal description of
+the mechanisms for self-sustenance in turbulence, allow-
+ing the resulting flow field to display realistic turbulent
+dynamics. In particular, the model displays features con-
+sistent with turbulence in the transition region, namely
+long periods of turbulent behavior with infrequent quasi-
+laminarization events (also called quiescent [12] or hiber-
+nating [13] intervals) and ultimately full laminarization
+[12–14]. These quasi- and complete relaminarizations will
+be the extreme events considered in the present work, in
+which we use time series from the MFE model as “data”
+with which to develop a data-driven model.
+In recent years, several attempts have been made to
+reproduce the dynamics of the MFE model (and other
+flow systems) through data-driven techniques based on
+
+2
+neural networks (NNs). Neural networks are a powerful
+data-driven modeling technique that has been shown to
+accurately recreate the dynamics of systems such as the
+viscous Burgers equation[15], the Kuramoto-Sivashinksy
+equation[16, 17], and Kolmogorov flow[18].
+Srinivasan
+et al. [19] developed both feedforward neural networks
+(FNNs) and long short term memory (LSTM) networks
+to recreate the MFE model as discrete-time maps. While
+the FNNs were unable to reproduce the model, LSTMs
+were able to accurately reconstruct long-time behaviors
+of the full-field velocity statistics. This problem was re-
+visited by Eivazi et al. [20], where the reconstruction via
+a LSTM network was compared to predictions generated
+via a Koopman-based framework with nonlinear forcing.
+Their work demonstrated that the Koopman framework
+could reproduce short-time and long-time statistics as
+well or better than the LSTM networks. Pandey et al.
+[21] introduced the use of reservoir computing in the form
+of an echo state network (ESN), to reproduce the MFE
+model as a discrete-time map, and provided comparisons
+to both a FNN and a LSTM network. The LSTM net-
+work and the ESN were shown to perform similarly, with
+both adequately capturing the full-field velocity statis-
+tics, while again the FNN was shown to perform appre-
+ciably worse. Racca and Magri [22] specifically examined
+the ability of an ESN to forecast the occurrence of an
+extreme event within a future time window. They deter-
+mined that their data-driven model could accurately fore-
+cast extreme event episodes far into the future without
+incorrectly predicting false quasi-laminarization events.
+Pershin et al. [23] assessed the ability of an ESN to fore-
+cast time until full laminarization.
+They showed that
+their model could adequately reproduce the lifetime dis-
+tribution of the MFE data, correctly predicting the prob-
+ability of an arbitrary MFE time series remaining in the
+turbulent state some time in the future. These studies
+only successfully modeled the MFE equations through
+the use of non-Markovian models, which forecast the fu-
+ture state through input of the current and past states.
+As the MFE model is itself Markovian, we will instead
+endeavor to model the MFE data with a Markovian dy-
+namical system.
+Specifically, we will use a recently developed method
+that will be denoted here as Charts and Atlases for Non-
+linear Data-Driven Dynamics on Manifolds (CANDy-
+Man) [24, 25].
+CANDyMan operates by decomposing
+the data distribution in state space into separate regions
+called charts with a clustering algorithm, learning lo-
+cal dynamical models in each chart using FNNs, then
+stitching together the charts to create a single atlas con-
+taining the global dynamical model. This technique has
+been previously applied to dimensional reduction prob-
+lems, accurately learning reduced order dynamical mod-
+els whose dimension is equal to the intrinsic dimension-
+ality of the system. The use of multiple charts allows
+low-dimensional manifolds embedded in high dimensional
+space to be broken down into locally low dimensional
+structures, capturing the dynamics of a system with the
+minimal number of dimensions, in a way that single chart
+methods cannot. Here, we do not perform dimension re-
+duction, but rather utilize the clustering of data to break
+down the dynamical system into separate regions rep-
+resenting extreme and non-extreme states. By learning
+the dynamics in the extreme region separately and in-
+dependently from the non-extreme regions, CANDyMan
+inherently overcomes the imbalance of extreme vs non-
+extreme information and thus the limited influence of
+extreme events in data driven model training.
+Here, we will use CANDyMan to reconstruct the dy-
+namics of the MFE model. A data set containing time
+series of the MFE amplitudes will be decomposed using
+k-means clustering into atlases containing between one
+and six charts. We will train deep neural networks to
+reconstruct the time evolution of the MFE amplitudes
+within each of the charts, then stitch them together to
+create six global models. To assess the accuracy of the
+models, we will first consider their ability to reconstruct
+the turbulent flow field. Next, we will analyze their per-
+formance in reproducing short-time and long-time statis-
+tics. Finally, we will assess the extreme event forecasting
+of the data-driven models by determining the statistical
+accuracy of forecasting extreme event occurrences and
+comparing predicted laminarization lifetime distribution
+to the true data.
+II.
+FORMULATION
+The MFE model is a severely truncated Fourier
+Galerkin approximation to the Navier-Stokes equations
+(NSE) for flow between two free-slip walls and driven
+by a spatially sinusoidal body force. The flow is com-
+posed of nine spatial Fourier modes ui(x), describing
+the basic profile, streaks, and vortices, as well as inter-
+actions between them. The velocity field at position x
+and time t is given by a superposition of the nine modes
+as u(x, t) =
+9�
+i=1
+ai(t)ui(x). The mode amplitudes ai(t)
+satisfy a system of nine ordinary differential equations
+(ODEs), generated through Galerkin projection, whose
+explicit form is given in Moehlis et al. [10]. Our study
+considers a domain of size Lx × Ly × Lz, with infinite,
+parallel walls at y = −Ly/2 and y = Ly/2 and peri-
+odic boundaries x = 0, x = Lx, z = 0, and z = Lz; x,
+y, and z are the streamwise, wall-normal, and spanwise
+coordinates, respectively. The domain size of Lx = 4π,
+Ly = 2, Lz = 2π was used, with a channel Reynolds
+number of 400; these parameters produce turbulent be-
+havior of suitable length for data-driven model develop-
+ment [19].
+As training data, we generated 100 unique time se-
+ries from a fourth-order Runge-Kutta integration of the
+MFE equation. Each time series encompasses the tran-
+sient turbulent state, consisting of turbulent intervals
+interspersed with quasi-laminarization events, with ter-
+minal laminarization occurring at long time.
+We will
+
+3
+FIG. 1. Evolution of three amplitudes, a1, a6, a9 and corre-
+sponding kinetic energy from one time series of the MFE data
+set.
+often characterize the flow using the total kinetic en-
+ergy (KE), given by KE
+=
+1
+2
+�9
+i=1 a2
+i .
+Therefore,
+the turbulent state is low energy while the laminar
+is high energy.
+Every time series collapses to the
+known laminar fixed point ai = δi1.
+To generate the
+time series, initial conditions of eight of the amplitudes
+were given as follows:
+(a1, a2, a3, a5, a6, a7, a8, a9) =
+(1, 0.07066, −0.07076, 0, 0, 0, 0, 0). The initial value of a4
+was randomly generated in the range [−0.1, 0.1]. These
+initial conditions were previously demonstrated to gen-
+erate chaotic dynamical data with quasi-laminarization
+events [19]. Amplitudes and KE from a randomly chosen
+time series are shown in Fig. 1. We will report all results
+in units ˜t = t/τL, where τL is the Lyapunov time for the
+system; in the original nondimensionalization τL ≈ 41
+[26].
+In this study, we examined the behavior of multi-chart
+models with between two and six charts, as well as a stan-
+dard approach with one global model – the “one-chart”
+limit of CANDyMan. The dynamical system data is first
+clustered into k charts via k-means clustering, which par-
+titions a data set into k clusters, minimizing the within-
+cluster variance [27, 28].
+Other clustering techniques,
+such as k-nearest neighbors [29] or single-linkage cluster-
+ing [30], could be used, provided the clustering technique
+produces charts that encompass contiguous regions of the
+state space. The clusters are then expanded so that they
+overlap, by locating the kNN nearest neighbors to each
+data point in a cluster by Euclidean distance and adding
+these to the original cluster. This creates an overlap re-
+gion between neighboring clusters, providing transition
+regions in which the dynamics are described by multiple
+charts and allowing for the movement into and out of the
+region to be handled by the separate local models.
+Then, in each augmented chart, we generated discrete-
+time models of the form a(j)(t + τ) = F (j)(a(j)(t); θ(j)),
+where a(j)(t) ∈ R9 is the representation of the state in
+chart j, the discrete time step is τ = 0.5, and F (j) is the
+corresponding discrete-time map, which takes the form
+of a FNN. The quantities θ(j) are the neural network
+weights for F (j), which are learned from the data using a
+standard stochastic gradient descent method and trained
+to minimize the loss function L(j) = ⟨||a(j)(t)−˜a(j)(t)||2⟩,
+where ⟨·⟩ is the average over the training data. To en-
+sure that the comparison between different numbers of
+charts was standardized, each global model contains the
+same number of total neurons, NT = 1800; a system of
+k charts would then use NN = NT /k neurons in each
+local model, each containing four fully-connected hidden
+layers of NN/6, NN/3, NN/3, and NN/6 of the total
+number of local neurons, respectively. Each neural net-
+work was trained using a learning rate scheduler with an
+initial learning rate of 0.01, decaying at a rate of 0.9 every
+2000 steps. Each model was then trained for 100 epochs,
+which was found to accurately reproduce the training
+data while avoiding overfitting.
+III.
+RESULTS AND DISCUSSION
+A.
+Distribution of data into clusters
+Insight into the number of charts necessary for prop-
+erly reconstructing the MFE data can be gained by ob-
+serving the clustering of the training data set.
+Fig. 2
+shows how one trajectory from the data set is partitioned
+when we use different numbers of charts, in terms of (a-
+f) the time series of KE and (g-l) state space projec-
+tions onto amplitudes a1, a6, a9.
+With two charts, the
+data partitions into one cluster covering the low-energy
+(turbulent) non-extreme states and the second contain-
+ing the high-energy extreme (quasi-laminar, laminariz-
+ing) states. When three charts were used, the clusters
+are further segmented, with one covering the low-energy
+turbulent state, the second primarily consisting of the
+transition into quasi-laminarization and laminarization
+events, and the third consisting mainly of the high en-
+ergy components of these events.
+Clustering into four
+charts breaks down the low-energy region into two sepa-
+rate clusters that maintain relatively distinct. When the
+data was clustered into five or six charts, the distinction
+between the charts in the low-energy turbulent regime
+decreased and the charts containing the turbulent states
+were described by increasingly similar centroids.
+B.
+Trajectory predictions and time-averaged
+statistics
+The performance of the data-driven models was eval-
+uated on their ability to reconstruct the evolution of
+the MFE model amplitudes.
+Two test data sets were
+generated for comparison between the MFE model and
+the single- and multi-chart data-driven models. For tra-
+jectory predictions, 100 trajectories of MFE amplitudes
+were generated from randomized initial conditions and
+time-integrated for 10 Lyapunov times, with the initial
+conditions separately evolved forward using the gener-
+ated data-driven models for the same length of time; this
+will henceforth be denoted as data set A. The purpose of
+this data set is to determine the short-time precision of
+
+1.0
+0.5
+ai
+a6
+ag
+KE
+0.4
+0.5
+0.3E
+e
+K
+0.2
+0.0
+0.1
+0.5
+0.0
+0
+50
+100
+150
+200
+250
+七4
+g)
+h)
+i)
+j)
+k)
+l)
+FIG. 2. Clustering of a randomly selected trajectory of kinetic
+energy (a-f) and the projection of the clustering of the first,
+sixth, and ninth MFE amplitude (g-l) for one to six charts,
+color coded by cluster.
+the predictions generated by the single- and multi-chart
+models, regardless of any observed or predicted laminar-
+ization. For time-averaged statistics, 100 trajectories of
+MFE amplitudes were generated from randomized initial
+conditions and time-integrated for 100 Lyapunov times
+or until a laminarization event occurred, with the initial
+conditions separately evolved forward using the gener-
+ated data-driven models and the same ending criteria;
+this will henceforth be denoted as data set B. The pur-
+pose of this data set is to assess the accuracy of the pre-
+dicted long-time turbulent state statistics, and as such
+removes any observed or predicted laminarization.
+The data-driven models are first evaluated on their
+ability to reconstruct the velocity statistics of the tur-
+bulent regime, the essential function of the MFE model.
+Using data set B, we project the amplitudes on to the
+spatial Fourier modes of the MFE model and compare
+the accuracy of the predicted velocity statistics in the
+turbulent state to the exact solution. The mean stream-
+wise velocity and Reynolds shear stress were calculated
+for each data set, shown in Fig. 3. As the figure shows,
+the single-chart model captures well the form of the ve-
+locity statistics, but fails to accurately capture the exact
+values. The three-chart model creates much better pre-
+dictions, correctly capturing the flow profile.
+Now we turn to the prediction of trajectories. To quan-
+tify the performance of the trajectory predictions, we an-
+alyzed the data-driven models’ ability accurately forecast
+the evolution of MFE amplitudes. Using data set A, the
+error in the predictions, E(t), was then calculated for
+each time series, averaged, and normalized, such that
+E(t) =
+||a(t)−˜a(t)||2
+D
+.
+Here, D is the average L2-norm
+FIG. 3. Mean streamwise velocity (solid line) and Reynolds
+shear stress (dashed line) of the full field of the testing data
+and of the reconstruction of the MFE model by the single-
+and three-chart model.
+FIG. 4. Ensemble averaged short-time error tracking of the
+reconstruction of the MFE model by the single- and three-
+chart model.
+between randomly chosen time instants in the turbulent
+state. Fig. 4 shows E(t) for the single- and three-chart
+models as a function of time.
+Both models create ac-
+curate predictions for ∼ 0.5τL, with the error remaining
+close to 0. After this, the error in the predictions of the
+single-chart model grows much more rapidly than the
+three-chart model, indicating that the forecasting ability
+is much stronger in the multi-chart model.
+C.
+Prediction of extreme events
+Now we examine the ability of the data driven-model
+to correctly capture the structure of the extreme events.
+An extreme event can be identified by a growth in the
+first MFE amplitude, which represents the mean shear,
+with a corresponding decrease in the remaining eight am-
+plitudes, which capture the turbulent fluctuations.
+In
+Fig. 5.a, we show the joint probability density function
+(PDF) of a1 and a3 for data set B. The extreme events
+can be seen as the long tail extending to the right to-
+ward the laminar state a1 = 1, a3 = 0. The prediction of
+the single-chart model, shown in 5.b fails to accurately
+capture the structure of the extreme events, with the
+
+2.0
+Single-chart model
+Three-chartmodel
+1.5
+E
+0.5
+0.0
+0
+2
+4
+6
+8
+10
++u'v'
+-0.006
+-0.005
+-0.004
+-0.003
+-0.002
+-0.001
+0.000
+Streamwisemeanvelocity
+1.0
+Reynoldsshearstress
+0.5
+Exactsolution
+0.0
+Single-chartmodel
+Three-chart model
+-0.5
+-1.0
+-0.4
+-0.2
+0.0
+0.2
+0.4
+nk= 1
+k=2
+k=3
+0.5
+a)
+b)
+c)
+0.0
+k=4
+k=5
+k=6
+0.5
+(p
+e)
+f)
+皖A·A9
+0.0
+0
+100
+0
+100
+0
+100
+tk=1
+k=2
+k=3
+a6
+ag
+a6
+ag
+a6
+ag
+k= 4
+k=5
+k=6
+1
+a6
+ag
+a6
+a6
+ag5
+a)
+b)
+c)
+FIG. 5. (a) Joint probability density function of a1 and a3;
+(b) and (c), predictions of the MFE model by the single- and
+multi-chart model, respectively. Note the logarithmic scale.
+tail almost entirely absent. By contrast, the three-chart
+model, shown in 5.c, captures the structure of the ex-
+treme events well, accurately reproducing the shape of
+the joint probability density function.
+We now examine the ability of the single- and multi-
+chart models to forecast an extreme event, defined by the
+kinetic energy of the time series increasing to KE > 0.1.
+To analyze the ability to predict quasi-laminarization
+events, each time series in data set A was segmented
+into time windows of duration 0.5τL and analyzed for the
+presence of an extreme event (i.e., KE exceeding 0.1 in
+the window). The exact solution and data-driven mod-
+els were then compared to determine if each predicted
+whether an extreme event occurred. If an extreme event
+occurred in both the exact solution and the model pre-
+dictions, this was labeled as a true positive (T P). If the
+exact solution exhibited an extreme event, but the data-
+driven model failed to forecast one, this was labeled as a
+false negative (FN). If the model predicted an extreme
+event when the exact solution showed none, it was iden-
+tified as a false positive (FP).
+[22] The total number
+of each identification type in each window was tabulated
+and the F-score, F, was calculated in each window, where
+F = (1 + F P +F N
+2T P
+)−1.
+Fig. 6 shows the F-score as a function of prediction
+time for the single- and multi-chart models, as well as a
+comparison to results from Racca and Magri [22] using an
+echo state network. The multi-chart model outperforms
+the single-chart model, more accurately forecasting ex-
+treme events at all prediction times.
+The multi-chart
+model performs particularly well, with correct extreme
+event predictions 1.5 τL out.
+Our data-driven model
+compares favorably to the (non-Markovian) echo state
+network developed by Racca and Magri [22], matching
+its predictive capabilities at all prediction times.
+Finally, we determine the ability of the data-driven
+models to forecast the lifetime of the turbulence before
+permanent laminarization. At long times, all time series
+FIG. 6. Ensemble averaged short-time error tracking of the
+reconstruction of the MFE model by the single- and three-
+chart model.
+generated by the MFE model at the given parameters
+collapse to the laminar fixed point; the lifetime of each
+time series is dependent on the initial condition, with the
+probability of remaining in the turbulent state approach-
+ing zero at long times. At Re ≲ 320, the probability that
+a given time series remains in the turbulent state for a
+duration t, known as the survival function S(t), takes
+the form [10, 23] S(t; Re) = exp
+�
+t−t0
+τS(Re)
+�
+, where t0 is the
+time delay caused by the approach to the attractor and
+1/τS(Re) is the Re-dependent decay rate. At Re ≳ 320,
+the distribution, particularly at long lifetimes, is known
+to deviate from an exponential decay, requiring increased
+time to laminarize.
+Here, we define a laminarization event as a high-energy
+state (KE > 0.1) for which the kinetic energy over 1 τL
+levels off. The survival function, S(t), is shown in Fig. 7
+for the full system and the one- and three-chart mod-
+els. The full system has a mean lifetime of 251 τL. The
+one-chart model produces poor predictions of the life-
+time distribution, vastly underestimating the lifetimes of
+the turbulent state, with a mean lifetime of 29τL. The
+three-chart model produces a much more accurate rep-
+resentation of the lifetime distribution.
+The predicted
+distribution closely matches the exact solution for ˜t up
+to about 250, while overestimating the lifetimes at longer
+times, and predicts an average lifetime of 298 τL, overes-
+timating the true result by less than 20%. It should be
+emphasized that we are measuring time here in units of
+Lyapunov time, so the inaccuracy of S(t) in the three-
+chart model only arises at extremely long times.
+IV.
+CONCLUSION
+In this paper, we have applied the CANDyMan [24]
+technique towards data-driven modeling of a dynamical
+system with extreme events: the MFE model [10] for tur-
+bulent shear flow. We have shown that clustering data
+sets and training multiple local data-driven models allows
+unique features of distinct data regimes (e.g. extreme
+events) to be separately and more accurately captured by
+
+1.0
+0.8
+0.6
+Single-chart model
+0.4
+Three-chart model
+Racca&Magri(2022)
+0.2
+2
+3
+5Three-chartmode
+10-3
+0.25
+10-4
+0
+10-5
+10-6
+-0.25
+10-7
+0
+0.5
+1
+a1b)
+Single-chart model
+10-2
+0.25
+10-3
+10-4
+0
+e
+10-5
+10-6
+-0.25
+10-7
+0
+0.5
+1
+a1a
+Exactsolution
+10-3
+0.25
+10-4
+0
+10-5
+10-6
+-0.25
+10-7
+0
+0.5
+1
+a16
+FIG. 7. Lifetime distribution of data and the reconstruction
+from the MFE model by the single- and three-chart models.
+a multi-chart global model than in a conventional data-
+driven model.
+Thus, multi-chart models were able to
+more accurately reproduce the evolution of this system,
+reducing forecasting error and improving reconstruction
+of the structure and frequency of extreme events. Im-
+portantly, multi-chart models dramatically improved pre-
+dictions of extreme event occurrences compared to the
+single-chart models used previously. Finally, we demon-
+strated the ability of multi-chart models to accurately
+reconstruct the lifetime distribution of turbulent states,
+producing accurate results hundreds of Lyapunov times
+in the future.
+Now that we have seen that CANDyMan has improved
+the performance of data-driven models forecasting a low-
+dimensional dynamical system with extreme events, fu-
+ture investigations should determine its applicability to
+higher-dimensional systems.
+As has been previously
+shown, the use of a charting technique such as CANDy-
+Man allows improved dimension reduction through the
+use of autoencoder neural networks, capturing the intrin-
+sic dimensionality of dynamical systems [24]. For high-
+dimensional dynamical systems with intermittency, such
+as turbulent fluid flows, the application of CANDyMan
+could not only aid in improved dimension reduction, but
+also produce more accurate forecasting than conventional
+single-chart techniques.
+ACKNOWLEDGMENTS
+This work was supported by ONR N00014-18-1-2865
+(Vannevar Bush Faculty Fellowship). We gratefully ac-
+knowledge Daniel Floryan for helpful discussions.
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+and Probability, Vol. 5.1,, edited by L. M. L. Cam and
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+tics (Wiley-Blackwell, Hoboken, NJ, 2011).
+
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf,len=443
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='11818v1  [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='flu-dyn]  27 Jan 2023  Predicting extreme events in a data-driven model of turbulent shear flow using an  atlas of charts  Andrew J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' Fox and Michael D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' Graham∗  Department of Chemical and Biological Engineering,  University of Wisconsin-Madison,  Madison, WI 53706, USA  (Dated: January 30, 2023)  Dynamical systems with extreme events are difficult to capture with data-driven modeling, due to  the relative scarcity of data within extreme events compared to the typical dynamics of the system,  and the strong dependence of the long-time occurrence of extreme events on short-time conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='  A recently developed technique [Floryan, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' & Graham, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' Data-driven discovery of intrinsic  dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' Nat Mach Intell 4, 1113–1120 (2022)], here denoted as Charts and Atlases for Nonlinear  Data-Driven Dynamics on Manifolds, or CANDyMan, overcomes these difficulties by decomposing  the time series into separate charts based on data similarity, learning dynamical models on each  chart via individual time-mapping neural networks, then stitching the charts together to create a  single atlas to yield a global dynamical model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' We apply CANDyMan to a nine-dimensional model  of turbulent shear flow between infinite parallel free-slip walls under a sinusoidal body force [Moehlis,  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=', Faisst, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' & Eckhardt, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' A low-dimensional model for turbulent shear flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' New J Phys 6,  56 (2004)], which undergoes extreme events in the form of intermittent quasi-laminarization and  long-time full laminarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='  We demonstrate that the CANDyMan method allows the trained  dynamical models to more accurately forecast the evolution of the model coefficients, reducing the  error in the predictions as the model evolves forward in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' The technique exhibits more accurate  predictions of extreme events, capturing the frequency of quasi-laminarization events and predicting  the time until full laminarization more accurately than a single neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='  INTRODUCTION  Real world dynamical systems often produce unusual  behaviors in the form extreme events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='  These extreme  events are characterized by a dissimilarity to the typical  dynamics of the system, usually greater in scope or scale,  that occur relatively infrequently compared to the typ-  ical dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' Common examples include rogue waves  in the ocean [1], extreme weather patterns such as hur-  ricanes and tornadoes [2, 3], and intermittency in turbu-  lent flows [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' While extreme events are a consequence of  the same dynamical system that governs the non-extreme  state, they are often difficult to forecast using data-driven  modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' The relative scarcity of data within extreme  events both limits the overall observations of the extreme  events on which to train the model and reduces the rel-  ative influence of extreme event behavior on data-driven  model training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' Thus, creating a data-driven model that  can accurately capture extreme events remains a active  challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='  Recent studies have proposed various techniques for  analyzing and forecasting the occurrence of extreme  events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='  Guth and Sapsis [5] developed a probabilistic  framework for the use of indicator observables as predic-  tors of the extreme events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' Ragone and Bouchet [6] sup-  plemented climate model simulations with a rare event  algorithm to examine and more accurately capture the  increasing frequency of extreme heatwaves in Europe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='  Blanchard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' [7] built a machine learning framework to  ∗ mdgraham@wisc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='edu  correct a biased climate model to produce better forecasts  of extreme events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' Mendez and Farazmand [8] applied  probabilistic models toward predicting indirect spread-  ing of wildfires by wind to improve forecasts of new wild-  fire locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' Gom´e et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' [9] applied a rare-event algo-  rithm to analyze the transition between states in turbu-  lent pressure-driven flow and more efficiently predict pas-  sage time between states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' While these studies improved  predictions of extreme events, they primarily corrected  and supplemented the forecasts of existing models;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' we  will instead aim to develop an improved model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='  One attractive test case of a dynamical system with ex-  treme events is the nine-dimensional model for turbulent  flow developed by Moehlist, Faisst, and Eckhardt (MFE)  [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' The MFE model, an extension of a model by Waleffe  [11], governs the evolution of nine amplitudes of spatial  Fourier modes describing a turbulent shear flow between  walls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' These nine modes provide a minimal description of  the mechanisms for self-sustenance in turbulence, allow-  ing the resulting flow field to display realistic turbulent  dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' In particular, the model displays features con-  sistent with turbulence in the transition region, namely  long periods of turbulent behavior with infrequent quasi-  laminarization events (also called quiescent [12] or hiber-  nating [13] intervals) and ultimately full laminarization  [12–14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' These quasi- and complete relaminarizations will  be the extreme events considered in the present work, in  which we use time series from the MFE model as “data”  with which to develop a data-driven model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='  In recent years, several attempts have been made to  reproduce the dynamics of the MFE model (and other  flow systems) through data-driven techniques based on  2  neural networks (NNs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' Neural networks are a powerful  data-driven modeling technique that has been shown to  accurately recreate the dynamics of systems such as the  viscous Burgers equation[15], the Kuramoto-Sivashinksy  equation[16, 17], and Kolmogorov flow[18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='  Srinivasan  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' [19] developed both feedforward neural networks  (FNNs) and long short term memory (LSTM) networks  to recreate the MFE model as discrete-time maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' While  the FNNs were unable to reproduce the model, LSTMs  were able to accurately reconstruct long-time behaviors  of the full-field velocity statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' This problem was re-  visited by Eivazi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' [20], where the reconstruction via  a LSTM network was compared to predictions generated  via a Koopman-based framework with nonlinear forcing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='  Their work demonstrated that the Koopman framework  could reproduce short-time and long-time statistics as  well or better than the LSTM networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' Pandey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='  [21] introduced the use of reservoir computing in the form  of an echo state network (ESN), to reproduce the MFE  model as a discrete-time map, and provided comparisons  to both a FNN and a LSTM network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' The LSTM net-  work and the ESN were shown to perform similarly, with  both adequately capturing the full-field velocity statis-  tics, while again the FNN was shown to perform appre-  ciably worse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' Racca and Magri [22] specifically examined  the ability of an ESN to forecast the occurrence of an  extreme event within a future time window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' They deter-  mined that their data-driven model could accurately fore-  cast extreme event episodes far into the future without  incorrectly predicting false quasi-laminarization events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='  Pershin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' [23] assessed the ability of an ESN to fore-  cast time until full laminarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='  They showed that  their model could adequately reproduce the lifetime dis-  tribution of the MFE data, correctly predicting the prob-  ability of an arbitrary MFE time series remaining in the  turbulent state some time in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' These studies  only successfully modeled the MFE equations through  the use of non-Markovian models, which forecast the fu-  ture state through input of the current and past states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='  As the MFE model is itself Markovian, we will instead  endeavor to model the MFE data with a Markovian dy-  namical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='  Specifically, we will use a recently developed method  that will be denoted here as Charts and Atlases for Non-  linear Data-Driven Dynamics on Manifolds (CANDy-  Man) [24, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='  CANDyMan operates by decomposing  the data distribution in state space into separate regions  called charts with a clustering algorithm, learning lo-  cal dynamical models in each chart using FNNs, then  stitching together the charts to create a single atlas con-  taining the global dynamical model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' This technique has  been previously applied to dimensional reduction prob-  lems, accurately learning reduced order dynamical mod-  els whose dimension is equal to the intrinsic dimension-  ality of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' The use of multiple charts allows  low-dimensional manifolds embedded in high dimensional  space to be broken down into locally low dimensional  structures, capturing the dynamics of a system with the  minimal number of dimensions, in a way that single chart  methods cannot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' Here, we do not perform dimension re-  duction, but rather utilize the clustering of data to break  down the dynamical system into separate regions rep-  resenting extreme and non-extreme states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' By learning  the dynamics in the extreme region separately and in-  dependently from the non-extreme regions, CANDyMan  inherently overcomes the imbalance of extreme vs non-  extreme information and thus the limited influence of  extreme events in data driven model training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='  Here, we will use CANDyMan to reconstruct the dy-  namics of the MFE model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' A data set containing time  series of the MFE amplitudes will be decomposed using  k-means clustering into atlases containing between one  and six charts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' We will train deep neural networks to  reconstruct the time evolution of the MFE amplitudes  within each of the charts, then stitch them together to  create six global models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' To assess the accuracy of the  models, we will first consider their ability to reconstruct  the turbulent flow field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' Next, we will analyze their per-  formance in reproducing short-time and long-time statis-  tics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' Finally, we will assess the extreme event forecasting  of the data-driven models by determining the statistical  accuracy of forecasting extreme event occurrences and  comparing predicted laminarization lifetime distribution  to the true data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='  FORMULATION  The MFE model is a severely truncated Fourier  Galerkin approximation to the Navier-Stokes equations  (NSE) for flow between two free-slip walls and driven  by a spatially sinusoidal body force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' The flow is com-  posed of nine spatial Fourier modes ui(x), describing  the basic profile, streaks, and vortices, as well as inter-  actions between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' The velocity field at position x  and time t is given by a superposition of the nine modes  as u(x, t) =  9�  i=1  ai(t)ui(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' The mode amplitudes ai(t)  satisfy a system of nine ordinary differential equations  (ODEs), generated through Galerkin projection, whose  explicit form is given in Moehlis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' Our study  considers a domain of size Lx × Ly × Lz, with infinite,  parallel walls at y = −Ly/2 and y = Ly/2 and peri-  odic boundaries x = 0, x = Lx, z = 0, and z = Lz;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' x,  y, and z are the streamwise, wall-normal, and spanwise  coordinates, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' The domain size of Lx = 4π,  Ly = 2, Lz = 2π was used, with a channel Reynolds  number of 400;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' these parameters produce turbulent be-  havior of suitable length for data-driven model develop-  ment [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='  As training data, we generated 100 unique time se-  ries from a fourth-order Runge-Kutta integration of the  MFE equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' Each time series encompasses the tran-  sient turbulent state, consisting of turbulent intervals  interspersed with quasi-laminarization events, with ter-  minal laminarization occurring at long time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='  We will  3  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' Evolution of three amplitudes, a1, a6, a9 and corre-  sponding kinetic energy from one time series of the MFE data  set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='  often characterize the flow using the total kinetic en-  ergy (KE), given by KE  =  1  2  �9  i=1 a2  i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='  Therefore,  the turbulent state is low energy while the laminar  is high energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='  Every time series collapses to the  known laminar fixed point ai = δi1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='  To generate the  time series, initial conditions of eight of the amplitudes  were given as follows:  (a1, a2, a3, a5, a6, a7, a8, a9) =  (1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='07066, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='07076, 0, 0, 0, 0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' The initial value of a4  was randomly generated in the range [−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' These  initial conditions were previously demonstrated to gen-  erate chaotic dynamical data with quasi-laminarization  events [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' Amplitudes and KE from a randomly chosen  time series are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' We will report all results  in units ˜t = t/τL, where τL is the Lyapunov time for the  system;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' in the original nondimensionalization τL ≈ 41  [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='  In this study, we examined the behavior of multi-chart  models with between two and six charts, as well as a stan-  dard approach with one global model – the “one-chart”  limit of CANDyMan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' The dynamical system data is first  clustered into k charts via k-means clustering, which par-  titions a data set into k clusters, minimizing the within-  cluster variance [27, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='  Other clustering techniques,  such as k-nearest neighbors [29] or single-linkage cluster-  ing [30], could be used, provided the clustering technique  produces charts that encompass contiguous regions of the  state space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' The clusters are then expanded so that they  overlap, by locating the kNN nearest neighbors to each  data point in a cluster by Euclidean distance and adding  these to the original cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' This creates an overlap re-  gion between neighboring clusters, providing transition  regions in which the dynamics are described by multiple  charts and allowing for the movement into and out of the  region to be handled by the separate local models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='  Then, in each augmented chart, we generated discrete-  time models of the form a(j)(t + τ) = F (j)(a(j)(t);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' θ(j)),  where a(j)(t) ∈ R9 is the representation of the state in  chart j, the discrete time step is τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='5, and F (j) is the  corresponding discrete-time map, which takes the form  of a FNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' The quantities θ(j) are the neural network  weights for F (j), which are learned from the data using a  standard stochastic gradient descent method and trained  to minimize the loss function L(j) = ⟨||a(j)(t)−˜a(j)(t)||2⟩,  where ⟨·⟩ is the average over the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' To en-  sure that the comparison between different numbers of  charts was standardized, each global model contains the  same number of total neurons, NT = 1800;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' a system of  k charts would then use NN = NT /k neurons in each  local model, each containing four fully-connected hidden  layers of NN/6, NN/3, NN/3, and NN/6 of the total  number of local neurons, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' Each neural net-  work was trained using a learning rate scheduler with an  initial learning rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='01, decaying at a rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='9 every  2000 steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' Each model was then trained for 100 epochs,  which was found to accurately reproduce the training  data while avoiding overfitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='  RESULTS AND DISCUSSION  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='  Distribution of data into clusters  Insight into the number of charts necessary for prop-  erly reconstructing the MFE data can be gained by ob-  serving the clustering of the training data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' 2  shows how one trajectory from the data set is partitioned  when we use different numbers of charts, in terms of (a-  f) the time series of KE and (g-l) state space projec-  tions onto amplitudes a1, a6, a9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='  With two charts, the  data partitions into one cluster covering the low-energy  (turbulent) non-extreme states and the second contain-  ing the high-energy extreme (quasi-laminar, laminariz-  ing) states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' When three charts were used, the clusters  are further segmented, with one covering the low-energy  turbulent state, the second primarily consisting of the  transition into quasi-laminarization and laminarization  events, and the third consisting mainly of the high en-  ergy components of these events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='  Clustering into four  charts breaks down the low-energy region into two sepa-  rate clusters that maintain relatively distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' When the  data was clustered into five or six charts, the distinction  between the charts in the low-energy turbulent regime  decreased and the charts containing the turbulent states  were described by increasingly similar centroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='  Trajectory predictions and time-averaged  statistics  The performance of the data-driven models was eval-  uated on their ability to reconstruct the evolution of  the MFE model amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='  Two test data sets were  generated for comparison between the MFE model and  the single- and multi-chart data-driven models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' For tra-  jectory predictions, 100 trajectories of MFE amplitudes  were generated from randomized initial conditions and  time-integrated for 10 Lyapunov times, with the initial  conditions separately evolved forward using the gener-  ated data-driven models for the same length of time;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' this  will henceforth be denoted as data set A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' The purpose of  this data set is to determine the short-time precision of  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='5  ai  a6  ag  KE  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='3E  e  K  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='0  0  50  100  150  200  250  七4  g)  h)  i)  j)  k)  l)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' Clustering of a randomly selected trajectory of kinetic  energy (a-f) and the projection of the clustering of the first,  sixth, and ninth MFE amplitude (g-l) for one to six charts,  color coded by cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='  the predictions generated by the single- and multi-chart  models, regardless of any observed or predicted laminar-  ization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' For time-averaged statistics, 100 trajectories of  MFE amplitudes were generated from randomized initial  conditions and time-integrated for 100 Lyapunov times  or until a laminarization event occurred, with the initial  conditions separately evolved forward using the gener-  ated data-driven models and the same ending criteria;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='  this will henceforth be denoted as data set B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' The pur-  pose of this data set is to assess the accuracy of the pre-  dicted long-time turbulent state statistics, and as such  removes any observed or predicted laminarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='  The data-driven models are first evaluated on their  ability to reconstruct the velocity statistics of the tur-  bulent regime, the essential function of the MFE model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='  Using data set B, we project the amplitudes on to the  spatial Fourier modes of the MFE model and compare  the accuracy of the predicted velocity statistics in the  turbulent state to the exact solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' The mean stream-  wise velocity and Reynolds shear stress were calculated  for each data set, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' As the figure shows,  the single-chart model captures well the form of the ve-  locity statistics, but fails to accurately capture the exact  values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' The three-chart model creates much better pre-  dictions, correctly capturing the flow profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='  Now we turn to the prediction of trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' To quan-  tify the performance of the trajectory predictions, we an-  alyzed the data-driven models’ ability accurately forecast  the evolution of MFE amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' Using data set A, the  error in the predictions, E(t), was then calculated for  each time series, averaged, and normalized, such that  E(t) =  ||a(t)−˜a(t)||2  D  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='  Here, D is the average L2-norm  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' Mean streamwise velocity (solid line) and Reynolds  shear stress (dashed line) of the full field of the testing data  and of the reconstruction of the MFE model by the single-  and three-chart model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' Ensemble averaged short-time error tracking of the  reconstruction of the MFE model by the single- and three-  chart model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='  between randomly chosen time instants in the turbulent  state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' 4 shows E(t) for the single- and three-chart  models as a function of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='  Both models create ac-  curate predictions for ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='5τL, with the error remaining  close to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' After this, the error in the predictions of the  single-chart model grows much more rapidly than the  three-chart model, indicating that the forecasting ability  is much stronger in the multi-chart model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='  Prediction of extreme events  Now we examine the ability of the data driven-model  to correctly capture the structure of the extreme events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='  An extreme event can be identified by a growth in the  first MFE amplitude, which represents the mean shear,  with a corresponding decrease in the remaining eight am-  plitudes, which capture the turbulent fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='  In  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='a, we show the joint probability density function  (PDF) of a1 and a3 for data set B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' The extreme events  can be seen as the long tail extending to the right to-  ward the laminar state a1 = 1, a3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' The prediction of  the single-chart model, shown in 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='b fails to accurately  capture the structure of the extreme events, with the  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='0  Single-chart model  Three-chartmodel  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='5  E  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
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+page_content='001  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='000  Streamwisemeanvelocity  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='0  Reynoldsshearstress  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='5  Exactsolution  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='0  Single-chartmodel  Three-chart model  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
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+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='4  nk= 1  k=2  k=3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='5  a)  b)  c)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='0  k=4  k=5  k=6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='5  (p  e)  f)  皖A·A9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='0  0  100  0  100  0  100  tk=1  k=2  k=3  a6  ag  a6  ag  a6  ag  k= 4  k=5  k=6  1  a6  ag  a6  a6  ag5  a)  b)  c)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' (a) Joint probability density function of a1 and a3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='  (b) and (c), predictions of the MFE model by the single- and  multi-chart model, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' Note the logarithmic scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='  tail almost entirely absent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' By contrast, the three-chart  model, shown in 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='c, captures the structure of the ex-  treme events well, accurately reproducing the shape of  the joint probability density function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='  We now examine the ability of the single- and multi-  chart models to forecast an extreme event, defined by the  kinetic energy of the time series increasing to KE > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='  To analyze the ability to predict quasi-laminarization  events, each time series in data set A was segmented  into time windows of duration 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='5τL and analyzed for the  presence of an extreme event (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=', KE exceeding 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='1 in  the window).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' The exact solution and data-driven mod-  els were then compared to determine if each predicted  whether an extreme event occurred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' If an extreme event  occurred in both the exact solution and the model pre-  dictions, this was labeled as a true positive (T P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' If the  exact solution exhibited an extreme event, but the data-  driven model failed to forecast one, this was labeled as a  false negative (FN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' If the model predicted an extreme  event when the exact solution showed none, it was iden-  tified as a false positive (FP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='  [22] The total number  of each identification type in each window was tabulated  and the F-score, F, was calculated in each window, where  F = (1 + F P +F N  2T P  )−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' 6 shows the F-score as a function of prediction  time for the single- and multi-chart models, as well as a  comparison to results from Racca and Magri [22] using an  echo state network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' The multi-chart model outperforms  the single-chart model, more accurately forecasting ex-  treme events at all prediction times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='  The multi-chart  model performs particularly well, with correct extreme  event predictions 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='5 τL out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='  Our data-driven model  compares favorably to the (non-Markovian) echo state  network developed by Racca and Magri [22], matching  its predictive capabilities at all prediction times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='  Finally, we determine the ability of the data-driven  models to forecast the lifetime of the turbulence before  permanent laminarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' At long times, all time series  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' Ensemble averaged short-time error tracking of the  reconstruction of the MFE model by the single- and three-  chart model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='  generated by the MFE model at the given parameters  collapse to the laminar fixed point;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' the lifetime of each  time series is dependent on the initial condition, with the  probability of remaining in the turbulent state approach-  ing zero at long times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' At Re ≲ 320, the probability that  a given time series remains in the turbulent state for a  duration t, known as the survival function S(t), takes  the form [10, 23] S(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' Re) = exp  �  t−t0  τS(Re)  �  , where t0 is the  time delay caused by the approach to the attractor and  1/τS(Re) is the Re-dependent decay rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' At Re ≳ 320,  the distribution, particularly at long lifetimes, is known  to deviate from an exponential decay, requiring increased  time to laminarize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='  Here, we define a laminarization event as a high-energy  state (KE > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='1) for which the kinetic energy over 1 τL  levels off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' The survival function, S(t), is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' 7  for the full system and the one- and three-chart mod-  els.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' The full system has a mean lifetime of 251 τL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' The  one-chart model produces poor predictions of the life-  time distribution, vastly underestimating the lifetimes of  the turbulent state, with a mean lifetime of 29τL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' The  three-chart model produces a much more accurate rep-  resentation of the lifetime distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='  The predicted  distribution closely matches the exact solution for ˜t up  to about 250, while overestimating the lifetimes at longer  times, and predicts an average lifetime of 298 τL, overes-  timating the true result by less than 20%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' It should be  emphasized that we are measuring time here in units of  Lyapunov time, so the inaccuracy of S(t) in the three-  chart model only arises at extremely long times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='  CONCLUSION  In this paper, we have applied the CANDyMan [24]  technique towards data-driven modeling of a dynamical  system with extreme events: the MFE model [10] for tur-  bulent shear flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' We have shown that clustering data  sets and training multiple local data-driven models allows  unique features of distinct data regimes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' extreme  events) to be separately and more accurately captured by  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='6  Single-chart model  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='4  Three-chart model  Racca&Magri(2022)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='2  2  3  5Three-chartmode  10-3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='25  10-4  0  10-5  10-6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='25  10-7  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='5  1  a1b)  Single-chart model  10-2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='25  10-3  10-4  0  e  10-5  10-6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='25  10-7  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='5  1  a1a  Exactsolution  10-3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='25  10-4  0  10-5  10-6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='25  10-7  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='5  1  a16  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' Lifetime distribution of data and the reconstruction  from the MFE model by the single- and three-chart models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='  a multi-chart global model than in a conventional data-  driven model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='  Thus, multi-chart models were able to  more accurately reproduce the evolution of this system,  reducing forecasting error and improving reconstruction  of the structure and frequency of extreme events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' Im-  portantly, multi-chart models dramatically improved pre-  dictions of extreme event occurrences compared to the  single-chart models used previously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' Finally, we demon-  strated the ability of multi-chart models to accurately  reconstruct the lifetime distribution of turbulent states,  producing accurate results hundreds of Lyapunov times  in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='  Now that we have seen that CANDyMan has improved  the performance of data-driven models forecasting a low-  dimensional dynamical system with extreme events, fu-  ture investigations should determine its applicability to  higher-dimensional systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='  As has been previously  shown, the use of a charting technique such as CANDy-  Man allows improved dimension reduction through the  use of autoencoder neural networks, capturing the intrin-  sic dimensionality of dynamical systems [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' For high-  dimensional dynamical systems with intermittency, such  as turbulent fluid flows, the application of CANDyMan  could not only aid in improved dimension reduction, but  also produce more accurate forecasting than conventional  single-chart techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='  ACKNOWLEDGMENTS  This work was supported by ONR N00014-18-1-2865  (Vannevar Bush Faculty Fellowship).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' We gratefully ac-  knowledge Daniel Floryan for helpful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='  [1] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' Dysthe, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' Krogstad, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content=' Muller, Oceanic rogue  waves, Annual Review of Fluid Mechanics 40, 287 (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
+page_content='  [2] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFKT4oBgHgl3EQfdC4X/content/2301.11818v1.pdf'}
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diff --git a/J9FAT4oBgHgl3EQfvh5f/content/tmp_files/2301.08676v1.pdf.txt b/J9FAT4oBgHgl3EQfvh5f/content/tmp_files/2301.08676v1.pdf.txt
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+Prepared for submission to JHEP
+Boundary Symmetry Breaking in CFT and the
+String Order Parameter
+Riccarda Bonsignori,a Luca Capizzi,b Pantelis Panopoulosa
+aRudjer Boskovic Institute, Division of Theoretical Physics, Bijenička 54, 10000 Zagreb, Croatia
+bSISSA and INFN Sezione di Trieste, via Bonomea 265, I-34136 Trieste, Italy
+E-mail: rbonsign@irb.hr, lcapizzi@sissa.it, Pantelis.Panopoulos@irb.hr
+Abstract:
+We consider the ground state of a one-dimensional critical quantum system
+carrying a global symmetry in the bulk, which is explicitly broken by its boundary con-
+ditions. We probe the system via a string-order parameter, showing how it detects the
+symmetry breaking pattern. We give a precise characterization of the mechanism depicted
+above in Boundary CFT, and we find a general logarithmic scaling for the order parameter.
+As a first example we analyze the breaking of a U(1) symmetry for complex free theories in-
+duced by a boundary pairing term. Moreover, we give predictions for the breaking of U(N)
+in free theories, arising from a boundary mixing. We test our predictions with numerical
+calculations for some lattice realizations of free fermionic system with boundary symmetry
+breaking, finding a good agreement.
+Keywords: Boundary Conformal Field Theory, String Order Parameter, Charged Parti-
+tion Function
+arXiv:2301.08676v1  [hep-th]  20 Jan 2023
+
+Contents
+1
+Introduction
+1
+2
+Definitions and Techniques
+3
+2.1
+String order parameter
+3
+2.2
+BCFT description
+5
+3
+Dirac fermions
+7
+3.1
+U(1)-symmetry breaking due to a boundary pairing term
+7
+3.2
+U(N)-symmetry breaking via boundary scattering matrix
+10
+4
+Complex bosons
+13
+4.1
+U(1)-symmetry breaking terms in complex bosons
+13
+4.2
+U(N)-symmetry breaking via boundary scattering matrix
+14
+5
+Lattice results
+16
+5.1
+U(1)-symmetry breaking by a boundary pairing: doubling trick
+17
+5.2
+Conformal defect and U(2) symmetry breaking
+20
+6
+Conclusions and Outlooks
+22
+1
+Introduction
+Symmetry is a cornerstone of modern physics.
+Its importance emerges in all branches
+of physics among which the Condensed Matter Theory [1, 2] (ferromagnetism, supercon-
+ductivity, superfluidity) and High-Energy Physics [3] (formulation of the Standard Model,
+Quantum gravity etc.). There are many cases where symmetry is responsible for phase
+transitions in statistical systems [4, 5]. In particular, behind plenty of classical and quan-
+tum systems showing a (second order) phase transition there is an underlying symmetry
+that can be preserved or spontaneously broken: this mechanism has been a guideline for
+the characterization of the possible phases of the system.
+As an illustrative example, we consider a toy model for ferromagnetism, namely the
+classical Ising model enjoying global symmetries. It has been shown that in d ≥ 2 dimen-
+sions, the model undergoes a phase transition at a critical value of the temperature T = Tc.
+Moreover, for T > Tc the Z2 symmetry of the model is unbroken, and the system has a
+paramagnetic behavior, while for T < Tc the system undergoes to a Spontaneous Symmetry
+Breaking (SSB) characterizing the ferromagnetic phase. The breaking of the symmetry can
+be spotted via the probing of the magnetization, described in terms of a local field σ(x).
+In particular, its expectation value ⟨σ(x)⟩ vanishes at T > Tc and it gets a finite value for
+T < Tc. The critical point T = Tc deserves special attention. Indeed, while at this point the
+– 1 –
+
+symmetry remains unbroken and thus ⟨σ(x)⟩ = 0, an additional application of a magnetic
+field in this critical temperature can have non-trivial effects even far from that point, due to
+the slow (algebraic) decay of the correlation functions, a distinct trait of the critical phase.
+Historically, the importance of localized perturbations for critical systems has not been
+taken for granted, until the discovery of the Kondo effect [6, 7], which is associated with
+anomalous transport properties of low-temperature metals in the presence of impurities.
+Despite the progresses in this field, both experimentally and theoretically (e.g. Boundary
+Conformal Field Theory (BCFT) formulation [9, 30, 35]), a comprehensive theory of the
+symmetry and its breaking pattern seems to remain an open problem for such systems. The
+main motivation behind this work is to address more systematically this lack of approach.
+We are interested in the ground-state |Ω⟩ of the following Hamiltonian
+H = H0 + H1 .
+(1.1)
+The H0 is a bulk term involving short-range interactions, which we assume to be critical,
+and invariant under the action of a group G associated with a global symmetry. In turn,
+the H1 term is a perturbation localized in space (say defect/impurity) which explicitly
+spoils the G-invariance. More precisely, we consider a unitary representation of G, which
+associates to any g ∈ G a unitary operator ˆg of the Hilbert space, and we require that
+[H0, ˆg] = 0,
+[H1, ˆg] ̸= 0,
+∀g ∈ G.
+(1.2)
+Our main goal is to understand whether and how the ground state |Ω⟩ of (1.2) breaks the
+global G-invariance. For this purpose, we propose
+⟨Ω| ˆg |Ω⟩ ,
+(1.3)
+regarded as a function of G, to be a good non-local order parameter. For one-dimensional
+systems, ˆg can be naively regarded as a string operator inserted along the whole system,
+and for this reason, we call it string order parameter (following the terminology of [12]).
+We anticipate that, rather generically, the unbroken symmetry group is identified by
+H = {g ∈ G| | ⟨Ω| ˆg |Ω⟩ | = 1},
+(1.4)
+while for the other elements of the group one has | ⟨Ω| ˆg |Ω⟩ | < 1. Moreover, for a critical
+system defined on a finite-size region [0, L], we find that in the presence of scale-invariant
+symmetry-breaking boundary terms, the (log of the) order parameter shows the following
+logarithmic growth
+− log | ⟨Ω| ˆg |Ω⟩ | ∼ log L,
+(1.5)
+in the limit of large L. In particular the quantity
+lim
+L→∞ − log | ⟨Ω| ˆg |Ω⟩ |
+log L
+,
+g ∈ G
+(1.6)
+– 2 –
+
+is a universal continuous, but in general not smooth function of the group G, which vanishes
+precisely for g ∈ H, with H being the unbroken subgroup.
+We provide an accurate description of this mechanism in the context of BCFT, ex-
+ploiting the power of conformal symmetry. Therefore, we compute explicitly the string
+order parameter for a class of free fermionic and bosonic theories. In the first case, we
+consider a free Dirac fermion on [0, L] and we insert a pairing term at one boundary point,
+studying the symmetry breaking pattern U(1) → Z2. We extend our analysis by taking
+N copies of uncoupled the Dirac fields in the bulk, but coupled at one boundary point
+via a scattering matrix S. In this case, we find a novel non-trivial breaking of the U(N)
+symmetry, strongly related to the symmetries of S. By repeating the same approach we
+study the string order parameter for free complex massless bosons under U(1) and under
+U(N) symmetry seperately as well. Finally, we present two possible realizations of the
+boundary symmetry breaking mechanism for free fermionic systems on the lattice. For one
+of them, we also perform the numerical calculation of the string order parameter to test
+our analytical prediction.
+Our manuscript is organized as follows. In Sec. 2 we provide some general definitions
+and give a BCFT description of the string order parameter. In Sec. 3 we analyze in detail
+the free fermions. In Sec. 4 we repeat the same analysis for free bosons. In Sec. 5 we
+consider the lattice counterpart for fermions and present the numerical results. Finally, in
+Sec. 6 we gather our results and discuss some possible future directions.
+2
+Definitions and Techniques
+The purpose of this section is two-fold. First, once the non-local order parameter is intro-
+duced, we explain how and why it detects symmetry-breaking, providing some properties
+which are independent of the detail of the systems. Then, we specify the treatment to 1+1
+BCFT with symmetry breaking terms at the boundaries, and give a general derivation of
+the logarithmic growth in Eq. (1.5). At this point, we will not specialize to any specific
+theory, and we derive the first results employing only conformal symmetry and describing
+the boundary conditions (BCs) as boundary states, via a space-time duality.
+2.1
+String order parameter
+Let us first review what is a symmetry in a quantum system. Given a Hilbert space H , a
+unitary representation of a group G is a linear map
+G → GL(H ) ,
+g → ˆg
+(2.1)
+from the group onto the unitary operators of H . One requires that the map is homomor-
+phic, namely
+ˆ
+(g1g2) = ˆg1 ˆg2 .
+(2.2)
+Without loss of generality, here we assume that the map is injective, so that distinct elements
+of the group are represented by distinct operators. We now consider a (normalized) state
+– 3 –
+
+|Ω⟩ ∈ H . We say that |Ω⟩ is symmetric (invariant) under G iff
+ˆg |Ω⟩ = eiφ(g) |Ω⟩ ,
+∀g ∈ G
+(2.3)
+with eiφ(g) being a g-dependent phase factor. The requirement above severely constrains the
+expectation values of the observables, which is the main reason why the order parameters
+are useful to detect the breaking of symmetry.
+As a first example, we consider a quantum system carrying a representation of Z2.
+We denote by {1, τ} the generators of the group, and consider an observable σ, say the
+magnetization, odd under Z2
+ˆτσˆτ −1 = −σ,
+(2.4)
+which plays the role of the order parameter. Whenever a state |Ω⟩, say the (a) ground
+state, is invariant under Z2 one safely concludes that
+⟨Ω| σ |Ω⟩ = ⟨Ω| ˆτ −1σˆτ |Ω⟩ = − ⟨Ω| σ |Ω⟩ ,
+(2.5)
+which clearly implies that ⟨Ω| σ |Ω⟩ = 0. This means that, whenever ⟨Ω| σ |Ω⟩ ̸= 0 one can
+be sure that the state |Ω⟩ is not Z2 invariant. Unfortunately, in principle, the converse is
+not true. Indeed, there is no reason why ⟨Ω| σ |Ω⟩ = 0 should imply a Z2 symmetry for |Ω⟩.
+This is somehow the main disadvantage behind the usage of the usual order parameters.
+In contrast, as we will show below, if one considers ˆg itself as an order parameter, one
+can unambiguously understand if |Ω⟩ is symmetric. A first immediate observation is that,
+if |Ω⟩ is symmetric under g ∈ G (see Eq. (2.3)), then
+| ⟨Ω| ˆg |Ω⟩ | = | ⟨Ω| eiφ(g) |Ω⟩ | = 1.
+(2.6)
+Less trivially, one can show the converse, namely that | ⟨Ω| ˆg |Ω⟩ | = 1 implies that the
+symmetry has to be unbroken. To prove it, we employ the Cauchy-Schwarz inequality [14],
+which tells us
+| ⟨Ω| ˆg |Ω⟩ | ≤ | ⟨Ω| ˆg†ˆg |Ω⟩ | · | ⟨Ω|Ω⟩ | = 1,
+(2.7)
+where the inequality is strict unless |Ω⟩ and ˆg |Ω⟩ are proportional, which is exactly the
+notion of symmetry in Eq. (2.3). To summarize, so far we have that
+| ⟨Ω| ˆg |Ω⟩ | = 1
+if and only if
+ˆg |Ω⟩ = eiφ(g) |Ω⟩ .
+(2.8)
+This property suggests a way to characterize the subgroup H ⊆ G which leaves |Ω⟩ invariant
+as Eq. (1.4). We would like to stress explicitly that, in their simplicity, the last conclusions
+are very general and apply to both abelian and non-abelian symmetries. Notice that up to
+this point our discussion is general, since we did not specify whether the symmetry breaking
+pattern G → H, associated with the state |Ω⟩, arises from an explicit or spontaneous
+symmetry breaking. For the reasons depicted above, the investigation of | ⟨Ω| ˆg |Ω⟩ |, the
+non-local (string) order parameter, is the main goal of this work.
+– 4 –
+
+2.2
+BCFT description
+Let us proceed to the description of the string order parameter in the framework of BCFT
+[11, 15–18]. We consider the ground state |Ω⟩ of a one-dimensional critical quantum system
+on a finite size geometry, namely the interval [0, L]. We assume that the bulk is described
+by a CFT, and we impose conformal invariant BCs at x = 0, L [11, 30]. Introducing the
+Euclidean time, one can describe the state |Ω⟩ as a strip geometry, parametrized by the
+complex coordinate w satisfying
+Re(w) ∈ [0, L],
+Im(w) ∈ (−∞, ∞).
+(2.9)
+In particular, Re(w) represents the spatial position, and Im(w) corresponds to the Euclidean
+time. In this picture, the expectation values of the observables in the state |Ω⟩ are described
+via the insertion of fields in the strip geometry depicted in Figure 1.
+We now assume that the theory has a global symmetry in the bulk, characterized by
+a representation of a group G, which may be eventually broken by the choice of the BCs.
+We are interested in the symmetry breaking pattern, strictly related to the evaluation of
+the string order parameter ⟨Ω| ˆg |Ω⟩. Since the symmetry is global, the action of the group
+G is nontrivial at any spatial point x ∈ [0, L]. For this reason, it is natural to represent
+pictorially ˆg as a line operator extended over Im(w) = 0. Then, ⟨Ω| ˆg |Ω⟩ becomes a charged
+partition function given by the insertion of a charged line connecting w = 0 and w = L in
+the strip geometry.
+The last key ingredient is the specification of the BCs. We denote by b, b′ the type of
+BCs at x = 0, L respectively. In the strip geometry, these boundary points become lines
+extended over the euclidean time, Re(w) = 0, L respectively, and we associate the labels
+b, b′ to each of these lines. We assume that b′, corresponding to x = L, preserves explicitly
+the symmetry, and we do not characterize it further. Instead, we focus on BCs at x = 0
+which breaks G explicitly, and our goal is to identify the corresponding symmetry breaking
+pattern.
+At this point, we employ conformal symmetry to relate the strip geometry described
+above to another geometry and we do that to simplify the computation of the charged
+partition function. After a UV and IR regulation of the original geometry, keeping only the
+points ε < |w| < L, we apply the transformation [19–22]
+z = log w.
+(2.10)
+The new geometry is the rectangle
+Re(z) ∈ (log ε, log L),
+Im(z) ∈ [−π/2, π/2],
+(2.11)
+with BCs of type b along Im(z) = ±π/2, corresponding to boundary states. It is important
+to stress that the information about b′ is lost explicitly by the choice of IR regularization.
+However, this is not big deal, as we required that b′ is G invariant. Indeed, on the physical
+ground, we do not expect any contribution to the string order parameter from the point
+– 5 –
+
+Figure 1: The expectation value of the string order parameter. We represent the original
+strip geometry (coordinate w) and the rectangular one (coordinate z). The red/blue lines
+correspond to the BCs of type b, b′ respectively. The insertion of the symmetry operator ˆg
+is a green line.
+x = L (at least at leading order).
+Putting these information together, we express the charged partition function as a
+transition element in Euclidean time between the boundary state |b⟩, corresponding to b
+(see Ref.[11]), with itself. More precisely, we define
+Z(g) ≡ ⟨b| exp(−πH)ˆg |b⟩ ,
+(2.12)
+with H being the Hamiltonian in the rectangular geometry
+H =
+2π
+log(L/ε)(L0 + ¯L0),
+(2.13)
+and L0, ¯L0 the Virasoro generators. Then, we express the expectation value of ˆg as
+⟨Ω| ˆg |Ω⟩ = Z(g)
+Z(1),
+(2.14)
+where the denominator Z(1) arises only as a normalization constant (the uncharged parti-
+tion of the rectangle). We represent the construction above in Fig. 1, showing the insertion
+of ˆg in the two geometries. So far, the discussion is general and it applies to any BCFT
+carrying a global symmetry G. The precise evaluation of the charged partition function
+Z(g) requires the characterization of the boundary state |b⟩. Still, in general, one can argue
+that the logarithm of the partition function (free energy) is extensive in the large-size limit.
+– 6 –
+
+(2|g/2)
+2(g)
+Im
+10
+1
+11
+1
+0
+1
+Z=
+9
+1
+T
+I
+1
+3= |ml
+1
+1
+1
+b
+-
+L
+log -
+LIn particular, we have
+log Z(g) ∝ log L
+ε ,
+L/ε → ∞
+(2.15)
+up to a proportionality constant depending on both g ∈ G and the BCs b. Summarizing,
+we discover that the ratio
+− log | ⟨Ω| ˆg |Ω⟩ |
+log L/ε
+(2.16)
+is finite in the limit L → ∞, where the conventional minus sign ensures a positive value
+result. In the next section, we will carefully analyze that ratio for specific CFTs, relating
+its behavior to the symmetry breaking pattern.
+3
+Dirac fermions
+We now proceed with an application of the previous discussion to the system of free fermions.
+First, we discuss free fermions with U(1) symmetry and then we continue with U(N)
+generalization1.
+3.1
+U(1)-symmetry breaking due to a boundary pairing term
+We consider the theory of massless Dirac fermions in a finite-size geometry, taking BCs
+that break explicitly the U(1) symmetry. The model is described by two fields Ψ and Ψ†,
+that correspond to particle and antiparticles. In radial quantization, one decomposes Ψ in
+its left and right Laurent modes as follows
+Ψ(z) =
+�
+k∈Z+1/2
+Ψk
+zk+1/2 ,
+¯Ψ(¯z) =
+�
+k∈Z+1/2
+¯Ψk
+¯zk+1/2 ,
+(3.1)
+where the Neveu-Schwarz (NS) sector has been considered. For k > 0 the modes Ψk, ¯Ψk
+destroy a left/right moving particle, while Ψ−k, ¯Ψ−k are creation operators. Similar con-
+siderations hold for Ψ†, the antiparticle field, and we refer to its left/right modes with
+Ψ†
+k, ¯Ψ†
+k.
+The bulk action in Euclidean space of the finite [0, L] geometry reads
+Sbulk =
+� L
+0
+dx
+�
+dτ
+�
+Ψ† ¯∂Ψ + ¯Ψ†∂ ¯Ψ
+�
+.
+(3.2)
+enjoying a U(1) global symmetry
+Ψ → eiαΨ,
+¯Ψ → eiα ¯Ψ,
+Ψ† → e−iαΨ†,
+¯Ψ† → e−iα ¯Ψ†,
+(3.3)
+and it corresponds to the imbalance between particles and antiparticles. We want to break
+U(1) explicitly through the BCs at x = 0. We do so via the insertion of a pairing term at
+1For more details in the calculations of this section the reader may consult [17].
+– 7 –
+
+the boundary point, described by a boundary action
+Sboundary =
+�
+dτ
+�
+¯Ψ(x = 0, τ)Ψ(x = 0, τ) + (Ψ → Ψ†)
+�
+.
+(3.4)
+The boundary term is not U(1) invariant, since it transforms as Ψ¯Ψ → ei2αΨ¯Ψ. A residual
+Z2 is nevertheless preserved (associated with α = 0, π), and it describes the conservation
+of fermion parity. One thus expects that these BCs should induce an explicit symmetry
+breaking pattern
+U(1) → Z2,
+(3.5)
+on the ground state |Ω⟩.
+While these considerations are so far not rigorous, they can
+capture the key features of the system. In the following, we aim to compute the string-
+order parameter via the BCFT techniques (2.2) for the U(1) symmetry. The first quantity
+we need is the boundary state |b⟩ associated with the U(1) breaking BCs. We consider a
+coherent state in which pairs of particles (and antiparticles) with opposite momenta are
+generated above the ground states. Its explicit expression is
+|b⟩ =
+�
+k>0
+exp(iΨ−k ¯Ψ−k + (Ψ → Ψ†)) |0⟩ ,
+(3.6)
+with |0⟩ being the vacuum of the theory. The generator of the symmetry is
+Q =
+�
+k>0
+Ψ−kΨk + ¯Ψ−k ¯Ψk − (Ψ ↔ Ψ†),
+(3.7)
+and we parametrize a generic element of U(1) as
+ˆg = eiαQ,
+α ∈ [−π, π).
+(3.8)
+Finally, we remind the relation between the Virasoro modes and the fermionic modes
+L0 + ¯L0 =
+�
+k>0
+k(Ψ−kΨk + ¯Ψ−k ¯Ψk + (Ψ ↔ Ψ†)).
+(3.9)
+We proceed with the evaluation of the charged partition function Z(eiα), associated with
+the U(1) phase eiα. Putting the previous elements together into (2.12), we get
+Z(eiα) = ⟨b| eiαQqL0+¯L0 |b⟩ ,
+(3.10)
+where q is defined, for later convenience, as
+q ≡ exp
+�
+−
+2π2
+log(L/ε)
+�
+.
+(3.11)
+– 8 –
+
+We further decompose Z(eiα) as a product over the fermionic modes
+Z(eiα) =
+�
+k>0
+⟨0| exp(−i¯ΨkΨk) exp(iαΨ−kΨk + iα¯Ψ−k ¯Ψk)
+×qkΨ−kΨk+k ¯Ψ−k ¯Ψk exp(iΨ−k ¯Ψ−k) |0⟩ × (Ψ → Ψ†).
+(3.12)
+The building block we need to proceed with is the contribution coming from the mode k,
+evaluated as
+⟨0| exp(−i¯ΨkΨk) exp(iαΨ−kΨk + iα¯Ψ−k ¯Ψk)qkΨ−kΨk+k ¯Ψ−k ¯Ψk exp(iΨ−k ¯Ψ−k) |0⟩ =
+⟨0| exp(−i¯ΨkΨk) exp(iq2kei2αΨ−k ¯Ψ−k) |0⟩ = (1 + q2kei2α),
+(3.13)
+where the commutation relations of the modes, together with the property Ψk |0⟩ = ¯Ψk |0⟩ =
+0 (valid for k > 0), have been employed. Putting it all together, we reach to
+Z(eiα) =
+�
+k∈N−1/2
+(1 + q2kei2α)(1 + q2ke−i2α) =
+∞
+�
+m=1
+(1 + 2 cos(2α)q2m−1 + q4m−2). (3.14)
+Before proceeding further, we observe that
+Z(−1) = Z(1),
+(3.15)
+that implies (2.14)
+⟨Ω| eiπQ |Ω⟩ = Z(−1)
+Z(1) = 1.
+(3.16)
+This means that the fermion parity, generated by (−1)Q, is a symmetry of |Ω⟩, as expected.
+In addition, as we will show below, there are no additional symmetries, and Z2 is precisely
+the unbroken subgroup H (see Eq. (1.4)). We provide the explicit expression of Z(eiα) in
+the limit of L/ε → 1, converting the infinite product in an integral
+log Z(eiα) ≃
+� ∞
+0
+dk log(1+q2kei2α)+log(1+q2ke−i2α) =
+1
+2 log q
+�
+Li2(−ei2α) + Li2(−e−i2α)
+�
+.
+(3.17)
+Using the properties of the dilogarithm function, and the definition of q (3.11) we reach the
+final expression of the order parameter
+− log ⟨Ω| eiαQ |Ω⟩ = − log Z(eiα)
+Z(1) = α2
+2π2 log L/ε,
+α ∈ [−π/2, π/2],
+(3.18)
+whose values are periodic under α → α + π. Since it vanishes for eiα = ±1, we identify
+the unbroken group (1.4) with H = {1, −1}, which is nothing but the fermionic parity. As
+a last remark, we notice the presence of a cusp singularity for eiα = ±i, where the order
+parameter is continuous but not differentiable. In Fig 2, we show the behavior of (3.18) as
+a function of α.
+– 9 –
+
+Figure 2: String order parameter as function of α for the Dirac fermion. The function is
+periodic for α → α+π, and shows cusp singularities in correspondence of α = π
+2 +kπ, k ∈ Z.
+As expected, it vanishes at α = kπ, k ∈ Z and it signals the presence of an unbroken Z2
+subgroup.
+3.2
+U(N)-symmetry breaking via boundary scattering matrix
+Here, we consider N Dirac fermions coupled together via a boundary scattering matrix. In
+particular, we take the bulk action
+Sbulk =
+� L
+0
+dx
+�
+dτ
+�
+(Ψ†)j ¯∂Ψj + (¯Ψ†)j∂ ¯Ψj�
+,
+(3.19)
+where the sum over j = 1, . . . , N, the species index, is implicit. Then, we notice that the
+following U(N) symmetry is present
+Ψj → Uj′jΨj′,
+¯Ψj → Uj′j ¯Ψj′,
+(Ψ†)j → ¯Uj′j(Ψ†)j′,
+(¯Ψ†)j → ¯Uj′j(¯Ψ†)j′,
+(3.20)
+with U being a generic N × N unitary matrix and ¯U its conjugate. So far, the species are
+decoupled, so we couple them by the insertion of a boundary term at x = 0
+Sboundary =
+�
+dτ(¯Ψ†)j(x = 0, τ)Sjj′Ψj′(x = 0, τ) + (Ψ†)j(x = 0, τ)S†
+jj′ ¯Ψj′(x = 0, τ),
+(3.21)
+– 10 –
+
+0.12
+0.10
+(Z(eiα) /Z(1)
+0.08
+(log
+0.06
+log(
+0.04
+0.02
+0.00
+-4
+-2
+0
+2
+4
+αwhere S is a unitary N ×N matrix parametrizing the mixing. Notice that, while in general
+the U(N) symmetry is broken, the U(1) symmetry associated with the imbalance between
+particle and quasi-particle is conserved. A naive way to characterize the unbroken group
+H ⊂ U(N), is to identify the set of unitary transformations which leave the boundary term
+(3.21) invariant. We thus require that U ∈ H iff
+¯UljSlmUmj′ = Sjj′,
+(3.22)
+where repeated indices are summed over. Being U unitary, we can rephrase the condition
+above as U −1SU = S or, equivalently, [S, U] = 0.
+To establish the validity of the previous considerations, we aim to characterize the
+string order parameter systematically via BCFT methods.
+We start by identifying the
+boundary state |b⟩ of our model as [17]
+|b⟩ =
+�
+k∈N− 1
+2
+exp
+�
+iSjj′Ψ†j
+−k ¯Ψj′
+−k + (Ψ → Ψ†)
+�
+|0⟩ .
+(3.23)
+The total Hamiltonian is just given by the sum of the single-specie Hamiltonian, and it is
+L0 + ¯L0 =
+�
+k>0
+k
+�
+Ψj
+−kΨj
+k + ¯Ψj
+−k ¯Ψj
+k + (Ψ → Ψ†)
+�
+.
+(3.24)
+The last key ingredient is the action of the symmetry on the fermionic modes. We associate
+to any U ∈ U(N) an operator ˆU satisfying
+ˆUΨj
+−k = Uj′jΨj′
+−k ˆU,
+ˆU ¯Ψj
+−k = Uj′j ¯Ψj′
+−k ˆU,
+ˆU(Ψ†)j
+−k = ¯Uj′j(Ψ†)j′
+−k ˆU,
+ˆU(¯Ψ†)j
+−k = ¯Uj′j(¯Ψ†)j′
+−k ˆU,
+(3.25)
+which is equivalent to Eq. (3.20). Putting everything together, we compute the charged
+partition function
+Z(U) ≡ ⟨b| qL0+¯L0 ˆU |b⟩ =
+�
+k>0
+⟨0| exp(−iS†
+jj′′ ¯Ψj
+k(Ψ†)j
+′
+k ) ˆUqL0+¯L0 exp(iSjj′(Ψ†)j
+−k ¯Ψj
+′
+−k) |0⟩ × (Ψ → Ψ†).
+(3.26)
+Using the commutation relations (3.25) and the invariance of the vacuum |0⟩ under the
+U(N) symmetry, we get
+⟨0| exp(−iS†
+jj′′ ¯Ψj
+k(Ψ†)j
+′
+k ) ˆUqL0+¯L0 exp(iSjj′(Ψ†)j
+−k ¯Ψj
+′
+−k) |0⟩ =
+⟨0| exp(−iS†
+jj′′ ¯Ψj
+k(Ψ†)j
+′
+k ) exp(iq2k(U †SU)jj′(Ψ†)j
+−k ¯Ψj
+′
+−k) |0⟩ =
+det
+�
+1 + q2kS†U †SU
+�
+,
+(3.27)
+– 11 –
+
+where in the final step we applied the formula
+⟨0| exp
+�
+O′jj′Ψj
+k ¯Ψj′
+k
+�
+exp
+�
+O′jj′Ψj
+−k ¯Ψj′
+−k
+�
+|0⟩ = det
+�
+1 − O′O
+�
+,
+(3.28)
+proved in [17]. Summing over the modes k, we finally reach to
+Z(U) =
+�
+k∈N−1/2
+���det
+�
+1 + q2kS†U †SU
+����
+2
+.
+(3.29)
+To proceed further with the computation, it is convenient to introduce the N × N unitary
+matrix
+O ≡ S†U †SU.
+(3.30)
+In this way, we express
+det(1 + q2kO) =
+�
+λ∈Spec(O)
+(1 + q2kλ),
+det(1 + q2kO†) =
+�
+λ∈Spec(O)
+(1 + q2kλ−1),
+(3.31)
+with Spec(O) being the set of eigenvalues of O. After a bit of algebra, we finally reach an
+exact expression for the string-order parameter in the large L/ε limit
+− log ⟨Ω| ˆU |Ω⟩ = − log Z(U)
+Z(1) =
+1
+4π2 log L
+ε
+�
+λ∈Spec(O)
+(Li2(−λ) + Li2(−λ−1) − 2Li2(−1)).
+(3.32)
+At this point, we want to understand for which U ∈ U(N) the order parameter vanishes,
+providing a characterization of the unbroken group H. We first observe that for |λ| = 1 it
+holds
+Li2(−λ) + Li2(−λ−1) − 2Li2(−1) ≥ 0,
+(3.33)
+and the inequality is saturated only for λ = 1. This implies that − log ⟨Ω| ˆU |Ω⟩ = 0 iff
+every eigenvalue of O is 1. In other words, the order parameter vanishes when O = 1, a
+condition equivalent to (see Eq. (3.30))
+[S, U] = 0.
+(3.34)
+This is not particularly surprising, as the naive argument based on the invariance of the
+boundary action leads to the same conclusion. Nevertheless, it establishes its validity and
+allows us to identify the unbroken group as
+H = {U ∈ U(N) | [U, S] = 0}.
+(3.35)
+– 12 –
+
+4
+Complex bosons
+In this section, we generalize the same symmetry breaking patterns discussed for fermions
+in Sec. 3 to free complex massless bosons2. Although many analogies can be recognized,
+and the underlying physics is similar, the analytical predictions for the order parameters
+differ explicitly.
+4.1
+U(1)-symmetry breaking terms in complex bosons
+First, we consider the U(1)-symmetry action
+Sbulk =
+� L
+0
+dx
+�
+dτ ∂Φ ¯∂Φ†.
+(4.1)
+We expand the bosonic field in its Laurent modes as
+Φ(z) =
+�
+k∈Z
+Φk
+zk ,
+¯Φ(¯z) =
+�
+k∈Z
+¯Φk
+¯zk .
+(4.2)
+The bulk action is invariant under the U(1) symmetry
+(Φ, ¯Φ) → eiα(Φ, ¯Φ) ,
+(Φ†, ¯Φ†) → e−iα(Φ†, ¯Φ†).
+(4.3)
+We are interested in a boundary breaking term which breaks U(1) explicitly and preserves
+a Z2 symmetry, as in (3.5). In analogy with the fermionic case, we take
+Sboundary =
+�
+dτ
+�
+Φ(x = 0, τ)¯∂Φ(x = 0, τ) + (Φ → Φ†)
+�
+.
+(4.4)
+Our aim is to compute the string-order parameter via BCFT. To abridge words, using the
+experience from fermions, we need the boundary states, corresponding to the chosen BCs,
+the symmetry generator and the Hamiltonian of the system. These are given respectively
+by
+|b⟩ =
+�
+k>0
+exp
+�
+Φ−k ¯Φ−k + (Φ → Φ†)
+�
+|0⟩,
+(4.5)
+Q =
+�
+k>0
+�
+Φ−kΦk + ¯Φ−k ¯Φk − (Φ → Φ†)
+�
+,
+(4.6)
+and
+L0 + ¯L0 =
+�
+k>0
+k
+�
+Φ−kΦk + ¯Φ−k ¯Φk + (Φ → Φ†)
+�
+.
+(4.7)
+In terms of these quantities, the partition function can be expressed by (3.10), as for
+fermions, with q given by (3.11). We now proceed with the evaluation, decomposing the
+2For more details in the calculations of this section the reader may consult [18].
+– 13 –
+
+partition function as a product of bosonic modes
+Z(eiα) =
+�
+k>0
+⟨0| exp
+�¯ΦkΦk
+�
+exp
+�
+iαΦ−kΦk + iα¯Φ−k ¯Φk
+�
+× qkΦ−kΦk+k¯Φ−k ¯Φk exp
+�
+Φ−k ¯Φ−k
+�
+|0⟩ ×
+�
+Φ → Φ†�
+.
+(4.8)
+The contribution from the k-th mode of Φ is given by
+⟨0| exp
+�¯ΦkΦk
+�
+exp
+�
+iαΦ−kΦk + iα¯Φ−k ¯Φk
+�
+qkΦ−kΦk+k¯Φ−k ¯Φk exp
+�
+Φ−k ¯Φ−k
+�
+|0⟩ =
+⟨0| exp(¯ΦkΦk) exp(q2kei2αΦ−k ¯Φ−k) |0⟩ = (1 − q2kei2α)−1,
+(4.9)
+where we used the commutation relations and the properties Φk |0⟩ = 0 and ¯Φk |0⟩ = 0, for
+k > 0. Taking the contribution of Φ† and putting everything together, we obtain
+Z(eiα) =
+�
+k>0
+�
+1 − q2ke2iα�−1 �
+1 − q2ke−2iα�−1
+=
+∞
+�
+m=1
+�
+1 − 2 cos(2α)q2m + q4m�−1 .
+(4.10)
+We notice that the property
+Z(−1) = Z(1),
+(4.11)
+holds also here, so that a Z2 symmetry is unbroken. The L/ϵ → ∞ limit is obtained by
+converting the infinite product to an integral
+log Z(eiα) ≃ −
+� ∞
+0
+dk log
+�
+1 − q2kei2α�
++ log
+�
+1 − q2ke−i2α�
+= −
+1
+2 log q
+�
+Li2(ei2α) + Li2(e−i2α)
+�
+.
+(4.12)
+In the limit above, we express the order parameter as
+− log ⟨Ω| eiαQ |Ω⟩ = − log Z(eiα)
+Z(1) = log L
+ε
+� α
+2π − α2
+2π2
+�
+,
+α ∈ [0, π],
+(4.13)
+and its value is periodic under α → α + π. Finally, it is worth to recognize explicitly the
+presence of cusp singularities at α = 0 and α = π, which were absent in the fermionic
+counterpart. We show the α-dependence of the order parameter in Fig. 3.
+4.2
+U(N)-symmetry breaking via boundary scattering matrix
+We now consider the bosonic version of the model in 3.2. The bulk action is
+Sbulk =
+� L
+0
+dx
+�
+dτ ∂Φj ¯∂Φ†j.
+(4.14)
+– 14 –
+
+Figure 3: String order parameter as function of α for the complex boson. The function is
+periodic for α → α + π, and shows cusp singularities in correspondence of α = kπ, k ∈ Z.
+It vanishes at α = kπ, k ∈ Z and it signals the presence of unbroken Z2 subgroup.
+where the sum over the specie index j (j = 1, . . . , N) is implicit. The action is invariant
+under the following U(N) transformation
+Φj → Uj′jΦj′,
+¯Φj → Uj′j ¯Φj′,
+(Φ†)j → ¯Uj′j(Φ†)j′,
+(¯Φ†)j → ¯Uj′j(¯Φ†)j′.
+(4.15)
+We take the following boundary action
+Sboundary =
+�
+dτ
+�
+(¯Φ†)j(0, τ)S†
+jj′Φj
+′
+(0, τ) + (Φ†)j(0, τ)Sjj′(¯Φ)j
+′
+(0, τ)
+�
+,
+(4.16)
+where S is a N × N matrix satisfying similar properties with the fermionic case.
+The boundary state, the charge operator and the total Hamiltonian are respectively
+|b⟩ =
+�
+k>0
+exp
+�
+Sjj′Φ†j
+−k ¯Φj
+′
+−k + Φ → Φ†
+�
+,
+Q =
+�
+k>0
+�
+Φj
+−kΦj
+k + ¯Φj
+−k ¯Φj
+k − (Φ → Φ†)
+�
+,
+(4.17)
+L0 + ¯L0 =
+�
+k>0
+k
+�
+Φj
+−kΦj
+k + ¯Φj
+−k ¯Φj
+k + (Φ → Φ†)
+�
+.
+(4.18)
+– 15 –
+
+0.12
+0.10
+(Z(eiα) /Z(1)
+0.08
+0.06
+log(
+0.04
+0.02
+0.00
+-2
+0
+2
+4
+6
+αThe charged partition function reads
+Z(U) ≡ ⟨b| ˆUqL0+¯L0 |b⟩ =
+�
+k>0
+⟨0| exp
+�
+S†
+jj′ ¯Φj
+k(Φ†)j
+′
+k
+�
+ˆUqL0+¯L0 exp
+�
+Sjj′(Φ†)j
+−k ¯Φj
+′
+−k
+�
+|0⟩ ×
+�
+Φ → Φ†�
+=
+�
+k>0
+| det(1 − q2kS†U †SU)|−2,
+(4.19)
+where in the last line we applied the formula
+⟨0| exp
+�
+O′jj′Φj
+k ¯Φj′
+k
+�
+exp
+�
+O′jj′Φj
+−k ¯Φj′
+−k
+�
+|0⟩ = det
+�
+1 − O′O
+�−1 ,
+(4.20)
+proven in [18]. Using the definition (3.30) and the relations
+det(1 − q2kO)−1 =
+�
+λ∈Spec(O)
+(1 − q2kλ)−1,
+det(1 − q2kO†)−1 =
+�
+λ∈Spec(O)
+(1 − q2kλ−1)−1,
+(4.21)
+one finally obtains the expression of the string order parameter in the large L/ϵ limit
+−log ⟨Ω| U |Ω⟩ = − log Z(U)
+Z(1) =
+1
+4π2 log L
+ϵ
+�
+λ∈Spec(O)
+�
+Li2(λ) + Li2(λ−1) − 2Li2(1)
+�
+. (4.22)
+Here, as for fermions, one observes that the order parameter vanishes exactly for O = 1,
+which means [S, U] = 0.
+5
+Lattice results
+In this section, we provide some lattice realizations of free fermions with a boundary sym-
+metry breaking, whose scaling regime is captured by the BCFTs described above.
+For instance, while we are not aware of any lattice realization of the field theory described
+in Sec.3.1, characterized by the boundary-breaking of a U(1) symmetry, we consider a
+system that we conjecture to be its doubling. We describe that system, relating its prop-
+erties to those of the homogeneous counterpart. Finally, we evaluate numerically the order
+parameter in the lattice, and we compare it to the analytical predictions of Sec. 3.1.
+Then, we consider a Fermi chain with the insertion of a conformal defect, whose scat-
+tering properties do not explicitly depend on the incoming momentum [26–28]. This system
+can be regarded as a theory of two species of particles on the half-line coupled together at
+a boundary point, via the so-called unfolding procedure, and its underlying BCFT is de-
+scribed 3.2. In particular, the U(2) symmetry associated with the mixing of the two species
+is broken explicitly due to BCs, encoded in the scattering matrix of the defect.
+– 16 –
+
+5.1
+U(1)-symmetry breaking by a boundary pairing: doubling trick
+Let us first consider the homogeneous hamiltonian
+H = −
+�
+x
+[c†
+xcx+1 + h.c.],
+(5.1)
+describing free fermions hopping on the lattice.
+Here c†
+x, cx are fermionic creation and
+annihilation operators associated with the site x, verifying anticommutation relations
+{c†
+x, cx′} = δxx′,
+{cx, cx′} = 0,
+{c†
+x, c†
+x′} = 0.
+(5.2)
+We now describe the previous hamiltonian in terms of a new set of fermionic operators ax
+and a†
+x, defined by
+cx =
+�
+ax,
+if x ≤ 0
+(−1)xa†
+x,
+if x > 0,
+(5.3)
+that amounts to the exchange of the role of creation and annihilation operators in the right
+half of the system. The explicit expression of the hamiltonian after the mapping becomes
+H = −
+�
+��
+x≤0
+(a†
+xax+1 + h.c.) + a0a1 + a†
+1a†
+0 +
+�
+x≥1
+(a†
+xax+1 + h.c.)
+�
+� .
+(5.4)
+In these new variables, H is no longer homogeneous, and a pairing term appears as a
+localized defect between the sites x = 0, 1. Moreover, the U(1) symmetry
+ax → axeiθ,
+a†
+x → a†
+xe−iθ,
+(5.5)
+shared by the bulk terms of the hamiltonian, is broken explicitly due to the defect. Be-
+fore analyzing further the lattice system, we provide a heuristic argument to explain the
+relationship between the Hamiltonian (5.4) and the CFT in Sec 3.1. Let us consider the
+vacuum3 |0⟩ as the state satisfying
+cx |0⟩ = 0,
+∀x.
+(5.6)
+One can consider one-particle excitations of |0⟩, generated by linear combinations of {c†
+x}
+acting on the vacuum |0⟩. Since in the formulation Eq. (5.1) the system is homogeneous, an
+incoming wave-packet would reach the point x = 0 and propagate across it without being
+partially reflected. The same process can be described in the language of the fermions {ax}x
+given the formulation (5.4). We first notice that |0⟩ satisfies
+ax|0⟩ = 0
+if
+x > 0,
+a†
+x|0⟩ = 0
+if
+x ≤ 0,
+(5.7)
+3This is not the ground state of H. The latter will be denoted below with |Ω⟩.
+– 17 –
+
+namely that the left/right chain is completely empty/filled.
+An incoming right-moving
+excitation can be thus interpreted as a particle that hits the central point, is completely
+transmitted, and then becomes a hole. Similarly, if we considered a Fermi sea, the ex-
+citations would have been given by particles/holes which change their U(1) charge after
+the scattering at x = 0. This mechanism is nothing but an explicit symmetry breaking,
+and its origin can be traced in the term a0a1 + a†
+1a†
+0 of (5.4). A similar scenario has been
+depicted in Sec. 3.1. The crucial difference is that, while in the lattice system (5.4) one
+can recognize both left and right-moving incoming particles, the incoming particles of 3.1
+are just left-moving. Heuristically, we thus conjecture that (5.4) is a discretization of the
+QFT depicted above once the degrees of freedom are doubled.
+To better motivate this argument, in the following we analyze the order parameter
+associated with the U(1)-symmetry breaking. We firstly regularize the hamiltonian H in
+(5.1), keeping the size of the system finite x ∈ [−L + 1, L], and we consider its ground state
+|Ω⟩. In the language of the fermionic operators cx, it can be regarded as a Fermi sea at
+half-filling, namely, the total number of particles is N = L, and it satisfies
+�
+��
+x≤0
+c†
+xcx +
+�
+x>0
+c†
+xcx
+�
+� |Ω⟩ = N|Ω⟩.
+(5.8)
+While the hamiltonian is not changed after the mapping (5.3), and so its ground state
+|Ω⟩, the description in terms of the operators ax is less transparent, as particles and holes
+(antiparticles) are mixed among each other by the pairing. A preliminary observation is
+that
+�
+��
+x≤0
+a†
+xax −
+�
+x>0
+a†
+xax
+�
+� |Ω⟩ = N|Ω⟩,
+(5.9)
+namely, the imbalance of particles among the left/right half-chain is fixed, and it does not
+fluctuate. Then, we express the generator of the U(1) symmetry ax → axeiθ as
+Q =
+�
+x
+a†
+xax =
+�
+x≤0
+c†
+xcx −
+�
+x>0
+c†
+xcx = NL − NR,
+(5.10)
+with NL,R the number operator in the left/right chain before the mapping (5.3). Crucially,
+the ground state fluctuations of Q are strictly related to those of 2NL, as it holds
+Q |Ω⟩ = (NL − NR) |Ω⟩ = (2NL − N) |Ω⟩ ,
+(5.11)
+and N = L is just an additive constant. In this way, we can finally express the full-counting
+statistics of Q as
+⟨Ω| eiαQ |Ω⟩ ∼ ⟨Ω| ei2αNL |Ω⟩ ,
+(5.12)
+up to an irrelevant proportionality constant. In conclusion, Eq. (5.12) gives an effective way
+to characterize the order parameter of the U(1) breaking in Eq. (5.4) to the full counting
+statistics of a subsystem in the homogeneous chain (5.1), which is easier to compute.
+– 18 –
+
+We now show how to express ⟨Ω| ei2αNL |Ω⟩, employing standard techniques for free
+fermions [29]. We first construct the correlation matrix of the ground state, defined as
+Cxx′ ≡ ⟨Ω| c†
+xcx′ |Ω⟩ .
+(5.13)
+Since |Ω⟩ is a Fermi sea with N = L particles, one can express
+Cxx′ =
+L
+�
+j=1
+φj(x)φj(x′),
+x ∈ [−L + 1, L]
+(5.14)
+with φj(x) being the single-particle eigenfunction of the Hamiltonian (5.1).
+Then, we
+restrict the spatial indices to the left side of the chain, obtaining a L × L matrix CA
+satisfying
+(CA)xx′ = Cxx′,
+x ∈ [−L + 1, 0].
+(5.15)
+Given the eigenvalues of CA, denoted by Spec(CA) = {νj}j=1,...,L, one can show that the
+following relation holds
+⟨ei2αNL⟩ =
+L
+�
+j=1
+[νje2iα + (1 − νj)].
+(5.16)
+Making use of Eq. (5.12), one finally gets the string-order parameter as
+⟨Ω| eiαQ |Ω⟩ = ⟨ei2αNL⟩ =
+L
+�
+j=1
+[νje2iα + (1 − νj)].
+(5.17)
+An explicit analytical expression for the eigenvalues {νj}j is a hard task, and should rely on
+numerics. Nevertheless, simple field theoretical arguments can capture the correct leading
+behavior of the order parameter, as we explain below. The state |Ω⟩ can be recovered as a
+euclidean path integral of the massless Dirac fermions over the strip
+x ∈ [−L, L],
+τ ∈ (−∞, ∞),
+(5.18)
+with τ representing the Euclidean time, and we assume L to be much larger than the lattice
+size. Furthermore, the operator eiαNL is described by the insertion of two vertex operators
+in the path integral at τ = 0, as
+eiαNL ∼ Vα(x = −L)V−α(x = 0).
+(5.19)
+It is known [23] that the bulk scaling dimension ∆α of Vα is
+∆α =
+� α
+2π
+�2
+,
+α ∈ [−π, π].
+(5.20)
+Moreover, the insertion of the boundary operator Vα(x = −L) does not play a role, since
+the BCs at x = −L are U(1) symmetric, and from now on we just discard it. Making use
+– 19 –
+
+Figure 4: Order parameter as function of L for different values of α. The symbols are the
+numerical data obtained from Eq.(5.17). The solid lines show the curve (α2/π2) log L+b0+
+b1L−1, where the first term is the analytical prediction given in Eq. (5.22) and coefficients
+b0, b1 are obtained fitting the data.
+of scale-invariance, we finally reach to
+⟨Ω| ei2αNL |Ω⟩ ∼ ⟨Ω| V−2α(x = 0) |Ω⟩ ∼
+1
+L∆2α =
+1
+Lα2/π2 ,
+α ∈ [−π/2, π/2],
+(5.21)
+where we employed scaling arguments on the bulk field. In conclusion, thanks to (5.12), we
+finally obtain
+− log | ⟨Ω| eiαQ |Ω⟩ | ≃ α2
+π2 log L,
+α ∈ [−π/2, π/2]
+(5.22)
+which is the main result of this section. We emphasize that the formula we obtained is just
+twice the prediction in Section 3.1. This is somehow expected, as we conjectured that the
+Hamiltonian (5.4) describes a doubling of the system in Sec. 3.1.
+5.2
+Conformal defect and U(2) symmetry breaking
+Let us consider a system made of two species of spinless fermions on the lattice x ∈ [0, L−1],
+coupled together at the boundary point x = 0 through a conformal defect.
+– 20 –
+
+α = 0.5
+1.2
+α = 0.8
+α=1
+1.0
+α = 1.2
+C
+1.5
+2a)
+0.8
+0.6
+0.4
+0.2
+0.0
+0
+25
+50
+75
+100
+125
+150
+175
+200
+LThe hamiltonian is
+H = −
+�
+x,j
+[(c†
+x)jcj
+x+1 + h.c.] −
+�
+jj′
+Sjj′(c†
+0)j(c0)j′,
+(5.23)
+where (c†
+x)j, (cx)j are fermionic operators verifying anticommutation relations
+{(c†
+x)j, (cx′)j′} = δxx′δjj′,
+{(cx)j, (cx′)j′} = 0,
+{(c†
+x)j, (c†
+x′)j′} = 0,
+j = 1, 2,
+(5.24)
+and S is the following 2 × 2 matrix
+S =
+�√
+1 − λ2
+λ
+λ
+−
+√
+1 − λ2
+�
+,
+λ ∈ [0, 1].
+(5.25)
+It is possible to show that the scattering matrix, induced by the boundary term in the
+hamiltonian, is exactly S [24, 25, 28]. Since the transmission/reflection probability does not
+depend on the incoming momenta of the particle, and it is given by λ2, 1 − λ2 respectively,
+the defect is scale-invariant, and it is described by a BCFT. It is easy to show that the
+bulk terms on the hamiltonian are invariant under a U(2) symmetry which mixes the two
+species as
+cj
+x → Uj′jcj′
+x ,
+(c†
+x)j → ¯Uj′j(c†
+x)j′,
+(5.26)
+with U a generic 2x2 unitary matrix, and the sum over j′ is implicit. However, the presence
+of the defect breaks explicitly the symmetry. Indeed, after a generic U(2) transformation,
+the boundary term is mapped onto
+− Sjj′U j1jUj2j′(c†
+0)j1(c0)j2,
+(5.27)
+and it is invariant only if U †SU = S, namely [U, S] = 0. While the condition above was
+already derived in CFT, it is instructive to notice that it holds in the lattice model too.
+For the sake of completeness, we characterize explicitly the unbroken subgroup H defined
+by
+H = {U ∈ U(2)|[S, H] = 0} .
+(5.28)
+Since S is hermitian and unitary, its eigenvalues have to be real phases, and they are just
+±1. In the basis in which S is diagonal, also U ∈ H has to be diagonal, since S and H
+commute, and so they share the same eigenspaces. This means that, given a matrix D
+which diagonalizes S, say
+S = D
+�
+1 0
+0 −1
+�
+D−1,
+D =
+�
+�
+λ
+√
+2−2
+√
+1−λ2
+λ
+√
+2+2
+√
+1−λ2
+√
+1−
+√
+1−λ2
+√
+2
+−
+√
+1+
+√
+1−λ2
+√
+2
+�
+� ,
+(5.29)
+– 21 –
+
+we identify the unbroken group as
+H =
+�
+U = D
+�
+eiα1
+0
+0
+eiα2
+�
+D−1, α1, α2 ∈ R
+�
+.
+(5.30)
+6
+Conclusions and Outlooks
+In this work, we considered the effect of an explicit symmetry breaking in a one-dimensional
+critical system induced by a localized impurity. We develop a general formalism to probe
+the symmetry breaking that can be applied to any system described by a BCFT. In par-
+ticular, we show that − log | ⟨Ω| ˆg |Ω⟩ |, which is a non-local order parameter, is generically
+logarithmic growing in the system size, and it vanishes precisely for the unbroken elements
+of the group. We provide exact calculations for free theories, fermions and bosons, with
+both abelian and non-abelian broken symmetries. For instance, we first consider the sym-
+metry breaking of a complex theory in the presence of a boundary pairing term, which has
+a residual Z2 symmetry. Then, we consider a theory made of many species of particles
+coupled together at a boundary point, and a non-trivial breaking of a U(N) symmetry is
+investigated.
+A natural generalization would be the calculation of the order parameter for rational
+minimal models (Ising, Potts model, Tricritical Ising, and so on)[11]. Indeed, their scale-
+invariant BCs are known, and they are mapped onto the so-called Cardy states.
+The
+evaluation of the overlap in Eq. (2.12) might be more involved for interacting CFTs, but
+we think it should be expressed as Virasoro characters, whose explicit expressions are known.
+One may also wonder what happens when the theory is not described by a BCFT,
+or due to a finite correlation length in the bulk ξ or in the presence of BCs which are
+not scale invariant [30–37]. In the first case, the boundary effect is localized in a typical
+distance ξ and we expect that the order parameter goes to a constant when the system size
+is increased. It might be interesting to study this scenario in Integrable Field Theories,
+providing exact results for massive theories. In the second case, the BCs are expected to
+flow to some scale-invariant ones via RG flow. Then, for a big enough size, the leading
+growth of the order parameter would be logarithmic, with a universal prefactor depending
+only on the IR fixed point. We hope to come back to these problems in the future.
+Finally, we mention that while our approach involves a genuine non-local probe of
+the system, one might be interested in the local observables. In particular, it has been
+proposed [38] that a subsystem measure dubbed as entanglement-asymmetry might be a
+good probe of the symmetry breaking. An interesting direction would be the computation
+of the entanglement asymmetry for an interval attached to the boundary point in BCFT.
+– 22 –
+
+Acknowledgments
+RB acknowledges support from the Croatian Science Foundation (HrZZ) project No. IP-
+2019-4-3321. LC acknowledges support from ERC under Consolidator grant number 771536
+(NEMO). The work of P. Panopoulos was supported by the Croatian Science Foundation
+Project "New Geometries for Gravity and Spacetime" (IP-2018-01-7615).
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+– 24 –
+
diff --git a/J9FAT4oBgHgl3EQfvh5f/content/tmp_files/load_file.txt b/J9FAT4oBgHgl3EQfvh5f/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..2f2f98010d66036c7223b39a8fb511c921e02c1e
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+++ b/J9FAT4oBgHgl3EQfvh5f/content/tmp_files/load_file.txt
@@ -0,0 +1,713 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf,len=712
+page_content='Prepared for submission to JHEP  Boundary Symmetry Breaking in CFT and the  String Order Parameter  Riccarda Bonsignori,a Luca Capizzi,b Pantelis Panopoulosa  aRudjer Boskovic Institute, Division of Theoretical Physics, Bijenička 54, 10000 Zagreb, Croatia  bSISSA and INFN Sezione di Trieste, via Bonomea 265, I-34136 Trieste, Italy  E-mail: rbonsign@irb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='hr, lcapizzi@sissa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='it, Pantelis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='Panopoulos@irb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='hr  Abstract:  We consider the ground state of a one-dimensional critical quantum system  carrying a global symmetry in the bulk, which is explicitly broken by its boundary con-  ditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' We probe the system via a string-order parameter, showing how it detects the  symmetry breaking pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' We give a precise characterization of the mechanism depicted  above in Boundary CFT, and we find a general logarithmic scaling for the order parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='  As a first example we analyze the breaking of a U(1) symmetry for complex free theories in-  duced by a boundary pairing term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' Moreover, we give predictions for the breaking of U(N)  in free theories, arising from a boundary mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' We test our predictions with numerical  calculations for some lattice realizations of free fermionic system with boundary symmetry  breaking, finding a good agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='  Keywords: Boundary Conformal Field Theory, String Order Parameter, Charged Parti-  tion Function  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='08676v1  [hep-th]  20 Jan 2023  Contents  1  Introduction  1  2  Definitions and Techniques  3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='1  String order parameter  3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='2  BCFT description  5  3  Dirac fermions  7  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='1  U(1)-symmetry breaking due to a boundary pairing term  7  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='2  U(N)-symmetry breaking via boundary scattering matrix  10  4  Complex bosons  13  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='1  U(1)-symmetry breaking terms in complex bosons  13  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='2  U(N)-symmetry breaking via boundary scattering matrix  14  5  Lattice results  16  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='1  U(1)-symmetry breaking by a boundary pairing: doubling trick  17  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='2  Conformal defect and U(2) symmetry breaking  20  6  Conclusions and Outlooks  22  1  Introduction  Symmetry is a cornerstone of modern physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='  Its importance emerges in all branches  of physics among which the Condensed Matter Theory [1, 2] (ferromagnetism, supercon-  ductivity, superfluidity) and High-Energy Physics [3] (formulation of the Standard Model,  Quantum gravity etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' There are many cases where symmetry is responsible for phase  transitions in statistical systems [4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' In particular, behind plenty of classical and quan-  tum systems showing a (second order) phase transition there is an underlying symmetry  that can be preserved or spontaneously broken: this mechanism has been a guideline for  the characterization of the possible phases of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='  As an illustrative example, we consider a toy model for ferromagnetism, namely the  classical Ising model enjoying global symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' It has been shown that in d ≥ 2 dimen-  sions, the model undergoes a phase transition at a critical value of the temperature T = Tc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='  Moreover, for T > Tc the Z2 symmetry of the model is unbroken, and the system has a  paramagnetic behavior, while for T < Tc the system undergoes to a Spontaneous Symmetry  Breaking (SSB) characterizing the ferromagnetic phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' The breaking of the symmetry can  be spotted via the probing of the magnetization, described in terms of a local field σ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='  In particular, its expectation value ⟨σ(x)⟩ vanishes at T > Tc and it gets a finite value for  T < Tc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' The critical point T = Tc deserves special attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' Indeed, while at this point the  – 1 –  symmetry remains unbroken and thus ⟨σ(x)⟩ = 0, an additional application of a magnetic  field in this critical temperature can have non-trivial effects even far from that point, due to  the slow (algebraic) decay of the correlation functions, a distinct trait of the critical phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='  Historically, the importance of localized perturbations for critical systems has not been  taken for granted, until the discovery of the Kondo effect [6, 7], which is associated with  anomalous transport properties of low-temperature metals in the presence of impurities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='  Despite the progresses in this field, both experimentally and theoretically (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' Boundary  Conformal Field Theory (BCFT) formulation [9, 30, 35]), a comprehensive theory of the  symmetry and its breaking pattern seems to remain an open problem for such systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' The  main motivation behind this work is to address more systematically this lack of approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='  We are interested in the ground-state |Ω⟩ of the following Hamiltonian  H = H0 + H1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='1)  The H0 is a bulk term involving short-range interactions, which we assume to be critical,  and invariant under the action of a group G associated with a global symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' In turn,  the H1 term is a perturbation localized in space (say defect/impurity) which explicitly  spoils the G-invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' More precisely, we consider a unitary representation of G, which  associates to any g ∈ G a unitary operator ˆg of the Hilbert space, and we require that  [H0, ˆg] = 0,  [H1, ˆg] ̸= 0,  ∀g ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='2)  Our main goal is to understand whether and how the ground state |Ω⟩ of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='2) breaks the  global G-invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' For this purpose, we propose  ⟨Ω| ˆg |Ω⟩ ,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='3)  regarded as a function of G, to be a good non-local order parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' For one-dimensional  systems, ˆg can be naively regarded as a string operator inserted along the whole system,  and for this reason, we call it string order parameter (following the terminology of [12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='  We anticipate that, rather generically, the unbroken symmetry group is identified by  H = {g ∈ G| | ⟨Ω| ˆg |Ω⟩ | = 1},  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='4)  while for the other elements of the group one has | ⟨Ω| ˆg |Ω⟩ | < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' Moreover, for a critical  system defined on a finite-size region [0, L], we find that in the presence of scale-invariant  symmetry-breaking boundary terms, the (log of the) order parameter shows the following  logarithmic growth  − log | ⟨Ω| ˆg |Ω⟩ | ∼ log L,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='5)  in the limit of large L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' In particular the quantity  lim  L→∞ − log | ⟨Ω| ˆg |Ω⟩ |  log L  ,  g ∈ G  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='6)  – 2 –  is a universal continuous, but in general not smooth function of the group G, which vanishes  precisely for g ∈ H, with H being the unbroken subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='  We provide an accurate description of this mechanism in the context of BCFT, ex-  ploiting the power of conformal symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' Therefore, we compute explicitly the string  order parameter for a class of free fermionic and bosonic theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' In the first case, we  consider a free Dirac fermion on [0, L] and we insert a pairing term at one boundary point,  studying the symmetry breaking pattern U(1) → Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' We extend our analysis by taking  N copies of uncoupled the Dirac fields in the bulk, but coupled at one boundary point  via a scattering matrix S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' In this case, we find a novel non-trivial breaking of the U(N)  symmetry, strongly related to the symmetries of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' By repeating the same approach we  study the string order parameter for free complex massless bosons under U(1) and under  U(N) symmetry seperately as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' Finally, we present two possible realizations of the  boundary symmetry breaking mechanism for free fermionic systems on the lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' For one  of them, we also perform the numerical calculation of the string order parameter to test  our analytical prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='  Our manuscript is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' 2 we provide some general definitions  and give a BCFT description of the string order parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' 3 we analyze in detail  the free fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' 4 we repeat the same analysis for free bosons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' 5 we  consider the lattice counterpart for fermions and present the numerical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' Finally, in  Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' 6 we gather our results and discuss some possible future directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='  2  Definitions and Techniques  The purpose of this section is two-fold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' First, once the non-local order parameter is intro-  duced, we explain how and why it detects symmetry-breaking, providing some properties  which are independent of the detail of the systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' Then, we specify the treatment to 1+1  BCFT with symmetry breaking terms at the boundaries, and give a general derivation of  the logarithmic growth in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' At this point, we will not specialize to any specific  theory, and we derive the first results employing only conformal symmetry and describing  the boundary conditions (BCs) as boundary states, via a space-time duality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='1  String order parameter  Let us first review what is a symmetry in a quantum system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' Given a Hilbert space H , a  unitary representation of a group G is a linear map  G → GL(H ) ,  g → ˆg  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='1)  from the group onto the unitary operators of H .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' One requires that the map is homomor-  phic, namely  ˆ  (g1g2) = ˆg1 ˆg2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='2)  Without loss of generality, here we assume that the map is injective, so that distinct elements  of the group are represented by distinct operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' We now consider a (normalized) state  – 3 –  |Ω⟩ ∈ H .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' We say that |Ω⟩ is symmetric (invariant) under G iff  ˆg |Ω⟩ = eiφ(g) |Ω⟩ ,  ∀g ∈ G  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='3)  with eiφ(g) being a g-dependent phase factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' The requirement above severely constrains the  expectation values of the observables, which is the main reason why the order parameters  are useful to detect the breaking of symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='  As a first example, we consider a quantum system carrying a representation of Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='  We denote by {1, τ} the generators of the group, and consider an observable σ, say the  magnetization, odd under Z2  ˆτσˆτ −1 = −σ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='4)  which plays the role of the order parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' Whenever a state |Ω⟩, say the (a) ground  state, is invariant under Z2 one safely concludes that  ⟨Ω| σ |Ω⟩ = ⟨Ω| ˆτ −1σˆτ |Ω⟩ = − ⟨Ω| σ |Ω⟩ ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='5)  which clearly implies that ⟨Ω| σ |Ω⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' This means that, whenever ⟨Ω| σ |Ω⟩ ̸= 0 one can  be sure that the state |Ω⟩ is not Z2 invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' Unfortunately, in principle, the converse is  not true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' Indeed, there is no reason why ⟨Ω| σ |Ω⟩ = 0 should imply a Z2 symmetry for |Ω⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='  This is somehow the main disadvantage behind the usage of the usual order parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='  In contrast, as we will show below, if one considers ˆg itself as an order parameter, one  can unambiguously understand if |Ω⟩ is symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' A first immediate observation is that,  if |Ω⟩ is symmetric under g ∈ G (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='3)), then  | ⟨Ω| ˆg |Ω⟩ | = | ⟨Ω| eiφ(g) |Ω⟩ | = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='6)  Less trivially, one can show the converse, namely that | ⟨Ω| ˆg |Ω⟩ | = 1 implies that the  symmetry has to be unbroken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' To prove it, we employ the Cauchy-Schwarz inequality [14],  which tells us  | ⟨Ω| ˆg |Ω⟩ | ≤ | ⟨Ω| ˆg†ˆg |Ω⟩ | · | ⟨Ω|Ω⟩ | = 1,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='7)  where the inequality is strict unless |Ω⟩ and ˆg |Ω⟩ are proportional, which is exactly the  notion of symmetry in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' To summarize, so far we have that  | ⟨Ω| ˆg |Ω⟩ | = 1  if and only if  ˆg |Ω⟩ = eiφ(g) |Ω⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='8)  This property suggests a way to characterize the subgroup H ⊆ G which leaves |Ω⟩ invariant  as Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' We would like to stress explicitly that, in their simplicity, the last conclusions  are very general and apply to both abelian and non-abelian symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' Notice that up to  this point our discussion is general, since we did not specify whether the symmetry breaking  pattern G → H, associated with the state |Ω⟩, arises from an explicit or spontaneous  symmetry breaking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' For the reasons depicted above, the investigation of | ⟨Ω| ˆg |Ω⟩ |, the  non-local (string) order parameter, is the main goal of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='  – 4 –  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='2  BCFT description  Let us proceed to the description of the string order parameter in the framework of BCFT  [11, 15–18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' We consider the ground state |Ω⟩ of a one-dimensional critical quantum system  on a finite size geometry, namely the interval [0, L].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' We assume that the bulk is described  by a CFT, and we impose conformal invariant BCs at x = 0, L [11, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' Introducing the  Euclidean time, one can describe the state |Ω⟩ as a strip geometry, parametrized by the  complex coordinate w satisfying  Re(w) ∈ [0, L],  Im(w) ∈ (−∞, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='9)  In particular, Re(w) represents the spatial position, and Im(w) corresponds to the Euclidean  time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' In this picture, the expectation values of the observables in the state |Ω⟩ are described  via the insertion of fields in the strip geometry depicted in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='  We now assume that the theory has a global symmetry in the bulk, characterized by  a representation of a group G, which may be eventually broken by the choice of the BCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='  We are interested in the symmetry breaking pattern, strictly related to the evaluation of  the string order parameter ⟨Ω| ˆg |Ω⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' Since the symmetry is global, the action of the group  G is nontrivial at any spatial point x ∈ [0, L].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' For this reason, it is natural to represent  pictorially ˆg as a line operator extended over Im(w) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' Then, ⟨Ω| ˆg |Ω⟩ becomes a charged  partition function given by the insertion of a charged line connecting w = 0 and w = L in  the strip geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='  The last key ingredient is the specification of the BCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' We denote by b, b′ the type of  BCs at x = 0, L respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' In the strip geometry, these boundary points become lines  extended over the euclidean time, Re(w) = 0, L respectively, and we associate the labels  b, b′ to each of these lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' We assume that b′, corresponding to x = L, preserves explicitly  the symmetry, and we do not characterize it further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' Instead, we focus on BCs at x = 0  which breaks G explicitly, and our goal is to identify the corresponding symmetry breaking  pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='  At this point, we employ conformal symmetry to relate the strip geometry described  above to another geometry and we do that to simplify the computation of the charged  partition function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' After a UV and IR regulation of the original geometry, keeping only the  points ε < |w| < L, we apply the transformation [19–22]  z = log w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='10)  The new geometry is the rectangle  Re(z) ∈ (log ε, log L),  Im(z) ∈ [−π/2, π/2],  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='11)  with BCs of type b along Im(z) = ±π/2, corresponding to boundary states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' It is important  to stress that the information about b′ is lost explicitly by the choice of IR regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='  However, this is not big deal, as we required that b′ is G invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' Indeed, on the physical  ground, we do not expect any contribution to the string order parameter from the point  – 5 –  Figure 1: The expectation value of the string order parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' We represent the original  strip geometry (coordinate w) and the rectangular one (coordinate z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' The red/blue lines  correspond to the BCs of type b, b′ respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' The insertion of the symmetry operator ˆg  is a green line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='  x = L (at least at leading order).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='  Putting these information together, we express the charged partition function as a  transition element in Euclidean time between the boundary state |b⟩, corresponding to b  (see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' [11]), with itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' More precisely, we define  Z(g) ≡ ⟨b| exp(−πH)ˆg |b⟩ ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='12)  with H being the Hamiltonian in the rectangular geometry  H =  2π  log(L/ε)(L0 + ¯L0),  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='13)  and L0, ¯L0 the Virasoro generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' Then, we express the expectation value of ˆg as  ⟨Ω| ˆg |Ω⟩ = Z(g)  Z(1),  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='14)  where the denominator Z(1) arises only as a normalization constant (the uncharged parti-  tion of the rectangle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' We represent the construction above in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' 1, showing the insertion  of ˆg in the two geometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' So far, the discussion is general and it applies to any BCFT  carrying a global symmetry G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' The precise evaluation of the charged partition function  Z(g) requires the characterization of the boundary state |b⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' Still, in general, one can argue  that the logarithm of the partition function (free energy) is extensive in the large-size limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='  – 6 –  (2|g/2)  2(g)  Im  10  1  11  1  0  1  Z=  9  1  T  I  1  3= |ml  1  1  1  b L  log -  LIn particular, we have  log Z(g) ∝ log L  ε ,  L/ε → ∞  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='15)  up to a proportionality constant depending on both g ∈ G and the BCs b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' Summarizing,  we discover that the ratio  − log | ⟨Ω| ˆg |Ω⟩ |  log L/ε  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='16)  is finite in the limit L → ∞, where the conventional minus sign ensures a positive value  result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' In the next section, we will carefully analyze that ratio for specific CFTs, relating  its behavior to the symmetry breaking pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='  3  Dirac fermions  We now proceed with an application of the previous discussion to the system of free fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='  First, we discuss free fermions with U(1) symmetry and then we continue with U(N)  generalization1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='1  U(1)-symmetry breaking due to a boundary pairing term  We consider the theory of massless Dirac fermions in a finite-size geometry, taking BCs  that break explicitly the U(1) symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' The model is described by two fields Ψ and Ψ†,  that correspond to particle and antiparticles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' In radial quantization, one decomposes Ψ in  its left and right Laurent modes as follows  Ψ(z) =  �  k∈Z+1/2  Ψk  zk+1/2 ,  ¯Ψ(¯z) =  �  k∈Z+1/2  ¯Ψk  ¯zk+1/2 ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='1)  where the Neveu-Schwarz (NS) sector has been considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' For k > 0 the modes Ψk, ¯Ψk  destroy a left/right moving particle, while Ψ−k, ¯Ψ−k are creation operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' Similar con-  siderations hold for Ψ†, the antiparticle field, and we refer to its left/right modes with  Ψ†  k, ¯Ψ†  k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='  The bulk action in Euclidean space of the finite [0, L] geometry reads  Sbulk =  � L  0  dx  �  dτ  �  Ψ† ¯∂Ψ + ¯Ψ†∂ ¯Ψ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='2)  enjoying a U(1) global symmetry  Ψ → eiαΨ,  ¯Ψ → eiα ¯Ψ,  Ψ† → e−iαΨ†,  ¯Ψ† → e−iα ¯Ψ†,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='3)  and it corresponds to the imbalance between particles and antiparticles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' We want to break  U(1) explicitly through the BCs at x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' We do so via the insertion of a pairing term at  1For more details in the calculations of this section the reader may consult [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='  – 7 –  the boundary point, described by a boundary action  Sboundary =  �  dτ  �  ¯Ψ(x = 0, τ)Ψ(x = 0, τ) + (Ψ → Ψ†)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='4)  The boundary term is not U(1) invariant, since it transforms as Ψ¯Ψ → ei2αΨ¯Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' A residual  Z2 is nevertheless preserved (associated with α = 0, π), and it describes the conservation  of fermion parity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' One thus expects that these BCs should induce an explicit symmetry  breaking pattern  U(1) → Z2,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='5)  on the ground state |Ω⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='  While these considerations are so far not rigorous, they can  capture the key features of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' In the following, we aim to compute the string-  order parameter via the BCFT techniques (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='2) for the U(1) symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' The first quantity  we need is the boundary state |b⟩ associated with the U(1) breaking BCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' We consider a  coherent state in which pairs of particles (and antiparticles) with opposite momenta are  generated above the ground states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' Its explicit expression is  |b⟩ =  �  k>0  exp(iΨ−k ¯Ψ−k + (Ψ → Ψ†)) |0⟩ ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='6)  with |0⟩ being the vacuum of the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' The generator of the symmetry is  Q =  �  k>0  Ψ−kΨk + ¯Ψ−k ¯Ψk − (Ψ ↔ Ψ†),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='7)  and we parametrize a generic element of U(1) as  ˆg = eiαQ,  α ∈ [−π, π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='8)  Finally, we remind the relation between the Virasoro modes and the fermionic modes  L0 + ¯L0 =  �  k>0  k(Ψ−kΨk + ¯Ψ−k ¯Ψk + (Ψ ↔ Ψ†)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='9)  We proceed with the evaluation of the charged partition function Z(eiα), associated with  the U(1) phase eiα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' Putting the previous elements together into (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='12), we get  Z(eiα) = ⟨b| eiαQqL0+¯L0 |b⟩ ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='10)  where q is defined, for later convenience, as  q ≡ exp  �  −  2π2  log(L/ε)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='11)  – 8 –  We further decompose Z(eiα) as a product over the fermionic modes  Z(eiα) =  �  k>0  ⟨0| exp(−i¯ΨkΨk) exp(iαΨ−kΨk + iα¯Ψ−k ¯Ψk)  ×qkΨ−kΨk+k ¯Ψ−k ¯Ψk exp(iΨ−k ¯Ψ−k) |0⟩ × (Ψ → Ψ†).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='12)  The building block we need to proceed with is the contribution coming from the mode k,  evaluated as  ⟨0| exp(−i¯ΨkΨk) exp(iαΨ−kΨk + iα¯Ψ−k ¯Ψk)qkΨ−kΨk+k ¯Ψ−k ¯Ψk exp(iΨ−k ¯Ψ−k) |0⟩ =  ⟨0| exp(−i¯ΨkΨk) exp(iq2kei2αΨ−k ¯Ψ−k) |0⟩ = (1 + q2kei2α),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='13)  where the commutation relations of the modes, together with the property Ψk |0⟩ = ¯Ψk |0⟩ =  0 (valid for k > 0), have been employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' Putting it all together, we reach to  Z(eiα) =  �  k∈N−1/2  (1 + q2kei2α)(1 + q2ke−i2α) =  ∞  �  m=1  (1 + 2 cos(2α)q2m−1 + q4m−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='14)  Before proceeding further, we observe that  Z(−1) = Z(1),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='15)  that implies (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='14)  ⟨Ω| eiπQ |Ω⟩ = Z(−1)  Z(1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='16)  This means that the fermion parity, generated by (−1)Q, is a symmetry of |Ω⟩, as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='  In addition, as we will show below, there are no additional symmetries, and Z2 is precisely  the unbroken subgroup H (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='4)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' We provide the explicit expression of Z(eiα) in  the limit of L/ε → 1, converting the infinite product in an integral  log Z(eiα) ≃  � ∞  0  dk log(1+q2kei2α)+log(1+q2ke−i2α) =  1  2 log q  �  Li2(−ei2α) + Li2(−e−i2α)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='17)  Using the properties of the dilogarithm function, and the definition of q (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='11) we reach the  final expression of the order parameter  − log ⟨Ω| eiαQ |Ω⟩ = − log Z(eiα)  Z(1) = α2  2π2 log L/ε,  α ∈ [−π/2, π/2],  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='18)  whose values are periodic under α → α + π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' Since it vanishes for eiα = ±1, we identify  the unbroken group (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='4) with H = {1, −1}, which is nothing but the fermionic parity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' As  a last remark, we notice the presence of a cusp singularity for eiα = ±i, where the order  parameter is continuous but not differentiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' In Fig 2, we show the behavior of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='18) as  a function of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='  – 9 –  Figure 2: String order parameter as function of α for the Dirac fermion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' The function is  periodic for α → α+π, and shows cusp singularities in correspondence of α = π  2 +kπ, k ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='  As expected, it vanishes at α = kπ, k ∈ Z and it signals the presence of an unbroken Z2  subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='2  U(N)-symmetry breaking via boundary scattering matrix  Here, we consider N Dirac fermions coupled together via a boundary scattering matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' In  particular, we take the bulk action  Sbulk =  � L  0  dx  �  dτ  �  (Ψ†)j ¯∂Ψj + (¯Ψ†)j∂ ¯Ψj�  ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='19)  where the sum over j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' , N, the species index, is implicit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' Then, we notice that the  following U(N) symmetry is present  Ψj → Uj′jΨj′,  ¯Ψj → Uj′j ¯Ψj′,  (Ψ†)j → ¯Uj′j(Ψ†)j′,  (¯Ψ†)j → ¯Uj′j(¯Ψ†)j′,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='20)  with U being a generic N × N unitary matrix and ¯U its conjugate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' So far, the species are  decoupled, so we couple them by the insertion of a boundary term at x = 0  Sboundary =  �  dτ(¯Ψ†)j(x = 0, τ)Sjj′Ψj′(x = 0, τ) + (Ψ†)j(x = 0, τ)S†  jj′ ¯Ψj′(x = 0, τ),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='21)  – 10 –  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='12  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='10  (Z(eiα) /Z(1)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='08  (log  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='06  log(  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='00  4  2  0  2  4  αwhere S is a unitary N ×N matrix parametrizing the mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' Notice that, while in general  the U(N) symmetry is broken, the U(1) symmetry associated with the imbalance between  particle and quasi-particle is conserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' A naive way to characterize the unbroken group  H ⊂ U(N), is to identify the set of unitary transformations which leave the boundary term  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='21) invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' We thus require that U ∈ H iff  ¯UljSlmUmj′ = Sjj′,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='22)  where repeated indices are summed over.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' Being U unitary, we can rephrase the condition  above as U −1SU = S or, equivalently, [S, U] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='  To establish the validity of the previous considerations, we aim to characterize the  string order parameter systematically via BCFT methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='  We start by identifying the  boundary state |b⟩ of our model as [17]  |b⟩ =  �  k∈N− 1  2  exp  �  iSjj′Ψ†j  −k ¯Ψj′  −k + (Ψ → Ψ†)  �  |0⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='23)  The total Hamiltonian is just given by the sum of the single-specie Hamiltonian, and it is  L0 + ¯L0 =  �  k>0  k  �  Ψj  −kΨj  k + ¯Ψj  −k ¯Ψj  k + (Ψ → Ψ†)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='24)  The last key ingredient is the action of the symmetry on the fermionic modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' We associate  to any U ∈ U(N) an operator ˆU satisfying  ˆUΨj  −k = Uj′jΨj′  −k ˆU,  ˆU ¯Ψj  −k = Uj′j ¯Ψj′  −k ˆU,  ˆU(Ψ†)j  −k = ¯Uj′j(Ψ†)j′  −k ˆU,  ˆU(¯Ψ†)j  −k = ¯Uj′j(¯Ψ†)j′  −k ˆU,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='25)  which is equivalent to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' Putting everything together, we compute the charged  partition function  Z(U) ≡ ⟨b| qL0+¯L0 ˆU |b⟩ =  �  k>0  ⟨0| exp(−iS†  jj′′ ¯Ψj  k(Ψ†)j  ′  k ) ˆUqL0+¯L0 exp(iSjj′(Ψ†)j  −k ¯Ψj  ′  −k) |0⟩ × (Ψ → Ψ†).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='26)  Using the commutation relations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='25) and the invariance of the vacuum |0⟩ under the  U(N) symmetry, we get  ⟨0| exp(−iS†  jj′′ ¯Ψj  k(Ψ†)j  ′  k ) ˆUqL0+¯L0 exp(iSjj′(Ψ†)j  −k ¯Ψj  ′  −k) |0⟩ =  ⟨0| exp(−iS†  jj′′ ¯Ψj  k(Ψ†)j  ′  k ) exp(iq2k(U †SU)jj′(Ψ†)j  −k ¯Ψj  ′  −k) |0⟩ =  det  �  1 + q2kS†U †SU  �  ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='27)  – 11 –  where in the final step we applied the formula  ⟨0| exp  �  O′jj′Ψj  k ¯Ψj′  k  �  exp  �  O′jj′Ψj  −k ¯Ψj′  −k  �  |0⟩ = det  �  1 − O′O  �  ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='28)  proved in [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' Summing over the modes k, we finally reach to  Z(U) =  �  k∈N−1/2  ���det  �  1 + q2kS†U †SU  ����  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='29)  To proceed further with the computation, it is convenient to introduce the N × N unitary  matrix  O ≡ S†U †SU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='30)  In this way, we express  det(1 + q2kO) =  �  λ∈Spec(O)  (1 + q2kλ),  det(1 + q2kO†) =  �  λ∈Spec(O)  (1 + q2kλ−1),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='31)  with Spec(O) being the set of eigenvalues of O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' After a bit of algebra, we finally reach an  exact expression for the string-order parameter in the large L/ε limit  − log ⟨Ω| ˆU |Ω⟩ = − log Z(U)  Z(1) =  1  4π2 log L  ε  �  λ∈Spec(O)  (Li2(−λ) + Li2(−λ−1) − 2Li2(−1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='32)  At this point, we want to understand for which U ∈ U(N) the order parameter vanishes,  providing a characterization of the unbroken group H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' We first observe that for |λ| = 1 it  holds  Li2(−λ) + Li2(−λ−1) − 2Li2(−1) ≥ 0,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='33)  and the inequality is saturated only for λ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' This implies that − log ⟨Ω| ˆU |Ω⟩ = 0 iff  every eigenvalue of O is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' In other words, the order parameter vanishes when O = 1, a  condition equivalent to (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='30))  [S, U] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='34)  This is not particularly surprising, as the naive argument based on the invariance of the  boundary action leads to the same conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' Nevertheless, it establishes its validity and  allows us to identify the unbroken group as  H = {U ∈ U(N) | [U, S] = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='35)  – 12 –  4  Complex bosons  In this section, we generalize the same symmetry breaking patterns discussed for fermions  in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' 3 to free complex massless bosons2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' Although many analogies can be recognized,  and the underlying physics is similar, the analytical predictions for the order parameters  differ explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='1  U(1)-symmetry breaking terms in complex bosons  First, we consider the U(1)-symmetry action  Sbulk =  � L  0  dx  �  dτ ∂Φ ¯∂Φ†.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='1)  We expand the bosonic field in its Laurent modes as  Φ(z) =  �  k∈Z  Φk  zk ,  ¯Φ(¯z) =  �  k∈Z  ¯Φk  ¯zk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='2)  The bulk action is invariant under the U(1) symmetry  (Φ, ¯Φ) → eiα(Φ, ¯Φ) ,  (Φ†, ¯Φ†) → e−iα(Φ†, ¯Φ†).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='3)  We are interested in a boundary breaking term which breaks U(1) explicitly and preserves  a Z2 symmetry, as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' In analogy with the fermionic case, we take  Sboundary =  �  dτ  �  Φ(x = 0, τ)¯∂Φ(x = 0, τ) + (Φ → Φ†)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='4)  Our aim is to compute the string-order parameter via BCFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' To abridge words, using the  experience from fermions, we need the boundary states, corresponding to the chosen BCs,  the symmetry generator and the Hamiltonian of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' These are given respectively  by  |b⟩ =  �  k>0  exp  �  Φ−k ¯Φ−k + (Φ → Φ†)  �  |0⟩,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='5)  Q =  �  k>0  �  Φ−kΦk + ¯Φ−k ¯Φk − (Φ → Φ†)  �  ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='6)  and  L0 + ¯L0 =  �  k>0  k  �  Φ−kΦk + ¯Φ−k ¯Φk + (Φ → Φ†)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='7)  In terms of these quantities, the partition function can be expressed by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='10), as for  fermions, with q given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' We now proceed with the evaluation, decomposing the  2For more details in the calculations of this section the reader may consult [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='  – 13 –  partition function as a product of bosonic modes  Z(eiα) =  �  k>0  ⟨0| exp  �¯ΦkΦk  �  exp  �  iαΦ−kΦk + iα¯Φ−k ¯Φk  �  × qkΦ−kΦk+k¯Φ−k ¯Φk exp  �  Φ−k ¯Φ−k  �  |0⟩ ×  �  Φ → Φ†�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='8)  The contribution from the k-th mode of Φ is given by  ⟨0| exp  �¯ΦkΦk  �  exp  �  iαΦ−kΦk + iα¯Φ−k ¯Φk  �  qkΦ−kΦk+k¯Φ−k ¯Φk exp  �  Φ−k ¯Φ−k  �  |0⟩ =  ⟨0| exp(¯ΦkΦk) exp(q2kei2αΦ−k ¯Φ−k) |0⟩ = (1 − q2kei2α)−1,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='9)  where we used the commutation relations and the properties Φk |0⟩ = 0 and ¯Φk |0⟩ = 0, for  k > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' Taking the contribution of Φ† and putting everything together, we obtain  Z(eiα) =  �  k>0  �  1 − q2ke2iα�−1 �  1 − q2ke−2iα�−1  =  ∞  �  m=1  �  1 − 2 cos(2α)q2m + q4m�−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='10)  We notice that the property  Z(−1) = Z(1),  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='11)  holds also here, so that a Z2 symmetry is unbroken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' The L/ϵ → ∞ limit is obtained by  converting the infinite product to an integral  log Z(eiα) ≃ −  � ∞  0  dk log  �  1 − q2kei2α�  + log  �  1 − q2ke−i2α�  = −  1  2 log q  �  Li2(ei2α) + Li2(e−i2α)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='12)  In the limit above, we express the order parameter as  − log ⟨Ω| eiαQ |Ω⟩ = − log Z(eiα)  Z(1) = log L  ε  � α  2π − α2  2π2  �  ,  α ∈ [0, π],  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='13)  and its value is periodic under α → α + π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' Finally, it is worth to recognize explicitly the  presence of cusp singularities at α = 0 and α = π, which were absent in the fermionic  counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' We show the α-dependence of the order parameter in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='2  U(N)-symmetry breaking via boundary scattering matrix  We now consider the bosonic version of the model in 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' The bulk action is  Sbulk =  � L  0  dx  �  dτ ∂Φj ¯∂Φ†j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='14)  – 14 –  Figure 3: String order parameter as function of α for the complex boson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' The function is  periodic for α → α + π, and shows cusp singularities in correspondence of α = kπ, k ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='  It vanishes at α = kπ, k ∈ Z and it signals the presence of unbroken Z2 subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='  where the sum over the specie index j (j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' , N) is implicit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' The action is invariant  under the following U(N) transformation  Φj → Uj′jΦj′,  ¯Φj → Uj′j ¯Φj′,  (Φ†)j → ¯Uj′j(Φ†)j′,  (¯Φ†)j → ¯Uj′j(¯Φ†)j′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='15)  We take the following boundary action  Sboundary =  �  dτ  �  (¯Φ†)j(0, τ)S†  jj′Φj  ′  (0, τ) + (Φ†)j(0, τ)Sjj′(¯Φ)j  ′  (0, τ)  �  ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='16)  where S is a N × N matrix satisfying similar properties with the fermionic case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='  The boundary state, the charge operator and the total Hamiltonian are respectively  |b⟩ =  �  k>0  exp  �  Sjj′Φ†j  −k ¯Φj  ′  −k + Φ → Φ†  �  ,  Q =  �  k>0  �  Φj  −kΦj  k + ¯Φj  −k ¯Φj  k − (Φ → Φ†)  �  ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='17)  L0 + ¯L0 =  �  k>0  k  �  Φj  −kΦj  k + ¯Φj  −k ¯Φj  k + (Φ → Φ†)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='18)  – 15 –  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='12  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='10  (Z(eiα) /Z(1)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='06  log(  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='00  2  0  2  4  6  αThe charged partition function reads  Z(U) ≡ ⟨b| ˆUqL0+¯L0 |b⟩ =  �  k>0  ⟨0| exp  �  S†  jj′ ¯Φj  k(Φ†)j  ′  k  �  ˆUqL0+¯L0 exp  �  Sjj′(Φ†)j  −k ¯Φj  ′  −k  �  |0⟩ ×  �  Φ → Φ†�  =  �  k>0  | det(1 − q2kS†U †SU)|−2,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='19)  where in the last line we applied the formula  ⟨0| exp  �  O′jj′Φj  k ¯Φj′  k  �  exp  �  O′jj′Φj  −k ¯Φj′  −k  �  |0⟩ = det  �  1 − O′O  �−1 ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='20)  proven in [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' Using the definition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='30) and the relations  det(1 − q2kO)−1 =  �  λ∈Spec(O)  (1 − q2kλ)−1,  det(1 − q2kO†)−1 =  �  λ∈Spec(O)  (1 − q2kλ−1)−1,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='21)  one finally obtains the expression of the string order parameter in the large L/ϵ limit  −log ⟨Ω| U |Ω⟩ = − log Z(U)  Z(1) =  1  4π2 log L  ϵ  �  λ∈Spec(O)  �  Li2(λ) + Li2(λ−1) − 2Li2(1)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='22)  Here, as for fermions, one observes that the order parameter vanishes exactly for O = 1,  which means [S, U] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='  5  Lattice results  In this section, we provide some lattice realizations of free fermions with a boundary sym-  metry breaking, whose scaling regime is captured by the BCFTs described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='  For instance, while we are not aware of any lattice realization of the field theory described  in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='1, characterized by the boundary-breaking of a U(1) symmetry, we consider a  system that we conjecture to be its doubling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' We describe that system, relating its prop-  erties to those of the homogeneous counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' Finally, we evaluate numerically the order  parameter in the lattice, and we compare it to the analytical predictions of Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='  Then, we consider a Fermi chain with the insertion of a conformal defect, whose scat-  tering properties do not explicitly depend on the incoming momentum [26–28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' This system  can be regarded as a theory of two species of particles on the half-line coupled together at  a boundary point, via the so-called unfolding procedure, and its underlying BCFT is de-  scribed 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' In particular, the U(2) symmetry associated with the mixing of the two species  is broken explicitly due to BCs, encoded in the scattering matrix of the defect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='  – 16 –  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='1  U(1)-symmetry breaking by a boundary pairing: doubling trick  Let us first consider the homogeneous hamiltonian  H = −  �  x  [c†  xcx+1 + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='],  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='1)  describing free fermions hopping on the lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='  Here c†  x, cx are fermionic creation and  annihilation operators associated with the site x, verifying anticommutation relations  {c†  x, cx′} = δxx′,  {cx, cx′} = 0,  {c†  x, c†  x′} = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='2)  We now describe the previous hamiltonian in terms of a new set of fermionic operators ax  and a†  x, defined by  cx =  �  ax,  if x ≤ 0  (−1)xa†  x,  if x > 0,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='3)  that amounts to the exchange of the role of creation and annihilation operators in the right  half of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' The explicit expression of the hamiltonian after the mapping becomes  H = −  �  ��  x≤0  (a†  xax+1 + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=') + a0a1 + a†  1a†  0 +  �  x≥1  (a†  xax+1 + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=')  �  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='4)  In these new variables, H is no longer homogeneous, and a pairing term appears as a  localized defect between the sites x = 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' Moreover, the U(1) symmetry  ax → axeiθ,  a†  x → a†  xe−iθ,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='5)  shared by the bulk terms of the hamiltonian, is broken explicitly due to the defect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' Be-  fore analyzing further the lattice system, we provide a heuristic argument to explain the  relationship between the Hamiltonian (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='4) and the CFT in Sec 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' Let us consider the  vacuum3 |0⟩ as the state satisfying  cx |0⟩ = 0,  ∀x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='6)  One can consider one-particle excitations of |0⟩, generated by linear combinations of {c†  x}  acting on the vacuum |0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' Since in the formulation Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='1) the system is homogeneous, an  incoming wave-packet would reach the point x = 0 and propagate across it without being  partially reflected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' The same process can be described in the language of the fermions {ax}x  given the formulation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' We first notice that |0⟩ satisfies  ax|0⟩ = 0  if  x > 0,  a†  x|0⟩ = 0  if  x ≤ 0,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='7)  3This is not the ground state of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' The latter will be denoted below with |Ω⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='  – 17 –  namely that the left/right chain is completely empty/filled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='  An incoming right-moving  excitation can be thus interpreted as a particle that hits the central point, is completely  transmitted, and then becomes a hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' Similarly, if we considered a Fermi sea, the ex-  citations would have been given by particles/holes which change their U(1) charge after  the scattering at x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' This mechanism is nothing but an explicit symmetry breaking,  and its origin can be traced in the term a0a1 + a†  1a†  0 of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' A similar scenario has been  depicted in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' The crucial difference is that, while in the lattice system (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='4) one  can recognize both left and right-moving incoming particles, the incoming particles of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='1  are just left-moving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' Heuristically, we thus conjecture that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='4) is a discretization of the  QFT depicted above once the degrees of freedom are doubled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='  To better motivate this argument, in the following we analyze the order parameter  associated with the U(1)-symmetry breaking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' We firstly regularize the hamiltonian H in  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='1), keeping the size of the system finite x ∈ [−L + 1, L], and we consider its ground state  |Ω⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' In the language of the fermionic operators cx, it can be regarded as a Fermi sea at  half-filling, namely, the total number of particles is N = L, and it satisfies  �  ��  x≤0  c†  xcx +  �  x>0  c†  xcx  �  � |Ω⟩ = N|Ω⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='8)  While the hamiltonian is not changed after the mapping (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='3), and so its ground state  |Ω⟩, the description in terms of the operators ax is less transparent, as particles and holes  (antiparticles) are mixed among each other by the pairing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' A preliminary observation is  that  �  ��  x≤0  a†  xax −  �  x>0  a†  xax  �  � |Ω⟩ = N|Ω⟩,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='9)  namely, the imbalance of particles among the left/right half-chain is fixed, and it does not  fluctuate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' Then, we express the generator of the U(1) symmetry ax → axeiθ as  Q =  �  x  a†  xax =  �  x≤0  c†  xcx −  �  x>0  c†  xcx = NL − NR,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='10)  with NL,R the number operator in the left/right chain before the mapping (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' Crucially,  the ground state fluctuations of Q are strictly related to those of 2NL, as it holds  Q |Ω⟩ = (NL − NR) |Ω⟩ = (2NL − N) |Ω⟩ ,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='11)  and N = L is just an additive constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' In this way, we can finally express the full-counting  statistics of Q as  ⟨Ω| eiαQ |Ω⟩ ∼ ⟨Ω| ei2αNL |Ω⟩ ,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='12)  up to an irrelevant proportionality constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' In conclusion, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='12) gives an effective way  to characterize the order parameter of the U(1) breaking in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='4) to the full counting  statistics of a subsystem in the homogeneous chain (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='1), which is easier to compute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='  – 18 –  We now show how to express ⟨Ω| ei2αNL |Ω⟩, employing standard techniques for free  fermions [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' We first construct the correlation matrix of the ground state, defined as  Cxx′ ≡ ⟨Ω| c†  xcx′ |Ω⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='13)  Since |Ω⟩ is a Fermi sea with N = L particles, one can express  Cxx′ =  L  �  j=1  φj(x)φj(x′),  x ∈ [−L + 1, L]  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='14)  with φj(x) being the single-particle eigenfunction of the Hamiltonian (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='  Then, we  restrict the spatial indices to the left side of the chain, obtaining a L × L matrix CA  satisfying  (CA)xx′ = Cxx′,  x ∈ [−L + 1, 0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='15)  Given the eigenvalues of CA, denoted by Spec(CA) = {νj}j=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=',L, one can show that the  following relation holds  ⟨ei2αNL⟩ =  L  �  j=1  [νje2iα + (1 − νj)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='16)  Making use of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='12), one finally gets the string-order parameter as  ⟨Ω| eiαQ |Ω⟩ = ⟨ei2αNL⟩ =  L  �  j=1  [νje2iα + (1 − νj)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='17)  An explicit analytical expression for the eigenvalues {νj}j is a hard task, and should rely on  numerics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' Nevertheless, simple field theoretical arguments can capture the correct leading  behavior of the order parameter, as we explain below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' The state |Ω⟩ can be recovered as a  euclidean path integral of the massless Dirac fermions over the strip  x ∈ [−L, L],  τ ∈ (−∞, ∞),  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='18)  with τ representing the Euclidean time, and we assume L to be much larger than the lattice  size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' Furthermore, the operator eiαNL is described by the insertion of two vertex operators  in the path integral at τ = 0, as  eiαNL ∼ Vα(x = −L)V−α(x = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='19)  It is known [23] that the bulk scaling dimension ∆α of Vα is  ∆α =  � α  2π  �2  ,  α ∈ [−π, π].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='20)  Moreover, the insertion of the boundary operator Vα(x = −L) does not play a role, since  the BCs at x = −L are U(1) symmetric, and from now on we just discard it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' Making use  – 19 –  Figure 4: Order parameter as function of L for different values of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' The symbols are the  numerical data obtained from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='(5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' The solid lines show the curve (α2/π2) log L+b0+  b1L−1, where the first term is the analytical prediction given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='22) and coefficients  b0, b1 are obtained fitting the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='  of scale-invariance, we finally reach to  ⟨Ω| ei2αNL |Ω⟩ ∼ ⟨Ω| V−2α(x = 0) |Ω⟩ ∼  1  L∆2α =  1  Lα2/π2 ,  α ∈ [−π/2, π/2],  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='21)  where we employed scaling arguments on the bulk field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' In conclusion, thanks to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='12), we  finally obtain  − log | ⟨Ω| eiαQ |Ω⟩ | ≃ α2  π2 log L,  α ∈ [−π/2, π/2]  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='22)  which is the main result of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' We emphasize that the formula we obtained is just  twice the prediction in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' This is somehow expected, as we conjectured that the  Hamiltonian (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='4) describes a doubling of the system in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='2  Conformal defect and U(2) symmetry breaking  Let us consider a system made of two species of spinless fermions on the lattice x ∈ [0, L−1],  coupled together at the boundary point x = 0 through a conformal defect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='  – 20 –  α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='2  α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='8  α=1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='0  α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='2  C  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='5  2a)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='0  0  25  50  75  100  125  150  175  200  LThe hamiltonian is  H = −  �  x,j  [(c†  x)jcj  x+1 + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='] −  �  jj′  Sjj′(c†  0)j(c0)j′,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='23)  where (c†  x)j, (cx)j are fermionic operators verifying anticommutation relations  {(c†  x)j, (cx′)j′} = δxx′δjj′,  {(cx)j, (cx′)j′} = 0,  {(c†  x)j, (c†  x′)j′} = 0,  j = 1, 2,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='24)  and S is the following 2 × 2 matrix  S =  �√  1 − λ2  λ  λ  −  √  1 − λ2  �  ,  λ ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='25)  It is possible to show that the scattering matrix, induced by the boundary term in the  hamiltonian, is exactly S [24, 25, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' Since the transmission/reflection probability does not  depend on the incoming momenta of the particle, and it is given by λ2, 1 − λ2 respectively,  the defect is scale-invariant, and it is described by a BCFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' It is easy to show that the  bulk terms on the hamiltonian are invariant under a U(2) symmetry which mixes the two  species as  cj  x → Uj′jcj′  x ,  (c†  x)j → ¯Uj′j(c†  x)j′,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='26)  with U a generic 2x2 unitary matrix, and the sum over j′ is implicit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' However, the presence  of the defect breaks explicitly the symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' Indeed, after a generic U(2) transformation,  the boundary term is mapped onto  − Sjj′U j1jUj2j′(c†  0)j1(c0)j2,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='27)  and it is invariant only if U †SU = S, namely [U, S] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' While the condition above was  already derived in CFT, it is instructive to notice that it holds in the lattice model too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='  For the sake of completeness, we characterize explicitly the unbroken subgroup H defined  by  H = {U ∈ U(2)|[S, H] = 0} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='28)  Since S is hermitian and unitary, its eigenvalues have to be real phases, and they are just  ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' In the basis in which S is diagonal, also U ∈ H has to be diagonal, since S and H  commute, and so they share the same eigenspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' This means that, given a matrix D  which diagonalizes S, say  S = D  �  1 0  0 −1  �  D−1,  D =  �  �  λ  √  2−2  √  1−λ2  λ  √  2+2  √  1−λ2  √  1−  √  1−λ2  √  2  −  √  1+  √  1−λ2  √  2  �  � ,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='29)  – 21 –  we identify the unbroken group as  H =  �  U = D  �  eiα1  0  0  eiα2  �  D−1, α1, α2 ∈ R  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='30)  6  Conclusions and Outlooks  In this work, we considered the effect of an explicit symmetry breaking in a one-dimensional  critical system induced by a localized impurity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' We develop a general formalism to probe  the symmetry breaking that can be applied to any system described by a BCFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' In par-  ticular, we show that − log | ⟨Ω| ˆg |Ω⟩ |, which is a non-local order parameter, is generically  logarithmic growing in the system size, and it vanishes precisely for the unbroken elements  of the group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' We provide exact calculations for free theories, fermions and bosons, with  both abelian and non-abelian broken symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' For instance, we first consider the sym-  metry breaking of a complex theory in the presence of a boundary pairing term, which has  a residual Z2 symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' Then, we consider a theory made of many species of particles  coupled together at a boundary point, and a non-trivial breaking of a U(N) symmetry is  investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='  A natural generalization would be the calculation of the order parameter for rational  minimal models (Ising, Potts model, Tricritical Ising, and so on)[11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' Indeed, their scale-  invariant BCs are known, and they are mapped onto the so-called Cardy states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='  The  evaluation of the overlap in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='12) might be more involved for interacting CFTs, but  we think it should be expressed as Virasoro characters, whose explicit expressions are known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='  One may also wonder what happens when the theory is not described by a BCFT,  or due to a finite correlation length in the bulk ξ or in the presence of BCs which are  not scale invariant [30–37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' In the first case, the boundary effect is localized in a typical  distance ξ and we expect that the order parameter goes to a constant when the system size  is increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' It might be interesting to study this scenario in Integrable Field Theories,  providing exact results for massive theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' In the second case, the BCs are expected to  flow to some scale-invariant ones via RG flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' Then, for a big enough size, the leading  growth of the order parameter would be logarithmic, with a universal prefactor depending  only on the IR fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' We hope to come back to these problems in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='  Finally, we mention that while our approach involves a genuine non-local probe of  the system, one might be interested in the local observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' In particular, it has been  proposed [38] that a subsystem measure dubbed as entanglement-asymmetry might be a  good probe of the symmetry breaking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' An interesting direction would be the computation  of the entanglement asymmetry for an interval attached to the boundary point in BCFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='  – 22 –  Acknowledgments  RB acknowledges support from the Croatian Science Foundation (HrZZ) project No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' IP-  2019-4-3321.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' LC acknowledges support from ERC under Consolidator grant number 771536  (NEMO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' The work of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' Panopoulos was supported by the Croatian Science Foundation  Project "New Geometries for Gravity and Spacetime" (IP-2018-01-7615).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='  References  [1] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' Shankar, Quantum Field Theory and Condensed Matter: An Introduction, Cambridge  Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content=' Press (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
+page_content='  [2] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FAT4oBgHgl3EQfvh5f/content/2301.08676v1.pdf'}
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+arXiv:2301.04026v1  [hep-ph]  10 Jan 2023
+Backreaction effect and plasma oscillation in pair production for rapidly oscillating
+electric fields
+R. Z. Jiang,1, 2 C. Gong,3 Z. L. Li,1, 2, ∗ and Y. J. Li1, 2, †
+1School of Science, China University of Mining and Technology, Beijing 100083, China
+2State Key Laboratory for GeoMechanics and Deep Underground Engineering,
+China University of Mining and Technology, Beijing 100083, China
+3North China Electric Power University, Baoding 071003, China
+(Dated: January 11, 2023)
+The backreaction effect and plasma oscillation in pair production for rapidly oscillating electric
+fields are investigated by solving quantum Vlasov equation. Contrary to previously thought, it is
+found that the backreaction effect can be neglected in the pair production for a rapidly oscillating
+but weak electric field, particularly, for a subcritical external electric field with frequency chirp. In
+some cases the oscillation period of created electron-positron plasma can be described by a simple
+formula constructed based on the Langmiur oscillation frequency, but it is impossible in general
+case because the plasma oscillation period directly depend not only on the final number density
+of created particles, but also on the external electric field parameters. Moreover, it is found that
+the momentum spectrum presents complex interferences after considering the backreaction. These
+results give us the safety range of external electric fields for taking no account of the backreaction
+effect and deepen our understanding of the pair production with the backreaction effect.
+I.
+INTRODUCTION
+Since Dirac proposed the relativistic wave equation
+and predicted the existence of positrons [1], many re-
+searches have been done on how to produce electron-
+positron pairs from vacuum. Sauter [2] found that the
+electron-positron pairs can be produced from vacuum
+in a strong static electric field by tunneling mechanism.
+And then, Schwinger [3] calculated the pair produc-
+tion rate in a constant electric field by the proper-time
+method and gave out the critical electric field strength
+Ecr = m2/e ≈ 1.3×1018V/m (the natural units ℏ = c = 1
+are used).
+As a result of these pioneering works, the
+phenomenon of electron-positron pairs produced from
+the vacuum in intense external fields is also known as
+the Sauter-Schwinger effect [4, 5].
+The laser intensity
+corresponding to the critical electric field strength is
+I ≈ 1029W/cm2, which is much larger than the possi-
+bility of the current laser facilities [6–8], so it is not yet
+possible to produce observable electron pairs in experi-
+ments. However, the already operating X-ray free elec-
+tron laser (XFEL) at DESY in Hamburg is expected to
+achieve subcritical fields of E ≈ 0.01 − 0.1Ecr [9].
+To reduce the threshold of pair production, several ap-
+proaches have been proposed to produce observable elec-
+tron pairs under subcritical electric field conditions. One
+method that can effectively enhance the pair production
+is the dynamically assisted Schwinger mechanism [10–
+14], in which the electric field adopts a combination of
+a low-frequency strong field and a high-frequency weak
+field. Another approach is to use frequency chirp to in-
+crease the pair yield by increasing the effective frequency
+∗ Corresponding author. zlli@cumtb.edu.cn
+† Corresponding author. lyj@aphy.iphy.ac.cn
+of the electric field.
+In Refs. [15–19], authors studied
+the effect of spatially homogeneous electric fields with
+frequency chirp on the pair production by the quantum
+kinetic equation and found that for some chirp param-
+eters the pair yield could be improved three to four or-
+ders of magnitude.
+Moreover, the enhancement effect
+of pair production in a spatially inhomogeneous electric
+field with the frequency chirp were also found [20, 21].
+Previous studies have shown that the electrons and
+positrons produced from vacuum can be accelerated in an
+external field and form a new electric field, which is called
+the internal electric field [22–25]. The internal electric
+field can affect the generation of particle pairs together
+with the external electric field, that is the backreaction.
+For subcritical external field strength, the number of pro-
+duced particle pairs is relatively less, and the backreac-
+tion is generally considered to be negligible. However, in
+the study of pair production in a frequency chirp elec-
+tric field, if the field lasts for a long time, the effective
+frequency will be very large and may even exceed the
+primary frequency, which can greatly enhance the multi-
+photon pair production process and produce a large num-
+ber of particle pairs. Therefore, whether the backreaction
+effect in this case can be negligible is a problem worth re-
+searching. In this paper, we will figure out the parameter
+scope of external electric fields in which the backreaction
+effect cannot be ignored.
+In addition, when the backreaction effect is considered
+a plasma oscillation will occur. In Ref. [22], the plasma
+oscillation in pair production for a sinusoidal electric field
+is studied by quantum Vlasov equation (QVE) and a ul-
+trarelativistic formula is given to estimate the oscillation
+period of the plasma. The ultrarelativistic formula shows
+that the oscillation period only depends on the maxima of
+particle number density and has no direct relation with
+the field parameters.
+In this work, we will study the
+plasma oscillation for a rapidly oscillating electric field
+
+2
+by QVE, explore the determining factors of the oscilla-
+tion period, and check whether this formula still holds.
+The structure of this paper is as follows: Section II
+briefly introduce the theoretical framework of quantum
+dynamics-based backreaction effects. Section III is our
+numerical results: Subsection III A discusses the effect of
+the backreaction effect on the final particle number den-
+sity; subsection III B gives the effect of the backreaction
+effect on the momentum spectra; subsection III C study
+the relation between the plasma oscillation period and
+the number density of produced particle pairs. Section
+IV is a summary and discussion.
+II.
+THEORETICAL FORMALISM
+The magnetic effects can be neglected when a standing-
+wave field produced by two counter-propagating laser
+pulses is considered.
+The spatial scales for electron-
+positron pairs production is on the same order of magni-
+tude as the Compton wavelength of the electron, which
+is far smaller than the focusing radius of the laser, so it
+can be assumed that the laser is spatially homogeneous.
+By using the temporal gauge, A0 = 0, the spatially ho-
+mogeneous and time-dependent four-vector potential can
+be written as Aµ = (0, 0, 0, A(t)), and the external elec-
+tric field is Eext = −dA(t)/dt. The form of the external
+electric field we used is
+Eext (t) = E0e− t2
+2τ2 cos (ωt) ,
+(1)
+where E0 is the electric field amplitude, ω is the laser
+frequency, and τ is the pulse duration.
+The key quantity to study the electron-positron pair
+production with the QVE is to obtain the momentum
+distribution function f(k, t). For spatially homogeneous
+and time-dependent electric fields, ignoring collisions be-
+tween particles, the distribution function is determined
+by df(k, t)/dt = S(k, t), where S(k, t) denotes the source
+term of pair production. When the external field strength
+is relatively large, the backreaction brought by the in-
+ternal electric field is gradually reflected, so the to-
+tal electric field Etot(t) should be modified as the sum
+of the external field and the internal field Eint(t), i.e.
+Etot(t) = Eext(t) + Eint(t). After considering the influ-
+ence of the backreaction, we can get the coupled equa-
+tions of the distribution function and the internal electric
+field
+˙f(k, t) = eEtot(t)ε⊥
+2Ω2(k, t)
+� t
+−∞
+dt′ eEtot(t)ε⊥
+Ω2(k, t′) [1 − 2f(k, t′)]
+× cos[2
+� t
+t′ dt′′Ω(k, t′′)],
+(2)
+˙Eint (t) = −4e
+� d3k
+(2π)3
+� k∥ (t)
+Ω (k, t)f (k, t)
++ Ω (k, t)
+eEtot (t)
+˙f (k, t) − e ˙Etot (t) ε2
+⊥
+8Ω5 (k, t)
+�
+, (3)
+where the dot on the letter represents the first order
+time derivative, |e| is the renormalized charge of the
+electron [23], k =
+�
+k⊥, k∥
+�
+is the canonical momentum,
+k∥ (t) = k∥ − eA (t) is defined as the kinetic momentum
+along the external electric field, ε⊥ =
+�
+m2 + k2
+⊥ is the
+transverse energy squared, m is the mass of the electron,
+and Ω (k, t) =
+�
+ε2
+⊥ + k2
+∥ (t) is the total energy squared.
+The quantum statistics effect and the non-Markov effect
+on the pair production can be seen from [1 − 2f(k, t′)]
+in Eq. (2). The first term on the right hand side of Eq.
+(3) represents the conduction current, the second term is
+the polarization current, and the third term is the charge
+renormalization part added to eliminate the divergence
+of the polarization current.
+For the convenience of numerical calculation, two aux-
+iliary variables u (k, t) and v (k, t) are introduced
+u (k, t) = � t
+t0 dt′W (k, t′) [1 − 2f (k, t′)]
+× cos
+�
+2
+� t
+t′ dt′′Ω (k, t′′)
+�
+,
+v (k, t) =
+� t
+t0 dt′W (k, t′) [1 − 2f (k, t′)]
+× sin
+�
+2
+� t
+t′ dt′′Ω (k, t′′)
+�
+,
+(4)
+then equation (2) can be equivalently transformed into
+the following first-order differential equations
+˙f (k, t) = 1
+2W (k, t) u (k, t) ,
+˙u (k, t) = W (k, t) [1 − 2f (k, t)] − 2Ω (k, t) v (k, t) ,
+˙v (k, t) = 2Ω (k, t) u (k, t) ,
+(5)
+where W (k, t) = eEtot (t) ε⊥/Ω2 (k, t).
+It can be seen from Eq. (3) that to obtain the inter-
+nal electric field at time t, the momentum distribution
+function at the same time must be integrated, but the
+internal electric field at the same time must be known
+to calculate the momentum distribution function at time
+t. To solve this contradiction, we calculate the internal
+electric field by iteration. First, we use the internal elec-
+tric field at the previous time (t − ∆t) to calculate the
+total electric field at time t by
+Etot(t) = Eext(t) + El
+int(t),
+(6)
+where l = 1, 2, · · · is the number of iterations. Note that
+for the first iteration (l = 1), the internal electric field
+E1
+int(t) = Eint(t − ∆t). Using Eq. (6), the momentum
+distribution function at time t can be obtained by solving
+Eq. (5). Then the internal electric field at time t can be
+solved by Eq. (3). This is the first iteration. Replacing
+the internal electric field in Eq. (6) with the new one, the
+second iteration begins. When the internal electric field
+satisfies our preset control condition |El+1
+int (t)−El
+int(t)| <
+ε, where ε is a very small number, it can be considered
+that the real internal electric field at time t has been
+
+3
+obtained.
+Plugging it into the total electric field and
+solving Eq. (5), the momentum distribution function at
+time t can be solved as well.
+The initial state of the system is a vacuum state with-
+out particles, so the single particle distribution func-
+tion and auxiliary function satisfy the initial conditions
+f (k, −∞) = u (k, −∞) = v (k, −∞) = 0.
+The initial
+condition of the internal electric field is Eint (−∞) = 0.
+After obtaining the single-particle distribution function,
+integrating it in the full momentum space can obtain the
+resulting particle number density
+n (t) = 2
+�
+d3k
+(2π)3 f (k, t).
+(7)
+The coefficient 2 comes from the spin degeneracy of the
+fermions.
+III.
+NUMERICAL RESULTS AND ANALYSIS
+A.
+Particle number density
+After considering the backreaction effect, when the ex-
+ternal electric field vanishes the total electric field is not
+zero due to the existence of the internal electric field and
+the particle number density still changes with time. So it
+is not easy to obtain a definite particle number density.
+However, in a wide range of external field parameters,
+the oscillation of particle number density induced by the
+internal electric field is very very small, for example the
+second case we considered below, it is reasonable to define
+this relatively stable number density as the real particle
+number density. The detailed explanation is as follows.
+In Fig. 1, we show the number density of created pairs
+nb in the presence and n0 in the absence of backreaction
+for two supercritical electric fields. The field frequencies
+in Fig. 1 (a) and (b) are 0.15m and 0.35m, respectively.
+Other field parameters are E0 = 4.0Ecr and τ = 12.0/m.
+It can be seen that the particle number density is invari-
+able when the backreaction is not considered, but the sit-
+uation is different when the backreaction is considered.
+For such a supercritical external electric field, when the
+field frequency is relatively small, such as ω = 0.15m in
+Fig. 1(a), the number density gradually increases with
+time, which indicates that the internal electric field in-
+duces the pair production. However, when the frequency
+increases to a certain value, such as ω = 0.35m in Fig.
+1(b), the particle number density remains nearly constant
+because the internal electric field is not strong enough to
+stimulate sufficient particle pairs. Enlarging the curve
+of nb in Fig. 1(b), one can see that the particle num-
+ber density oscillates with time due to the existence of
+the internal electric field, but its variation range does
+not exceed 2.0 × 10−4m−3. Therefore, the relatively sta-
+ble number density can be considered as the real particle
+number density within the range of error allowed. Our
+following calculation always meets this condition.
+-50
+0
+50
+100
+150
+200
+0.0
+0.2
+0.4
+0.6
+-50
+0
+50
+100
+150
+200
+0.0
+0.1
+0.2
+0.3
+0.4
+50
+100
+150
+200
+0.3600
+0.3602
+0.3604
+ n
+0
+ n
+b
+t [m
+-1
+]
+(a)
+ n
+0
+ n
+b
+(b)
+t [m
+-1
+]
+ n
+b
+FIG. 1. The particle number density as a function of time
+with (solid black lines) and without backreaction (dashed red
+lines). The external electric field parameters in (a) are E0 =
+4.0Ecr, ω = 0.15m and τ = 12.0/m, and in (b) are E0 =
+4.0Ecr, ω = 0.35m and τ = 12.0/m.
+The particle number density varying with the field fre-
+quency for different external electric field amplitudes is
+shown in Fig. 2. It can be seen that for the subcritical
+field with E0 = 0.1Ecr, see Fig. 2(a), the number density
+with and without the backreaction effect is almost the
+same, and the multi-photon absorption is obvious [26].
+At the frequency ω = 2m/N0 with the number of ab-
+sorbed photons N0, the particle number density increases
+greatly. In the figure, 1−, 2−, 3−, 4−, and 5−photon ab-
+sorption are marked by vertical lines.
+However, when
+E0 = 2.0Ecr, 3.0Ecr, and 4.0Ecr, see Fig. 2(b), the mul-
+tiphoton pair production is not obvious, and simply in-
+creasing the frequency of the external field does not en-
+hance the pair production but suppress it.
+According
+to the Keldysh adiabatic parameter [27] γ = mω/eE0,
+we can know that in our calculation γ ∼ O (1). In this
+range, the tunneling pair production coexists with the
+multiphoton pair production, and their competitive rela-
+tion is unfavourable for the pair production. In addition,
+although the photon energy is high, the number density
+of produced particles is also affected by other field pa-
+rameters. When E0 = 2.0Ecr, 3.0Ecr, the backreaction
+effect is still insignificant, but when E0 = 4.0Ecr, the
+backreaction effect of the internal electric field begins to
+affect the particle number density, see ω ≤ 0.85m.
+To further investigate the region where the backre-
+action effect cannot be ignored, we define the differ-
+
+4
+2/5
+2/4
+2/3
+2/2
+2/1
+10
+-11
+10
+-8
+10
+-5
+10
+-2
+0.5
+1.0
+1.5
+2.0
+2.5
+0.0
+0.1
+0.2
+0.3
+0.4
+n [m
+-3
+]
+w [m]
+ n
+b
+(a)
+ n
+0
+n [m
+-3
+]
+w [m]
+ n
+b
+ , E
+0
+=2 E
+cr
+ n
+0
+ , E
+0
+=2 E
+cr
+ n
+b
+ , E
+0
+=3 E
+cr
+ n
+0
+ , E
+0
+=3 E
+cr
+ n
+b
+ , E
+0
+=4 E
+cr
+ n
+0
+ , E
+0
+=4 E
+cr
+(b)
+FIG. 2. Particle number density changing with the field fre-
+quency for different field amplitudes. In (a), E0 = 0.1Ecr,
+and in (b), E0 = 2.0Ecr, 3.0Ecr, 4.0Ecr, the value of τ in both
+figures is 12.0/m.
+ence degree of pair number density (DDOPND) as δ =
+|nb − n0| /n0 × 100% and study its changes with the ex-
+ternal field parameters E0 and ω, see Fig. 3. Although
+the supercritical field strength is used in the calculation,
+which is far away from the current experimental condi-
+tions, some interesting results can still be obtained. In
+our results, for E0 ≤ 2.0Ecr, the maximum value of δ is
+about 1.93% at E0 = 2.0Ecr, ω = 0.5m and the backre-
+action effect can be ignored. For 2.0Ecr < E0 ≤ 3.0Ecr,
+the value of δ is generally within 5%, and in the region
+where ω is small, the DDOPND value may exceed 5%.
+When the electric field amplitude E0 is large and the fre-
+quency ω is small, i.e., the Keldysh parameter γ is small,
+the DDOPND is large and the backreaction effect is more
+obvious. This also shows that generally the backreaction
+effect can be neglected in the study of pair production in
+a high-frequency but weak electric field. Therefore, it is
+safe to ignore the backreaction effect when study the pair
+production in a subcritical external field with frequency
+chirp. This result is beyond our expectation. In the very
+beginning, we thought that the particle number density
+would be improved greatly by the high-frequency photon
+absorption process and the backreaction effect became
+significant. However, the actual result is not what we
+previously thought because of the competitive relation
+between the tunneling pair production and the multi-
+photon absorption. In fact, the change of DDOPND with
+E0 and ω is complex. For example, when E0 = 7.0Ecr
+and ω = 0.8m, δ ≈ 4.37%, the particle number density
+with and without backreaction are about 0.970m3 and
+0.929m3, respectively. The influence of the backreaction
+effect is almost unnecessary. Whereas, when E0 = 7.0Ecr
+and ω = 0.9m, δ ≈ 23.67%, the number density with and
+without backreaction are about 0.956m3 and 0.773m3,
+the backreaction effect is very obvious.
+2
+4
+6
+8
+10
+0.5
+1.0
+1.5
+w [m]
+E
+0
+ [E
+cr
+]
+0.000
+7.083
+14.17
+21.25
+28.33
+34.00
+d
+FIG. 3.
+The difference degree of pair number density
+(DDOPND) δ as a function of the external field amplitude
+and frequency. The value of τ in this figure is 12.0/m. In the
+calculation, the interval of E0 is 0.1Ecr, and the interval of ω
+is 0.05m.
+B.
+Momentum spectrum
+As shown in Fig. 4, we compare the momentum dis-
+tribution functions of created particles with and without
+backreaction, denoted by fb and f0, respectively. In Fig
+4(a), the momentum distribution f0 is smooth because
+the particles are mainly generated by the main peak of
+the electric field, but the situation is different for slightly
+larger values of ω. As shown in Fig. 4(b), although the
+sub-maximal peak of the electric field cannot produce
+sufficient particle pairs, the momentum distribution func-
+tion f0 shows obvious oscillations because of the infield
+interference. Furthermore, without the backreaction, the
+momentum distribution functions f0 is symmetric about
+the zero kinetic momentum in both cases. However, with
+the backreaction, the momentum spectrum presents ir-
+regular oscillations and the symmetry is broken. Another
+phenomenon is that the backreaction effect is observed
+in the left side of the momentum distribution functions
+while on the right side they are approximately identical
+to that without backreaction. In fact, the change of the
+momentum spectrum with backreaction is very complex
+and sensitive to the parameters of the external electric
+field.
+
+5
+-30
+-20
+-10
+0
+0.0
+0.1
+0.2
+-20
+-10
+0
+10
+0.0
+0.1
+0.2
+k
+//
+ [m]
+ 
+b
+(k
+//
+)
+ 
+0
+(k
+//
+)
+(a)
+k
+//
+ [m]
+ 
+b
+(k
+//
+)
+ 
+0
+(k
+//
+)
+(b)
+FIG. 4. Comparison of the momentum distribution functions
+between with and without backreaction at t = 220.0/m. The
+parameters of the external electric field in (a) are E0 = 2.0Ecr,
+ω = 0.1m, τ = 12.0/m, and in (b) are E0 = 2.0Ecr, ω =
+0.15m, τ = 12.0/m.
+C.
+Plasma oscillation period in pair production
+We show the internal electric field evolution with time
+for different external field amplitudes in Fig. 5. One can
+see when the external electric field exists (t < 40/m), the
+amplitude of internal electric field increases with E0, and
+its frequency is the same as that of the external electric
+field. For instance, when E0 = 7Ecr, the peak value of
+the internal electric field is about 0.2Ecr. The motion
+of particles is mainly dominated by the external electric
+field. However, when the external electric field is turned
+off, the relation between the internal electric field and
+the external field parameters becomes complicated. For
+example, the amplitude of the internal electric field does
+not increase monotonically with the E0. Moreover, we al-
+ready know that when the external electric field is strong
+enough, a large number of real electron-positron pairs
+will be generated, and these particle pairs will maintain a
+dynamic balance under the action of the internal electric
+field. Therefore, the motion of created electron-positron
+pairs forms plasma oscillation. It is worth noting that
+the period of plasma oscillation will have some relation
+with the number density of created particles and the pa-
+rameters of the external electric field. Previous studies
+have shown that the frequency of plasma oscillation de-
+creases gradually with time and tends to a stable value
+at the end [28]. For the external electric field with high
+intensity and high frequency, the frequency of plasma os-
+cillation can reach the stable state quickly.
+0
+50
+100
+150
+200
+-0.2
+-0.1
+0.0
+0.1
+0.2
+E
+int
+(t) [E
+cr
+]
+t [m
+-1
+]
+ E
+0
+=6E
+cr
+ E
+0
+=7E
+cr
+ E
+0
+=8E
+cr
+FIG. 5. The internal electric field changes with time for E0 =
+6.0Ecr, 7.0Ecr and 8.0Ecr respectively. Other field parameters
+are ω = 0.9m and τ = 12.0/m.
+Assuming that the external electric field disappears at
+t0, then the internal electric field at t > t0 can be ap-
+proximately expressed as
+Eint (t) = Eint0cos
+�2π
+T t + ϕ
+�
+,
+(8)
+where Eint0 is the amplitude of the internal electric field,
+T is the period of oscillation, and ϕ is the phase. In this
+subsection, we mainly study the relationship between the
+oscillation period T of the internal electric field and the
+number density of created particles n, the parameters of
+the external electric field E0, ω and τ.
+In the follow-
+ing research, we study the relationship between T and n
+when E0, ω, and τ change, respectively.
+First, we keep ω and τ unchanged, and E0 ranges from
+5.0Ecr to 14.0Ecr with an interval of 1.0Ecr. The oscil-
+lation period changing with the particle number density
+is shown in Fig. 6. The black dots represent the numer-
+ical results where the oscillation period of internal elec-
+tric field T is estimated by Eq. (8). From the results, we
+can see that the particle number density always increases
+with the external field amplitude E0 and the oscillation
+period T decreases with the increase of the number den-
+sity. The red lines are the fitting curves corresponding to
+the fitting formula
+T (n) =
+α
+√n + β,
+(9)
+where n is the particle number density, α and β are fit-
+ting parameters. Note that this formula is constructed
+based on the frequency of Langmuir oscillation [29] in
+plasma physics. Although it is a little different from the
+ultrarelativistic formula given in Ref. [22], both of them
+
+6
+have the same varying tendency.
+In (a), (b), (c), and
+(d) of Fig.
+6, the goodness of fit R2 is about 0.998,
+0.999, 0.998, and 0.999, respectively.
+This shows that
+the formula above can fit the numerical results very well.
+Furthermore, since the fitting parameters α and β are
+almost independent of E0, the oscillation period T is not
+directly related to E0. Finally, we emphasize that the ex-
+ternal electric field we considered here is a high-frequency
+strong field. For other cases, the fitting formula (9) may
+fail.
+0
+1
+2
+3
+20
+30
+40
+50
+0
+1
+2
+3
+4
+20
+30
+40
+50
+0
+1
+2
+3
+20
+30
+40
+50
+0
+1
+2
+3
+20
+30
+40
+50
+60
+70
+T [m
+-1
+]
+n [m
+-3
+]
+ numerical results
+ a=21.25,
+ b =12.29
+(a)
+T [m
+-1
+]
+n [m
+-3
+]
+ numerical results
+ a=21.54,
+ b =11.40
+(b)
+T [m
+-1
+]
+n [m
+-3
+]
+ numerical results
+ a=21.35,
+ b =12.39
+(c)
+T [m
+-1
+]
+n [m
+-3
+]
+ numerical results
+ a=23.77,
+ b =9.48
+(d)
+FIG. 6.
+The oscillation period of electron-positron plasma
+varying with the particle number density. The ω and τ are
+fixed and the value of E0 ranges from 5.0Ecr to 14.0Ecr with
+an interval of 1.0Ecr. Other external electric field parameters
+are (a) ω = 1.5m, τ = 18.0/m; (b) ω = 1.5m, τ = 30.0/m;
+(c) ω = 1.7m, τ = 12.0/m; (d) ω = 2.0m, τ = 12.0/m.
+When E0 and τ are kept unchanged and ω is changed,
+or keep E0 and ω constant and change τ, the above fit-
+ting formula will be invalid, see Fig.
+7.
+The relation
+between the oscillation period and the number density
+of created pairs is very complex and not monotonically
+decreasing. In Fig. 7(a), we change the value of ω and
+mark each data point with the frequency of the external
+electric field. It can be seen that the particle number den-
+sity generally decreases with the increase of the frequency
+ω. In the high-frequency region, such as ω ≥ 1.6m for
+E0 = 6.5Ecr, ω ≥ 1.5m for E0 = 7.5Ecr, and ω ≥ 1.6m
+for E0 = 8.5Ecr, the oscillation period decreases with
+the increase of particle number density. This behavior is
+somewhat similar to that in Fig. 6. But when the fre-
+quency takes other values, the relation between them ex-
+hibits a very complex oscillations. In particular, we find
+that in the low frequency region (such as ω = 0.6m) the
+fitting formula (9) still fails even if the field frequency and
+the pulse duration are fixed and only the field amplitude
+changes, because the number density does not decrease
+monotonically with the field amplitude. In Fig. 7(b), we
+change the value of τ and mark each data point with the
+value of τ. It can be seen that the relationship between
+the particle number density and the pulse duration τ is
+not monotonic. As the pulse duration becomes large, the
+number density of created pairs may not increase. Thus,
+the fitting formula (9) also does not work here. In addi-
+tion, from Fig. 7(a), we can see that for E0 = 6.5Ecr the
+number density of created particles at ω = 1.1m almost
+equals that at ω = 1.0m, but their oscillation periods
+have a big difference. That is to say, the particle num-
+ber density for different field frequencies can be equal,
+see Fig. 2, but the same number density corresponds to
+different oscillation periods, which shows that the oscil-
+lation periods directly depend on not only the particle
+number density but also the field frequency. Similarly,
+from Fig. 7(b), we can find that the oscillation periods
+also directly depend on the pulse duration, because the
+different number density corresponding to different pulse
+durations gives the same oscillation period, see the data
+points marked with the pulse duration 12 and 13.
+0.5
+1.0
+1.5
+2.0
+33
+36
+39
+42
+45
+48
+T [m
+-1
+]
+n [m
+-3
+]
+ E
+0
+=6.5 E
+cr
+ E
+0
+=7.5 E
+cr
+ E
+0
+=8.5 E
+cr
+0.5
+0.6
+0.7
+1.0
+1.2
+1.4
+1.5
+1.6
+1.8
+2.0
+0.5
+0.6
+0.7
+1.0
+1.2
+1.4
+1.5
+1.6
+2.0
+1.9
+1.8
+1.7
+1.6
+1.5
+1.4
+1.3
+1.2
+1.1
+1.0
+0.7
+0.6
+0.5
+2.0
+1.8
+(a)
+0.8
+0.85
+0.95
+1.05
+32
+34
+36
+38
+10
+T [m
+-1
+]
+n [m
+-3
+]
+8
+9
+11
+12
+13
+14
+15
+16
+17
+20
+18(21)
+19
+22
+23
+24
+28
+25
+40
+36
+32
+(b)
+FIG. 7.
+The oscillation period of electron-positron plasma
+varying with the particle number density. In (a), E0 = 6.5Ecr,
+7.5Ecr, 8.5Ecr, and τ = 12.0/m are fixed, and ω ranges from
+0.5m to 2.0m. The particle number density changes with ω.
+In (b), E0 = 8.0Ecr and ω = 1.5m are constant, and the
+particle number density changes with τ.
+We further explore the relation between the oscilla-
+tion period and the particle number density for fixing
+E0 = 7.0Ecr and keeping ωτ = 20.0, which ensures
+that the number of cycles in the external electric field
+is constant, see Fig.
+8.
+To separate the number den-
+sity the value of ω ranges from 0.4m to 1.65m with a
+variable interval.
+From the figure, we find that with
+the frequency increasing (the pulse duration decreasing
+correspondingly), the particle number density always de-
+creases. When the value of ω is relatively small, the par-
+
+7
+ticle number density changes faster with ω. For example,
+when ω changes from 0.41m to 0.4m, the particle num-
+ber density changes 0.1817m−3, while when ω changes
+from 1.5m to 1.4m, the number density only changes
+0.0725m−3. This result is also reflected in Fig. 4(b). Be-
+sides this, the oscillation period tends to decrease with
+the increase of the number density. Therefore, we can try
+to fit it with Eq. (9). The fitted curve is represented by
+the solid red line in Fig. 8, and the goodness of fit R2 is
+about 0.89. This suggests that considering the product
+of ω and τ as a whole may be more helpful to explore the
+relation between the oscillation period and the number
+density of produced pairs.
+1
+2
+3
+32
+34
+36
+38
+40
+T [m
+-1
+]
+n [m
+-3
+]
+ numerical results
+  a=8.24, b=28.86
+FIG. 8.
+The oscillation period of electron-positron plasma
+changing with the particle number density. Here E0 = 7.0Ecr,
+ωτ = 20.0 is kept to ensure that the number of cycles in the
+external electric field is constant. The value of ω ranges from
+0.4m to 1.65m and τ changes accordingly.
+It is noted that the approximate expression of the in-
+ternal electric field Eq. (8) is under the premise that the
+external field is zero, but the plasma oscillation period is
+still directly related to the external field parameters E0,
+ω and τ, which also embodies the non-Markov effect in
+the electron-positron pair production.
+IV.
+CONCLUSION AND DISCUSSION
+In summary, the backreaction effect and plasma oscil-
+lation in the pair production for a high-frequency electric
+field are investigated by the QVE.
+First, the parameter region of the external electric field
+where the backreaction effect cannot be ignored is ex-
+plored by calculating and analyzing the difference degree
+of pair number density with and without the backreac-
+tion. It is found that the backreaction effect can be ne-
+glected in the pair production for a high-frequency but
+weak electric field. In other words, the backreaction ef-
+fect can be ignored in the pair production for a subcritical
+external field with frequency chirp. This result is beyond
+what we previously thought, because the great improve-
+ment of particle number density by the high-frequency
+photon absorption process is expected to make the back-
+reaction effect become obvious. The reason are as follows.
+When the external electric field strength is smaller than
+the critical field strength, no matter how large the field
+frequency is, the backreaction effect can be neglected.
+Thus, to study the backreaction effect, the external elec-
+tric field strength should be larger than the critical one.
+However, for a high-frequency and strong electric field,
+the pair production is dominated by tunneling pair pro-
+duction and the multiphoton absorption mechanism at
+the same time, which will suppress the pair production
+because of the competitive relation between these two
+mechanisms.
+The influence of backreaction on the momentum spec-
+trum is also considered. It is found that the change of
+the momentum spectrum is very complex and sensitive
+to the parameters of the external electric field.
+Finally, the relationship between the plasma oscilla-
+tion period and the number density of produced particle
+pairs is studied. When the frequency and duration of the
+external electric field are kept constant and only the field
+strength is changed, the relation between the oscillation
+period and the particle number density is obvious and can
+be fitted by a simple formula constructed based on the
+Langmiur oscillation frequency. However, when the field
+frequency or the pulse duration changes, the relationship
+between them is very complex. One way to ensure that
+the oscillation frequency is regular is to keep the number
+of cycles in the external electric field unchanged when
+changing the field frequency. Furthermore, we find that
+the plasma oscillation period not only directly depend
+on the final number density of created particles, but also
+depend on the external field parameters, such as the field
+strength, the frequency, and the pulse duration, which is
+different from the common result in plasma physics. The
+reason for this is that the external electric field has left
+an imprint on the internal field by the non-Markov effect
+in pair production.
+Our results clarify the question whether it is reasonable
+to study the pair production for a subcritical electric field
+with frequency chirp, and deepen the understanding of
+the influence factors of the plasma oscillation period.
+ACKNOWLEDGMENTS
+The work is supported by the National Natural Science
+Foundation of China (NSFC) under Grants No. 11974419
+and No. 11705278, and by the Fundamental Research
+Funds for the Central Universities (20226943).
+
+8
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+
diff --git a/JdE2T4oBgHgl3EQfpQgx/content/tmp_files/load_file.txt b/JdE2T4oBgHgl3EQfpQgx/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..38565830191b0e276018a6c7601843e6f6607d95
--- /dev/null
+++ b/JdE2T4oBgHgl3EQfpQgx/content/tmp_files/load_file.txt
@@ -0,0 +1,827 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf,len=826
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='04026v1  [hep-ph]  10 Jan 2023  Backreaction effect and plasma oscillation in pair production for rapidly oscillating  electric fields  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' Jiang,1, 2 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' Gong,3 Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' Li,1, 2, ∗ and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' Li1, 2, †  1School of Science, China University of Mining and Technology, Beijing 100083, China  2State Key Laboratory for GeoMechanics and Deep Underground Engineering,  China University of Mining and Technology, Beijing 100083, China  3North China Electric Power University, Baoding 071003, China  (Dated: January 11, 2023)  The backreaction effect and plasma oscillation in pair production for rapidly oscillating electric  fields are investigated by solving quantum Vlasov equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' Contrary to previously thought, it is  found that the backreaction effect can be neglected in the pair production for a rapidly oscillating  but weak electric field, particularly, for a subcritical external electric field with frequency chirp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' In  some cases the oscillation period of created electron-positron plasma can be described by a simple  formula constructed based on the Langmiur oscillation frequency, but it is impossible in general  case because the plasma oscillation period directly depend not only on the final number density  of created particles, but also on the external electric field parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' Moreover, it is found that  the momentum spectrum presents complex interferences after considering the backreaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' These  results give us the safety range of external electric fields for taking no account of the backreaction  effect and deepen our understanding of the pair production with the backreaction effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='  INTRODUCTION  Since Dirac proposed the relativistic wave equation  and predicted the existence of positrons [1], many re-  searches have been done on how to produce electron-  positron pairs from vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' Sauter [2] found that the  electron-positron pairs can be produced from vacuum  in a strong static electric field by tunneling mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='  And then, Schwinger [3] calculated the pair produc-  tion rate in a constant electric field by the proper-time  method and gave out the critical electric field strength  Ecr = m2/e ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='3×1018V/m (the natural units ℏ = c = 1  are used).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='  As a result of these pioneering works, the  phenomenon of electron-positron pairs produced from  the vacuum in intense external fields is also known as  the Sauter-Schwinger effect [4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='  The laser intensity  corresponding to the critical electric field strength is  I ≈ 1029W/cm2, which is much larger than the possi-  bility of the current laser facilities [6–8], so it is not yet  possible to produce observable electron pairs in experi-  ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' However, the already operating X-ray free elec-  tron laser (XFEL) at DESY in Hamburg is expected to  achieve subcritical fields of E ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='01 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='1Ecr [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='  To reduce the threshold of pair production, several ap-  proaches have been proposed to produce observable elec-  tron pairs under subcritical electric field conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' One  method that can effectively enhance the pair production  is the dynamically assisted Schwinger mechanism [10–  14], in which the electric field adopts a combination of  a low-frequency strong field and a high-frequency weak  field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' Another approach is to use frequency chirp to in-  crease the pair yield by increasing the effective frequency  ∗ Corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' zlli@cumtb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='cn  † Corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' lyj@aphy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='iphy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='cn  of the electric field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='  In Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' [15–19], authors studied  the effect of spatially homogeneous electric fields with  frequency chirp on the pair production by the quantum  kinetic equation and found that for some chirp param-  eters the pair yield could be improved three to four or-  ders of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='  Moreover, the enhancement effect  of pair production in a spatially inhomogeneous electric  field with the frequency chirp were also found [20, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='  Previous studies have shown that the electrons and  positrons produced from vacuum can be accelerated in an  external field and form a new electric field, which is called  the internal electric field [22–25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' The internal electric  field can affect the generation of particle pairs together  with the external electric field, that is the backreaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='  For subcritical external field strength, the number of pro-  duced particle pairs is relatively less, and the backreac-  tion is generally considered to be negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' However, in  the study of pair production in a frequency chirp elec-  tric field, if the field lasts for a long time, the effective  frequency will be very large and may even exceed the  primary frequency, which can greatly enhance the multi-  photon pair production process and produce a large num-  ber of particle pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' Therefore, whether the backreaction  effect in this case can be negligible is a problem worth re-  searching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' In this paper, we will figure out the parameter  scope of external electric fields in which the backreaction  effect cannot be ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='  In addition, when the backreaction effect is considered  a plasma oscillation will occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' [22], the plasma  oscillation in pair production for a sinusoidal electric field  is studied by quantum Vlasov equation (QVE) and a ul-  trarelativistic formula is given to estimate the oscillation  period of the plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' The ultrarelativistic formula shows  that the oscillation period only depends on the maxima of  particle number density and has no direct relation with  the field parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='  In this work, we will study the  plasma oscillation for a rapidly oscillating electric field  2  by QVE, explore the determining factors of the oscilla-  tion period, and check whether this formula still holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='  The structure of this paper is as follows: Section II  briefly introduce the theoretical framework of quantum  dynamics-based backreaction effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' Section III is our  numerical results: Subsection III A discusses the effect of  the backreaction effect on the final particle number den-  sity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' subsection III B gives the effect of the backreaction  effect on the momentum spectra;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' subsection III C study  the relation between the plasma oscillation period and  the number density of produced particle pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' Section  IV is a summary and discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='  THEORETICAL FORMALISM  The magnetic effects can be neglected when a standing-  wave field produced by two counter-propagating laser  pulses is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='  The spatial scales for electron-  positron pairs production is on the same order of magni-  tude as the Compton wavelength of the electron, which  is far smaller than the focusing radius of the laser, so it  can be assumed that the laser is spatially homogeneous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='  By using the temporal gauge, A0 = 0, the spatially ho-  mogeneous and time-dependent four-vector potential can  be written as Aµ = (0, 0, 0, A(t)), and the external elec-  tric field is Eext = −dA(t)/dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' The form of the external  electric field we used is  Eext (t) = E0e− t2  2τ2 cos (ωt) ,  (1)  where E0 is the electric field amplitude, ω is the laser  frequency, and τ is the pulse duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='  The key quantity to study the electron-positron pair  production with the QVE is to obtain the momentum  distribution function f(k, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' For spatially homogeneous  and time-dependent electric fields, ignoring collisions be-  tween particles, the distribution function is determined  by df(k, t)/dt = S(k, t), where S(k, t) denotes the source  term of pair production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' When the external field strength  is relatively large, the backreaction brought by the in-  ternal electric field is gradually reflected, so the to-  tal electric field Etot(t) should be modified as the sum  of the external field and the internal field Eint(t), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='  Etot(t) = Eext(t) + Eint(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' After considering the influ-  ence of the backreaction,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' we can get the coupled equa-  tions of the distribution function and the internal electric  field  ˙f(k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' t) = eEtot(t)ε⊥  2Ω2(k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' t)  � t  −∞  dt′ eEtot(t)ε⊥  Ω2(k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' t′) [1 − 2f(k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' t′)]  × cos[2  � t  t′ dt′′Ω(k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' t′′)],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='  (2)  ˙Eint (t) = −4e  � d3k  (2π)3  � k∥ (t)  Ω (k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' t)f (k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' t)  + Ω (k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' t)  eEtot (t)  ˙f (k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' t) − e ˙Etot (t) ε2  ⊥  8Ω5 (k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' t)  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' (3)  where the dot on the letter represents the first order  time derivative,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' |e| is the renormalized charge of the  electron [23],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' k =  �  k⊥,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' k∥  �  is the canonical momentum,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='  k∥ (t) = k∥ − eA (t) is defined as the kinetic momentum  along the external electric field,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' ε⊥ =  �  m2 + k2  ⊥ is the  transverse energy squared,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' m is the mass of the electron,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='  and Ω (k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' t) =  �  ε2  ⊥ + k2  ∥ (t) is the total energy squared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='  The quantum statistics effect and the non-Markov effect  on the pair production can be seen from [1 − 2f(k, t′)]  in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' The first term on the right hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='  (3) represents the conduction current, the second term is  the polarization current, and the third term is the charge  renormalization part added to eliminate the divergence  of the polarization current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='  For the convenience of numerical calculation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' two aux-  iliary variables u (k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' t) and v (k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' t) are introduced  u (k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' t) = � t  t0 dt′W (k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' t′) [1 − 2f (k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' t′)]  × cos  �  2  � t  t′ dt′′Ω (k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' t′′)  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='  v (k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' t) =  � t  t0 dt′W (k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' t′) [1 − 2f (k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' t′)]  × sin  �  2  � t  t′ dt′′Ω (k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' t′′)  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='  (4)  then equation (2) can be equivalently transformed into  the following first-order differential equations  ˙f (k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' t) = 1  2W (k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' t) u (k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' t) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='  ˙u (k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' t) = W (k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' t) [1 − 2f (k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' t)] − 2Ω (k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' t) v (k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' t) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='  ˙v (k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' t) = 2Ω (k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' t) u (k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' t) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='  (5)  where W (k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' t) = eEtot (t) ε⊥/Ω2 (k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='  It can be seen from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' (3) that to obtain the inter-  nal electric field at time t, the momentum distribution  function at the same time must be integrated, but the  internal electric field at the same time must be known  to calculate the momentum distribution function at time  t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' To solve this contradiction, we calculate the internal  electric field by iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' First, we use the internal elec-  tric field at the previous time (t − ∆t) to calculate the  total electric field at time t by  Etot(t) = Eext(t) + El  int(t),  (6)  where l = 1, 2, · · · is the number of iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' Note that  for the first iteration (l = 1), the internal electric field  E1  int(t) = Eint(t − ∆t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' (6), the momentum  distribution function at time t can be obtained by solving  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' Then the internal electric field at time t can be  solved by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' This is the first iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' Replacing  the internal electric field in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' (6) with the new one, the  second iteration begins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' When the internal electric field  satisfies our preset control condition |El+1  int (t)−El  int(t)| <  ε, where ε is a very small number, it can be considered  that the real internal electric field at time t has been  3  obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='  Plugging it into the total electric field and  solving Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' (5), the momentum distribution function at  time t can be solved as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='  The initial state of the system is a vacuum state with-  out particles, so the single particle distribution func-  tion and auxiliary function satisfy the initial conditions  f (k, −∞) = u (k, −∞) = v (k, −∞) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='  The initial  condition of the internal electric field is Eint (−∞) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='  After obtaining the single-particle distribution function,  integrating it in the full momentum space can obtain the  resulting particle number density  n (t) = 2  �  d3k  (2π)3 f (k, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='  (7)  The coefficient 2 comes from the spin degeneracy of the  fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='  NUMERICAL RESULTS AND ANALYSIS  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='  Particle number density  After considering the backreaction effect, when the ex-  ternal electric field vanishes the total electric field is not  zero due to the existence of the internal electric field and  the particle number density still changes with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' So it  is not easy to obtain a definite particle number density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='  However, in a wide range of external field parameters,  the oscillation of particle number density induced by the  internal electric field is very very small, for example the  second case we considered below, it is reasonable to define  this relatively stable number density as the real particle  number density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' The detailed explanation is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' 1, we show the number density of created pairs  nb in the presence and n0 in the absence of backreaction  for two supercritical electric fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' The field frequencies  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' 1 (a) and (b) are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='15m and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='35m, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='  Other field parameters are E0 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='0Ecr and τ = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='0/m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='  It can be seen that the particle number density is invari-  able when the backreaction is not considered, but the sit-  uation is different when the backreaction is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='  For such a supercritical external electric field, when the  field frequency is relatively small, such as ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='15m in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' 1(a), the number density gradually increases with  time, which indicates that the internal electric field in-  duces the pair production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' However, when the frequency  increases to a certain value, such as ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='35m in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='  1(b), the particle number density remains nearly constant  because the internal electric field is not strong enough to  stimulate sufficient particle pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' Enlarging the curve  of nb in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' 1(b), one can see that the particle num-  ber density oscillates with time due to the existence of  the internal electric field, but its variation range does  not exceed 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='0 × 10−4m−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' Therefore, the relatively sta-  ble number density can be considered as the real particle  number density within the range of error allowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' Our  following calculation always meets this condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='  50  0  50  100  150  200  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='6  50  0  50  100  150  200  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='4  50  100  150  200  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='3600  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='3602  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='3604  n  0  n  b  t [m  1  ]  (a)  n  0  n  b  (b)  t [m  1  ]  n  b  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' The particle number density as a function of time  with (solid black lines) and without backreaction (dashed red  lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' The external electric field parameters in (a) are E0 =  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='0Ecr, ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='15m and τ = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='0/m, and in (b) are E0 =  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='0Ecr, ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='35m and τ = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='0/m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='  The particle number density varying with the field fre-  quency for different external electric field amplitudes is  shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' It can be seen that for the subcritical  field with E0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='1Ecr, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' 2(a), the number density  with and without the backreaction effect is almost the  same, and the multi-photon absorption is obvious [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='  At the frequency ω = 2m/N0 with the number of ab-  sorbed photons N0, the particle number density increases  greatly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' In the figure, 1−, 2−, 3−, 4−, and 5−photon ab-  sorption are marked by vertical lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='  However, when  E0 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='0Ecr, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='0Ecr, and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='0Ecr, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' 2(b), the mul-  tiphoton pair production is not obvious, and simply in-  creasing the frequency of the external field does not en-  hance the pair production but suppress it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='  According  to the Keldysh adiabatic parameter [27] γ = mω/eE0,  we can know that in our calculation γ ∼ O (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' In this  range, the tunneling pair production coexists with the  multiphoton pair production, and their competitive rela-  tion is unfavourable for the pair production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' In addition,  although the photon energy is high, the number density  of produced particles is also affected by other field pa-  rameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' When E0 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='0Ecr, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='0Ecr, the backreaction  effect is still insignificant, but when E0 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='0Ecr, the  backreaction effect of the internal electric field begins to  affect the particle number density, see ω ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='85m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='  To further investigate the region where the backre-  action effect cannot be ignored, we define the differ-  4  2/5  2/4  2/3  2/2  2/1  10  11  10  8  10  5  10  2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='4  n [m  3  ]  w [m]  n  b  (a)  n  0  n [m  3  ]  w [m]  n  b  , E  0  =2 E  cr  n  0  , E  0  =2 E  cr  n  b  , E  0  =3 E  cr  n  0  , E  0  =3 E  cr  n  b  , E  0  =4 E  cr  n  0  , E  0  =4 E  cr  (b)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' Particle number density changing with the field fre-  quency for different field amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' In (a), E0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='1Ecr,  and in (b), E0 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='0Ecr, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='0Ecr, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='0Ecr, the value of τ in both  figures is 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='0/m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='  ence degree of pair number density (DDOPND) as δ =  |nb − n0| /n0 × 100% and study its changes with the ex-  ternal field parameters E0 and ω, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' Although  the supercritical field strength is used in the calculation,  which is far away from the current experimental condi-  tions, some interesting results can still be obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' In  our results, for E0 ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='0Ecr, the maximum value of δ is  about 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='93% at E0 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='0Ecr, ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='5m and the backre-  action effect can be ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' For 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='0Ecr < E0 ≤ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='0Ecr,  the value of δ is generally within 5%, and in the region  where ω is small, the DDOPND value may exceed 5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='  When the electric field amplitude E0 is large and the fre-  quency ω is small, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=', the Keldysh parameter γ is small,  the DDOPND is large and the backreaction effect is more  obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' This also shows that generally the backreaction  effect can be neglected in the study of pair production in  a high-frequency but weak electric field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' Therefore, it is  safe to ignore the backreaction effect when study the pair  production in a subcritical external field with frequency  chirp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' This result is beyond our expectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' In the very  beginning, we thought that the particle number density  would be improved greatly by the high-frequency photon  absorption process and the backreaction effect became  significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' However, the actual result is not what we  previously thought because of the competitive relation  between the tunneling pair production and the multi-  photon absorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' In fact, the change of DDOPND with  E0 and ω is complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' For example, when E0 = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='0Ecr  and ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='8m, δ ≈ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='37%, the particle number density  with and without backreaction are about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='970m3 and  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='929m3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' The influence of the backreaction  effect is almost unnecessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' Whereas, when E0 = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='0Ecr  and ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='9m, δ ≈ 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='67%, the number density with and  without backreaction are about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='956m3 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='773m3,  the backreaction effect is very obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='  2  4  6  8  10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='5  w [m]  E  0  [E  cr  ]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='000  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='083  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='17  21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='25  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='33  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='00  d  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='  The difference degree of pair number density  (DDOPND) δ as a function of the external field amplitude  and frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' The value of τ in this figure is 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='0/m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' In the  calculation, the interval of E0 is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='1Ecr, and the interval of ω  is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='05m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='  Momentum spectrum  As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' 4, we compare the momentum dis-  tribution functions of created particles with and without  backreaction, denoted by fb and f0, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' In Fig  4(a), the momentum distribution f0 is smooth because  the particles are mainly generated by the main peak of  the electric field, but the situation is different for slightly  larger values of ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' 4(b), although the  sub-maximal peak of the electric field cannot produce  sufficient particle pairs, the momentum distribution func-  tion f0 shows obvious oscillations because of the infield  interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' Furthermore, without the backreaction, the  momentum distribution functions f0 is symmetric about  the zero kinetic momentum in both cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' However, with  the backreaction, the momentum spectrum presents ir-  regular oscillations and the symmetry is broken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' Another  phenomenon is that the backreaction effect is observed  in the left side of the momentum distribution functions  while on the right side they are approximately identical  to that without backreaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' In fact, the change of the  momentum spectrum with backreaction is very complex  and sensitive to the parameters of the external electric  field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='  5  30  20  10  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='2  20  10  0  10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='2  k  //  [m]  b  (k  //  )  0  (k  //  )  (a)  k  //  [m]  b  (k  //  )  0  (k  //  )  (b)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' Comparison of the momentum distribution functions  between with and without backreaction at t = 220.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='0/m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' The  parameters of the external electric field in (a) are E0 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='0Ecr,  ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='1m, τ = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='0/m, and in (b) are E0 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='0Ecr, ω =  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='15m, τ = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='0/m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='  Plasma oscillation period in pair production  We show the internal electric field evolution with time  for different external field amplitudes in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' One can  see when the external electric field exists (t < 40/m), the  amplitude of internal electric field increases with E0, and  its frequency is the same as that of the external electric  field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' For instance, when E0 = 7Ecr, the peak value of  the internal electric field is about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='2Ecr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' The motion  of particles is mainly dominated by the external electric  field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' However, when the external electric field is turned  off, the relation between the internal electric field and  the external field parameters becomes complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' For  example, the amplitude of the internal electric field does  not increase monotonically with the E0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' Moreover, we al-  ready know that when the external electric field is strong  enough, a large number of real electron-positron pairs  will be generated, and these particle pairs will maintain a  dynamic balance under the action of the internal electric  field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' Therefore, the motion of created electron-positron  pairs forms plasma oscillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' It is worth noting that  the period of plasma oscillation will have some relation  with the number density of created particles and the pa-  rameters of the external electric field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' Previous studies  have shown that the frequency of plasma oscillation de-  creases gradually with time and tends to a stable value  at the end [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' For the external electric field with high  intensity and high frequency, the frequency of plasma os-  cillation can reach the stable state quickly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='  0  50  100  150  200  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='2  E  int  (t) [E  cr  ]  t [m  1  ]  E  0  =6E  cr  E  0  =7E  cr  E  0  =8E  cr  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' The internal electric field changes with time for E0 =  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='0Ecr, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='0Ecr and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='0Ecr respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' Other field parameters  are ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='9m and τ = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='0/m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='  Assuming that the external electric field disappears at  t0, then the internal electric field at t > t0 can be ap-  proximately expressed as  Eint (t) = Eint0cos  �2π  T t + ϕ  �  ,  (8)  where Eint0 is the amplitude of the internal electric field,  T is the period of oscillation, and ϕ is the phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' In this  subsection, we mainly study the relationship between the  oscillation period T of the internal electric field and the  number density of created particles n, the parameters of  the external electric field E0, ω and τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='  In the follow-  ing research, we study the relationship between T and n  when E0, ω, and τ change, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='  First, we keep ω and τ unchanged, and E0 ranges from  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='0Ecr to 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='0Ecr with an interval of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='0Ecr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' The oscil-  lation period changing with the particle number density  is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' The black dots represent the numer-  ical results where the oscillation period of internal elec-  tric field T is estimated by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' From the results, we  can see that the particle number density always increases  with the external field amplitude E0 and the oscillation  period T decreases with the increase of the number den-  sity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' The red lines are the fitting curves corresponding to  the fitting formula  T (n) =  α  √n + β,  (9)  where n is the particle number density, α and β are fit-  ting parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' Note that this formula is constructed  based on the frequency of Langmuir oscillation [29] in  plasma physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' Although it is a little different from the  ultrarelativistic formula given in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' [22], both of them  6  have the same varying tendency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='  In (a), (b), (c), and  (d) of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='  6, the goodness of fit R2 is about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='998,  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='999, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='998, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='999, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='  This shows that  the formula above can fit the numerical results very well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='  Furthermore, since the fitting parameters α and β are  almost independent of E0, the oscillation period T is not  directly related to E0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' Finally, we emphasize that the ex-  ternal electric field we considered here is a high-frequency  strong field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' For other cases, the fitting formula (9) may  fail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='  0  1  2  3  20  30  40  50  0  1  2  3  4  20  30  40  50  0  1  2  3  20  30  40  50  0  1  2  3  20  30  40  50  60  70  T [m  1  ]  n [m  3  ]  numerical results  a=21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='25,  b =12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='29  (a)  T [m  1  ]  n [m  3  ]  numerical results  a=21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='54,  b =11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='40  (b)  T [m  1  ]  n [m  3  ]  numerical results  a=21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='35,  b =12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='39  (c)  T [m  1  ]  n [m  3  ]  numerical results  a=23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='77,  b =9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='48  (d)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='  The oscillation period of electron-positron plasma  varying with the particle number density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' The ω and τ are  fixed and the value of E0 ranges from 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='0Ecr to 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='0Ecr with  an interval of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='0Ecr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' Other external electric field parameters  are (a) ω = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='5m, τ = 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='0/m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' (b) ω = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='5m, τ = 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='0/m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='  (c) ω = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='7m, τ = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='0/m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' (d) ω = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='0m, τ = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='0/m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='  When E0 and τ are kept unchanged and ω is changed,  or keep E0 and ω constant and change τ, the above fit-  ting formula will be invalid, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='  The relation  between the oscillation period and the number density  of created pairs is very complex and not monotonically  decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' 7(a), we change the value of ω and  mark each data point with the frequency of the external  electric field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' It can be seen that the particle number den-  sity generally decreases with the increase of the frequency  ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' In the high-frequency region, such as ω ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='6m for  E0 = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='5Ecr, ω ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='5m for E0 = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='5Ecr, and ω ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='6m  for E0 = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='5Ecr, the oscillation period decreases with  the increase of particle number density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' This behavior is  somewhat similar to that in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' But when the fre-  quency takes other values, the relation between them ex-  hibits a very complex oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' In particular, we find  that in the low frequency region (such as ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='6m) the  fitting formula (9) still fails even if the field frequency and  the pulse duration are fixed and only the field amplitude  changes, because the number density does not decrease  monotonically with the field amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' 7(b), we  change the value of τ and mark each data point with the  value of τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' It can be seen that the relationship between  the particle number density and the pulse duration τ is  not monotonic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' As the pulse duration becomes large, the  number density of created pairs may not increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' Thus,  the fitting formula (9) also does not work here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' In addi-  tion, from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' 7(a), we can see that for E0 = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='5Ecr the  number density of created particles at ω = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='1m almost  equals that at ω = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='0m, but their oscillation periods  have a big difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' That is to say, the particle num-  ber density for different field frequencies can be equal,  see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' 2, but the same number density corresponds to  different oscillation periods, which shows that the oscil-  lation periods directly depend on not only the particle  number density but also the field frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' Similarly,  from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' 7(b), we can find that the oscillation periods  also directly depend on the pulse duration, because the  different number density corresponding to different pulse  durations gives the same oscillation period, see the data  points marked with the pulse duration 12 and 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='0  33  36  39  42  45  48  T [m  1  ]  n [m  3  ]  E  0  =6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='5 E  cr  E  0  =7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='5 E  cr  E  0  =8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='5 E  cr  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='7  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='8  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='7  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='6  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='7  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='8  (a)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='85  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='95  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='05  32  34  36  38  10  T [m  1  ]  n [m  3  ]  8  9  11  12  13  14  15  16  17  20  18(21)  19  22  23  24  28  25  40  36  32  (b)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='  The oscillation period of electron-positron plasma  varying with the particle number density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' In (a), E0 = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='5Ecr,  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='5Ecr, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='5Ecr, and τ = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='0/m are fixed, and ω ranges from  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='5m to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='0m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' The particle number density changes with ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='  In (b), E0 = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='0Ecr and ω = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='5m are constant, and the  particle number density changes with τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='  We further explore the relation between the oscilla-  tion period and the particle number density for fixing  E0 = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='0Ecr and keeping ωτ = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='0, which ensures  that the number of cycles in the external electric field  is constant, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='  To separate the number den-  sity the value of ω ranges from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='4m to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='65m with a  variable interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='  From the figure, we find that with  the frequency increasing (the pulse duration decreasing  correspondingly), the particle number density always de-  creases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' When the value of ω is relatively small, the par-  7  ticle number density changes faster with ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' For example,  when ω changes from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='41m to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='4m, the particle num-  ber density changes 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='1817m−3, while when ω changes  from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='5m to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='4m, the number density only changes  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='0725m−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' This result is also reflected in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' 4(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' Be-  sides this, the oscillation period tends to decrease with  the increase of the number density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' Therefore, we can try  to fit it with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' The fitted curve is represented by  the solid red line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' 8, and the goodness of fit R2 is  about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' This suggests that considering the product  of ω and τ as a whole may be more helpful to explore the  relation between the oscillation period and the number  density of produced pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='  1  2  3  32  34  36  38  40  T [m  1  ]  n [m  3  ]  numerical results  a=8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='24, b=28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='86  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='  The oscillation period of electron-positron plasma  changing with the particle number density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' Here E0 = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='0Ecr,  ωτ = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='0 is kept to ensure that the number of cycles in the  external electric field is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' The value of ω ranges from  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='4m to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='65m and τ changes accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='  It is noted that the approximate expression of the in-  ternal electric field Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' (8) is under the premise that the  external field is zero, but the plasma oscillation period is  still directly related to the external field parameters E0,  ω and τ, which also embodies the non-Markov effect in  the electron-positron pair production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='  CONCLUSION AND DISCUSSION  In summary, the backreaction effect and plasma oscil-  lation in the pair production for a high-frequency electric  field are investigated by the QVE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='  First, the parameter region of the external electric field  where the backreaction effect cannot be ignored is ex-  plored by calculating and analyzing the difference degree  of pair number density with and without the backreac-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' It is found that the backreaction effect can be ne-  glected in the pair production for a high-frequency but  weak electric field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' In other words, the backreaction ef-  fect can be ignored in the pair production for a subcritical  external field with frequency chirp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' This result is beyond  what we previously thought, because the great improve-  ment of particle number density by the high-frequency  photon absorption process is expected to make the back-  reaction effect become obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' The reason are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='  When the external electric field strength is smaller than  the critical field strength, no matter how large the field  frequency is, the backreaction effect can be neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='  Thus, to study the backreaction effect, the external elec-  tric field strength should be larger than the critical one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='  However, for a high-frequency and strong electric field,  the pair production is dominated by tunneling pair pro-  duction and the multiphoton absorption mechanism at  the same time, which will suppress the pair production  because of the competitive relation between these two  mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='  The influence of backreaction on the momentum spec-  trum is also considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' It is found that the change of  the momentum spectrum is very complex and sensitive  to the parameters of the external electric field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='  Finally, the relationship between the plasma oscilla-  tion period and the number density of produced particle  pairs is studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' When the frequency and duration of the  external electric field are kept constant and only the field  strength is changed, the relation between the oscillation  period and the particle number density is obvious and can  be fitted by a simple formula constructed based on the  Langmiur oscillation frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' However, when the field  frequency or the pulse duration changes, the relationship  between them is very complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' One way to ensure that  the oscillation frequency is regular is to keep the number  of cycles in the external electric field unchanged when  changing the field frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' Furthermore, we find that  the plasma oscillation period not only directly depend  on the final number density of created particles, but also  depend on the external field parameters, such as the field  strength, the frequency, and the pulse duration, which is  different from the common result in plasma physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' The  reason for this is that the external electric field has left  an imprint on the internal field by the non-Markov effect  in pair production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='  Our results clarify the question whether it is reasonable  to study the pair production for a subcritical electric field  with frequency chirp, and deepen the understanding of  the influence factors of the plasma oscillation period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='  ACKNOWLEDGMENTS  The work is supported by the National Natural Science  Foundation of China (NSFC) under Grants No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' 11974419  and No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' 11705278, and by the Fundamental Research  Funds for the Central Universities (20226943).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
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+page_content=' Dancus, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
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+page_content='  Ghenuche, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' Ghita, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' Kisyov, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' Nastasa, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' Ong,  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' Rotaru, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' Sangwan, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content='-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' S¨oderstr¨om, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' Stutman,  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' Suliman, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' Tesileanu, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' Tudor, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' Tsoneva, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfpQgx/content/2301.04026v1.pdf'}
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diff --git a/K9AzT4oBgHgl3EQfyf6u/content/tmp_files/2301.01755v1.pdf.txt b/K9AzT4oBgHgl3EQfyf6u/content/tmp_files/2301.01755v1.pdf.txt
new file mode 100644
index 0000000000000000000000000000000000000000..9234dc36187f983bd933d93ba9f16eaca84f4344
--- /dev/null
+++ b/K9AzT4oBgHgl3EQfyf6u/content/tmp_files/2301.01755v1.pdf.txt
@@ -0,0 +1,2585 @@
+1
+Past, Present, and Future of Autonomous Drone
+Racing: A Survey
+Drew Hanover1, Antonio Loquercio3, Leonard Bauersfeld1, Angel Romero1, Robert Penicka2,
+Yunlong Song1, Giovanni Cioffi1, Elia Kaufmann1 and Davide Scaramuzza1
+a) Number of Papers on Drone Racing
+'15
+0
+'16
+1
+'17
+4
+'18
+15
+'19
+69
+Year
+'20
+95
+'21
+143
+'22
+188
+b) Onboard View
+c) Drone Racing
+Fig. 1: Drone racing is a sport rapidly gaining popularity where opponents compete on a pre-defined race track consisting of a series of gates. Autonomous
+drone racing research aims to build algorithms that can outperform human pilots in such competitions. a) The task of autonomous drone racing has gained
+a substantial amount of interest from the research community in the last few years, as indicated by the increasing number of related publications per year.
+b) Autonomous drones rely on visual and inertial sensors to estimate their own states, as well as their opponents’ states. c) Agile maneuvers are required to
+overtake opponents and win the race.
+Abstract—Over the last decade, the use of autonomous drone
+systems for surveying, search and rescue, or last-mile delivery
+has increased exponentially. With the rise of these applications
+comes the need for highly robust, safety-critical algorithms which
+can operate drones in complex and uncertain environments.
+Additionally, flying fast enables drones to cover more ground
+which in turn increases productivity and further strengthens
+their use case. One proxy for developing algorithms used in
+high-speed navigation is the task of autonomous drone racing,
+where researchers program drones to fly through a sequence of
+gates and avoid obstacles as quickly as possible using onboard
+sensors and limited computational power. Speeds and accelera-
+tions exceed over 80 kph and 4 g respectively, raising significant
+challenges across perception, planning, control, and state estima-
+tion. To achieve maximum performance, systems require real-
+time algorithms that are robust to motion blur, high dynamic
+range, model uncertainties, aerodynamic disturbances, and often
+unpredictable opponents. This survey covers the progression of
+autonomous drone racing across model-based and learning-based
+approaches. We provide an overview of the field, its evolution
+over the years, and conclude with the biggest challenges and
+open questions to be faced in the future.
+I. INTRODUCTION
+Throughout history, humans have been obsessed with racing
+competitions, where physical and mental fitness are put to the
+1D. Hanover, L. Bauersfeld, A. Romero, Y. Song, G. Cioffi, E. Kaufmann
+and D. Scaramuzza are with the Robotics and Perception Group, University
+of Zurich, Switzerland (http://rpg.ifi.uzh.ch). 2R. Penicka is with the Multi-
+robot Systems Group, Czech Technical University in Prague, Czech Republic.
+3A. Loquercio is with UC Berkeley. This work was supported by the
+Swiss National Science Foundation (SNSF) through the National Centre of
+Competence in Research (NCCR) Robotics, the Czech Science Foundation
+(GACR) under research projects No. 23-06162M, the European Union’s
+Horizon 2020 Research and Innovation Programme under grant agreement
+No. 871479 (AERIAL-CORE), and the European Research Council (ERC)
+under grant agreement No. 864042 (AGILEFLIGHT).
+test. The earliest mention of a formal race dates all the way
+back to 3000 BC in ancient Egypt where the Pharaoh was
+thought to have ran a race at the Sed festival to demonstrate
+his physical fitness, indicating his ability to rule over the
+kingdom [1], [2]. As time has progressed, humans have
+moved from racing on-foot to using chariots, cars, planes, and
+more recently quadcopters [3]. Although the vessel frequently
+changes, one thing that has remained constant since the early
+days of racing has been the recurring theme of using the
+task as a catalyst for scientific and engineering development.
+Recently, we have seen a push to remove humans from the
+loop, automating the highly complex task of racing in order to
+push vehicle performance beyond what a human can achieve.
+A. Why Drone Racing?
+Drone racing is a popular sport with high-profile interna-
+tional competitions. In a traditional drone race, each vehicle
+is controlled by a human pilot, who receives a first-person-
+view (FPV) live stream from an onboard camera and flies
+the drone via a radio transmitter. An onboard image from the
+drone can be seen in Fig. 1. Human drone pilots need years of
+training to master the advanced navigation and control skills
+that are required to be successful in international competitions.
+Such skills would also be valuable to autonomous systems that
+must quickly and safely fly through complex environments,
+in applications such as disaster response, aerial delivery, and
+inspection of complex structures. For example, in search-and-
+rescue scenarios, drones must be able to rapidly navigate in
+complex environments in order to maximize their spatial cov-
+erage. Put more simply, drones that can fly fast, fly farther [4].
+Automating inspection tasks can save lives while being
+more productive than manual inspection. According to a recent
+arXiv:2301.01755v1  [cs.RO]  4 Jan 2023
+
+2
+survey on unmanned aerial vehicle (UAV) use in bridge
+inspection [5], most drones used for inspection tasks rely on
+GPS navigation with the biggest limiting factor on inspection
+efficiency being the drones endurance and mobility. Addition-
+ally, the authors note that the most popular drones used for
+surveying by several US Departments of Transportation are
+not fully autonomous and require expert human pilots [5].
+The commercial and safety advantages of highly agile drone
+systems is clear, however research into autonomous drone
+racing can also help us gain new understandings on how
+the visual processing and control by human pilots works, as
+demonstrated in [6].
+Over the last five years, several projects have been launched
+to encourage rapid progress within the field, such as DARPA’s
+Fast Lightweight Autonomy (FLA)
+[8] and European Re-
+search Council’s AgileFlight [9]. With funding pools of
+over $1 million for each of these projects and significant
+commercial potential, a large incentive exists for researchers
+and entrepreneurs alike to explore new paradigms in agile
+flight research. Competitions such as the IROS’16-19 Au-
+tonomous Drone Racing series [10], NeurIPS 2019’s Game
+of Drones [11], and the 2019 AlphaPilot Challenge [12], [13]
+provided further opportunity for researchers to compare their
+methodologies against one another in a competitive fashion.
+A depiction of the progress made from these competitions can
+be seen in Fig. 2.
+Drone racing is a challenging benchmark which can help re-
+searchers to gauge progress on complex perception, planning,
+and control algorithms. Autonomous drones in a racing setting
+must be able to perceive, reason, plan, and act on the tens of
+milliseconds scale, all onboard a computationally limited plat-
+form. Apart from being very challenging to solve, the drone
+racing task offers a single measure of the progress of the state-
+of-the-art in autonomous flying robotics: lap time. Solving this
+problem requires algorithms to be efficient, lightweight, and
+provide optimal decision and control behaviors all in real-time.
+Additionally, we see nearly exponential growth of the number
+of papers in the field year over year as seen in Fig 1.
+To the best of the authors’ knowledge, this is the first
+survey on the state of the art in autonomous drone racing.
+This overview will be useful to researchers who wish to make
+connections between existing works, learn about the strengths
+and weaknesses of current and past approaches, and identify
+directions moving forward which should progress the field in
+a meaningful way.
+B. Task Specification
+The drone racing task is to fly a quadrotor through a
+sequence of gates in a given order in minimum time while
+avoiding collisions. Humans are astonishingly good at this
+task, flying at speeds well over 100kph with only a first-person
+view camera as their sensory input. Beyond this, expert pilots
+can adapt to new race tracks quickly in a matter of minutes,
+however the sensorimotor skills required by professional drone
+pilots take years of training to acquire.
+For an autonomous drone to successfully complete this task,
+it must be able to detect opponents and waypoints along the
+track, calculate their location and orientation in 3-dimensional
+space, and compute an action that enables navigation through
+the track as quickly as possible while still controlling a highly
+nonlinear system at the limits. This is challenging in three
+different aspects: Perception, Planning, and Control. Poor
+design in any of these aspects can make the difference between
+winning or losing the race, which can be decided by less than
+a tenth of a second.
+The paper is structured as follows: First, the modeling pro-
+cedure of the drone including aerodynamics, batteries, motors,
+cameras, and the system nonlinearities are discussed in detail
+in Sect. II. A classical robotics pipeline is then introduced in
+Sect. III, with a deep dive into literature relevant to agile flight
+split into Perception, Planning, and Control subsections. After-
+wards, we delve into learning-based methods for Perception,
+Planning, and Control which rely on recent advancements from
+the machine learning community in Sect. IV. Then, a discus-
+sion of the development of simulation tools which can enable
+rapid development for agile flight applications in Sect. V. A
+history of drone racing competitions and the methods used for
+each are included in Sect. VI. Next, a summary of open source
+code bases, hardware platforms, and datasets for researchers
+is provided in Sect. VII. Finally, a forward-looking discussion
+on the Opportunities and Challenges for future researchers
+interested in autonomous drone racing in Sect. VIII.
+II. DRONE MODELING
+To further advance research on fast and agile flight, it is
+important to have accurate models that capture the complex
+nonlinear dynamics of multicopter vehicles at the limit of their
+performance envelope.
+This section reviews different dynamics modeling ap-
+proaches from classic, first-principles modeling to pure data-
+driven models in the context of drone racing. For the vehicle
+dynamics, the key aspects that need to be modeled are the
+kinematics, aerodynamics, the electric motors, and the battery.
+In addition to the vehicle dynamics models discussed in this
+section, many difficulties for autonomous drone racing models
+are introduced by the onboard sensors, whose characteristics
+need to be modeled. For example, IMUs are subject to bias
+and noise, and the intrinsic as well as extrinsic parameters of
+onboard sensors change over time as hard crashes may lead
+to miscalibration.
+A. Kinematics
+Typically, the vehicle is assumed to be a 6 degree-of-
+freedom rigid body of mass m with a (diagonal) inertia matrix
+J
+= diag(Jx, Jy, Jz). Given a dynamic state x ∈ R17
+the equations describing its evolution in time are commonly
+written as:
+˙x = f(x, u) =
+�
+���������
+˙pW B
+˙qW B
+˙vW
+˙ωB
+˙Ω
+�
+���������
+=
+�
+����������
+vW
+qW B
+�
+0
+ωB/2
+�
+1
+m
+�
+qW B ⊙ f
+�
++ gW
+J −1�
+τ − ωB × JωB
+�
+1
+τΩ (Ωss − Ω)
+�
+����������
+,
+(1)
+
+3
+The first autonomous drone racing challenge at
+IROS
+2016,
+Daejong
+Korea.
+Slow
+moving
+quadrotors cautiously navi-gated the course shown
+above using only onboard sensors. Team KIRD
+from KAIST placed 1st, reaching a top speed of
+0.6 m/s.
+IROS ADR I
+In the third iteration of the ADR challenge held in
+Madrid, teams began implementing learning based
+methods with optimal control techniques. The
+Robotics
+and
+Perception
+Group
+from
+the
+University of Zurich successfully completed the
+course the fastest with speeds up to 2.0 m/s.
+IROS ADR III
+The following year, another autonomous drone
+racing
+competition
+took
+place
+in
+Vancouver,
+Canada. Similar to the year prior, teams used
+classical methods to navigate a challenging course
+with compute done onboard and the INAOE team
+from Mexico won with a speed of 0.7 m/s.
+IROS ADR II
+In summer of 2022, the Robotics and Perception
+Group of the University of Zurich hosted a drone
+racing
+competition
+to
+face
+their
+autonomous
+drones off against some of the best human FPV
+pilots in the world. Speeds exceeded 20 m/s,
+relying only on onboard sensing.
+Swiss Drone Days
+2017
+2016
+2018
+2022
+2020
+Lockheed Martin sponsored a $1M prize to teams
+who could successfully navigate a challenging
+drone racing course completely autonomously. The
+MAVLAB team from TU Delft won, with top
+speeds approaching 10 m/s – a significant jump
+over previous competitions.
+AlphaPilot
+vmax = 0.6 m/s
+vmax = 0.7 m/s
+vmax = 2.0 m/s
+vmax = 10 m/s
+vmax = 22 m/s
+Fig. 2: History of drone racing competitions that use real drones for navigating the race track, IROS ADR II photo credit [7].
+where pW B ∈ R3 is the position of the center of mass given
+in the world frame, qW B ∈ SO(3) is a quaternion defining
+the rotation of the body frame relative to the world (vehicle
+attitude), vW ∈ R3 is the velocity of the vehicle in the world
+frame, ωB ∈ R3 are the bodyrates of the vehicle, Ω ∈ R4 are
+the motor speeds, gW = [0, 0, −9.81 m/s2]⊺ denotes earth’s
+gravity, and u ∈ R4 is the input. Depending on the control
+modality the input can be single rotor thrusts or collective
+thrust and body rates. In this setting, the task of the model is
+to calculate the total force f and total torque τ that acts on
+the drone as accurately as possible. Note the quaternion-vector
+product denoted by ⊙ representing a rotation of the vector by
+the quaternion as in q ⊙ f = q · [0, f ⊺]⊺ · ¯q, where ¯q is the
+quaternion’s conjugate. Those forces and torques, collectively
+referred to as wrench, are determined by the aerodynamics
+of the platform as well as the vehicles’ actuators, e.g. the
+propellers.
+B. Aerodynamics
+This section discusses the different approaches to modeling
+the aerodynamics of the drone and its propellers. The most
+widely used modeling assumption is that the propeller thrust
+and drag torque are proportional to the square of the rotational
+speed [14]–[18] and that the body drag is negligible. These as-
+sumptions quickly break down at the high speeds encountered
+in drone racing as this model neglects (a) linear rotor drag [19],
+[20], (b) dynamic lift [19], (c) rotor-to-rotor [21]–[23], (d)
+rotor-to-body [21]–[23] interactions and (e) aerodynamic body
+drag [20], [22].
+The accuracy of the propeller model can be improved
+by leveraging blade-element-momentum theory, where the
+propeller is modeled as a rotating wing. Such first-principle ap-
+proaches [24]–[28] have been shown to provide very accurate
+models of the wrench generated by a single propeller as they
+properly capture effects (a) and (b). Implemented efficiently, a
+Blade Element Momentum (BEM) model can be run in real-
+time [29] and has been successfully used to test algorithms in
+simulation [30], [31].
+Accounting for the remaining open points (c)-(e), the aero-
+dynamics of the drone body as well as any interaction effects
+need to be calculated, which requires a full Computational
+Fluid Dynamics (CFD) simulation [21]–[23], [32], [33]. Due
+to the extreme computational demands, this is impractical
+in drone racing. To still get close to the accuracy of CFD
+methods while retaining the computational simplicity of the
+previously mentioned methods, data-driven approaches are
+employed [29], [34]–[38]. In the early works [36], [37],
+the whole vehicle dynamics model was learned from data.
+In a similar fashion [35] uses a combination of polynomi-
+als—identified from wind-tunnel flight data—to represent the
+vehicle dynamics. In [29], [34], it has been shown that higher
+modeling accuracies can be achieved when combining a first-
+principle model with a data-driven component. Such com-
+
+4
+bination of first-principle and data-driven models also leads
+to improved generalization performance, as shown in [29],
+which combines a BEM model with a temporal convolutional
+network [39] to regress the residual wrench. With this, a high-
+fidelity model is obtained [29] that is still accurate at the
+speeds and accelerations encountered in drone racing.
+C. Motor and Battery Models
+The previous section outlines different approaches to how
+the aerodynamic wrench can be estimated based on the state of
+the vehicle. However, for all such models the rotational speed
+of the propeller is assumed to be known. On most multicopters,
+the motors are not equipped with closed-loop motor speed
+control but controlled by a ’throttle’ command which controls
+the duty cycle of a PWM (pulse-width-modulation) signal
+applied to the motors. The actual rotational speed that the
+motor achieves is a function of the throttle command as well
+as other parameters such as the battery voltage and the drag
+torque of the rotor [4]. Therefore, in order to have a dynamics
+model for the motors we need a model of the battery to
+calculate the voltage applied to the motors. Most literature
+on battery modeling relies on so-called Peukert models [40],
+but for lithium-polymer batteries in drone racing this is hardly
+applicable because the battery discharge current often exceeds
+100 A (e.g. 50-100 C) [41], [42]. Graybox battery models
+for the voltage that are based on a one-time-constant (OTC)
+equivalent circuit [43], [44] are much more suitable for drone
+racing tasks as shown in [4], because they are applicable to the
+extremely high loads experienced during a racing scenario. In
+combination with either a polynomial or a constant-efficiency
+motor model such OTC models can be used to accurately
+simulate the battery voltage during agile flight [4]. Given
+a simulation of the battery voltage, one can measure the
+performance characteristics of a given motor-propeller com-
+bination to determine the mapping of throttle command and
+voltage to resulting steady-state propeller speed Ωss. When
+the highest model fidelity is desired, a more sophisticated
+motor simulation [45] can further improve the accuracy, which
+can be desirable if the controller directly outputs single-rotor
+thrusts instead of the more commonly used collective-thrust
+and bodyrates control modality.
+D. Camera and IMU Modeling
+Drone racing pushes not only the mechanical and electrical
+components of drones to their limits, but is also highly
+demanding in terms of sensor performance. For an in-depth
+overview of the many different sensor options for drone racing
+the reader is referred to [46]. The most common sensors
+aboard autonomous drones are monocular or stereo cameras
+combined with IMUs (inertial measurement units) thanks
+to their low cost, low weight, and mechanical robustness.
+Typically, the algorithms processing the visual information
+assume a pinhole camera [47] mounted in the center of gravity
+of the vehicle. In reality, the camera is usually never mounted
+directly at the center of gravity of the vehicle, meaning that a
+transformation between the camera frame and the body frame
+of the drone is required.
+The camera and IMU must be calibrated before using
+their measurements. The camera intrinsic calibration uses the
+assumption of a pinhole model [47] and estimates the focal
+length, image center and distortion parameters. The IMU
+intrinsic calibration estimates the noise characteristic of the
+sensor. The camera-IMU extrinsic calibration estimates the
+relative position and orientation of the two sensors as well
+as the time offset. Kalibr [48] is a widespread tool to perform
+these calibrations.
+The biggest source of measurement error of the sensors
+onboard a drone are not the sensors themselves but the strong
+high-frequency vibrations introduced by the fast-spinning pro-
+pellers. The vibrations lead to aliasing effects on the IMU
+measurements and introduce additional motion blur on the
+camera images. The structural vibrations and their effect on
+the measurements are extremely difficult to model and correct
+for. Therefore, a suitable hardware design is imperative which
+dampens the mount of the camera and the IMU with respect
+to the vehicle frame.
+III. CLASSICAL PERCEPTION, PLANNING, AND CONTROL
+PIPELINE
+Sensors
+Hardware
+Perception
+Software
+Planning
+Control
+Drone
+Fig. 3: Architecture 1: A classic architecture for an autonomous system
+programmed using model-based approaches
+Since the inception of the field of mobile robotics, a
+common architecture has been primarily used to achieve
+autonomous navigation capabilities across a wide variety of
+systems. In a traditional robotics software stack, the naviga-
+tion task is broken into three main components: Perception,
+Planning, and Control. A diagram of this architecture can be
+seen in Fig. 3. In this section, we cover recent research in
+each of these areas relating specifically to agile flight and
+autonomous drone racing. All of the approaches detailed in
+this section rely on first principles modeling and optimization
+techniques.
+A. Perception
+The perception block estimates the vehicle state and per-
+ceives the environment using onboard sensors. The most
+common solution for state estimation of flying vehicles is
+visual-inertial odometry (VIO) thanks to its low cost and low
+weight requirements. Most of the time, a layout or map of
+the race track is known apriori, and thus this section focuses
+exclusively on VIO methods. VIO uses camera and IMU
+measurements to estimate the state ˆx (position, orientation,
+and velocity) of the drone platform. The inertial measurements
+are integrated to obtain relative position, orientation, and
+velocity estimates in a short time, e.g. between two camera
+images. However, the integration for a longer time, e.g. few
+seconds, accumulates large drift due to scale factor errors,
+axis misalignment errors, and time-varying biases [49] that
+commonly affect off-the-self IMU measurements. The camera
+measurements provide rich information about the environment
+
+5
+at a lower rate, usually around 30 Hz, than IMU measurements.
+Differently from the IMU measurements, the camera measure-
+ments are affected by environmental conditions. The quality of
+information that they provide for state estimation degrades in
+the case of poor illumination conditions, textureless scenes,
+and motion blur. For this reason, the camera and inertial
+measurements complement each other and are the standard
+choice for state estimation of flying vehicles [50]. In this
+section, we first give an overview of VIO with a focus on
+the methods that can be applied for online state estimation
+of aerial robotic vehicles. Second, we give an overview of
+recent VIO algorithms that include drone dynamics in the
+estimation process. Third, we conclude with a discussion on
+the application of classical VIO methods to drone racing tasks.
+1) VIO: VIO is the most common solution for state esti-
+mation of aerial vehicles [50] using only onboard sensing and
+computing thanks to its favorable trade-off between accuracy
+and computational requirements. VIO algorithms are usually
+composed of two main blocks: the frontend and the backend.
+The frontend uses camera images to estimate the motion of
+the sensor. Two main approaches exist in the literature: direct
+methods and feature-based methods. Direct methods [51], [52]
+work directly on the raw pixel intensities. These methods
+commonly extract image patches and estimate the camera
+trajectory by tracking the motion of such patches through
+consecutive images. The tracking is achieved by minimizing a
+photometric error defined on the raw pixel intensities [51]. On
+the contrary, feature-based methods [53]–[55] extract points
+of interest, commonly known as visual features or keypoints,
+from the raw image pixels. The camera trajectory is estimated
+by tracking these points through consecutive images. Feature-
+based methods are more mature and robust than direct meth-
+ods; however, the latter achieve higher reliability in low-texture
+environments. Hybrid methods, which combine keypoints and
+patches of raw pixel intensities to estimate the camera motion,
+also exist [56]. We refer the reader to the work in [47] for a
+tutorial on the VIO frontend.
+The backend fuses the output of the fronted with the inertial
+measurements. Two categories exist in the literature according
+to how the sensor fusion problem is solved: filtering methods
+and fixed-lag smoothing methods. Filtering methods are based
+on an Extended Kalman Filter (EKF). These methods propa-
+gate the state of the system using the inertial measurements
+and fuse the camera measurements in the update step. The
+pioneer filter-based VIO algorithm is the Multi-State Con-
+straint Kalman Filter (MSCKF) originally proposed in [53].
+Since then, many different versions of MSCKF have been
+developed [57]. Fixed-lag smoothing methods [54], [55], also
+referred to as sliding window estimators, solve a non-linear
+optimization problem where the variables to be optimized are
+a window of the recent robot states. The cost function to
+minimize contains visual, inertial, and past states marginal-
+ization residuals. Fixed-lag smoothing methods accumulate
+less linearization error than filtering methods but are more
+computationally demanding. We refer the reader to the work
+in [58] for a tutorial on the VIO backend.
+Recent works [59], [60] have proposed to include event
+cameras [61] in VIO for flying vehicles. Event cameras do
+not capture images at a fixed rate but they asynchronously
+measure per-pixel brightness changes. The output of these
+cameras is a stream of events that consists of time, location,
+and the sign of brightness change. Their main properties are
+low latency, high temporal resolution (in the order of µs), and
+high dynamic range (140 dB compared to 60 dB of standard
+cameras). Thanks to these properties, event cameras are a great
+complementary sensor to standard cameras. Including event
+data in VIO algorithms achieves increased robustness against
+motion blur as demonstrated in [59], [60]. UltimateSLAM
+[59] is a VIO algorithm that combines both standard and
+event cameras in a fixed-lag smoother-based VIO algorithm.
+In [60], a revised version of UltimeSLAM was proposed to
+demonstrate autonomous quadrotor flight despite rotor failure
+with onboard sensors.
+2) Drone dynamics in VIO: The drone dynamics are used to
+define additional error terms in the VIO backend in [62], [63].
+In [62], the authors propose VIMO which is a VIO algorithm
+that includes error terms on the drone transitional dynamics
+in a fixed-lag smoother-based backend [55]. These error terms
+on the drone dynamics are derived through the preintegration
+theory [64]. In addition to the drone state, VIMO is able to
+estimate the external force acting on the drone platform. The
+external force is modeled as a random variable distributed
+according to a zero-mean gaussian distribution. This choice
+allows the estimation of impulse-like forces acting on the
+drone platform.
+The work in [63] extends VIMO by introducing a different
+model of the external force. The external force is modeled
+as a gaussian distribution whose mean value is equal to the
+difference between the commanded collective thrust and the
+force, up to the mass, detected by the accelerometer. This is
+a suitable design choice in the case when the external force
+affects the drone platform for long periods of time.
+3) Discussions: The work in [65] presents a benchmark
+comparison between a number of VIO solutions on the EuRoC
+dataset [66]. The EuRoC dataset contains camera- and IMU-
+data recorded onboard a drone flying in indoor environments.
+The drone moves with average linear and angular velocities
+up to 0.9 m/s and 0.75 rad/s, respectively. These values are
+far below the ones reached in drone racing. The conclusions
+of [65] show that state-of-the-art VIO algorithms provide
+reliable solutions for estimating the state of the drone at limited
+speeds. However, these classical VIO methods are not able to
+provide accurate state estimates for drone racing tasks. VIO
+methods accumulate large drift in scenarios characterized by
+motion blur, low texture, and high dynamic range [67]. These
+scenarios are the norm in drone racing.
+To help the research in VIO algorithms for drone racing
+tasks, the work in [68] proposes the UZH-FPV Drone Racing
+Dataset. This dataset contains images recorded from standard
+cameras, event camera data, and IMU data recorded onboard
+a quadrotor flown by a human pilot. All the flights include
+visual challenges that are similar to the ones present in drone
+racing competitions.
+Successful state estimation solutions for drone racing [67],
+[69] reduce the drift accumulated in VIO by localizing to a
+prior map of the track. This is a viable solution when a map
+
+6
+of the track in the form of gate positions is known beforehand.
+The localization process is based on the detection of the gates.
+In [70], it was proposed a gate detector that uses an RGB
+camera to identify the gates based on their color. All the other
+gate detection methods existing in the literature are based on
+deep learning techniques [71]. We review them in Sec. IV. The
+known gate positions and the detections in the onboard images
+are used to estimate the relative pose between the camera and
+the gate using the Perspective-n-Point algorithm (PnP) [72].
+This relative pose is used to constrain the VIO backend and
+consequently reduce the drift. There is significant room for
+innovation on this front, as the VIO-PnP paradigm has existed
+for several years with little innovation. Other approaches used
+in early drone racing competitions relied on the technique of
+visual servoing via stereo cameras [7], but this solution was
+found to be sensitive to indoor lighting changes and needed
+to be hand-tuned for every flight.
+Recent works [73]–[75] proposed vision-based odometry
+algorithms that are learned end-to-end. In theory, these meth-
+ods could be specialized to drone racing tasks and potentially
+outperform classical VIO approaches. However, they are in
+the early development phase and how to customize them for
+the drone racing task is still an open research question. In
+addition, they currently have high computational costs that
+make them impractical for online state estimation onboard
+drones. We refer the reader to Sec. IV for a detailed review
+of VIO methods based on deep learning.
+B. Planning
+Once
+a
+state
+estimate
+ˆx
+has
+been
+obtained
+from
+the
+perception
+module,
+the
+next
+step
+in
+the
+classical
+pipeline
+is
+to
+plan
+a
+feasible,
+time-optimal
+trajectory
+τref = (xref, uref)k, ∀k ∈ 0 . . . N, which respects the phys-
+ical limits of the platform as well as the constraints imposed
+by the environment. This requires predicting the drone’s future
+states such that minimum lap time is reached without crashing.
+Trajectory planning has matured over the last decade from
+works mostly verified in simulation to works shown in both
+controlled lab environments and unknown unstructured envi-
+ronments. Here, we categorize these methods in polynomial
+and spline trajectory planning, optimization-based planning,
+search-based planning, and sampling-based planning.
+1) Polynomial and Spline Trajectories: The Polynomial
+and Spline methods leverage the differential flatness prop-
+erty [76], [77] of quadrotors and represent a trajectory as a
+continuous-time polynomial or spline. This property simplifies
+the full-state trajectory planning to a variant where only four
+flat outputs need to be planned (typically 3D position and
+heading). By taking their high order derivatives, these flat
+outputs can represent a dynamically feasible trajectory with
+their respective control inputs. This property is used by many
+polynomial and spline methods that are nowadays among the
+most used for general quadrotor flight.
+The widely used polynomial trajectories [76], [77] minimize
+snap (4th order position derivative) of a trajectory. Different
+methods opted for minimizing jerk (3rd order position deriva-
+tive) for planning a trajectory [78]. However, the trajectories
+that result from having jerk as the primary objective have been
+shown to minimize the aggressiveness of the control inputs
+[78], which is fundamentally different from minimizing the lap
+times, where extremely aggressive trajectories are generally
+required. Richter et al. [79], therefore, extended the objective
+by jointly optimizing both the snap of a trajectory and the
+total time through a user-specified penalty on time. Recently,
+Han [80] proposed a polynomial-based trajectory planning
+method for drone racing. It jointly optimizes control effort,
+regularized time, and penalizes the dynamic feasibility and
+collisions.
+Because of their numerical stability, other methods make
+use of B-splines for representing trajectories [81], [82] in-
+stead of high order polynomial representations that are nu-
+merically sensitive. These methods jointly optimize different
+objectives, simultaneously smoothness, dynamic feasibility,
+collision avoidance, safety [82] and vision-based target track-
+ing [83].
+Although both polynomial and spline trajectories are widely
+used due to their computational efficiency, polynomial-based
+trajectories (and their derivatives) are smooth by definition.
+Therefore, only smooth control inputs can be sampled from
+them.
+Yet, a time-optimal trajectory is not smooth, but rather
+attempts to keep the acceleration to the possible maximum at
+all times [84]. This means that our planned actuation needs to
+be able to maintain a certain value during extended periods of
+time, while also being able to have quick changes. Therefore
+polynomials cannot represent true time-optimal trajectories.
+2) Optimization-based: Optimization-based trajectory plan-
+ning enables us to independently select the optimal sequence
+of states and inputs at every time step, which inherently con-
+siders time minimization while complying with quadrotor dy-
+namics and input constraints. Optimization-based approaches
+have been extensively considered in the literature, ranging
+from exploiting point-mass models [85], simplified quadrotor
+models [86], [87], and full-state quadrotor models [84], [88].
+Time-optimality of a trajectory could also be accomplished
+by using a specific path parameterization that maximizes
+velocity over a given path [89]. This method was shown for
+quadrotors in [90] for minimizing time of flight considering
+both translational and rotational quadrotor dynamics. However,
+the method only creates a velocity profile over a given path
+which is not further optimized.
+Apart from time optimality, complying with intermediate
+waypoint constraints is another requirement for path planning
+in autonomous drone racing. A common practice of solving
+a trajectory optimization problem with waypoint constraints
+is allocating waypoints to specific time steps and minimizing
+the spatial distance between these waypoints and the position
+at the corresponding allocated time steps on the reference
+trajectory (e.g. [91], [92]). The time allocation of the way-
+points is, however, non-trivial and difficult to determine. This
+is tackled in [88], but the work uses body rates and collec-
+tive thrust as control inputs and does not represent realistic
+actuator saturation. Recent work [84] introduces a comple-
+mentary progress constraints (CPC) approach, which considers
+true actuator saturation, uses single rotor thrusts as control
+
+7
+inputs, and exploits quaternions to create full, singularity-
+free representations of the orientation space with consistent
+linearization characteristics. While the above methods create
+time-optimal trajectories passing through given gates, they are
+computationally costly and hence intractable in real-time.
+3) Search-based: Search-based planning methods [93], [94]
+rely on discretized state and time spaces. They solve the
+trajectory planning through graph search algorithms such as
+A*. The search graph is built using minimum-time motion
+primitives with discretized velocity, acceleration, or jerk input.
+The algorithms then use trajectories of a simpler model, e.g.
+with velocity input, as heuristics for the search with a more
+complex model. Search-based planning methods can optimize
+the flight time up to discretization, but they suffer from
+the curse of dimensionality which renders them increasingly
+computationally demanding for increasing complexity of the
+quadrotor model. Furthermore, the employed per-axis dynamic
+limits (velocity, acceleration, jerk) does not represent the
+true quadrotor model, which further decreases the quality of
+found plans. Finally, although searching for minimum time
+trajectories, the methods are currently limited to planning
+between two states which is not suitable for multi-waypoint
+drone racing.
+4) Sampling-based:
+Sampling-based
+methods
+like
+RRT*
+[95]
+can
+be
+used
+for
+planning
+trajectories
+for
+linearized
+quadrotor
+models.
+Several
+time-minimizing
+approaches [67], [96] use a point-mass model for high-level
+time-optimal trajectory planning. In [96], an additional
+trajectory smoothing step is performed where the generated
+trajectory is connected with high-order polynomials by
+leveraging the differential flatness property of the quadrotor.
+However, these point-mass approaches need to relax the
+single actuator constraints and instead limit the per-axis
+acceleration, which results in trajectories that are conservative
+and sub-optimal given a minimum time objective. In [97], the
+authors use minimum-jerk motion primitives for connecting
+randomly sampled states inside RRT* to plan a collision-free
+trajectory. Since the authors use polynomials, this approach
+can only generate smooth control inputs, meaning that they
+cannot rapidly switch from full thrust to zero thrust (i.e.
+bang-bang) if required.
+The first method for planning minimum-time trajectories in
+a cluttered environment for the full quadrotor model was pro-
+posed in [31]. It uses a hierarchical sampling-based approach
+with an incrementally more complex quadrotor model to guide
+the sampling. The authors showed that the method outperforms
+both polynomial and search-based methods in minimizing
+trajectory time. Yet, the method is offline and intractable in
+real-time. Most recently, the authors of [98] proposed an online
+replanning approach that plans minimum-time trajectories for
+a point-mass model. The paths of replanned trajectories are
+then consequently used by Model Predictive Contouring Con-
+trol [99] with a full quadrotor model to maximize the progress
+along the path. This method is capable of outperforming
+other classical approaches due to the replanning capability and
+progress maximization with a full quadrotor model.
+5) Discussion: A planned trajectory can be understood as
+an intermediate representation that, given information about
+2016
+2017
+2018
+2019
+2020
+2021
+2022
+0
+10
+20
+30
+[46]
+[46]
+[12]
+[34]
+[103], [104]
+[105]
+Year
+Top Speeds on Autonomous Hardware [m/s]
+Fig. 4: Top speeds demonstrated on autonomous drones over time from both
+literature and competition data
+the robot’s dynamics and the environment, helps guide the
+platform through the race track and ultimately perform the task
+at hand. One might argue if this intermediate representation is
+needed at all, since ultimately, what we are looking for is a
+policy that maps sensor information and current environment
+knowledge to the actuation space. This is generally achieved
+with learning-based approaches, discussed in Section IV,
+which bypass the planning stage and directly convert sensor
+observations to actuation commands [100]–[102].
+One of the biggest benefits of explicit planning is modu-
+larity. This means that the developed algorithms can be used
+off-the-shelf for different drone tasks outside racing, such as
+search and rescue, which is not the case for single-purpose
+learned approaches. However, explicit planning suffers from
+the disconnection (or an open loop) between the planning and
+the deployment stage. Unexpected deviations from the plan,
+be it in the time domain (like unmodeled system delays) or in
+the state-space domain (like state estimation drifts or jumps in
+the VIO pipeline), can lead to compound errors and ultimately,
+a complete system failure.
+This can be tackled with more complex control approaches
+that do some part of the replanning online [98].
+C. Control
+Over the last decade, significant advancements have been
+made in agile multicopters control. Every year, increasing top
+speeds are demonstrated in the literature as shown in Figure 4.
+Controllers must be able to make real-time decisions in the
+face of poor sensor information and model mismatch. Control
+inputs, u(t), can come in a variety of modalities for quadrotor
+control, such as velocity and heading, body rates and collective
+thrust, or direct rotor thrust commands [30]. Typically, a high-
+level controller computes a desired virtual input such as body
+rates and collective thrust, which is then passed down to a
+low-level flight controller that directly controls the individual
+rotors on the multicopter.
+
+8
+Commonly used open source controllers such as PixHawk1
+or BetaFlight2 are widely available to the drone racing com-
+munity. BetaFlight is the most commonly used low-level
+controller for agile drone flight and has been widely adopted
+by the First Person View (FPV) racing community.
+In the following sections, we provide an overview of suc-
+cessful approaches to achieving high speeds in both simulation
+and real-world applications. We sort the approaches into
+model-based control and coupled perception and control.
+1) Model-Based Control: In model-based control, an ex-
+plicit model of the dynamic system is used to calculate control
+commands which satisfy a given objective such as minimizing
+time or tracking error. Models enable the prediction of future
+states of the drone and provide information about the system’s
+stability properties. In [106], Geometric Tracking control is
+introduced on the Special Euclidean group SE(3) and com-
+pletely avoids singularities commonly associated with Euler
+angle formulations on SO(3). This nonlinear controller showed
+the ability to execute acrobatic maneuvers in simulation and
+was the first to demonstrate recovery from an inverted initial
+attitude. The dynamic model of a quadrotor is shown to be
+differentially flat when choosing its position and heading as
+flat outputs in [77]. In this work, many agile maneuvers are
+performed onboard real drones with speeds up to 2.6 m/s.
+The previous work is extended in [20], proving that the
+dynamics model of a quadrotor subject to linear rotor drag
+is also differentially flat. The inclusion of the aerodynamic
+model within the nonlinear controller lead to demonstrated
+flight speeds up to 4.0 m/s while reducing tracking error by
+50% onboard a real drone.
+The differential flatness method is further extended by
+in [107] by cascading an Incremental Nonlinear Dynamic
+Inversion (INDI) controller with the differential flatness con-
+troller described in [77] but neglects the aerodynamic model
+addition from [20]. The INDI controller is designed to track
+the angular acceleration commands ˙Ω from the given reference
+trajectory. Top speeds of nearly 13 m/s and accelerations over
+2g are demonstrated onboard a real quadrotor. The controller
+shows robustness against large aerodynamic disturbances in
+part due to the INDI controller.
+An investigation of the performance of nonlinear model pre-
+dictive control (NMPC) against differential flatness methods is
+available in [104]. Cascaded controllers of INDI-NMPC and
+INDI-differential flatness are shown to track aggressive racing
+trajectories which achieving speeds of around 20m/s and
+accelerations of over 4g. While differential flatness methods
+are computationally efficient controllers and relatively easy to
+implement, they are outperformed on racing tasks by NMPC.
+An excellent overview of MPC methods applied to micro
+aerial vehicles can be found in [108]. Because quadrotors
+are highly nonlinear systems, nonlinear MPC is often used
+as the tool of choice for agile maneuvers. The debate of
+linear versus nonlinear MPC is thoroughly discussed in [109].
+Model Predictive Path Integral (MPPI) control is a sampling-
+based optimal control method that has found excellent suc-
+1https://pixhawk.org/
+2https://github.com/betaflight/betaflight
+cess on the AutoRally project, a 1/5th scale ground vehicle
+designed to drive as fast as possible on loose dirt surfaces
+[110], [111]. An introduction to MPPI can be found in
+https://autorally.github.io/. The MPPI approach can be used
+on agile quadrotors to navigate complex forest environments,
+however, analysis was only performed in simulation [110].
+Most of the successful demonstrations of MPPI come from
+ground robots [110], [111]. Because MPPI is a sampling based
+algorithm, scaling to higher-dimension state spaces like those
+of a quadrotor can lead to performance issues as shown in
+[103].
+Nonlinear MPC methods are also used in [34] where a
+nominal quadrotor model is augmented with a data-driven
+model composed of Gaussian Processes and used directly
+within the MPC formulation. The authors found that the
+Gaussian-Process model could capture highly nonlinear aero-
+dynamic behavior which is difficult to model in practice as
+described in Sec. II. The additional terms introduced by the
+Gaussian-Process added computational overhead to the MPC
+solve times, but it was still able to run onboard a Jetson TX2
+computer.
+Similar to [107], authors in [103] question whether or not
+it is necessary to explicitly model the additional aerodynamic
+terms from [34] due to the added computational and modeling
+complexity. Instead, they propose to learn residual model
+dynamics online using a cascaded adaptive nonlinear model
+predictive control architecture. Aggressive flight approaching
+20m/s and over 4g acceleration is demonstrated on real rac-
+ing quadrotors. Additionally, completely unknown payloads
+can be introduced to the system, with minimal degradation
+in tracking performance. The adaptive inner loop controller
+added minimal computational overhead and improved tracking
+performance over the Gaussian Process MPC by 70% on a
+series of high-speed flights onboard a racing quadrotor [34],
+[103].
+Contouring control methods can deal with competing op-
+timization goals such as trajectory tracking accuracy and
+minimum flight times [112]. These methods minimize a cost
+function which makes trade-offs between these competing
+objectives. In [113], Nonlinear Model Predictive Contouring
+Control (MPCC) is applied to control small model racecars.
+MPCC was then extended to agile quadrotor flight in
+[99].
+Although the velocities achieved by the MPCC controller were
+lower than that of [103], [104], the lap times for the same race
+track were actually lower due to the ability of the controller to
+find a new time-allocation that takes into account the current
+state of the platform at every timestep. The work is further
+extended to solve the time-allocation problem online, and to
+re-plan online [98] while also controlling near the limit of the
+flight system. Similar work uses tunneling constraints in the
+MPCC formulation in [114],
+2) Perception Aware Model Predictive Control: When con-
+trol is coupled with perception, an optimization problem that
+constrains or penalizes the movement of the drone to ensure
+that an area of interest is always kept within the field of view of
+the camera can be solved. This is integral to the drone-racing
+problem because, to navigate a challenging race course, the
+gates that define the course layout must be kept in view of
+
+9
+the onboard cameras at all times. Coupling the perception and
+control problem can alleviate issues in state estimation because
+the racing gates are usually feature-rich.
+The goal is as follows: navigate a trajectory with low
+tracking error while keeping a point of interest in view while
+minimizing motion blur for maximum feature detection and
+tracking. The first instance applied to agile quadrotors was
+Perception-Aware MPC (PAMPC) introduced in [115]. In this
+work, a nonlinear program is optimized using a sequential
+quadratic programming approximation in real time. The cost
+function contains both vehicle dynamic terms as well as
+perception awareness terms such as keeping an area of interest
+in the center of the camera frame.
+This technique is applied to the drone racing problem in
+[116], where an MPPI controller is designed with a Deep
+Optical Flow (DOF) component which predicts the movement
+of relevant pixels (i.e. gates). The perception constraints are
+introduced into a nonlinear optimization problem and deployed
+in a drone-racing simulator. The approach was not demon-
+strated onboard real hardware. In [117], a perception-aware
+MPC based on Differential Flatness was used to ensure that
+a minimum number of features are tracked between control
+updates and thus guarantee localization. To achieve this, a
+Perception Chance Constraint within the MPC formulation is
+introduced to ensure that at least n number of landmarks are
+within the field-of-view of the camera at all times with some
+bounded probability.
+3) Discussion: The performance of model-based controllers
+degrades when the model they operate on is inaccurate [103].
+For drones, defining a good-enough model is an arduous
+process due to highly complex aerodynamic forces, which can
+be difficult to capture accurately within a real-time capable
+model. In addition, the tuning process of many model-based
+controllers can be arduous, and requires a high level of domain
+expertise to achieve satisfactory performance
+In any optimal control problem, a cost function that the
+user wants to optimize must be defined. Traditionally, con-
+venient mathematical functions leveraging convex costs are
+used because these functions are easy to optimize and there
+is a large toolchain available for optimizing such problems
+such as Acados [118], CVXGEN [119], HPIPM [120], or
+Mosek [121]. In many drone racing papers, the optimal control
+problem is formulated as follows:
+min
+u xT
+NQxN +
+N−1
+�
+k=0
+xT
+k Qxk + uT
+k Ruk ,
+(2)
+subject to:
+xk+1 = f RK4(xk, uk, δt) ,
+x0 = xinit ,
+umin ≤ uk ≤ umax ,
+where the state is given by xk, the control input is given by
+uk, the state cost matrix is given by Q, and the control cost
+matrix is given by R. The optimization problem is constrained
+by the dynamics of the system given by f(xk, uk, δt) where
+δt is a finite time step. The nonlinear dynamics are typically
+propagated forward using an integrator such 4th order Runge-
+Kutta, RK4. Additionally, the problem is subject to the thrust
+limits of the platform. umin and umax, and some initial
+condition of the system x0. In this formulation, a reference
+position and control are provided by a high-level planner and
+the goal of the controller is to track the given reference, but
+this objective is ill defined for the drone racing problem: in
+drone racing, we wish to complete the track in as little time
+as possible; therefore, our objective can be better formulated
+as follows:
+min
+u
+T
+�
+k=0
+δt ,
+(3)
+subject to:
+xk+1 = f RK4(xk, uk, δt) ,
+x0 = xinit ,
+umin ≤ uk ≤ umax ,
+where T is the number of discrete time steps it takes to
+complete the race. This approach requires a time-horizon
+which predicts all the way until the end of the task which
+is intractable to optimize online.
+Reinforcement learning (RL) methods [30], [101] can opti-
+mize a proxy of this cost function, however do so in an offline
+fashion, requiring large amounts of training experience to
+approximate the value function. RL methods do not necessarily
+depend on a high-level planner to provide a reference to track.
+We will discuss some recent approaches using reinforcement
+learning methods in the following section.
+IV. LEARNING-BASED APPROACHES
+In this section, we present various learning-based ap-
+proaches for drone racing. These approaches replace the plan-
+ner, controller, and/or perception stack with a neural network.
+Learning-based methods have gained significant traction in the
+last few years, given their ability to cope with both high-
+dimensional (e.g. images) or low-dimensional (e.g. states)
+input data, their representation power, and the ease to develop
+and deploy them on hardware.
+The biggest challenge for learning-based methods is col-
+lecting enough data to effectively train the neural network for
+the task at hand. There are currently two possibilities for data
+gathering. The first, mostly popular in the initial stages of
+learning-based robotics [69], [122]–[125] is to collect data
+in the real world. The data is then annotated by a human
+or an automated process, and used for training. The second,
+much more popular in recent years and currently achieving
+the best results, consists of using simulation for collecting
+training data [30], [100], [126]–[128]. Both approaches have
+their advantages and limitations, which we will discuss in
+the following sections. Surveys covering existing methods for
+learning-based flight already exist [129], [130]. In contrast to
+them, we cover the most recent advances and give a broader
+discussion on the comparison between learning-based and
+traditional methods for drone racing.
+A. Learned Perception
+Sensors
+Hardware
+Planning
+Control
+Drone
+Software
+Fig. 5: Architecture 2: Learned Perception
+
+10
+For learned perception modules, the goal of the network
+is to use images from an RGB, depth, or event camera to
+detect landmarks within the environment and output useful
+representations such as waypoints, or the location of gates on
+the race track. A depiction of this architecture can be seen in
+Figure 5. An overview of deep learning methods for vision-
+based navigation specific to the drone racing task can be found
+in [130].
+In [124], a dataset of images is collected from a forward-
+facing camera mounted on a drone labeled with the relative
+position to the closest gate. This dataset is used to train a net-
+work which predicts from an image both the next gate location
+and its uncertainty. Predictions are then fused with a visual-
+inertial odometry system in an Extended Kalman Filter (EKF)
+to predict the position of the drone on the track. Similarly
+in [12], a Convolutional Neural Network (CNN) is used to
+detect gate corners in the AlphaPilot challenge. Once the gate
+corners are detected, classical computer vision algorithms like
+PnP can be used to find the coordinates of the gate in the
+camera frame. Using an EKF, the gate corner locations can be
+fused with a traditional VIO pipeline to improve the estimates
+of the drone’s location and orientation [12].
+Oftentimes, perception networks consume precious re-
+sources onboard computationally limited drones. To minimize
+the network processing time, [71], [131] proposed optimized
+architectures for gate detection on real-world data. A similar
+optimization went into “GateNet” [132] a CNN to detect gate
+center locations, distance, and orientation relative to the drone.
+The same authors developed a follow-up work denoted as
+”Pencil-Net” to do gate detection using a lightweight CNN
+in [133]. Most learning-based perception networks can suffer
+from poor generalization when deployed in environments that
+were not included in the training data.To reduce deployment
+sensitivity to lighting conditions or background content, virtual
+gates can be added to real-world backgrounds [134].
+Up until recently, RGB and depth cameras were used
+exclusively in the drone racing task, however, these sensor
+modalities can be sensitive to changes in the environment such
+as illumination changes. To overcome this, [135] proposed
+using event cameras coupled with a sparse CNN, recurrent
+modules, and a You Only Look Once (YOLO) object detector
+to detect gates. The use of event cameras overcomes poten-
+tial issues with motion blur induced by rapid movement of
+the drone and is a promising path forwards for high-speed
+navigation.
+Overall, deep learning methods for gate detection are the de-
+facto standard in all drone racing systems. However, such gate
+detectors are always coupled with traditional visual-inertial
+odometry systems which explicitly estimate the metric state
+of the drone. These approaches are discussed in Sec. III. It
+is interesting to notice that learning-based odometry systems,
+such as [73]–[75] have not yet replaced traditional methods.
+This is particularly surprising since deep visual odometry
+systems can specialize to a particular environment, which
+can be useful for drone racing since the race track is fixed
+and known in advance. A disadvantage of these methods is
+the high computational cost that makes them impractical for
+online applications. Research in end-to-end visual odometry
+is moving forward at a fast pace [75]. We foresee that in the
+near future, researchers will be able to apply these methods
+to the drone racing task.
+B. Learned Planning & Perception
+Sensors
+Hardware
+Control
+Drone
+Software
+Fig. 6: Architecture 3: Learned Planning and Perception
+A tightly-coupled planning and perception stack (Figure 6)
+is a very attractive algorithmic perspective. First, it greatly
+simplifies the perception task: an explicit notion of a map
+or globally-consistent metric state is not required. Second, it
+largely reduces computational costs, both in the pre-training
+and evaluation stages. Finally, it can leverage large amounts
+of data, collected either in simulation or the real world, to
+become robust against noise in perception or dynamics. Yet,
+an interesting observation is that these methods still work
+best when coupled with an explicit estimator of the metric
+state. In contrast to traditional methods, a locally consistent
+odometry system is sufficient [69], [126], [127], waving away
+the complexities of full-slam methods (e.g. loop-closure).
+In [69], a coupled perception and planning stack for drone
+racing is trained using real-world flight demonstrations. While
+good performance is indicated on the racing task as well as
+robustness against drift in state estimation, the method requires
+re-training for each new environment. Therefore, in the follow-
+up work [126], data generated entirely from simulation is
+used to train the perception-planning stack, waiving the labor
+and time-consuming requirement of data collection in the real
+world. A similar pipeline was used for high-speed autonomous
+flight through complex environments in [127], which proposes
+to train a neural network in simulation to map noisy sensory
+observations to collision-free trajectories directly.
+Several other works apply a similar stacked perception
+and planning pipeline for other autonomous drone racing
+tasks [122], [123], [125], [136]. We point the interested
+reader to existing surveys on the role of learning in drone
+navigation [129].
+A few works also studied the problem of planning using
+data-driven methods, decoupling it from the perception prob-
+lem. An interesting approach demonstrated in the NeurIPS
+Game of Drones competition [137] used an off-the-shelf
+reinforcement learning algorithm in place of a classic model-
+based planner for drone racing [138]. More recently, a novel
+multimodal learning-based trajectory planning framework was
+introduced in [139], which can generate collision-free trajec-
+tories that avoid a dynamic obstacle while maximizing its
+presence in the field of view (FOV).
+The big advantage of these methods is that they require
+less computational effort than traditional methods, possibly
+enabling online re-planning. In addition, they are much more
+robust to system latencies and sensor noise, which can be eas-
+ily accounted for by identifying them on physical drones and
+then adding them to the training environments [30]. However,
+
+11
+the major limitation of these methods is their sample complex-
+ity. If the training data comes from a simulator, significant
+simulation engineering is required to enable generalization.
+Conversely, if data come from the real world, generalization
+is easier, but the data collection process is very slow, tedious,
+and expensive.
+C. Learned Control
+Sensors
+Hardware
+Perception
+Software
+Planning
+Drone
+Fig. 7: Architecture 4: Learned Control
+Data-driven control, like reinforcement learning, allows
+overcoming many limitations of prior model-based controller
+designs by learning effective controllers directly from expe-
+rience. For example, control of a physical quadrotor using
+reinforcement learning was demonstrated by [140], where a
+neural network policy was used for waypoints tracking and
+vehicle recovery from harsh initialization. The neural network
+policy takes about 7 µs to generate the control command
+given the state, while a linear MPC requires about 1000 µs.
+Recently, [30] demonstrated high-speed trajectory tracking
+using learning-based control. They additionally showed that
+learned policies can be made robust to sensor noise and
+system latency by training with simulated sensor noise and
+latencies. Model-free RL was also applied to low-level attitude
+control [141], in which a learned low-level controller trained
+with PPO outperformed a fully tuned PID controller on almost
+every metric. Similarly, [142] used model-based RL for low-
+level control of an a priori unknown dynamic system.
+With any learning-based controller, it can be difficult to
+provide robustness guarantees as with traditional methods such
+as the Linear Quadratic Regulator (LQR). However, it is
+possible to make the planner and controller robust to system
+latencies, model uncertainties, and sensor noise by identifying
+them on physical drones and then adding them into the sim-
+ulation environments used to gather training data [30]. While
+a learning-based controller may provide superior performance
+to classical methods, it may be the case that they cannot be
+used in practice due to the inability to provide an analysis of
+the controller’s stability properties. These properties are often
+required in safety-critical systems such as flight controllers
+for aircraft. Recent works have attempted to address this
+using Lyapunov-stable neural network design for the control
+of quadrotors [143]. This work shows that it is possible to have
+a learning-based controller with guarantees that can also out-
+perform classical LQR methods. Building upon this concept,
+reachability analysis, and safety checks can be embedded in a
+learned Safety Layer [144].
+None of the systems discussed so far deal with the chal-
+lenging problem of adapting to new and uncertain environ-
+ments. The field of adaptive control has studied this problem
+extensively [145]–[147], however, we have seen a recent
+push to use advancements in machine learning within the
+adaptive control framework. A method to learn parametric
+uncertainty functions is introduced in [148]. These uncertainty
+functions could be learned offline using data captured from
+agile flight experiments, and then embedded within an adaptive
+controller to adjust controller parameters online during flight.
+Results indicate that highly accurate trajectory tracking can
+be achieved with this approach, even in the face of strong
+wing gusts exceeding 6.5 m/s. More recently, learning-based
+controllers have shown the ability to adapt zero-shot to large
+variations in hardware and external disturbances [149]. We see
+this as a promising area of research and one that is integral for
+reliable performance in changing environmental conditions.
+D. Learned Planning & Control
+Sensors
+Hardware
+Perception
+Software
+Drone
+Fig. 8: Architecture 5: Learned Planning & Control
+The second paradigm of learned control is to produce the
+control command directly from state inputs without requiring
+a high-level trajectory planner, as shown in the architecture
+diagram of Figure 8. In autonomous drone racing, this was
+proposed by [101], where a neural network policy is trained
+with reinforcement learning to fly through a race track in
+simulation in near-minimum time. Major advantages of the
+reinforcement-learning-based method are its capability to han-
+dle large track changes and the scalability to tackle large-scale
+random track layouts while retaining computational efficiency.
+In [102], deep reinforcement learning is combined with clas-
+sical topological path planning to train robust neural network
+controllers for minimum-time quadrotor flight in cluttered
+environments. The learned policy solves the planning and
+control problem simultaneously, forgoing the need for explicit
+trajectory planning and control.
+These methods inherit the classic advantage of policy learn-
+ing: to achieve robustness to system latencies, model uncer-
+tainties, and sensor noise, one can identify them on physical
+drones and then add them into the simulation environments
+used to gather training data [30]. In addition, they do not
+require an external controller to track the plan. This eliminates
+the discrepancy between the planning and deployment stage,
+which is one of the main limitations of traditional planning
+methods (Sec. III-B). Some of the limitations of traditional
+planning still remain such as the requirement of a globally-
+consistent state estimation and a map of the environment. Also,
+they have not yet been demonstrated in sparse long-horizon
+planning problems, e.g. flying through a maze at high speeds,
+where their performance would likely drop due to sample
+complexity.
+E. End-to-End Flight
+Sensors
+Hardware
+Drone
+Software
+Fig. 9: Architecture 7: End to End Learning
+
+12
+Expert pilots take raw sensory images from a first-person-
+view camera stream and map directly to control commands.
+In this section, we explore approaches emulating this holistic
+navigation paradigm in autonomous drones.
+Two families of approaches can be used to pursue an end-to-
+end navigation paradigm. The first is substituting each of the
+perception, planning, and control blocks with a neural network.
+This structure is followed by [150], [151], where the authors
+train a perception-planning network and a control network
+using imitation learning. The perception network takes raw
+images as input and predicts waypoints to the next gate.
+The control network uses such predictions with ground-truth
+velocity and attitude information to predict control commands
+for tracking the waypoints. They showed improvements over
+pure end-to-end approaches, which directly map pixels to con-
+trol commands and were able to show competitive lap times
+on par with intermediate human pilots within the Sim4CV
+simulator [152]. Yet, the division into independent blocks leads
+to compounding errors and latencies, which negatively affect
+performance when flying at high speeds [127].
+The second family of approaches directly maps sensor
+observation to commands without any modularity. This design
+is used by [153], which to date remains the only example of
+the completely end-to-end racing system. Indeed, other end-
+to-end systems generally require an inner-loop controller and
+inertial information to be executed. For instance, [154] trains
+an end-to-end CNN to directly predict roll, pitch, yaw, and
+altitude from camera images. Similarly, [155] uses a neural
+network to predict commands directly from vision. To improve
+sample complexity, they use contrastive learning to extract
+robust feature representations from images and leverage a two-
+stage learning-by-cheating framework. Given the absence of
+any division between perception, planning, and control, this
+family of approaches is potentially more robust to sensor noise
+and latencies. Yet, these policies are extremely data-hungry,
+which hinders their generalization in environments different
+from the training ones.
+Independently of the design paradigm they follow, end-
+to-end navigation algorithms are currently bound to simula-
+tion. The reasons why no method was successfully deployed
+in the real world include weak generalization to unseen
+environments, large computational complexity, and inferior
+performance to other modular methods. Another interesting
+observation is that humans can pilot a drone exclusively
+from visual observations. Conversely, except for [153], end-
+to-end systems still rely on the state extracted from other
+measurement modalities, e.g. an IMU. The question of whether
+autonomous drones can race in the real world at high-speed
+without any inertial information remains open. We provide
+more details on this question in Section VIII.
+F. Discussion
+Data-driven approaches are revolutionizing the research in
+autonomous drone racing, ranging from improving the system
+model to end-to-end control of the vehicle. Currently, the best-
+performing algorithms for drone racing include a learning-
+based component [12], [13], and this trend is unlikely to
+change in the coming years. Indeed, compared to classical
+model-driven design, they can process high-dimensional sen-
+sory inputs directly, can be made robust to any modeling
+uncertainty (e.g. latency) by simply incorporating it in the
+training pipeline, and require far less engineering effort for
+tuning and deploying them [30].
+A trend that is recently gaining more and more popularity is
+training policies in simulation and deploying them in the real
+world [69], [126], [127]. Leveraging years of advancement in
+drone modeling technology (See Section II), the simulation
+of drone dynamics is extremely realistic and fast. Conversely,
+the simulation of sensor measurements (e.g. cameras, IMUs,
+lidars) is either inaccurate or very computationally expensive.
+Therefore, researchers generally aim to abstract observations
+using classical perception algorithms (see Section III-A) to
+train their model in a timely fashion and favor simulation-to-
+real-world transfer [127]. However, this hinders the deploy-
+ment of completely end-to-end systems on real-world robots,
+which, as human pilots, only rely on a stream of color images.
+We refer the reader to Sec. VIII for the implication of this
+research.
+While simulators may get better and faster in the near future,
+recent advances in real-world training [156], [157] and fine-
+tuning [158], [159] offer a potential alternative for zero-shot
+simulation to reality transfer for sensorimotor policies. So far,
+these works have been limited to legged locomotion. Extension
+to agile drones could lead to the successful deployment of
+end-to-end policies, possibly improving the state of the art in
+racing performance.
+V. DRONE RACING SIMULATORS
+One tool that has drastically accelerated the progress of
+research in autonomous drone flight is the use of simulation
+environments that attempt to recreate the conditions that
+real drones experience when flying. Over the years, several
+simulation environments have been developed for the use of
+general research.
+In 2016, the widely used RotorS simulation environment
+was published, which extends the capabilities of the popular
+Gazebo simulation engine to multi-rotors [17]. Gazebo uses
+the Bullet physics engine for basic dynamic simulation and
+contact forces. Linear drag on the body of the multicopter is
+simulated based on the cross-sectional area and linear velocity
+of the simulated object. The RotorS extension features many
+easy-to-use plugins for developing multi-rotors, however, it
+distinctly lacks the photorealistic details needed to simulate
+accurate behavior of estimation and perception pipelines.
+AirSim was introduced by Microsoft in 2018 as a photo-
+realistic simulator for the control of drones [15]. It is built on
+the Unreal graphics engine and features easy-to-use plugins
+for popular flight controllers such as PX43, ArduPilot4, and
+others. It was used in the 2019 NeurIPS Game of Drones
+challenge [137]. Because of the photorealism of AirSim, it
+is possible to simulate the entire perception and estimation
+pipeline with good possibility of transfer to real-world drone
+3https://px4.io/
+4https://ardupilot.org/
+
+13
+systems. Additionally, AirSim comes pre-packaged with an
+OpenAI-Gym environment for training Reinforcement Learn-
+ing algorithms. Organizations such as Bell, Airtonomy, and
+NASA are using AirSim to generate training data for learning-
+based perception models.
+FlightGoggles [160] was developed as another photorealistic
+simulator and was used as the primary simulation environment
+for the Lockheed Martin AlphaPilot challenge. FlightGoggles
+contains two separate components: a photorealistic rendering
+engine built with Unity3D and a dynamic simulation im-
+plemented in C++. FlightGoggles provides an interface with
+real-world vehicles using a motion capture system; such an
+interface allows rendering simulated images that correspond
+to the position of physical vehicles in the real world.
+A recent simulator focused on Safe RL was proposed
+in [161]. It uses Gazebo and the Pybullet physics engine as
+the backend. Leaderboards for several safety-focused training
+environments exist, encouraging researchers to submit their
+approaches and compete with other researchers around the
+world.
+Finally, Flightmare [16] is a simulation environment fea-
+turing photorealistic graphics provided by the Unity engine.
+The physics engine is decoupled and can be swapped out
+with various engines for user-defined levels of simulation
+fidelity. Similar to FlightGoggles, Flightmare can also provide
+hardware-in-the-loop simulation functions where a virtual,
+synthetic camera image can be provided to the drone for use
+in control and estimation [162].
+VI. COMPETITIONS
+To gauge the progress of the field as a whole, several
+drone racing competitions have taken place since 2016. We
+include a graphical overview of these events in Figure 2. The
+Autonomous Drone Racing (ADR) competition was an annual
+competition which took place during the IROS conference
+between 2016 and 2019. In 2016, 11 teams competed in
+autonomous drone racing and were tasked to navigate a series
+of gates in sequence. The positions of the gates were not
+known to the participating teams ahead of time, therefore
+teams flew very cautiously identifying the next waypoints
+online. Each team was given 30 minutes prior to the official
+competition to fly the course as many times as they wished.
+The winning team, from KAIST, made it through 10 of the 26
+gates in 1 minute and 26 seconds. For comparison, a human
+was able to complete the entire 26-gate course in 1 minute 31
+seconds. A survey summarizing the approaches used for these
+early competitions can be found in [163]. The following year,
+a similar competition took place during IROS in Vancouver,
+Canada, with better results. This time, 14 teams participated
+and were given a CAD drawing of the course prior to the
+event with locations and dimensions of all gates. Only 5 teams
+participated in the final in-person event, with the winning team
+making it through 9 out of 13 gates in over 3 minutes. A
+summary of the winning approaches can be found in [10].
+Two more ADR competitions took place at IROS 2018 and
+2019, with drones navigating courses faster and more reliably.
+In 2019, Lockheed Martin sponsored the AlphaPilot AI
+Drone Racing Innovation Challenge where a 1 million dollar
+grand prize was awarded to the winning team [164]. The
+competition took place first in a virtual qualifying round
+which used the FlightGoggles simulation environment [160].
+Nine teams out of more than 400 worldwide qualified for the
+final challenge which included navigating a new track in a
+time-trial setting against an expert human pilot. Ultimately,
+professional pilot Gabriel Kocher, from the Drone Racing
+League, manually piloted his drone through the course in only
+6 seconds. It took 11 seconds to the winner, MAVLab from
+TU Delft, and 15 seconds to the second-place winner, UZH-
+RPG from the University of Zurich, to complete the course
+autonomously. The two different approaches are documented
+in [12], [13]. Further comments are provided by the winner in
+[165]. Perez et al. provides an interesting overview of the types
+of hardware used for some of the drone racing competitions
+mentioned so far [46].
+In 2019, the Game of Drones competition took place at the
+NeurIPS conference. This competition was purely simulation
+based and used the AirSim simulation environment built by
+Microsoft [11], [15], [137]. Participants in the Game of Drones
+competition raced against simulated opponents in a head-to-
+head fashion, similar to how humans compete in FPV drone
+racing. Teams raced against a single simulated opponent,
+navigating through a complex series of gates in three different
+tiers: Planning Only, Perception Only, and Perception with
+Planning.
+In 2022, at the Swiss Drone Days event in Zurich, Switzer-
+land, three of the world’s best human pilots competed against
+researchers from the Robotics and Perception Group of the
+University of Zurich. Flight speeds exceeding 100 kph were
+demonstrated by the autonomous drones. When relying on
+motion capture, the autonomous drones were able to achieve
+significantly faster laptimes than the expert human pilots.
+They additionally demonstrated it was possible to win races
+without motion capture, using only onboard computing and
+sensors to navigate the race track. IEEE Spectrum author Evan
+Ackermann discusses the multi-day event in [105].
+VII. DATASETS AND OPEN SOURCE CODE
+In this section, we provide an overview of the existing open
+source code bases as well as useful datasets for autonomous
+drone racing. We first discuss datasets, and then group the
+existing open source code bases by their use-cases in table I.
+In 2018, researchers from MIT released a large scale dataset
+for perception during aggressive UAV flight [168]. This dataset
+contains over 10 hours of flight data which includes simulated
+stereo and downward-facing camera images at 120 Hz, real-
+world IMU data at 100 Hz, motor speed data at 190 Hz, and
+motion capture data at 360 Hz. The sensor suite was chosen
+such that algorithms like Visual-Inertial Odometry (VIO) or
+Simultaneous Localization and Mapping (SLAM) could be
+evaluated on the dataset.
+In 2019, the UZH-FPV Drone Racing Dataset was released,
+which contains many agile maneuvers flown by a professional
+racing pilot [68]. The dataset includes indoors and outdoors
+real-world camera images, inertial measurements, event cam-
+era data, and ground truth poses provided by an advanced
+
+14
+TABLE I: Open Source Software and Datasets
+Name and Reference
+Category
+Year
+Link
+PAMPC [115]
+Controller
+2018
+https://github.com/uzh-rpg/rpg mpc
+Deep Drone Acrobatics [100]
+Controller
+2019
+https://github.com/uzh-rpg/deep drone acrobatics
+Data Driven MPC [34]
+Controller
+2020
+https://github.com/uzh-rpg/data driven mpc
+High MPC [166]
+Controller
+2022
+https://github.com/uzh-rpg/high mpc
+AutoTune [167]
+Controller Tuner
+2022
+https://github.com/uzh-rpg/mh autotune
+Blackbird [168]
+Dataset
+2018
+https://github.com/mit-aera/Blackbird-Dataset
+UZH-FPV [68]
+Dataset
+2019
+https://fpv.ifi.uzh.ch/
+NeuroBEM [29]
+Dataset
+2020
+https://rpg.ifi.uzh.ch/NeuroBEM.html
+Eye Gaze Drone Racing [6]
+Dataset
+2021
+https://osf.io/gvdse/
+Time-optimal Planning for Quadrotor Waypoint Flight [84]
+Planner
+2021
+https://github.com/uzh-rpg/rpg time optimal
+Minimum-Time Quadrotor Waypoint Flight in Cluttered Environments [31]
+Planner
+2022
+https://github.com/uzh-rpg/sb min time quadrotor planning
+RotorS [17]
+Simulator
+2016
+https://github.com/ethz-asl/rotors simulator
+AirSim [137]
+Simulator
+2018
+https://microsoft.github.io/AirSim/
+FlightGoggles [160]
+Simulator
+2019
+https://github.com/mit-aera/FlightGoggles
+Flightmare [16]
+Simulator
+2020
+https://uzh-rpg.github.io/flightmare/
+Learning to fly—a gym environment with pybullet physics for reinforcement learning of multi-agent quadcopter control [161]
+Simulator
+2021
+https://github.com/utiasDSL/gym-pybullet-drones
+Sim 2 Real Domain Randomization [126]
+Sim2Real Transfer
+2019
+https://github.com/uzh-rpg/sim2real drone racing
+RPG Quadrotor Control [20]
+Software Stack
+2017
+https://github.com/uzh-rpg/rpg quadrotor control
+Agilicious [162]
+Software Stack
+2022
+https://github.com/uzh-rpg/agilicious
+Kalibr [48]
+Camera Calibration
+2022
+https://github.com/ethz-asl/kalibr
+motion capture system (a total station) providing millimeter-
+level accuracy. Similar to the authors in [168], the authors of
+this dataset hope to push the state of the art in state estimation
+during aggressive motion and have created competitions to
+allow researchers to compete against one another on this agile
+flight benchmark.5
+Research on how expert human pilots focus on their targets
+during flying and provide a dataset that contains flight trajec-
+tories, videos, and data from the pilots is examined in [6]
+NeuroBEM [29] is a hybrid aerodynamic quadrotor model
+which
+combines
+blade-element-momentum-theory
+models
+with learned aerodynamic representations from highly ag-
+gressive maneuvers. While the model is fit to the specific
+quadrotor platform defined in [162], the approach can be used
+for any quadrotor platform and provides over 50% reduction in
+model prediction errors compared to traditional, exclusively-
+first-principles approaches.
+A significant amount of autonomous drone racing research
+has been open sourced to the community, making implemen-
+tation less daunting for newcomers to the field. A collection of
+all known drone racing repositories has been provided to the
+reader in Table I. These code bases range across controllers,
+planners, sensor calibration, and even entire software stacks
+dedicated to drone racing. We encourage both newcomers and
+experienced researchers to check out the extensive amount of
+open source code bases available and contribute back to the
+community.
+VIII. OPEN RESEARCH QUESTIONS AND CHALLENGES
+In this section, we examine some of the biggest chal-
+lenges that the field of autonomous drone racing is facing.
+Autonomous drone racing is a field that is growing rapidly.
+To quantify the rate of growth, we examined the number
+of papers that mentioned the key-phrase ”autonomous drone
+racing” since 2015. The data, indicated in Fig. 10, shows
+exponential-like growth of the field. Thus, it is appropriate
+to discuss where future opportunities exist for incoming and
+experienced researchers alike.
+5https://fpv.ifi.uzh.ch/uzh/uzh-fpv-leader-board/
+2015 2016 2017 2018 2019 2020 2021 2022
+0
+50
+100
+150
+200
+Year
+Number of Papers
+Fig. 10: The number of papers related to ”autonomous drone racing” cataloged
+on Google Scholar.
+A. Challenge 1: Enabling VIO for High-Speed Applications
+In its current form, online, robust, and accurate state
+estimation is highly beneficial when pushing autonomous
+drones to their limits. Currently, classical state estimation
+approaches based on visual-inertial odometry cannot cope
+with the perceptual challenges present in drone racing tasks.
+Motion blur, low texture, and high dynamic range are some
+reasons why classical VIO algorithms accumulate large errors
+in localization. The miscalibration of intrinsic and extrinsic
+camera parameters can lead to improper estimates of the
+camera pose on a drone. This is due to local movements of the
+camera frame relative to the drone body, as well as changes
+in temperature and pressure. VIO drift can render the state
+estimates unusable unless corrected through localizations to
+a prior map. New sensor modalities, such as event cameras,
+could potentially alleviate this issue. Although event-aided
+VIO algorithms for drones have been proposed to improve
+robustness to motion blur, they have not been demonstrated at
+high speeds as seen in drone racing. Future research in agile
+
+15
+flight may focus on finding new event representations that are
+computationally efficient and compatible with classical VIO
+formulations. One example is to exploit direct methods [169].
+Other promising sensor modalities are motor speed controllers
+and force sensors. These sensor measurements could be used
+to include more advanced drone models in VIO, e.g. modeling
+aerodynamics effects, in order to limit the drift that accu-
+mulates where camera measurements are degraded. One of
+the main consequences of motion blur, low texture, and high
+dynamic range is unreliable feature extraction and matching.
+This consequently degrades the performance of the visual
+frontend. Deep learning methods have the potential to solve
+this problem. What hinders the application of these methods to
+drone racing at the moment is their computational cost. Future
+research should work on lightweight neural networks that can
+provide inference at a high rate. Neural networks could also
+be used to remove non-zero mean noise and constant errors
+from the inertial measurements. A potentially fruitful area of
+research is in combining neural networks for input processing
+with a geometry-based VIO backend. This could lead to the
+next step in the research on VIO for drone racing. Current
+works [75], [170] have shown that this direction outperforms
+end-to-end visual-based odometry methods.
+B. Challenge 2: Flying from Purely Vision
+State-of-the-art autonomous navigation methods rely on
+visual and inertial information, usually combined with classic
+perception algorithms. Conversely, expert human pilots rely
+on nothing more than a first-person-view video stream, which
+they use to identify goals and estimate the ego-motion of
+the drone. Building systems that, similarly to human pilots,
+only rely on visual information is very interesting from a
+scientific perspective. Indeed, since simulating RGB is yet
+very challenging, solving this question might require lifelong
+learning algorithms operating in the real world. In addition,
+eliminating inertial information might have some engineering
+advantages too, e.g., data throughput, power consumption, and
+lower cost. Seminal works in this direction try to understand
+how humans solve this task [6], [171]. They found that expert
+pilots can control drones despite a 200ms latency, which is
+compensated by the human brain. Taking inspiration from
+biology, a recent work [172] shows that it is possible to fly with
+camera images and an onboard gyroscope (e.g., removing the
+accelerometer), as long as the system never hovers. However,
+the above questions still remain mostly open and a good
+avenue for research at the intersection of computer vision,
+neuroscience, and biology.
+C. Challenge 3: Multiplayer Racing
+Much of the work done up until this point on autonomous
+drone racing has focused on time-optimal flight without con-
+sidering how a capable opponent might impact the compe-
+tition dynamics. In FPV races, pilots can compete against
+up to 5 opponents simultaneously, bringing about the need
+to anticipate how their opponents might behave. Humans
+are astonishingly capable of recognizing opportunities for
+overtaking and executing complex maneuvers in the face of
+large aerodynamic disturbances caused by flying close to
+another drone. Achieving such capabilities requires an agent
+to estimate their opponent’s state using only onboard visual
+sensors. However, these observations in drone racing are
+sparse because the camera faces forward along the heading
+axis, meaning that the only time an opponent is observable
+is when the ego-agent is behind them. Sophisticated motion
+and planning models which can propagate predictions of the
+opponents’ states and racing lines through time are necessary
+to anticipate collisions or overtaking opportunities. An initial
+study [173] examined how game-theoretic planners can lead
+to highly competitive behavior in two-player drone racing,
+however, this work was confined to racing on a 2D plane.
+The work was further extended to 3D spaces in [174], but
+there is a significant opportunity for researchers to explore
+the competitive nature of drone racing and develop interesting
+racing strategies that lead to time-optimal agents that are able
+to deal with complex opponent behavior.
+D. Challenge 4: Transfer to Real-World Applications
+Drone racing, while an extraordinarily challenging research
+environment, is ultimately not the end goal. Opportunities
+exist for technology transfer between the drone racing research
+community to real-world applications such as search and res-
+cue, inspection, agriculture, videography, delivery, passenger
+air vehicles, law enforcement, and defense. Until this point,
+the rate of technology transfer has been slow due to challenges
+in flight certification and a lack of generalization between
+environments. However, commercial applications that leverage
+the full agility of the platform have much to gain. Drones
+that fly fast, fly farther, therefore increase the productivity of
+drones in every commercial sector [4]. As it stands today,
+drone racing algorithms can be difficult to directly transfer
+to a new environment due to overfitting and minimal safety
+guarantees. Calculating time-optimal paths that are safety
+critical currently takes too long for deployment in emergency
+scenarios. Existing works that leverage expert perception mod-
+els to sense and plan around obstacles can be sensitive to
+changes in the environment and lead to crashes. Beyond this,
+we often do not have a known map ahead of time for real
+world applications, requiring researchers to think about how to
+simultaneously estimate the state of the drone while mapping
+the environment. Building algorithms that can continually
+improve from their own experience is key in enabling this
+transfer. While recent advances in reinforcement learning
+research point to the feasibility of this path [158], [159], [175],
+it is unclear when and how such recent approaches would
+be applicable to drones or similarly agile platforms in the
+real world. Collecting data for continual RL onboard a drone
+is notoriously difficult. This is because the drone does not
+have the luxury of remaining in contact with the ground like
+legged robots and cars, and thus has to immediately know how
+to hover otherwise a crash will occur. One interesting area
+that may be useful for continual RL in drones is the notion
+of “safe-RL”. The goal of safe RL is to enable exploration
+without ever incurring catastrophic failure of the system. Initial
+work on this topic can be found in [176]. A survey paper
+
+16
+covering safe RL methods can be found in [177]. Furthermore,
+a thorough review paper on continual, or life-long RL can be
+found in [178].
+IX. SUMMARY AND CONCLUSIONS
+In this survey, we provided a comprehensive overview of
+the task of autonomous drone racing across model-based and
+learning-based approaches. A history of all recent autonomous
+drone racing events was given, along with a list of all open-
+source code bases, datasets, and simulators. These resources
+can be used to greatly reduce the learning curve and time
+needed when it comes to getting started with autonomous
+drone racing. With these resources and a list of open chal-
+lenges for the field, researchers should have the tools to push
+the limits.
+X. CONTRIBUTIONS
+Drew Hanover initiated the idea of this paper, created the
+paper structure, and contributed to all sections of this paper
+while coordinating efforts amongst the co-authors. Antonio
+Loquercio contributed to the paper structure and the learning-
+based sections. Leonard Bauersfeld authored the Drone Mod-
+eling section and created the graphics seen throughout. Angel
+Romero contributed to the Classical Planning and Control
+sections. Giovanni Cioffi contributed to the Classical Percep-
+tion and Challenges sections. Yunlong Song contributed to
+the Simulators and Learning-Based Planning/Control sections
+Robert Penicka contributed to both Classical and Learning-
+Based Planning sections. Elia Kaufmann contributed to the
+paper structure and throughout the Learning-Based sections.
+Davide Scaramuzza contributed to the general paper structure
+and revised the paper thoroughly and critically.
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf,len=2147
+page_content="1  Past, Present, and Future of Autonomous Drone  Racing: A Survey  Drew Hanover1, Antonio Loquercio3, Leonard Bauersfeld1, Angel Romero1, Robert Penicka2,  Yunlong Song1, Giovanni Cioffi1, Elia Kaufmann1 and Davide Scaramuzza1  a) Number of Papers on Drone Racing  '15  0  '16  1  '17  4  '18  15  '19  69  Year  '20  95  '21  143  '22  188  b) Onboard View  c) Drone Racing  Fig." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' 1: Drone racing is a sport rapidly gaining popularity where opponents compete on a pre-defined race track consisting of a series of gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Autonomous  drone racing research aims to build algorithms that can outperform human pilots in such competitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' a) The task of autonomous drone racing has gained  a substantial amount of interest from the research community in the last few years, as indicated by the increasing number of related publications per year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  b) Autonomous drones rely on visual and inertial sensors to estimate their own states, as well as their opponents’ states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' c) Agile maneuvers are required to  overtake opponents and win the race.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  Abstract—Over the last decade, the use of autonomous drone  systems for surveying, search and rescue, or last-mile delivery  has increased exponentially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' With the rise of these applications  comes the need for highly robust, safety-critical algorithms which  can operate drones in complex and uncertain environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  Additionally, flying fast enables drones to cover more ground  which in turn increases productivity and further strengthens  their use case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' One proxy for developing algorithms used in  high-speed navigation is the task of autonomous drone racing,  where researchers program drones to fly through a sequence of  gates and avoid obstacles as quickly as possible using onboard  sensors and limited computational power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Speeds and accelera-  tions exceed over 80 kph and 4 g respectively, raising significant  challenges across perception, planning, control, and state estima-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' To achieve maximum performance, systems require real-  time algorithms that are robust to motion blur, high dynamic  range, model uncertainties, aerodynamic disturbances, and often  unpredictable opponents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' This survey covers the progression of  autonomous drone racing across model-based and learning-based  approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' We provide an overview of the field, its evolution  over the years, and conclude with the biggest challenges and  open questions to be faced in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' INTRODUCTION  Throughout history, humans have been obsessed with racing  competitions, where physical and mental fitness are put to the  1D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Hanover, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Bauersfeld, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Romero, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Song, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Cioffi, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Kaufmann  and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Scaramuzza are with the Robotics and Perception Group, University  of Zurich, Switzerland (http://rpg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='ifi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='uzh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='ch).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' 2R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Penicka is with the Multi-  robot Systems Group, Czech Technical University in Prague, Czech Republic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  3A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Loquercio is with UC Berkeley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' This work was supported by the  Swiss National Science Foundation (SNSF) through the National Centre of  Competence in Research (NCCR) Robotics, the Czech Science Foundation  (GACR) under research projects No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' 23-06162M, the European Union’s  Horizon 2020 Research and Innovation Programme under grant agreement  No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' 871479 (AERIAL-CORE), and the European Research Council (ERC)  under grant agreement No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' 864042 (AGILEFLIGHT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' The earliest mention of a formal race dates all the way  back to 3000 BC in ancient Egypt where the Pharaoh was  thought to have ran a race at the Sed festival to demonstrate  his physical fitness, indicating his ability to rule over the  kingdom [1], [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' As time has progressed, humans have  moved from racing on-foot to using chariots, cars, planes, and  more recently quadcopters [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Although the vessel frequently  changes, one thing that has remained constant since the early  days of racing has been the recurring theme of using the  task as a catalyst for scientific and engineering development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  Recently, we have seen a push to remove humans from the  loop, automating the highly complex task of racing in order to  push vehicle performance beyond what a human can achieve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Why Drone Racing?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  Drone racing is a popular sport with high-profile interna-  tional competitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' In a traditional drone race, each vehicle  is controlled by a human pilot, who receives a first-person-  view (FPV) live stream from an onboard camera and flies  the drone via a radio transmitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' An onboard image from the  drone can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Human drone pilots need years of  training to master the advanced navigation and control skills  that are required to be successful in international competitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  Such skills would also be valuable to autonomous systems that  must quickly and safely fly through complex environments,  in applications such as disaster response, aerial delivery, and  inspection of complex structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' For example, in search-and-  rescue scenarios, drones must be able to rapidly navigate in  complex environments in order to maximize their spatial cov-  erage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Put more simply, drones that can fly fast, fly farther [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  Automating inspection tasks can save lives while being  more productive than manual inspection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' According to a recent  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='01755v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='RO]  4 Jan 2023  2  survey on unmanned aerial vehicle (UAV) use in bridge  inspection [5], most drones used for inspection tasks rely on  GPS navigation with the biggest limiting factor on inspection  efficiency being the drones endurance and mobility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Addition-  ally, the authors note that the most popular drones used for  surveying by several US Departments of Transportation are  not fully autonomous and require expert human pilots [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  The commercial and safety advantages of highly agile drone  systems is clear, however research into autonomous drone  racing can also help us gain new understandings on how  the visual processing and control by human pilots works, as  demonstrated in [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  Over the last five years, several projects have been launched  to encourage rapid progress within the field, such as DARPA’s  Fast Lightweight Autonomy (FLA)  [8] and European Re-  search Council’s AgileFlight [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' With funding pools of  over $1 million for each of these projects and significant  commercial potential, a large incentive exists for researchers  and entrepreneurs alike to explore new paradigms in agile  flight research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Competitions such as the IROS’16-19 Au-  tonomous Drone Racing series [10], NeurIPS 2019’s Game  of Drones [11], and the 2019 AlphaPilot Challenge [12], [13]  provided further opportunity for researchers to compare their  methodologies against one another in a competitive fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  A depiction of the progress made from these competitions can  be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  Drone racing is a challenging benchmark which can help re-  searchers to gauge progress on complex perception, planning,  and control algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Autonomous drones in a racing setting  must be able to perceive, reason, plan, and act on the tens of  milliseconds scale, all onboard a computationally limited plat-  form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Apart from being very challenging to solve, the drone  racing task offers a single measure of the progress of the state-  of-the-art in autonomous flying robotics: lap time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Solving this  problem requires algorithms to be efficient, lightweight, and  provide optimal decision and control behaviors all in real-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  Additionally, we see nearly exponential growth of the number  of papers in the field year over year as seen in Fig 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  To the best of the authors’ knowledge, this is the first  survey on the state of the art in autonomous drone racing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  This overview will be useful to researchers who wish to make  connections between existing works, learn about the strengths  and weaknesses of current and past approaches, and identify  directions moving forward which should progress the field in  a meaningful way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Task Specification  The drone racing task is to fly a quadrotor through a  sequence of gates in a given order in minimum time while  avoiding collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Humans are astonishingly good at this  task, flying at speeds well over 100kph with only a first-person  view camera as their sensory input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Beyond this, expert pilots  can adapt to new race tracks quickly in a matter of minutes,  however the sensorimotor skills required by professional drone  pilots take years of training to acquire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  For an autonomous drone to successfully complete this task,  it must be able to detect opponents and waypoints along the  track, calculate their location and orientation in 3-dimensional  space, and compute an action that enables navigation through  the track as quickly as possible while still controlling a highly  nonlinear system at the limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' This is challenging in three  different aspects: Perception, Planning, and Control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Poor  design in any of these aspects can make the difference between  winning or losing the race, which can be decided by less than  a tenth of a second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  The paper is structured as follows: First, the modeling pro-  cedure of the drone including aerodynamics, batteries, motors,  cameras, and the system nonlinearities are discussed in detail  in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' A classical robotics pipeline is then introduced in  Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' III, with a deep dive into literature relevant to agile flight  split into Perception, Planning, and Control subsections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' After-  wards, we delve into learning-based methods for Perception,  Planning, and Control which rely on recent advancements from  the machine learning community in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Then, a discus-  sion of the development of simulation tools which can enable  rapid development for agile flight applications in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' A  history of drone racing competitions and the methods used for  each are included in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Next, a summary of open source  code bases, hardware platforms, and datasets for researchers  is provided in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Finally, a forward-looking discussion  on the Opportunities and Challenges for future researchers  interested in autonomous drone racing in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' DRONE MODELING  To further advance research on fast and agile flight, it is  important to have accurate models that capture the complex  nonlinear dynamics of multicopter vehicles at the limit of their  performance envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  This section reviews different dynamics modeling ap-  proaches from classic, first-principles modeling to pure data-  driven models in the context of drone racing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' For the vehicle  dynamics, the key aspects that need to be modeled are the  kinematics, aerodynamics, the electric motors, and the battery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  In addition to the vehicle dynamics models discussed in this  section, many difficulties for autonomous drone racing models  are introduced by the onboard sensors, whose characteristics  need to be modeled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' For example, IMUs are subject to bias  and noise, and the intrinsic as well as extrinsic parameters of  onboard sensors change over time as hard crashes may lead  to miscalibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Kinematics  Typically, the vehicle is assumed to be a 6 degree-of-  freedom rigid body of mass m with a (diagonal) inertia matrix  J  = diag(Jx, Jy, Jz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Given a dynamic state x ∈ R17  the equations describing its evolution in time are commonly  written as:  ˙x = f(x, u) =  �  ���������  ˙pW B  ˙qW B  ˙vW  ˙ωB  ˙Ω  �  ���������  =  �  ����������  vW  qW B  �  0  ωB/2  �  1  m  �  qW B ⊙ f  �  + gW  J −1�  τ − ωB × JωB  �  1  τΩ (Ωss − Ω)  �  ����������  ,  (1)  3  The first autonomous drone racing challenge at  IROS  2016,  Daejong  Korea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  Slow  moving  quadrotors cautiously navi-gated the course shown  above using only onboard sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Team KIRD  from KAIST placed 1st, reaching a top speed of  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='6 m/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  IROS ADR I  In the third iteration of the ADR challenge held in  Madrid, teams began implementing learning based  methods with optimal control techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' The  Robotics  and  Perception  Group  from  the  University of Zurich successfully completed the  course the fastest with speeds up to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='0 m/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  IROS ADR III  The following year, another autonomous drone  racing  competition  took  place  in  Vancouver,  Canada.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Similar to the year prior, teams used  classical methods to navigate a challenging course  with compute done onboard and the INAOE team  from Mexico won with a speed of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='7 m/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  IROS ADR II  In summer of 2022, the Robotics and Perception  Group of the University of Zurich hosted a drone  racing  competition  to  face  their  autonomous  drones off against some of the best human FPV  pilots in the world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Speeds exceeded 20 m/s,  relying only on onboard sensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  Swiss Drone Days  2017  2016  2018  2022  2020  Lockheed Martin sponsored a $1M prize to teams  who could successfully navigate a challenging  drone racing course completely autonomously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' The  MAVLAB team from TU Delft won, with top  speeds approaching 10 m/s – a significant jump  over previous competitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  AlphaPilot  vmax = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='6 m/s  vmax = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='7 m/s  vmax = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='0 m/s  vmax = 10 m/s  vmax = 22 m/s  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' 2: History of drone racing competitions that use real drones for navigating the race track, IROS ADR II photo credit [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  where pW B ∈ R3 is the position of the center of mass given  in the world frame, qW B ∈ SO(3) is a quaternion defining  the rotation of the body frame relative to the world (vehicle  attitude), vW ∈ R3 is the velocity of the vehicle in the world  frame, ωB ∈ R3 are the bodyrates of the vehicle, Ω ∈ R4 are  the motor speeds, gW = [0, 0, −9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='81 m/s2]⊺ denotes earth’s  gravity, and u ∈ R4 is the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Depending on the control  modality the input can be single rotor thrusts or collective  thrust and body rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' In this setting, the task of the model is  to calculate the total force f and total torque τ that acts on  the drone as accurately as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Note the quaternion-vector  product denoted by ⊙ representing a rotation of the vector by  the quaternion as in q ⊙ f = q · [0, f ⊺]⊺ · ¯q, where ¯q is the  quaternion’s conjugate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Those forces and torques, collectively  referred to as wrench, are determined by the aerodynamics  of the platform as well as the vehicles’ actuators, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' the  propellers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Aerodynamics  This section discusses the different approaches to modeling  the aerodynamics of the drone and its propellers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' The most  widely used modeling assumption is that the propeller thrust  and drag torque are proportional to the square of the rotational  speed [14]–[18] and that the body drag is negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' These as-  sumptions quickly break down at the high speeds encountered  in drone racing as this model neglects (a) linear rotor drag [19],  [20], (b) dynamic lift [19], (c) rotor-to-rotor [21]–[23], (d)  rotor-to-body [21]–[23] interactions and (e) aerodynamic body  drag [20], [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  The accuracy of the propeller model can be improved  by leveraging blade-element-momentum theory, where the  propeller is modeled as a rotating wing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Such first-principle ap-  proaches [24]–[28] have been shown to provide very accurate  models of the wrench generated by a single propeller as they  properly capture effects (a) and (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Implemented efficiently, a  Blade Element Momentum (BEM) model can be run in real-  time [29] and has been successfully used to test algorithms in  simulation [30], [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  Accounting for the remaining open points (c)-(e), the aero-  dynamics of the drone body as well as any interaction effects  need to be calculated, which requires a full Computational  Fluid Dynamics (CFD) simulation [21]–[23], [32], [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Due  to the extreme computational demands, this is impractical  in drone racing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' To still get close to the accuracy of CFD  methods while retaining the computational simplicity of the  previously mentioned methods, data-driven approaches are  employed [29], [34]–[38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' In the early works [36], [37],  the whole vehicle dynamics model was learned from data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  In a similar fashion [35] uses a combination of polynomi-  als—identified from wind-tunnel flight data—to represent the  vehicle dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' In [29], [34], it has been shown that higher  modeling accuracies can be achieved when combining a first-  principle model with a data-driven component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Such com-  4  bination of first-principle and data-driven models also leads  to improved generalization performance, as shown in [29],  which combines a BEM model with a temporal convolutional  network [39] to regress the residual wrench.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' With this, a high-  fidelity model is obtained [29] that is still accurate at the  speeds and accelerations encountered in drone racing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Motor and Battery Models  The previous section outlines different approaches to how  the aerodynamic wrench can be estimated based on the state of  the vehicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' However, for all such models the rotational speed  of the propeller is assumed to be known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' On most multicopters,  the motors are not equipped with closed-loop motor speed  control but controlled by a ’throttle’ command which controls  the duty cycle of a PWM (pulse-width-modulation) signal  applied to the motors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' The actual rotational speed that the  motor achieves is a function of the throttle command as well  as other parameters such as the battery voltage and the drag  torque of the rotor [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Therefore, in order to have a dynamics  model for the motors we need a model of the battery to  calculate the voltage applied to the motors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Most literature  on battery modeling relies on so-called Peukert models [40],  but for lithium-polymer batteries in drone racing this is hardly  applicable because the battery discharge current often exceeds  100 A (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' 50-100 C) [41], [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Graybox battery models  for the voltage that are based on a one-time-constant (OTC)  equivalent circuit [43], [44] are much more suitable for drone  racing tasks as shown in [4], because they are applicable to the  extremely high loads experienced during a racing scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' In  combination with either a polynomial or a constant-efficiency  motor model such OTC models can be used to accurately  simulate the battery voltage during agile flight [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Given  a simulation of the battery voltage, one can measure the  performance characteristics of a given motor-propeller com-  bination to determine the mapping of throttle command and  voltage to resulting steady-state propeller speed Ωss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' When  the highest model fidelity is desired, a more sophisticated  motor simulation [45] can further improve the accuracy, which  can be desirable if the controller directly outputs single-rotor  thrusts instead of the more commonly used collective-thrust  and bodyrates control modality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Camera and IMU Modeling  Drone racing pushes not only the mechanical and electrical  components of drones to their limits, but is also highly  demanding in terms of sensor performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' For an in-depth  overview of the many different sensor options for drone racing  the reader is referred to [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' The most common sensors  aboard autonomous drones are monocular or stereo cameras  combined with IMUs (inertial measurement units) thanks  to their low cost, low weight, and mechanical robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  Typically, the algorithms processing the visual information  assume a pinhole camera [47] mounted in the center of gravity  of the vehicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' In reality, the camera is usually never mounted  directly at the center of gravity of the vehicle, meaning that a  transformation between the camera frame and the body frame  of the drone is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  The camera and IMU must be calibrated before using  their measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' The camera intrinsic calibration uses the  assumption of a pinhole model [47] and estimates the focal  length, image center and distortion parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' The IMU  intrinsic calibration estimates the noise characteristic of the  sensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' The camera-IMU extrinsic calibration estimates the  relative position and orientation of the two sensors as well  as the time offset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Kalibr [48] is a widespread tool to perform  these calibrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  The biggest source of measurement error of the sensors  onboard a drone are not the sensors themselves but the strong  high-frequency vibrations introduced by the fast-spinning pro-  pellers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' The vibrations lead to aliasing effects on the IMU  measurements and introduce additional motion blur on the  camera images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' The structural vibrations and their effect on  the measurements are extremely difficult to model and correct  for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Therefore, a suitable hardware design is imperative which  dampens the mount of the camera and the IMU with respect  to the vehicle frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' CLASSICAL PERCEPTION, PLANNING, AND CONTROL  PIPELINE  Sensors  Hardware  Perception  Software  Planning  Control  Drone  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' 3: Architecture 1: A classic architecture for an autonomous system  programmed using model-based approaches  Since the inception of the field of mobile robotics, a  common architecture has been primarily used to achieve  autonomous navigation capabilities across a wide variety of  systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' In a traditional robotics software stack, the naviga-  tion task is broken into three main components: Perception,  Planning, and Control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' A diagram of this architecture can be  seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' In this section, we cover recent research in  each of these areas relating specifically to agile flight and  autonomous drone racing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' All of the approaches detailed in  this section rely on first principles modeling and optimization  techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Perception  The perception block estimates the vehicle state and per-  ceives the environment using onboard sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' The most  common solution for state estimation of flying vehicles is  visual-inertial odometry (VIO) thanks to its low cost and low  weight requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Most of the time, a layout or map of  the race track is known apriori, and thus this section focuses  exclusively on VIO methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' VIO uses camera and IMU  measurements to estimate the state ˆx (position, orientation,  and velocity) of the drone platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' The inertial measurements  are integrated to obtain relative position, orientation, and  velocity estimates in a short time, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' between two camera  images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' However, the integration for a longer time, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' few  seconds, accumulates large drift due to scale factor errors,  axis misalignment errors, and time-varying biases [49] that  commonly affect off-the-self IMU measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' The camera  measurements provide rich information about the environment  5  at a lower rate, usually around 30 Hz, than IMU measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  Differently from the IMU measurements, the camera measure-  ments are affected by environmental conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' The quality of  information that they provide for state estimation degrades in  the case of poor illumination conditions, textureless scenes,  and motion blur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' For this reason, the camera and inertial  measurements complement each other and are the standard  choice for state estimation of flying vehicles [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' In this  section, we first give an overview of VIO with a focus on  the methods that can be applied for online state estimation  of aerial robotic vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Second, we give an overview of  recent VIO algorithms that include drone dynamics in the  estimation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Third, we conclude with a discussion on  the application of classical VIO methods to drone racing tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  1) VIO: VIO is the most common solution for state esti-  mation of aerial vehicles [50] using only onboard sensing and  computing thanks to its favorable trade-off between accuracy  and computational requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' VIO algorithms are usually  composed of two main blocks: the frontend and the backend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  The frontend uses camera images to estimate the motion of  the sensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Two main approaches exist in the literature: direct  methods and feature-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Direct methods [51], [52]  work directly on the raw pixel intensities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' These methods  commonly extract image patches and estimate the camera  trajectory by tracking the motion of such patches through  consecutive images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' The tracking is achieved by minimizing a  photometric error defined on the raw pixel intensities [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' On  the contrary, feature-based methods [53]–[55] extract points  of interest, commonly known as visual features or keypoints,  from the raw image pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' The camera trajectory is estimated  by tracking these points through consecutive images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Feature-  based methods are more mature and robust than direct meth-  ods;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' however, the latter achieve higher reliability in low-texture  environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Hybrid methods, which combine keypoints and  patches of raw pixel intensities to estimate the camera motion,  also exist [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' We refer the reader to the work in [47] for a  tutorial on the VIO frontend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  The backend fuses the output of the fronted with the inertial  measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Two categories exist in the literature according  to how the sensor fusion problem is solved: filtering methods  and fixed-lag smoothing methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Filtering methods are based  on an Extended Kalman Filter (EKF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' These methods propa-  gate the state of the system using the inertial measurements  and fuse the camera measurements in the update step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' The  pioneer filter-based VIO algorithm is the Multi-State Con-  straint Kalman Filter (MSCKF) originally proposed in [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  Since then, many different versions of MSCKF have been  developed [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Fixed-lag smoothing methods [54], [55], also  referred to as sliding window estimators, solve a non-linear  optimization problem where the variables to be optimized are  a window of the recent robot states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' The cost function to  minimize contains visual, inertial, and past states marginal-  ization residuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Fixed-lag smoothing methods accumulate  less linearization error than filtering methods but are more  computationally demanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' We refer the reader to the work  in [58] for a tutorial on the VIO backend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  Recent works [59], [60] have proposed to include event  cameras [61] in VIO for flying vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Event cameras do  not capture images at a fixed rate but they asynchronously  measure per-pixel brightness changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' The output of these  cameras is a stream of events that consists of time, location,  and the sign of brightness change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Their main properties are  low latency, high temporal resolution (in the order of µs), and  high dynamic range (140 dB compared to 60 dB of standard  cameras).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Thanks to these properties, event cameras are a great  complementary sensor to standard cameras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Including event  data in VIO algorithms achieves increased robustness against  motion blur as demonstrated in [59], [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' UltimateSLAM  [59] is a VIO algorithm that combines both standard and  event cameras in a fixed-lag smoother-based VIO algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  In [60], a revised version of UltimeSLAM was proposed to  demonstrate autonomous quadrotor flight despite rotor failure  with onboard sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  2) Drone dynamics in VIO: The drone dynamics are used to  define additional error terms in the VIO backend in [62], [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  In [62], the authors propose VIMO which is a VIO algorithm  that includes error terms on the drone transitional dynamics  in a fixed-lag smoother-based backend [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' These error terms  on the drone dynamics are derived through the preintegration  theory [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' In addition to the drone state, VIMO is able to  estimate the external force acting on the drone platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' The  external force is modeled as a random variable distributed  according to a zero-mean gaussian distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' This choice  allows the estimation of impulse-like forces acting on the  drone platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  The work in [63] extends VIMO by introducing a different  model of the external force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' The external force is modeled  as a gaussian distribution whose mean value is equal to the  difference between the commanded collective thrust and the  force, up to the mass, detected by the accelerometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' This is  a suitable design choice in the case when the external force  affects the drone platform for long periods of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  3) Discussions: The work in [65] presents a benchmark  comparison between a number of VIO solutions on the EuRoC  dataset [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' The EuRoC dataset contains camera- and IMU-  data recorded onboard a drone flying in indoor environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  The drone moves with average linear and angular velocities  up to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='9 m/s and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='75 rad/s, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' These values are  far below the ones reached in drone racing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' The conclusions  of [65] show that state-of-the-art VIO algorithms provide  reliable solutions for estimating the state of the drone at limited  speeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' However, these classical VIO methods are not able to  provide accurate state estimates for drone racing tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' VIO  methods accumulate large drift in scenarios characterized by  motion blur, low texture, and high dynamic range [67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' These  scenarios are the norm in drone racing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  To help the research in VIO algorithms for drone racing  tasks, the work in [68] proposes the UZH-FPV Drone Racing  Dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' This dataset contains images recorded from standard  cameras, event camera data, and IMU data recorded onboard  a quadrotor flown by a human pilot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' All the flights include  visual challenges that are similar to the ones present in drone  racing competitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  Successful state estimation solutions for drone racing [67],  [69] reduce the drift accumulated in VIO by localizing to a  prior map of the track.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' This is a viable solution when a map  6  of the track in the form of gate positions is known beforehand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  The localization process is based on the detection of the gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  In [70], it was proposed a gate detector that uses an RGB  camera to identify the gates based on their color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' All the other  gate detection methods existing in the literature are based on  deep learning techniques [71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' We review them in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' The  known gate positions and the detections in the onboard images  are used to estimate the relative pose between the camera and  the gate using the Perspective-n-Point algorithm (PnP) [72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  This relative pose is used to constrain the VIO backend and  consequently reduce the drift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' There is significant room for  innovation on this front, as the VIO-PnP paradigm has existed  for several years with little innovation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Other approaches used  in early drone racing competitions relied on the technique of  visual servoing via stereo cameras [7], but this solution was  found to be sensitive to indoor lighting changes and needed  to be hand-tuned for every flight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  Recent works [73]–[75] proposed vision-based odometry  algorithms that are learned end-to-end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' In theory, these meth-  ods could be specialized to drone racing tasks and potentially  outperform classical VIO approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' However, they are in  the early development phase and how to customize them for  the drone racing task is still an open research question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' In  addition, they currently have high computational costs that  make them impractical for online state estimation onboard  drones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' We refer the reader to Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' IV for a detailed review  of VIO methods based on deep learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Planning  Once  a  state  estimate  ˆx  has  been  obtained  from  the  perception  module,  the  next  step  in  the  classical  pipeline  is  to  plan  a  feasible,  time-optimal  trajectory  τref = (xref, uref)k, ∀k ∈ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' N, which respects the phys-  ical limits of the platform as well as the constraints imposed  by the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' This requires predicting the drone’s future  states such that minimum lap time is reached without crashing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  Trajectory planning has matured over the last decade from  works mostly verified in simulation to works shown in both  controlled lab environments and unknown unstructured envi-  ronments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Here, we categorize these methods in polynomial  and spline trajectory planning, optimization-based planning,  search-based planning, and sampling-based planning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  1) Polynomial and Spline Trajectories: The Polynomial  and Spline methods leverage the differential flatness prop-  erty [76], [77] of quadrotors and represent a trajectory as a  continuous-time polynomial or spline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' This property simplifies  the full-state trajectory planning to a variant where only four  flat outputs need to be planned (typically 3D position and  heading).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' By taking their high order derivatives, these flat  outputs can represent a dynamically feasible trajectory with  their respective control inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' This property is used by many  polynomial and spline methods that are nowadays among the  most used for general quadrotor flight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  The widely used polynomial trajectories [76], [77] minimize  snap (4th order position derivative) of a trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Different  methods opted for minimizing jerk (3rd order position deriva-  tive) for planning a trajectory [78].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' However, the trajectories  that result from having jerk as the primary objective have been  shown to minimize the aggressiveness of the control inputs  [78], which is fundamentally different from minimizing the lap  times, where extremely aggressive trajectories are generally  required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Richter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' [79], therefore, extended the objective  by jointly optimizing both the snap of a trajectory and the  total time through a user-specified penalty on time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Recently,  Han [80] proposed a polynomial-based trajectory planning  method for drone racing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' It jointly optimizes control effort,  regularized time, and penalizes the dynamic feasibility and  collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  Because of their numerical stability, other methods make  use of B-splines for representing trajectories [81], [82] in-  stead of high order polynomial representations that are nu-  merically sensitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' These methods jointly optimize different  objectives, simultaneously smoothness, dynamic feasibility,  collision avoidance, safety [82] and vision-based target track-  ing [83].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  Although both polynomial and spline trajectories are widely  used due to their computational efficiency, polynomial-based  trajectories (and their derivatives) are smooth by definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  Therefore, only smooth control inputs can be sampled from  them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  Yet, a time-optimal trajectory is not smooth, but rather  attempts to keep the acceleration to the possible maximum at  all times [84].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' This means that our planned actuation needs to  be able to maintain a certain value during extended periods of  time, while also being able to have quick changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Therefore  polynomials cannot represent true time-optimal trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  2) Optimization-based: Optimization-based trajectory plan-  ning enables us to independently select the optimal sequence  of states and inputs at every time step, which inherently con-  siders time minimization while complying with quadrotor dy-  namics and input constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Optimization-based approaches  have been extensively considered in the literature, ranging  from exploiting point-mass models [85], simplified quadrotor  models [86], [87], and full-state quadrotor models [84], [88].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  Time-optimality of a trajectory could also be accomplished  by using a specific path parameterization that maximizes  velocity over a given path [89].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' This method was shown for  quadrotors in [90] for minimizing time of flight considering  both translational and rotational quadrotor dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' However,  the method only creates a velocity profile over a given path  which is not further optimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  Apart from time optimality, complying with intermediate  waypoint constraints is another requirement for path planning  in autonomous drone racing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' A common practice of solving  a trajectory optimization problem with waypoint constraints  is allocating waypoints to specific time steps and minimizing  the spatial distance between these waypoints and the position  at the corresponding allocated time steps on the reference  trajectory (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' [91], [92]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' The time allocation of the way-  points is, however, non-trivial and difficult to determine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' This  is tackled in [88], but the work uses body rates and collec-  tive thrust as control inputs and does not represent realistic  actuator saturation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Recent work [84] introduces a comple-  mentary progress constraints (CPC) approach, which considers  true actuator saturation, uses single rotor thrusts as control  7  inputs, and exploits quaternions to create full, singularity-  free representations of the orientation space with consistent  linearization characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' While the above methods create  time-optimal trajectories passing through given gates, they are  computationally costly and hence intractable in real-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  3) Search-based: Search-based planning methods [93], [94]  rely on discretized state and time spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' They solve the  trajectory planning through graph search algorithms such as  A*.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' The search graph is built using minimum-time motion  primitives with discretized velocity, acceleration, or jerk input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  The algorithms then use trajectories of a simpler model, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  with velocity input, as heuristics for the search with a more  complex model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Search-based planning methods can optimize  the flight time up to discretization, but they suffer from  the curse of dimensionality which renders them increasingly  computationally demanding for increasing complexity of the  quadrotor model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Furthermore, the employed per-axis dynamic  limits (velocity, acceleration, jerk) does not represent the  true quadrotor model, which further decreases the quality of  found plans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Finally, although searching for minimum time  trajectories, the methods are currently limited to planning  between two states which is not suitable for multi-waypoint  drone racing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  4) Sampling-based:  Sampling-based  methods  like  RRT*  [95]  can  be  used  for  planning  trajectories  for  linearized  quadrotor  models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  Several  time-minimizing  approaches [67], [96] use a point-mass model for high-level  time-optimal trajectory planning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' In [96], an additional  trajectory smoothing step is performed where the generated  trajectory is connected with high-order polynomials by  leveraging the differential flatness property of the quadrotor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  However, these point-mass approaches need to relax the  single actuator constraints and instead limit the per-axis  acceleration, which results in trajectories that are conservative  and sub-optimal given a minimum time objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' In [97], the  authors use minimum-jerk motion primitives for connecting  randomly sampled states inside RRT* to plan a collision-free  trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Since the authors use polynomials, this approach  can only generate smooth control inputs, meaning that they  cannot rapidly switch from full thrust to zero thrust (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  bang-bang) if required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  The first method for planning minimum-time trajectories in  a cluttered environment for the full quadrotor model was pro-  posed in [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' It uses a hierarchical sampling-based approach  with an incrementally more complex quadrotor model to guide  the sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' The authors showed that the method outperforms  both polynomial and search-based methods in minimizing  trajectory time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Yet, the method is offline and intractable in  real-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Most recently, the authors of [98] proposed an online  replanning approach that plans minimum-time trajectories for  a point-mass model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' The paths of replanned trajectories are  then consequently used by Model Predictive Contouring Con-  trol [99] with a full quadrotor model to maximize the progress  along the path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' This method is capable of outperforming  other classical approaches due to the replanning capability and  progress maximization with a full quadrotor model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  5) Discussion: A planned trajectory can be understood as  an intermediate representation that, given information about  2016  2017  2018  2019  2020  2021  2022  0  10  20  30  [46]  [46]  [12]  [34]  [103], [104]  [105]  Year  Top Speeds on Autonomous Hardware [m/s]  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' 4: Top speeds demonstrated on autonomous drones over time from both  literature and competition data  the robot’s dynamics and the environment, helps guide the  platform through the race track and ultimately perform the task  at hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' One might argue if this intermediate representation is  needed at all, since ultimately, what we are looking for is a  policy that maps sensor information and current environment  knowledge to the actuation space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' This is generally achieved  with learning-based approaches, discussed in Section IV,  which bypass the planning stage and directly convert sensor  observations to actuation commands [100]–[102].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  One of the biggest benefits of explicit planning is modu-  larity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' This means that the developed algorithms can be used  off-the-shelf for different drone tasks outside racing, such as  search and rescue, which is not the case for single-purpose  learned approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' However, explicit planning suffers from  the disconnection (or an open loop) between the planning and  the deployment stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Unexpected deviations from the plan,  be it in the time domain (like unmodeled system delays) or in  the state-space domain (like state estimation drifts or jumps in  the VIO pipeline), can lead to compound errors and ultimately,  a complete system failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  This can be tackled with more complex control approaches  that do some part of the replanning online [98].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Control  Over the last decade, significant advancements have been  made in agile multicopters control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Every year, increasing top  speeds are demonstrated in the literature as shown in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  Controllers must be able to make real-time decisions in the  face of poor sensor information and model mismatch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Control  inputs, u(t), can come in a variety of modalities for quadrotor  control, such as velocity and heading, body rates and collective  thrust, or direct rotor thrust commands [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Typically, a high-  level controller computes a desired virtual input such as body  rates and collective thrust, which is then passed down to a  low-level flight controller that directly controls the individual  rotors on the multicopter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  8  Commonly used open source controllers such as PixHawk1  or BetaFlight2 are widely available to the drone racing com-  munity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' BetaFlight is the most commonly used low-level  controller for agile drone flight and has been widely adopted  by the First Person View (FPV) racing community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  In the following sections, we provide an overview of suc-  cessful approaches to achieving high speeds in both simulation  and real-world applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' We sort the approaches into  model-based control and coupled perception and control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  1) Model-Based Control: In model-based control, an ex-  plicit model of the dynamic system is used to calculate control  commands which satisfy a given objective such as minimizing  time or tracking error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Models enable the prediction of future  states of the drone and provide information about the system’s  stability properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' In [106], Geometric Tracking control is  introduced on the Special Euclidean group SE(3) and com-  pletely avoids singularities commonly associated with Euler  angle formulations on SO(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' This nonlinear controller showed  the ability to execute acrobatic maneuvers in simulation and  was the first to demonstrate recovery from an inverted initial  attitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' The dynamic model of a quadrotor is shown to be  differentially flat when choosing its position and heading as  flat outputs in [77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' In this work, many agile maneuvers are  performed onboard real drones with speeds up to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='6 m/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  The previous work is extended in [20], proving that the  dynamics model of a quadrotor subject to linear rotor drag  is also differentially flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' The inclusion of the aerodynamic  model within the nonlinear controller lead to demonstrated  flight speeds up to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='0 m/s while reducing tracking error by  50% onboard a real drone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  The differential flatness method is further extended by  in [107] by cascading an Incremental Nonlinear Dynamic  Inversion (INDI) controller with the differential flatness con-  troller described in [77] but neglects the aerodynamic model  addition from [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' The INDI controller is designed to track  the angular acceleration commands ˙Ω from the given reference  trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Top speeds of nearly 13 m/s and accelerations over  2g are demonstrated onboard a real quadrotor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' The controller  shows robustness against large aerodynamic disturbances in  part due to the INDI controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  An investigation of the performance of nonlinear model pre-  dictive control (NMPC) against differential flatness methods is  available in [104].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Cascaded controllers of INDI-NMPC and  INDI-differential flatness are shown to track aggressive racing  trajectories which achieving speeds of around 20m/s and  accelerations of over 4g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' While differential flatness methods  are computationally efficient controllers and relatively easy to  implement, they are outperformed on racing tasks by NMPC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  An excellent overview of MPC methods applied to micro  aerial vehicles can be found in [108].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Because quadrotors  are highly nonlinear systems, nonlinear MPC is often used  as the tool of choice for agile maneuvers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' The debate of  linear versus nonlinear MPC is thoroughly discussed in [109].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  Model Predictive Path Integral (MPPI) control is a sampling-  based optimal control method that has found excellent suc-  1https://pixhawk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='org/  2https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='com/betaflight/betaflight  cess on the AutoRally project, a 1/5th scale ground vehicle  designed to drive as fast as possible on loose dirt surfaces  [110], [111].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' An introduction to MPPI can be found in  https://autorally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='io/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' The MPPI approach can be used  on agile quadrotors to navigate complex forest environments,  however, analysis was only performed in simulation [110].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  Most of the successful demonstrations of MPPI come from  ground robots [110], [111].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Because MPPI is a sampling based  algorithm, scaling to higher-dimension state spaces like those  of a quadrotor can lead to performance issues as shown in  [103].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  Nonlinear MPC methods are also used in [34] where a  nominal quadrotor model is augmented with a data-driven  model composed of Gaussian Processes and used directly  within the MPC formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' The authors found that the  Gaussian-Process model could capture highly nonlinear aero-  dynamic behavior which is difficult to model in practice as  described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' The additional terms introduced by the  Gaussian-Process added computational overhead to the MPC  solve times, but it was still able to run onboard a Jetson TX2  computer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  Similar to [107], authors in [103] question whether or not  it is necessary to explicitly model the additional aerodynamic  terms from [34] due to the added computational and modeling  complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Instead, they propose to learn residual model  dynamics online using a cascaded adaptive nonlinear model  predictive control architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Aggressive flight approaching  20m/s and over 4g acceleration is demonstrated on real rac-  ing quadrotors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Additionally, completely unknown payloads  can be introduced to the system, with minimal degradation  in tracking performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' The adaptive inner loop controller  added minimal computational overhead and improved tracking  performance over the Gaussian Process MPC by 70% on a  series of high-speed flights onboard a racing quadrotor [34],  [103].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  Contouring control methods can deal with competing op-  timization goals such as trajectory tracking accuracy and  minimum flight times [112].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' These methods minimize a cost  function which makes trade-offs between these competing  objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' In [113], Nonlinear Model Predictive Contouring  Control (MPCC) is applied to control small model racecars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  MPCC was then extended to agile quadrotor flight in  [99].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  Although the velocities achieved by the MPCC controller were  lower than that of [103], [104], the lap times for the same race  track were actually lower due to the ability of the controller to  find a new time-allocation that takes into account the current  state of the platform at every timestep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' The work is further  extended to solve the time-allocation problem online, and to  re-plan online [98] while also controlling near the limit of the  flight system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Similar work uses tunneling constraints in the  MPCC formulation in [114],  2) Perception Aware Model Predictive Control: When con-  trol is coupled with perception, an optimization problem that  constrains or penalizes the movement of the drone to ensure  that an area of interest is always kept within the field of view of  the camera can be solved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' This is integral to the drone-racing  problem because, to navigate a challenging race course, the  gates that define the course layout must be kept in view of  9  the onboard cameras at all times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Coupling the perception and  control problem can alleviate issues in state estimation because  the racing gates are usually feature-rich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  The goal is as follows: navigate a trajectory with low  tracking error while keeping a point of interest in view while  minimizing motion blur for maximum feature detection and  tracking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' The first instance applied to agile quadrotors was  Perception-Aware MPC (PAMPC) introduced in [115].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' In this  work, a nonlinear program is optimized using a sequential  quadratic programming approximation in real time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' The cost  function contains both vehicle dynamic terms as well as  perception awareness terms such as keeping an area of interest  in the center of the camera frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  This technique is applied to the drone racing problem in  [116], where an MPPI controller is designed with a Deep  Optical Flow (DOF) component which predicts the movement  of relevant pixels (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' gates).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' The perception constraints are  introduced into a nonlinear optimization problem and deployed  in a drone-racing simulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' The approach was not demon-  strated onboard real hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' In [117], a perception-aware  MPC based on Differential Flatness was used to ensure that  a minimum number of features are tracked between control  updates and thus guarantee localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' To achieve this, a  Perception Chance Constraint within the MPC formulation is  introduced to ensure that at least n number of landmarks are  within the field-of-view of the camera at all times with some  bounded probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  3) Discussion: The performance of model-based controllers  degrades when the model they operate on is inaccurate [103].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  For drones, defining a good-enough model is an arduous  process due to highly complex aerodynamic forces, which can  be difficult to capture accurately within a real-time capable  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' In addition, the tuning process of many model-based  controllers can be arduous, and requires a high level of domain  expertise to achieve satisfactory performance  In any optimal control problem, a cost function that the  user wants to optimize must be defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Traditionally, con-  venient mathematical functions leveraging convex costs are  used because these functions are easy to optimize and there  is a large toolchain available for optimizing such problems  such as Acados [118], CVXGEN [119], HPIPM [120], or  Mosek [121].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' In many drone racing papers, the optimal control  problem is formulated as follows:  min  u xT  NQxN +  N−1  �  k=0  xT  k Qxk + uT  k Ruk ,  (2)  subject to:  xk+1 = f RK4(xk, uk, δt) ,  x0 = xinit ,  umin ≤ uk ≤ umax ,  where the state is given by xk, the control input is given by  uk, the state cost matrix is given by Q, and the control cost  matrix is given by R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' The optimization problem is constrained  by the dynamics of the system given by f(xk, uk, δt) where  δt is a finite time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' The nonlinear dynamics are typically  propagated forward using an integrator such 4th order Runge-  Kutta, RK4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Additionally, the problem is subject to the thrust  limits of the platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' umin and umax, and some initial  condition of the system x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' In this formulation, a reference  position and control are provided by a high-level planner and  the goal of the controller is to track the given reference, but  this objective is ill defined for the drone racing problem: in  drone racing, we wish to complete the track in as little time  as possible;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' therefore, our objective can be better formulated  as follows:  min  u  T  �  k=0  δt ,  (3)  subject to:  xk+1 = f RK4(xk, uk, δt) ,  x0 = xinit ,  umin ≤ uk ≤ umax ,  where T is the number of discrete time steps it takes to  complete the race.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' This approach requires a time-horizon  which predicts all the way until the end of the task which  is intractable to optimize online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  Reinforcement learning (RL) methods [30], [101] can opti-  mize a proxy of this cost function, however do so in an offline  fashion, requiring large amounts of training experience to  approximate the value function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' RL methods do not necessarily  depend on a high-level planner to provide a reference to track.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  We will discuss some recent approaches using reinforcement  learning methods in the following section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' LEARNING-BASED APPROACHES  In this section, we present various learning-based ap-  proaches for drone racing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' These approaches replace the plan-  ner, controller, and/or perception stack with a neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  Learning-based methods have gained significant traction in the  last few years, given their ability to cope with both high-  dimensional (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' images) or low-dimensional (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' states)  input data, their representation power, and the ease to develop  and deploy them on hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  The biggest challenge for learning-based methods is col-  lecting enough data to effectively train the neural network for  the task at hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' There are currently two possibilities for data  gathering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' The first, mostly popular in the initial stages of  learning-based robotics [69], [122]–[125] is to collect data  in the real world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' The data is then annotated by a human  or an automated process, and used for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' The second,  much more popular in recent years and currently achieving  the best results, consists of using simulation for collecting  training data [30], [100], [126]–[128].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Both approaches have  their advantages and limitations, which we will discuss in  the following sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Surveys covering existing methods for  learning-based flight already exist [129], [130].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' In contrast to  them, we cover the most recent advances and give a broader  discussion on the comparison between learning-based and  traditional methods for drone racing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Learned Perception  Sensors  Hardware  Planning  Control  Drone  Software  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' 5: Architecture 2: Learned Perception  10  For learned perception modules, the goal of the network  is to use images from an RGB, depth, or event camera to  detect landmarks within the environment and output useful  representations such as waypoints, or the location of gates on  the race track.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' A depiction of this architecture can be seen in  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' An overview of deep learning methods for vision-  based navigation specific to the drone racing task can be found  in [130].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  In [124], a dataset of images is collected from a forward-  facing camera mounted on a drone labeled with the relative  position to the closest gate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' This dataset is used to train a net-  work which predicts from an image both the next gate location  and its uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Predictions are then fused with a visual-  inertial odometry system in an Extended Kalman Filter (EKF)  to predict the position of the drone on the track.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Similarly  in [12], a Convolutional Neural Network (CNN) is used to  detect gate corners in the AlphaPilot challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Once the gate  corners are detected, classical computer vision algorithms like  PnP can be used to find the coordinates of the gate in the  camera frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Using an EKF, the gate corner locations can be  fused with a traditional VIO pipeline to improve the estimates  of the drone’s location and orientation [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  Oftentimes, perception networks consume precious re-  sources onboard computationally limited drones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' To minimize  the network processing time, [71], [131] proposed optimized  architectures for gate detection on real-world data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' A similar  optimization went into “GateNet” [132] a CNN to detect gate  center locations, distance, and orientation relative to the drone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  The same authors developed a follow-up work denoted as  ”Pencil-Net” to do gate detection using a lightweight CNN  in [133].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Most learning-based perception networks can suffer  from poor generalization when deployed in environments that  were not included in the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='To reduce deployment  sensitivity to lighting conditions or background content, virtual  gates can be added to real-world backgrounds [134].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  Up until recently, RGB and depth cameras were used  exclusively in the drone racing task, however, these sensor  modalities can be sensitive to changes in the environment such  as illumination changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' To overcome this, [135] proposed  using event cameras coupled with a sparse CNN, recurrent  modules, and a You Only Look Once (YOLO) object detector  to detect gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' The use of event cameras overcomes poten-  tial issues with motion blur induced by rapid movement of  the drone and is a promising path forwards for high-speed  navigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  Overall, deep learning methods for gate detection are the de-  facto standard in all drone racing systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' However, such gate  detectors are always coupled with traditional visual-inertial  odometry systems which explicitly estimate the metric state  of the drone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' These approaches are discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' It  is interesting to notice that learning-based odometry systems,  such as [73]–[75] have not yet replaced traditional methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  This is particularly surprising since deep visual odometry  systems can specialize to a particular environment, which  can be useful for drone racing since the race track is fixed  and known in advance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' A disadvantage of these methods is  the high computational cost that makes them impractical for  online applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Research in end-to-end visual odometry  is moving forward at a fast pace [75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' We foresee that in the  near future, researchers will be able to apply these methods  to the drone racing task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Learned Planning & Perception  Sensors  Hardware  Control  Drone  Software  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' 6: Architecture 3: Learned Planning and Perception  A tightly-coupled planning and perception stack (Figure 6)  is a very attractive algorithmic perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' First, it greatly  simplifies the perception task: an explicit notion of a map  or globally-consistent metric state is not required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Second, it  largely reduces computational costs, both in the pre-training  and evaluation stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Finally, it can leverage large amounts  of data, collected either in simulation or the real world, to  become robust against noise in perception or dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Yet,  an interesting observation is that these methods still work  best when coupled with an explicit estimator of the metric  state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' In contrast to traditional methods, a locally consistent  odometry system is sufficient [69], [126], [127], waving away  the complexities of full-slam methods (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' loop-closure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  In [69], a coupled perception and planning stack for drone  racing is trained using real-world flight demonstrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' While  good performance is indicated on the racing task as well as  robustness against drift in state estimation, the method requires  re-training for each new environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Therefore, in the follow-  up work [126], data generated entirely from simulation is  used to train the perception-planning stack, waiving the labor  and time-consuming requirement of data collection in the real  world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' A similar pipeline was used for high-speed autonomous  flight through complex environments in [127], which proposes  to train a neural network in simulation to map noisy sensory  observations to collision-free trajectories directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  Several other works apply a similar stacked perception  and planning pipeline for other autonomous drone racing  tasks [122], [123], [125], [136].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' We point the interested  reader to existing surveys on the role of learning in drone  navigation [129].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  A few works also studied the problem of planning using  data-driven methods, decoupling it from the perception prob-  lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' An interesting approach demonstrated in the NeurIPS  Game of Drones competition [137] used an off-the-shelf  reinforcement learning algorithm in place of a classic model-  based planner for drone racing [138].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' More recently, a novel  multimodal learning-based trajectory planning framework was  introduced in [139], which can generate collision-free trajec-  tories that avoid a dynamic obstacle while maximizing its  presence in the field of view (FOV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  The big advantage of these methods is that they require  less computational effort than traditional methods, possibly  enabling online re-planning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' In addition, they are much more  robust to system latencies and sensor noise, which can be eas-  ily accounted for by identifying them on physical drones and  then adding them to the training environments [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' However,  11  the major limitation of these methods is their sample complex-  ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' If the training data comes from a simulator, significant  simulation engineering is required to enable generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  Conversely, if data come from the real world, generalization  is easier, but the data collection process is very slow, tedious,  and expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Learned Control  Sensors  Hardware  Perception  Software  Planning  Drone  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' 7: Architecture 4: Learned Control  Data-driven control, like reinforcement learning, allows  overcoming many limitations of prior model-based controller  designs by learning effective controllers directly from expe-  rience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' For example, control of a physical quadrotor using  reinforcement learning was demonstrated by [140], where a  neural network policy was used for waypoints tracking and  vehicle recovery from harsh initialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' The neural network  policy takes about 7 µs to generate the control command  given the state, while a linear MPC requires about 1000 µs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  Recently, [30] demonstrated high-speed trajectory tracking  using learning-based control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' They additionally showed that  learned policies can be made robust to sensor noise and  system latency by training with simulated sensor noise and  latencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Model-free RL was also applied to low-level attitude  control [141], in which a learned low-level controller trained  with PPO outperformed a fully tuned PID controller on almost  every metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Similarly, [142] used model-based RL for low-  level control of an a priori unknown dynamic system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  With any learning-based controller, it can be difficult to  provide robustness guarantees as with traditional methods such  as the Linear Quadratic Regulator (LQR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' However, it is  possible to make the planner and controller robust to system  latencies, model uncertainties, and sensor noise by identifying  them on physical drones and then adding them into the sim-  ulation environments used to gather training data [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' While  a learning-based controller may provide superior performance  to classical methods, it may be the case that they cannot be  used in practice due to the inability to provide an analysis of  the controller’s stability properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' These properties are often  required in safety-critical systems such as flight controllers  for aircraft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Recent works have attempted to address this  using Lyapunov-stable neural network design for the control  of quadrotors [143].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' This work shows that it is possible to have  a learning-based controller with guarantees that can also out-  perform classical LQR methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Building upon this concept,  reachability analysis, and safety checks can be embedded in a  learned Safety Layer [144].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  None of the systems discussed so far deal with the chal-  lenging problem of adapting to new and uncertain environ-  ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' The field of adaptive control has studied this problem  extensively [145]–[147], however, we have seen a recent  push to use advancements in machine learning within the  adaptive control framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' A method to learn parametric  uncertainty functions is introduced in [148].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' These uncertainty  functions could be learned offline using data captured from  agile flight experiments, and then embedded within an adaptive  controller to adjust controller parameters online during flight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  Results indicate that highly accurate trajectory tracking can  be achieved with this approach, even in the face of strong  wing gusts exceeding 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='5 m/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' More recently, learning-based  controllers have shown the ability to adapt zero-shot to large  variations in hardware and external disturbances [149].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' We see  this as a promising area of research and one that is integral for  reliable performance in changing environmental conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Learned Planning & Control  Sensors  Hardware  Perception  Software  Drone  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' 8: Architecture 5: Learned Planning & Control  The second paradigm of learned control is to produce the  control command directly from state inputs without requiring  a high-level trajectory planner, as shown in the architecture  diagram of Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' In autonomous drone racing, this was  proposed by [101], where a neural network policy is trained  with reinforcement learning to fly through a race track in  simulation in near-minimum time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Major advantages of the  reinforcement-learning-based method are its capability to han-  dle large track changes and the scalability to tackle large-scale  random track layouts while retaining computational efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  In [102], deep reinforcement learning is combined with clas-  sical topological path planning to train robust neural network  controllers for minimum-time quadrotor flight in cluttered  environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' The learned policy solves the planning and  control problem simultaneously, forgoing the need for explicit  trajectory planning and control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  These methods inherit the classic advantage of policy learn-  ing: to achieve robustness to system latencies, model uncer-  tainties, and sensor noise, one can identify them on physical  drones and then add them into the simulation environments  used to gather training data [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' In addition, they do not  require an external controller to track the plan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' This eliminates  the discrepancy between the planning and deployment stage,  which is one of the main limitations of traditional planning  methods (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' III-B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Some of the limitations of traditional  planning still remain such as the requirement of a globally-  consistent state estimation and a map of the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Also,  they have not yet been demonstrated in sparse long-horizon  planning problems, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' flying through a maze at high speeds,  where their performance would likely drop due to sample  complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' End-to-End Flight  Sensors  Hardware  Drone  Software  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' 9: Architecture 7: End to End Learning  12  Expert pilots take raw sensory images from a first-person-  view camera stream and map directly to control commands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  In this section, we explore approaches emulating this holistic  navigation paradigm in autonomous drones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  Two families of approaches can be used to pursue an end-to-  end navigation paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' The first is substituting each of the  perception, planning, and control blocks with a neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  This structure is followed by [150], [151], where the authors  train a perception-planning network and a control network  using imitation learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' The perception network takes raw  images as input and predicts waypoints to the next gate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  The control network uses such predictions with ground-truth  velocity and attitude information to predict control commands  for tracking the waypoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' They showed improvements over  pure end-to-end approaches, which directly map pixels to con-  trol commands and were able to show competitive lap times  on par with intermediate human pilots within the Sim4CV  simulator [152].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Yet, the division into independent blocks leads  to compounding errors and latencies, which negatively affect  performance when flying at high speeds [127].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  The second family of approaches directly maps sensor  observation to commands without any modularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' This design  is used by [153], which to date remains the only example of  the completely end-to-end racing system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Indeed, other end-  to-end systems generally require an inner-loop controller and  inertial information to be executed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' For instance, [154] trains  an end-to-end CNN to directly predict roll, pitch, yaw, and  altitude from camera images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Similarly, [155] uses a neural  network to predict commands directly from vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' To improve  sample complexity, they use contrastive learning to extract  robust feature representations from images and leverage a two-  stage learning-by-cheating framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Given the absence of  any division between perception, planning, and control, this  family of approaches is potentially more robust to sensor noise  and latencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Yet, these policies are extremely data-hungry,  which hinders their generalization in environments different  from the training ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  Independently of the design paradigm they follow, end-  to-end navigation algorithms are currently bound to simula-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' The reasons why no method was successfully deployed  in the real world include weak generalization to unseen  environments, large computational complexity, and inferior  performance to other modular methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Another interesting  observation is that humans can pilot a drone exclusively  from visual observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Conversely, except for [153], end-  to-end systems still rely on the state extracted from other  measurement modalities, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' an IMU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' The question of whether  autonomous drones can race in the real world at high-speed  without any inertial information remains open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' We provide  more details on this question in Section VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Discussion  Data-driven approaches are revolutionizing the research in  autonomous drone racing, ranging from improving the system  model to end-to-end control of the vehicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Currently, the best-  performing algorithms for drone racing include a learning-  based component [12], [13], and this trend is unlikely to  change in the coming years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Indeed, compared to classical  model-driven design, they can process high-dimensional sen-  sory inputs directly, can be made robust to any modeling  uncertainty (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' latency) by simply incorporating it in the  training pipeline, and require far less engineering effort for  tuning and deploying them [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  A trend that is recently gaining more and more popularity is  training policies in simulation and deploying them in the real  world [69], [126], [127].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Leveraging years of advancement in  drone modeling technology (See Section II), the simulation  of drone dynamics is extremely realistic and fast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Conversely,  the simulation of sensor measurements (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' cameras, IMUs,  lidars) is either inaccurate or very computationally expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  Therefore, researchers generally aim to abstract observations  using classical perception algorithms (see Section III-A) to  train their model in a timely fashion and favor simulation-to-  real-world transfer [127].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' However, this hinders the deploy-  ment of completely end-to-end systems on real-world robots,  which, as human pilots, only rely on a stream of color images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  We refer the reader to Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' VIII for the implication of this  research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  While simulators may get better and faster in the near future,  recent advances in real-world training [156], [157] and fine-  tuning [158], [159] offer a potential alternative for zero-shot  simulation to reality transfer for sensorimotor policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' So far,  these works have been limited to legged locomotion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Extension  to agile drones could lead to the successful deployment of  end-to-end policies, possibly improving the state of the art in  racing performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' DRONE RACING SIMULATORS  One tool that has drastically accelerated the progress of  research in autonomous drone flight is the use of simulation  environments that attempt to recreate the conditions that  real drones experience when flying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Over the years, several  simulation environments have been developed for the use of  general research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  In 2016, the widely used RotorS simulation environment  was published, which extends the capabilities of the popular  Gazebo simulation engine to multi-rotors [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Gazebo uses  the Bullet physics engine for basic dynamic simulation and  contact forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Linear drag on the body of the multicopter is  simulated based on the cross-sectional area and linear velocity  of the simulated object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' The RotorS extension features many  easy-to-use plugins for developing multi-rotors, however, it  distinctly lacks the photorealistic details needed to simulate  accurate behavior of estimation and perception pipelines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  AirSim was introduced by Microsoft in 2018 as a photo-  realistic simulator for the control of drones [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' It is built on  the Unreal graphics engine and features easy-to-use plugins  for popular flight controllers such as PX43, ArduPilot4, and  others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' It was used in the 2019 NeurIPS Game of Drones  challenge [137].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Because of the photorealism of AirSim, it  is possible to simulate the entire perception and estimation  pipeline with good possibility of transfer to real-world drone  3https://px4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='io/  4https://ardupilot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='org/  13  systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Additionally, AirSim comes pre-packaged with an  OpenAI-Gym environment for training Reinforcement Learn-  ing algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Organizations such as Bell, Airtonomy, and  NASA are using AirSim to generate training data for learning-  based perception models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  FlightGoggles [160] was developed as another photorealistic  simulator and was used as the primary simulation environment  for the Lockheed Martin AlphaPilot challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' FlightGoggles  contains two separate components: a photorealistic rendering  engine built with Unity3D and a dynamic simulation im-  plemented in C++.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' FlightGoggles provides an interface with  real-world vehicles using a motion capture system;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' such an  interface allows rendering simulated images that correspond  to the position of physical vehicles in the real world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  A recent simulator focused on Safe RL was proposed  in [161].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' It uses Gazebo and the Pybullet physics engine as  the backend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Leaderboards for several safety-focused training  environments exist, encouraging researchers to submit their  approaches and compete with other researchers around the  world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  Finally, Flightmare [16] is a simulation environment fea-  turing photorealistic graphics provided by the Unity engine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  The physics engine is decoupled and can be swapped out  with various engines for user-defined levels of simulation  fidelity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Similar to FlightGoggles, Flightmare can also provide  hardware-in-the-loop simulation functions where a virtual,  synthetic camera image can be provided to the drone for use  in control and estimation [162].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' COMPETITIONS  To gauge the progress of the field as a whole, several  drone racing competitions have taken place since 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' We  include a graphical overview of these events in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' The  Autonomous Drone Racing (ADR) competition was an annual  competition which took place during the IROS conference  between 2016 and 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' In 2016, 11 teams competed in  autonomous drone racing and were tasked to navigate a series  of gates in sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' The positions of the gates were not  known to the participating teams ahead of time, therefore  teams flew very cautiously identifying the next waypoints  online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Each team was given 30 minutes prior to the official  competition to fly the course as many times as they wished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  The winning team, from KAIST, made it through 10 of the 26  gates in 1 minute and 26 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' For comparison, a human  was able to complete the entire 26-gate course in 1 minute 31  seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' A survey summarizing the approaches used for these  early competitions can be found in [163].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' The following year,  a similar competition took place during IROS in Vancouver,  Canada, with better results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' This time, 14 teams participated  and were given a CAD drawing of the course prior to the  event with locations and dimensions of all gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Only 5 teams  participated in the final in-person event, with the winning team  making it through 9 out of 13 gates in over 3 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' A  summary of the winning approaches can be found in [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  Two more ADR competitions took place at IROS 2018 and  2019, with drones navigating courses faster and more reliably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  In 2019, Lockheed Martin sponsored the AlphaPilot AI  Drone Racing Innovation Challenge where a 1 million dollar  grand prize was awarded to the winning team [164].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' The  competition took place first in a virtual qualifying round  which used the FlightGoggles simulation environment [160].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  Nine teams out of more than 400 worldwide qualified for the  final challenge which included navigating a new track in a  time-trial setting against an expert human pilot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Ultimately,  professional pilot Gabriel Kocher, from the Drone Racing  League, manually piloted his drone through the course in only  6 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' It took 11 seconds to the winner, MAVLab from  TU Delft, and 15 seconds to the second-place winner, UZH-  RPG from the University of Zurich, to complete the course  autonomously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' The two different approaches are documented  in [12], [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Further comments are provided by the winner in  [165].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Perez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' provides an interesting overview of the types  of hardware used for some of the drone racing competitions  mentioned so far [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  In 2019, the Game of Drones competition took place at the  NeurIPS conference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' This competition was purely simulation  based and used the AirSim simulation environment built by  Microsoft [11], [15], [137].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Participants in the Game of Drones  competition raced against simulated opponents in a head-to-  head fashion, similar to how humans compete in FPV drone  racing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Teams raced against a single simulated opponent,  navigating through a complex series of gates in three different  tiers: Planning Only, Perception Only, and Perception with  Planning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  In 2022, at the Swiss Drone Days event in Zurich, Switzer-  land, three of the world’s best human pilots competed against  researchers from the Robotics and Perception Group of the  University of Zurich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Flight speeds exceeding 100 kph were  demonstrated by the autonomous drones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' When relying on  motion capture, the autonomous drones were able to achieve  significantly faster laptimes than the expert human pilots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  They additionally demonstrated it was possible to win races  without motion capture, using only onboard computing and  sensors to navigate the race track.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' IEEE Spectrum author Evan  Ackermann discusses the multi-day event in [105].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' DATASETS AND OPEN SOURCE CODE  In this section, we provide an overview of the existing open  source code bases as well as useful datasets for autonomous  drone racing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' We first discuss datasets, and then group the  existing open source code bases by their use-cases in table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  In 2018, researchers from MIT released a large scale dataset  for perception during aggressive UAV flight [168].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' This dataset  contains over 10 hours of flight data which includes simulated  stereo and downward-facing camera images at 120 Hz, real-  world IMU data at 100 Hz, motor speed data at 190 Hz, and  motion capture data at 360 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' The sensor suite was chosen  such that algorithms like Visual-Inertial Odometry (VIO) or  Simultaneous Localization and Mapping (SLAM) could be  evaluated on the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  In 2019, the UZH-FPV Drone Racing Dataset was released,  which contains many agile maneuvers flown by a professional  racing pilot [68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' The dataset includes indoors and outdoors  real-world camera images, inertial measurements, event cam-  era data, and ground truth poses provided by an advanced  14  TABLE I: Open Source Software and Datasets  Name and Reference  Category  Year  Link  PAMPC [115]  Controller  2018  https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='com/uzh-rpg/rpg mpc  Deep Drone Acrobatics [100]  Controller  2019  https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='com/uzh-rpg/deep drone acrobatics  Data Driven MPC [34]  Controller  2020  https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='com/uzh-rpg/data driven mpc  High MPC [166]  Controller  2022  https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='com/uzh-rpg/high mpc  AutoTune [167]  Controller Tuner  2022  https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='com/uzh-rpg/mh autotune  Blackbird [168]  Dataset  2018  https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='com/mit-aera/Blackbird-Dataset  UZH-FPV [68]  Dataset  2019  https://fpv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='ifi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='uzh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='ch/  NeuroBEM [29]  Dataset  2020  https://rpg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='ifi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='uzh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='ch/NeuroBEM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='html  Eye Gaze Drone Racing [6]  Dataset  2021  https://osf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='io/gvdse/  Time-optimal Planning for Quadrotor Waypoint Flight [84]  Planner  2021  https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='com/uzh-rpg/rpg time optimal  Minimum-Time Quadrotor Waypoint Flight in Cluttered Environments [31]  Planner  2022  https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='com/uzh-rpg/sb min time quadrotor planning  RotorS [17]  Simulator  2016  https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='com/ethz-asl/rotors simulator  AirSim [137]  Simulator  2018  https://microsoft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='io/AirSim/  FlightGoggles [160]  Simulator  2019  https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='com/mit-aera/FlightGoggles  Flightmare [16]  Simulator  2020  https://uzh-rpg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='io/flightmare/  Learning to fly—a gym environment with pybullet physics for reinforcement learning of multi-agent quadcopter control [161]  Simulator  2021  https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='com/utiasDSL/gym-pybullet-drones  Sim 2 Real Domain Randomization [126]  Sim2Real Transfer  2019  https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='com/uzh-rpg/sim2real drone racing  RPG Quadrotor Control [20]  Software Stack  2017  https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='com/uzh-rpg/rpg quadrotor control  Agilicious [162]  Software Stack  2022  https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='com/uzh-rpg/agilicious  Kalibr [48]  Camera Calibration  2022  https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='com/ethz-asl/kalibr  motion capture system (a total station) providing millimeter-  level accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Similar to the authors in [168], the authors of  this dataset hope to push the state of the art in state estimation  during aggressive motion and have created competitions to  allow researchers to compete against one another on this agile  flight benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='5  Research on how expert human pilots focus on their targets  during flying and provide a dataset that contains flight trajec-  tories, videos, and data from the pilots is examined in [6]  NeuroBEM [29] is a hybrid aerodynamic quadrotor model  which  combines  blade-element-momentum-theory  models  with learned aerodynamic representations from highly ag-  gressive maneuvers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' While the model is fit to the specific  quadrotor platform defined in [162], the approach can be used  for any quadrotor platform and provides over 50% reduction in  model prediction errors compared to traditional, exclusively-  first-principles approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  A significant amount of autonomous drone racing research  has been open sourced to the community, making implemen-  tation less daunting for newcomers to the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' A collection of  all known drone racing repositories has been provided to the  reader in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' These code bases range across controllers,  planners, sensor calibration, and even entire software stacks  dedicated to drone racing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' We encourage both newcomers and  experienced researchers to check out the extensive amount of  open source code bases available and contribute back to the  community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' OPEN RESEARCH QUESTIONS AND CHALLENGES  In this section, we examine some of the biggest chal-  lenges that the field of autonomous drone racing is facing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  Autonomous drone racing is a field that is growing rapidly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  To quantify the rate of growth, we examined the number  of papers that mentioned the key-phrase ”autonomous drone  racing” since 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' The data, indicated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' 10, shows  exponential-like growth of the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Thus, it is appropriate  to discuss where future opportunities exist for incoming and  experienced researchers alike.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  5https://fpv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='ifi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='uzh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='ch/uzh/uzh-fpv-leader-board/  2015 2016 2017 2018 2019 2020 2021 2022  0  50  100  150  200  Year  Number of Papers  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' 10: The number of papers related to ”autonomous drone racing” cataloged  on Google Scholar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Challenge 1: Enabling VIO for High-Speed Applications  In its current form, online, robust, and accurate state  estimation is highly beneficial when pushing autonomous  drones to their limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Currently, classical state estimation  approaches based on visual-inertial odometry cannot cope  with the perceptual challenges present in drone racing tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  Motion blur, low texture, and high dynamic range are some  reasons why classical VIO algorithms accumulate large errors  in localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' The miscalibration of intrinsic and extrinsic  camera parameters can lead to improper estimates of the  camera pose on a drone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' This is due to local movements of the  camera frame relative to the drone body, as well as changes  in temperature and pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' VIO drift can render the state  estimates unusable unless corrected through localizations to  a prior map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' New sensor modalities, such as event cameras,  could potentially alleviate this issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Although event-aided  VIO algorithms for drones have been proposed to improve  robustness to motion blur, they have not been demonstrated at  high speeds as seen in drone racing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Future research in agile  15  flight may focus on finding new event representations that are  computationally efficient and compatible with classical VIO  formulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' One example is to exploit direct methods [169].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  Other promising sensor modalities are motor speed controllers  and force sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' These sensor measurements could be used  to include more advanced drone models in VIO, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' modeling  aerodynamics effects, in order to limit the drift that accu-  mulates where camera measurements are degraded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' One of  the main consequences of motion blur, low texture, and high  dynamic range is unreliable feature extraction and matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  This consequently degrades the performance of the visual  frontend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Deep learning methods have the potential to solve  this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' What hinders the application of these methods to  drone racing at the moment is their computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Future  research should work on lightweight neural networks that can  provide inference at a high rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Neural networks could also  be used to remove non-zero mean noise and constant errors  from the inertial measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' A potentially fruitful area of  research is in combining neural networks for input processing  with a geometry-based VIO backend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' This could lead to the  next step in the research on VIO for drone racing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Current  works [75], [170] have shown that this direction outperforms  end-to-end visual-based odometry methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Challenge 2: Flying from Purely Vision  State-of-the-art autonomous navigation methods rely on  visual and inertial information, usually combined with classic  perception algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Conversely, expert human pilots rely  on nothing more than a first-person-view video stream, which  they use to identify goals and estimate the ego-motion of  the drone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Building systems that, similarly to human pilots,  only rely on visual information is very interesting from a  scientific perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Indeed, since simulating RGB is yet  very challenging, solving this question might require lifelong  learning algorithms operating in the real world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' In addition,  eliminating inertial information might have some engineering  advantages too, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=', data throughput, power consumption, and  lower cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Seminal works in this direction try to understand  how humans solve this task [6], [171].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' They found that expert  pilots can control drones despite a 200ms latency, which is  compensated by the human brain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Taking inspiration from  biology, a recent work [172] shows that it is possible to fly with  camera images and an onboard gyroscope (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=', removing the  accelerometer), as long as the system never hovers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' However,  the above questions still remain mostly open and a good  avenue for research at the intersection of computer vision,  neuroscience, and biology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Challenge 3: Multiplayer Racing  Much of the work done up until this point on autonomous  drone racing has focused on time-optimal flight without con-  sidering how a capable opponent might impact the compe-  tition dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' In FPV races, pilots can compete against  up to 5 opponents simultaneously, bringing about the need  to anticipate how their opponents might behave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Humans  are astonishingly capable of recognizing opportunities for  overtaking and executing complex maneuvers in the face of  large aerodynamic disturbances caused by flying close to  another drone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Achieving such capabilities requires an agent  to estimate their opponent’s state using only onboard visual  sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' However, these observations in drone racing are  sparse because the camera faces forward along the heading  axis, meaning that the only time an opponent is observable  is when the ego-agent is behind them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Sophisticated motion  and planning models which can propagate predictions of the  opponents’ states and racing lines through time are necessary  to anticipate collisions or overtaking opportunities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' An initial  study [173] examined how game-theoretic planners can lead  to highly competitive behavior in two-player drone racing,  however, this work was confined to racing on a 2D plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  The work was further extended to 3D spaces in [174], but  there is a significant opportunity for researchers to explore  the competitive nature of drone racing and develop interesting  racing strategies that lead to time-optimal agents that are able  to deal with complex opponent behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Challenge 4: Transfer to Real-World Applications  Drone racing, while an extraordinarily challenging research  environment, is ultimately not the end goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Opportunities  exist for technology transfer between the drone racing research  community to real-world applications such as search and res-  cue, inspection, agriculture, videography, delivery, passenger  air vehicles, law enforcement, and defense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Until this point,  the rate of technology transfer has been slow due to challenges  in flight certification and a lack of generalization between  environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' However, commercial applications that leverage  the full agility of the platform have much to gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Drones  that fly fast, fly farther, therefore increase the productivity of  drones in every commercial sector [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' As it stands today,  drone racing algorithms can be difficult to directly transfer  to a new environment due to overfitting and minimal safety  guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Calculating time-optimal paths that are safety  critical currently takes too long for deployment in emergency  scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Existing works that leverage expert perception mod-  els to sense and plan around obstacles can be sensitive to  changes in the environment and lead to crashes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Beyond this,  we often do not have a known map ahead of time for real  world applications, requiring researchers to think about how to  simultaneously estimate the state of the drone while mapping  the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Building algorithms that can continually  improve from their own experience is key in enabling this  transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' While recent advances in reinforcement learning  research point to the feasibility of this path [158], [159], [175],  it is unclear when and how such recent approaches would  be applicable to drones or similarly agile platforms in the  real world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Collecting data for continual RL onboard a drone  is notoriously difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' This is because the drone does not  have the luxury of remaining in contact with the ground like  legged robots and cars, and thus has to immediately know how  to hover otherwise a crash will occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' One interesting area  that may be useful for continual RL in drones is the notion  of “safe-RL”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' The goal of safe RL is to enable exploration  without ever incurring catastrophic failure of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Initial  work on this topic can be found in [176].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' A survey paper  16  covering safe RL methods can be found in [177].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Furthermore,  a thorough review paper on continual, or life-long RL can be  found in [178].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  IX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' SUMMARY AND CONCLUSIONS  In this survey, we provided a comprehensive overview of  the task of autonomous drone racing across model-based and  learning-based approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' A history of all recent autonomous  drone racing events was given, along with a list of all open-  source code bases, datasets, and simulators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' These resources  can be used to greatly reduce the learning curve and time  needed when it comes to getting started with autonomous  drone racing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' With these resources and a list of open chal-  lenges for the field, researchers should have the tools to push  the limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' CONTRIBUTIONS  Drew Hanover initiated the idea of this paper, created the  paper structure, and contributed to all sections of this paper  while coordinating efforts amongst the co-authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Antonio  Loquercio contributed to the paper structure and the learning-  based sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Leonard Bauersfeld authored the Drone Mod-  eling section and created the graphics seen throughout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Angel  Romero contributed to the Classical Planning and Control  sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Giovanni Cioffi contributed to the Classical Percep-  tion and Challenges sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Yunlong Song contributed to  the Simulators and Learning-Based Planning/Control sections  Robert Penicka contributed to both Classical and Learning-  Based Planning sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content=' Elia Kaufmann contributed to the  paper structure and throughout the Learning-Based sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='  Davide Scaramuzza contributed to the general paper structure  and revised the paper thoroughly and critically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
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+page_content=' Precup, “Towards continual  reinforcement learning: A review and perspectives,” arXiv preprint  arXiv:2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
+page_content='13490, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfyf6u/content/2301.01755v1.pdf'}
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+arXiv:2301.04484v1  [math.CV]  11 Jan 2023
+H¨older Regularity of the ¯∂−equation on the Polydisc
+Yu Jun Loo and Alexander Tumanov
+University of Illinois at Urbana-Champaign
+Abstract
+In this note, we show that the canonical solution operator to the ¯∂−equation in the polydisc preserves
+H¨older regularity. It is a well-known fact that such solution operators do not improve H¨older regularity,
+and as such, our solution operator is optimal in this regard.
+1
+Introduction
+It is a classical problem in complex analysis to describe solutions to the ¯∂−equation with estimates in
+prescribed normed function spaces. The most general result on problems of this type was given by Sergeev
+and Henkin in [SH], giving uniform estimates for the ¯∂−equation in any pseudoconvex polyhedron. Recently,
+the H¨older spaces Ck+α on product domains in Cn have been given some attention, and some results have
+been published on this matter. In [PZ1], [PZ2], a solution operator which loses arbitrarily small amounts of
+H¨older regularity was found, while in [Zhang] a solution operator which preserves H¨older regularity was found
+in the case n = 2. In the papers mentioned, the solution operators were based on Nijenhuis and Woolf’s
+formula in [NW]. This note seeks to improve on those results and show that optimal H¨older regularity can
+be achieved in the polydisc Dn ⊂ Cn. Indeed, the main theorem of the paper is as follows.
+Theorem 1. For any integer k ≥ 0, and 0 < α < 1, Let Zk+α
+(0,1)(Dn) ⊆ Ck+α
+(0,1)(Dn) denote the subspace of
+¯∂−closed, H¨older k + α, (0, 1)−forms on the polydisc. Then for all g ∈ Zk+α
+(0,1)(Dn), the equation ¯∂u = g
+admits a bounded linear solution operator
+H : Zk+α
+(0,1)(Dn) → Ck+α(Dn)
+For simplicity, we have restricted our attention to the case of the polydisc. However, Theorem 1 readily
+extends to the more general case of products of planar domains with smooth boundary.
+2
+Preliminary Results
+The proof of the main theorem rests on an analysis of Henkin’s weighted formula for solutions to the
+¯∂−equation on the polydisc, which was announced in his survey paper [Henkin] of 1985. The simplest case
+of this formula, obtained by setting all weights equal to 0, has the following form.
+1
+
+Theorem 2. Let Z(0,1)(¯Dn) ⊆ C(0,1)(¯Dn) denote the space of (uniformly) continuous, ¯∂−closed (0, 1)−forms
+on the polydisc. Fix g ∈ Z(0,1)(¯Dn). Then,
+u(z) = H[g](z) = −
+n−1
+�
+r=0
+�
+|J|=r
+(−1)c(n,r)
+�
+γJ(z)
+g(ζ) ∧ HJ(ζ, z)
+is a distributional solution to the equation ¯∂u = g. Here, c(n, r) is an integer depending only on the constants
+n, r, while the sum ranges over all ordered r-tuples J = (j1, ..., jr) such that {j1, ..., jr} is a size r subset
+of {1, ..., n}. The complement of J in {1, ..., n} is denoted by {k1, ..., kn−r}, while the region of integration
+γJ(z) is given by
+γJ(z) = {ζ ∈ Dn : ζj1 = zj1, ..., ζjr = zjr, |zj1| ≥ ... ≥ |zjr| ≥ |ζk1| = ... = |ζkn−r|}
+The kernel of integration is the n − r form
+HJ(ζ, z) =
+1
+(2πi)n−r ·
+n−r
+�
+s=1
+dζks
+ζks − zks
+A version of this formula with weights equal to 1 was used in [HP1] to solve an interpolation problem in the
+polydisc, while a proof of the weighted formula for the more general class of analytic polyhedra appears in
+[HP2]. According to [HP2], Henkin’s formula gives uniform estimates for the ¯∂−equation in the sup-norm.
+In addition, Henkin’s formula also yields a bounded solution operator that preserves H¨older regularity for
+α ∈ (0, 1). Stated precisely, we have the following theorem.
+Theorem 3. Let 0 < α < 1 and g ∈ Zα
+(0,1)(Dn) be a ¯∂−closed H¨older−α, (0, 1)−form in the distributional
+sense. Then Henkin’s solution operator to the ¯∂−equation, restricts to a bounded linear operator
+H : Zα
+(0,1)(Dn) → Cα(Dn)
+In fact, the solutions produced by Henkin’s formula agrees with the solutions produced by Nijenhuis and
+Woolf in [NW], and studied by Pan and Zhang in [PZ2], and [Zhang].
+Definition 1. (Nijenhui’s and Woolf’s Formula)
+Let k ≥ 0, k ∈ Z. For any f ∈ Ck+α(D), z ∈ D let
+Tj[f](z) =
+1
+2πi
+�
+D
+f(z1, ..., ζj, zj+1, ..., zn)
+ζj − zj
+dζj ∧ d¯ζj
+Sj[f](z) =
+1
+2πi
+�
+∂D
+f(z1, ..., ζj, zj+1, ..., zn)
+ζj − zj
+dζj
+Let
+˜S1 = id, ˜Sk = Sk−1...S1, for all 1 < k ≤ n
+For any g ∈ Z(0,1)(Dn), g = �
+i gid¯zi we define
+T [g] =
+n
+�
+j=1
+Tj ˜Sj[gj]
+2
+
+Remark 1. Direct estimates of the operator norm of T using Nijenhuis and Woolf’s formula suggest that
+it loses (arbitrarily small) amounts of H¨older regularity. This is due to the fact that the Cauchy integral
+operators Sj lose H¨older regularity in parameters, as observed in [Tumanov] and [PZ2]. It turns out, however,
+that T does not lose any H¨older regularity.
+In order to show that the solutions to the ¯∂−equation produced by the operators H and T agree, we require
+the following lemma.
+Lemma 1. Given u ∈ C(¯Dn) and g ∈ C(0,1)(¯Dn) such that ¯∂u = g there exists a linear operator Φ depending
+only on n such that
+u(z) = K[u] + Φ[g]
+Here,
+K[u](z) =
+1
+(2πi)n
+�
+(∂D)n
+u(ζ)
+�n
+j=1(ζj − zj)dζ1 ∧ ... ∧ dζn
+is the Cauchy torus integral
+Proof. We first introduce some notation to simplify the exposition. Let dnz = dz1 ∧ ... ∧ dzn. For any subset
+{j1, ..., js} of {1, ..., n}, let
+dn−s
+j1,...,js¯z =
+n�
+j=1,
+j̸∈{j1,...,js}
+d¯zj
+Let
+˜Dj =
+
+
+
+D if j ̸∈ {j1, ..., js}
+∂D otherwise
+(1)
+and define
+Dj1,...,js =
+n
+�
+j=1
+˜Dj
+On the set {1, ..., n}, we work modulo n so that n + r = r. In all that follows we let C be an arbitrary
+constant depending only on n, and Ψ be an arbitrary linear operator depending only on n. Both the symbols
+C and Ψ will act as local variables, and the same symbols will be used for different constants and operators
+to simplify the exposition. By the Bochner-Martinelli formula [Henkin], given uniformly continuous u, g on
+Dn, ¯∂u = g in the distributional sense, we have
+u(z) = C ·
+�
+∂Dn
+n
+�
+j=1
+(−1)j−1u(ζ)
+¯ζj − ¯zj
+|ζ − z|2n dn−1
+j
+¯ζ ∧ dnζ + Ψ[g] =: C ·
+n
+�
+j=1
+uj(z) + Ψ[g]
+Observe that
+¯ζj − ¯zj
+|ζ − z|2n =
+−1
+n − 1
+∂
+∂ζj+1
+¯ζj − ¯zj
+|ζ − z|2n−2(ζj+1 − zj+1)
+Therefore
+uj(z) = C · (−1)
+�
+Dj
+u(ζ)
+∂
+∂ζj+1
+¯ζj − ¯zj
+|ζ − z|2n−2(ζj+1 − zj+1)d¯ζj+1 ∧ dn−2
+j,j+1¯ζ ∧ dnζ + Ψ[g]
+For fixed z, our integrand is a uniformly continuous form in ζ with uniformly continuous differential. Thus
+3
+
+by Stokes’ theorem, we obtain
+uj(z) = C · (−1)
+�
+Dj,j+1
+u(ζ)
+¯ζj − ¯zj
+|ζ − z|2n−2(ζj+1 − zj+1)dn−2
+j,j+1 ¯ζ ∧ dnζ + Ψ[g]
+Here, the domain of integration is Dj,j+1 since the pullback of the form in the integral vanishes on the other
+components of ∂Dj. By repeating this procedure n − 2 more times, each time observing that
+¯ζj − ¯zj
+|ζ − z|2n−2s �s
+r=1(ζj+r − zj+r) = −
+1
+(n − s − 1)
+∂
+∂ζs+1
+¯ζj − ¯zj
+|ζ − z|2n−2s−2 �s+1
+r=1(ζj+r − zj+r)
+We obtain
+uj(z) = C · (−1)(j−1)(−1)(j−1)(n−j)(−1)(n−1)
+�
+(∂D)n
+j
+u(ζ)
+¯ζj − ¯zj
+|ζ − z|2 �
+r̸=j(ζr − zr)dnζ + Ψ[g]
+Here, (∂D)n
+j is the set (∂D)n with orientation induced by successive applications of the boundary operator be-
+ginning with the j’th copy of D in the product. It is clear from inspection that (∂D)n
+j = (−1)(j−1)(n−j+1)(∂D)n
+1.
+Therefore,
+uj(z) = C ·
+�
+T1
+u(ζ)
+¯ζj − ¯zj
+|ζ − z|2 �
+r̸=j(ζr − zr)dnζ + Ψ[g]
+Summing over j and noting that
+n
+�
+j=1
+¯ζj − ¯zj
+|ζ − z|2 �
+r̸=j(ζr − zr) =
+1
+�n
+r=1(ζr − zr)
+We see that
+u(z) = C · K[u](z) + Ψ[g]
+Since this equation holds for any u, by considering for example u = 1, we see that C = 1. By taking Φ = Ψ,
+this concludes the proof.
+Corollary 1. There is a canonical solution operator L : Z(0,1)(¯Dn) → C(¯Dn) to the ¯∂−equation such that
+KL = 0. Furthermore, L = H and for all non-negative integers k and 0 < α < 1 we have L|Zk+α
+(0,1)(¯Dn) = T .
+Proof. By Lemma 1, we need only show that KH = 0 and KT = 0. To see the former, fix g ∈ Z(0,1)(¯Dn),
+and let d2ζj = d¯ζj ∧ dζj. We note that for |z1| = ... = |zn| = 1, H is the sum of terms, each with a region of
+integration given by some
+γJ(z) = {ζ ∈ Dn : ζj1 = zj1, ..., ζjr = zjr, 1 ≥ |ζk1| = ... = |ζkn−r|}
+4
+
+Hence, K[H[g]] is given by K applied to the sum of terms of the form
+�
+|ζk1 |=...=|ζkn−r |≤1
+gks(ζk1, ...ζkn−r, zJ)
+(ζk1 − zk1)...(ζkn−r − zkn−r)dζk1 ∧ ... ∧ d2ζks ∧ ... ∧ dζkn−r
+=
+�
+|ζks |≤1
+� �
+|ζk1 |=...= �
+|ζks|=...|ζkn−r|≤1
+gks(ζk1, ...ζkn−r, zJ)
+(ζk1 − zk1)...
+�
+(ζks − zks)...(ζkn−r − zkn−r)
+dζk1... �
+dζks...dζkn−r
+�
+d2ζks
+(ζks − zks)
+= Tks[G]
+Here,
+G(z1, ..., ζks, ...., zn) =
+�
+|ζk1 |=...= �
+|ζks|=...|ζkn−r |≤1
+gks(ζk1, ...ζkn−r, zJ)
+(ζk1 − zk1)...
+�
+(ζks − zks)...(ζkn−r − zkn−r)
+dζk1... �
+dζks...dζkn−r
+But since SjTj = 0 for all j = 0, ...n and K = Sn...S1, as these operators commute, KTks[G] = 0. Hence
+KH[g] = 0. Since g was arbitrary, KH = 0 Likewise, since T [g] = �n
+j=1 Tj ˜Sj[gj], we have KT = 0
+Remark 2. In light of Corollary 1, we may view both H and T as different formulae for the same canonical
+solution operator. As such, Pan and Zhang’s analysis of the formula T carry over to Henkin’s formula. Hence
+we obtain the following proposition.
+Proposition 1. For all k ≥ 0, k ∈ Z and 0 < α < 1, Henkin’s canonical solution operator H is a bounded
+linear operator
+H : Zk+α
+(0,1)(Dn) → C(¯Dn)
+3
+Proof of Theorem 3
+Fix g ∈ Z(0,1)(¯Dn). In order to prove Theorem 3 we make a short digression. Fix a permutation σ of {1, ..., n}
+and let Dσ denote the open sectors {z ∈ Dn : |zσ(1)| > ... > |zσ(n)|}. For z ∈ Dσ, the only terms in H[g](z)
+which do not vanish, take the form
+�
+|ζk1 |=...=|ζkn−r|≤|zjr |
+gks(ζk1, ...ζkn−r, zJ)
+(ζk1 − zk1)...(ζkn−r − zkn−r)dζk1 ∧ ... ∧ d2ζks ∧ ... ∧ dζkn−r
+Here, g = �n
+i=1 gid¯ζj, J = (σ(j1), ...., σ(jr)), 1 ≤ j1 < ... < jn ≤ n
+We therefore consider the following linear operator
+Definition 2. Let a, b ∈ D, ζ, z ∈ Dq, ζ = (ζ1, ..., ζq), z = (z1, ..., zq), h ∈ C(Dq × D × D). We write d2ζj for
+d¯ζj ∧ dζj and define
+P[h](z, a, b) =
+�
+|ζ1|=...=|ζq|≤|a|
+h(ζ, a, b)
+(ζ1 − z1)...(ζq − zq)d2ζ1 ∧ ... ∧ dζq
+We have the following Lemmas:
+Lemma 2. For each s ∈ {0, ..., q} let Ds = {(z, a, b) ∈ Dq+2 : |z1| < ... < |zs| < |a| < |zs+1| < ... < |zq|},
+and t ∈ {1, ...q}. Then on any open sector Ds, P[h] is H¨older−α uniformly in the variable zt, with coefficient
+independent of z, a, b, and proportional to ∥h∥Cα(Dq+2)
+5
+
+Proof. We first consider the case t = 1. Observe that after ignoring sign changes,
+P[h](z, a, b) =
+�
+|ζ1|=...=|ζq|≤|a|
+h(ζ, a, b)
+(ζ1 − z1)...(ζq − zq)d2ζ1 ∧ dζ2 ∧ ... ∧ dζq
+=
+�
+|ζ1|≤|a|
+� �
+|ζ2|=...=|ζq|=|ζ1|
+h(ζ, a, b)
+(ζ2 − z2)...(ζq − zq)dζ2 ∧ ... ∧ dζq
+�
+d2ζ1
+(ζ1 − z1)
+=:
+�
+|ζ1|≤|a|
+H(ζ1, z2, ..., zq, a, b)
+d2ζ1
+(ζ1 − z1)
+Thus the theorem will hold provided we can show H is uniformly bounded with coefficient independent of
+z, a, b, and proportional to ∥h∥Cα(Dq+2). But this is clear since
+H(ζ1, z2, ..., zq, a, b) =
+�
+|ζ2|=...=|ζq|=|ζ1|
+h(ζ, a, b)
+(ζ2 − z2)...(ζq − zq)dζ2 ∧ ... ∧ dζq
+=
+�
+|ξ2|=...=|ξq|=1
+h(ζ1, |ζ1|ξ2, ...|ζ1|ξq, a, b)
+(ξ2 −
+z2
+|ζ1|)...(ξq −
+zq
+|ζ1|) dξ2 ∧ ... ∧ dξq
+Since the function (ζ1, ξ2, ..., ξq, a, b) �→ h(ζ1, |ζ1|ξ2, ...|ζ1|ξq, a, b) is Cα on the region of integration, its Cauchy
+torus integral is uniformly bounded, so H is uniformly bounded as well. Furthermore, ∥H∥L∞ ≲ ∥h∥Cα(Dp+q).
+Since P[h] is obtained by applying the Cauchy integral on a domain to H, for every 0 < ǫ < 1, P[h](z, a, b)
+is C1−ǫ in Ds uniformly in z1. In particular, it is Cα in Ds uniformly in z1.
+For t ̸= 1 we observe that ζt¯ζt = ζ1¯ζ1, Therefore d¯ζ1 = ζt
+ζ1 d¯ζt. Hence, after ignoring changes in sign,
+P[h](z, a, b) =
+�
+|ζ1|=...=|ζq|≤|a|
+ζtζ−1
+1 h(ζ, a, b)
+(ζ1 − z1)...(ζq − zq)dζ1 ∧ ... ∧ d2ζt ∧ ... ∧ dζq
+The same analysis as before shows that P[h](z, a, b) is Cα in zt uniformly with coefficient independent of
+z, a, b, and proportional to ∥h∥Cα(Dq+2).
+Lemma 3. On any open sector Ds, P[h] is H¨older−α uniformly in the parameters a, b with coefficient
+proportional to ∥h∥Cα(Dq+2).
+6
+
+Proof. We first show that P[h] is Cα uniformly in b. Indeed, let ǫ ∈ C.
+|P[h](z, a, b + ǫ) − P[h](z, a, b)| ≤
+�
+|ζ1|=...=|ζq|≤|a|
+∥h∥Cα(Dq+2)|ǫ|α
+|ζ1 − z1|...|ζq − zq||d2ζ1|...|dζq|
+≤ ∥h∥Cα(Dq+2)|ǫ|α
+� 1
+0
+� 2π
+0
+...
+� 2π
+0
+rq · dθ1...dθqdr
+|reiθ1 − z1|...|reiθq − zq|
+≲ ∥h∥Cα(Dq+2)|ǫ|α
+� 1
+0
+� 2π
+0
+...
+� 2π
+0
+rq · dθ1...dθqdr
+|rθ1 + |r − |z1|||...|rθq + |r − |zq|||
+≲ ∥h∥Cα(Dq+2)|ǫ|α
+� 1
+0
+�
+ln (rθ1 + |r − |z1||)
+�2π
+0 ...
+�
+ln (rθq + |r − |zq||)
+�2π
+0 dr
+≲ ∥h∥Cα(Dq+2)|ǫ|α
+� 1
+0
+| ln (|r − |z1||)|....| ln (|r − |zq||)|dr
+≲ ∥h∥Cα(Dq+2)|ǫ|α
+� 1
+0
+q
+�
+i=1
+| ln (|r − |zi||)|qdr
+≲ ∥h∥Cα(Dq+2)|ǫ|α
+� 1
+0
+| ln (r)|qdr
+≲ ∥h∥Cα(Dq+2)|ǫ|α
+Similarly, we can show P[h] is Cα uniformly in a.
+P[h](z, a + ǫ, b) − P[h](z, a, b) =
+�
+|ζ1|=...=|ζq|≤|a+ǫ|
+h(ζ, a + ǫ, b) − h(ζ, a, b)
+(ζ1 − z1)...(ζq − zq)
+d2ζ1 ∧ ... ∧ dζq
++
+�
+|a|≤|ζ1|=...=|ζq|≤|a+ǫ|
+h(ζ, a, b)
+(ζ1 − z1)...(ζq − zq)d2ζ1 ∧ ... ∧ dζq =: I1 + I2
+We denote these two terms I1 and I2 respectively and control them separately. Repeating the same analysis
+as we did for b, we see that |I1| ≲ ∥h∥Cα(Dq+2)|ǫ|α. Thus we need only control I2.
+|I2| =
+����
+� |a+ǫ|
+|a|
+� 2π
+0
+...
+� 2π
+0
+h(reiθ1, ..., eriθq, a, b)
+(reiθ1 − z1)...(reiθq − zq)rdθ1(ireiθ2dθ2)...(ireiθqdθq)dr
+����
+≲ |ǫ|
+����
+� 2π
+0
+...
+� 2π
+0
+h(|a|eiθ1...|a|eiθq, a, b)
+(|a|eiθ1 − z1)...(|a|eiθq − zq)|a|qdθ1(ieiθ2dθ2)...(ieiθqdθq)
+����
+≲ |ǫ|
+����
+�
+|ξ1|=1
+...
+�
+|ξ|=1
+h(|a|ξ, a, b)iξ1
+(ξ1 − z1
+|a|)...(ξq − |zq|
+|a| )
+dξ1 ∧ ... ∧ dξq
+����
+≲ |ǫ| · ∥h∥Cα(Dq+2)
+Here, the last inequality holds as the function (ζ, a, b) �→ h(|a|ξ, a, b)iξ1 is Cα, so that its Cauchy torus
+integral is bounded, depending only on ∥h∥Cα(Dq+2).
+Therefore P preserves H¨older−α regularity in the
+parameters a, b.
+These lemmas assemble into the following proposition.
+Proposition 2. Let 0 < α < 1, then for any Ds, P : Cα(Dq × D × D) → Cα(Ds) is a bounded linear
+operator.
+Proof. By the preceding two lemmas, P[h] is Cα in Ds uniformly in each variable zt and parameters a, b,
+7
+
+with coefficient proportional to ∥h∥Cα(Dq+2). Hence ∥P[h]∥Cα(Ds) ≲ ∥h∥Cα(Dq+2).
+Theorem 3 immediately follows from proposition 2.
+Proof. (Theorem 3) Observe that the closures of the finitely many open sectors Dσ cover the polydisc Dn.
+By Proposition 1, H[g] is uniformly continuous on the polydisc Dn. Therefore to show that H : Zα
+(0,1)(Dn) →
+Cα(Dn) is a bounded linear operator, we need only show that given Cα datum g, the solution H[g] is Cα on
+each of the open sectors Dσ, with coefficient proportional to ∥g∥Cα(Dn). But this follows immediately from
+Proposition 2.
+4
+Proof of Theorem 1
+Having shown Theorem 3, by Corollary 1, we obtain that for 0 < α < 1, H : Zα
+(0,1)(Dn) → Cα(Dn) is a
+bounded linear operator. To complete the proof of Theorem 1, we need only induct on k using Nijenhuis
+and Wolf’s formula T = H : Zk+α
+(0,1)(Dn) → Ck+α(Dn)
+Proof. (Theorem 1) As noted in [Vekua], we may differentiate the singular integral Tj with respect to zj and
+¯zj. From which we obtain
+∂
+∂zj
+Tj[g](z) = Πj[g](z) :=
+1
+2πi
+�
+D
+g(z1, ..., ζj, zj+1, ..., zn)
+(ζj − zj)2
+dζj ∧ d¯ζj
+and
+∂
+∂ ¯zj
+Tj[g](z) = g
+Suppose now T = H : Zk+α
+(0,1)(Dn) → Ck+α(Dn) is a bounded linear operator. Fix g ∈ Zk+1+α
+(0,1)
+(Dn).
+Then, as T is a solution operator to the ¯∂−equation, we have that ¯∂T [g] = g. In addition, for all j = 1, ..., n,
+∂
+∂zj Tj[g](z) = Πj[g](z) and
+∂
+∂zj Sj[g](z) = Sj[ ∂g
+∂zj ](z). Therefore by commuting the differentiation symbol
+with the operator when possible, we obtain the formula
+∂
+∂zj
+T [g](z) =
+�
+i̸=j
+Ti ˜Si[ ∂
+∂ζj
+gi] + Πj ˜Sj[gj]
+Since k + 1 ≥ 1, we may apply Stokes’ Theorem, to the term Πj ˜Sj[gj] as in [Vekua] to obtain
+Πj ˜Sj[gj] = Tj ˜Sj[ ∂
+∂ζj
+gj] −
+1
+2πi
+�
+∂D
+˜Sj[gj](z1, ..., zj−1, ζj, ...zn)
+ζj − zj
+d¯ζj = Tj ˜Sj[ ∂
+∂ζj
+gj] + Sj[ ˜Sj[gj] · ¯ζ2
+j ]
+From which it immediately follows that
+∂
+∂zj
+T [g](z) =
+�
+i
+Ti ˜Si[ ∂
+∂ζj
+gi] + Sj[ ˜Sj[gj] · ¯ζ2
+j ] = T [ ∂
+∂ζj
+g] + Sj[ ˜Sj[gj] · ¯ζ2
+j ]
+By the induction hypothesis, ∥T [ ∂
+∂ζj g]∥Ck+α(Dn) ≲ ∥ ∂
+∂ζj g∥Ck+α(D) ≲ ∥g∥Ck+1+α(D). Furthermore, for all j,
+gj ∈ Ck+1+α(Dn) we have ∥Sj[ ˜Sj[gj] · ¯ζ2
+j ]∥Ck+α ≲ ∥gj∥Ck+1+α as the Cauchy integral loses arbitrarily small
+amounts of H¨older regularity in parameters. This completes the induction.
+8
+
+References
+[Henkin]
+Henkin, G. M. The method of integral representations in complex analysis. (Russian) Current
+problems in mathematics. Fundamental directions, Vol. 7, 23–124, 258, Itogi Nauki i Tekhniki,
+Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, (1985).
+[HP1]
+Henkin, G. M. and Polyakov, P. L. Prolongement des fonctions holomorphes born´e d’une
+sous-vari´et´e du polydisue, C. R. Acad. Sci. Paris, Vol. 298 (1984), 221–224.
+[HP2]
+Henkin, G. M. and Polyakov, P. L. Integral formulas for solution of the ∂-equation, and
+interpolation problems in analytic polyhedra. (Russian) Trudy Moskov. Mat. Obshch. 53
+(1990), 130–170, 260–261; translation in Trans. Moscow Math. Soc. (1991), 135–175.
+[SH]
+Sergeev, A. G. and Henkin, G. M. Uniform estimates of the solutions of the ¯∂-equation in
+pseudoconvex polyhedra. (Russian) Mat. Sb. (N.S.) 112(154) (1980), no. 4(8), 522–567.
+[NW]
+Nijenhuis, A. and Woolf, W.B. Some integration problems in almost-complex and complex
+manifolds. Ann. of Math. (2) 77 (1963), 424–489.
+[PZ1]
+Pan, Y. and Zhang, Y. H¨older estimates for the ¯∂ problem for (p, q) forms on product domains.
+Internat. J. Math. 32 (2021), no. 3, Paper No. 2150014, 20 pp.
+[PZ2]
+Pan, Y. and Zhang, Y. Cauchy singular integral operator with parameters in log-H¨older spaces.
+(English summary) J. Anal. Math. 145 (2021), no. 1, 357–379.
+[Tumanov]
+Tumanov, A. On the propagation of extendibility of CR functions. Complex analysis and
+geometry (Trento, 1993), 479–498, Lecture Notes in Pure and Appl. Math., 173, Dekker, New
+York, (1996).
+[Vekua]
+Vekua, I. N. Generalized analytic functions. Pergamon Press, London-Paris-Frankfurt;
+Addison-Wesley Publishing Company, Inc., Reading, Mass. (1962) xxix+668 pp.
+[Zhang]
+Zhang, Y. Optimal H¨older regularity for the ¯∂ problem on product domains in C2. To appear
+at Proceedings of the AMS.
+9
+
diff --git a/K9E3T4oBgHgl3EQfYAqA/content/tmp_files/load_file.txt b/K9E3T4oBgHgl3EQfYAqA/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..ab46102269c9b5a626ce0ceaa03ab20cc2fa9bae
--- /dev/null
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@@ -0,0 +1,444 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf,len=443
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='04484v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='CV]  11 Jan 2023  H¨older Regularity of the ¯∂−equation on the Polydisc  Yu Jun Loo and Alexander Tumanov  University of Illinois at Urbana-Champaign  Abstract  In this note, we show that the canonical solution operator to the ¯∂−equation in the polydisc preserves  H¨older regularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' It is a well-known fact that such solution operators do not improve H¨older regularity,  and as such, our solution operator is optimal in this regard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='  1  Introduction  It is a classical problem in complex analysis to describe solutions to the ¯∂−equation with estimates in  prescribed normed function spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' The most general result on problems of this type was given by Sergeev  and Henkin in [SH], giving uniform estimates for the ¯∂−equation in any pseudoconvex polyhedron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' Recently,  the H¨older spaces Ck+α on product domains in Cn have been given some attention, and some results have  been published on this matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' In [PZ1], [PZ2], a solution operator which loses arbitrarily small amounts of  H¨older regularity was found, while in [Zhang] a solution operator which preserves H¨older regularity was found  in the case n = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' In the papers mentioned, the solution operators were based on Nijenhuis and Woolf’s  formula in [NW].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' This note seeks to improve on those results and show that optimal H¨older regularity can  be achieved in the polydisc Dn ⊂ Cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' Indeed, the main theorem of the paper is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' For any integer k ≥ 0, and 0 < α < 1, Let Zk+α  (0,1)(Dn) ⊆ Ck+α  (0,1)(Dn) denote the subspace of  ¯∂−closed, H¨older k + α, (0, 1)−forms on the polydisc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' Then for all g ∈ Zk+α  (0,1)(Dn), the equation ¯∂u = g  admits a bounded linear solution operator  H : Zk+α  (0,1)(Dn) → Ck+α(Dn)  For simplicity, we have restricted our attention to the case of the polydisc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' However, Theorem 1 readily  extends to the more general case of products of planar domains with smooth boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='  2  Preliminary Results  The proof of the main theorem rests on an analysis of Henkin’s weighted formula for solutions to the  ¯∂−equation on the polydisc, which was announced in his survey paper [Henkin] of 1985.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' The simplest case  of this formula, obtained by setting all weights equal to 0, has the following form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='  1  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' Let Z(0,1)(¯Dn) ⊆ C(0,1)(¯Dn) denote the space of (uniformly) continuous, ¯∂−closed (0, 1)−forms  on the polydisc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' Fix g ∈ Z(0,1)(¯Dn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' Then,  u(z) = H[g](z) = −  n−1  �  r=0  �  |J|=r  (−1)c(n,r)  �  γJ(z)  g(ζ) ∧ HJ(ζ, z)  is a distributional solution to the equation ¯∂u = g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' Here, c(n, r) is an integer depending only on the constants  n, r, while the sum ranges over all ordered r-tuples J = (j1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=', jr) such that {j1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=', jr} is a size r subset  of {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=', n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' The complement of J in {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=', n} is denoted by {k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=', kn−r}, while the region of integration  γJ(z) is given by  γJ(z) = {ζ ∈ Dn : ζj1 = zj1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=', ζjr = zjr, |zj1| ≥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' ≥ |zjr| ≥ |ζk1| = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' = |ζkn−r|}  The kernel of integration is the n − r form  HJ(ζ, z) =  1  (2πi)n−r ·  n−r  �  s=1  dζks  ζks − zks  A version of this formula with weights equal to 1 was used in [HP1] to solve an interpolation problem in the  polydisc, while a proof of the weighted formula for the more general class of analytic polyhedra appears in  [HP2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' According to [HP2], Henkin’s formula gives uniform estimates for the ¯∂−equation in the sup-norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='  In addition, Henkin’s formula also yields a bounded solution operator that preserves H¨older regularity for  α ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' Stated precisely, we have the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' Let 0 < α < 1 and g ∈ Zα  (0,1)(Dn) be a ¯∂−closed H¨older−α, (0, 1)−form in the distributional  sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' Then Henkin’s solution operator to the ¯∂−equation, restricts to a bounded linear operator  H : Zα  (0,1)(Dn) → Cα(Dn)  In fact, the solutions produced by Henkin’s formula agrees with the solutions produced by Nijenhuis and  Woolf in [NW], and studied by Pan and Zhang in [PZ2], and [Zhang].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' (Nijenhui’s and Woolf’s Formula)  Let k ≥ 0, k ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' For any f ∈ Ck+α(D), z ∈ D let  Tj[f](z) =  1  2πi  �  D  f(z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=', ζj, zj+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=', zn)  ζj − zj  dζj ∧ d¯ζj  Sj[f](z) =  1  2πi  �  ∂D  f(z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=', ζj, zj+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=', zn)  ζj − zj  dζj  Let  ˜S1 = id, ˜Sk = Sk−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='S1, for all 1 < k ≤ n  For any g ∈ Z(0,1)(Dn), g = �  i gid¯zi we define  T [g] =  n  �  j=1  Tj ˜Sj[gj]  2  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' Direct estimates of the operator norm of T using Nijenhuis and Woolf’s formula suggest that  it loses (arbitrarily small) amounts of H¨older regularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' This is due to the fact that the Cauchy integral  operators Sj lose H¨older regularity in parameters, as observed in [Tumanov] and [PZ2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' It turns out, however,  that T does not lose any H¨older regularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='  In order to show that the solutions to the ¯∂−equation produced by the operators H and T agree, we require  the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' Given u ∈ C(¯Dn) and g ∈ C(0,1)(¯Dn) such that ¯∂u = g there exists a linear operator Φ depending  only on n such that  u(z) = K[u] + Φ[g]  Here,  K[u](z) =  1  (2πi)n  �  (∂D)n  u(ζ)  �n  j=1(ζj − zj)dζ1 ∧ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' ∧ dζn  is the Cauchy torus integral  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' We first introduce some notation to simplify the exposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' Let dnz = dz1 ∧ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' ∧ dzn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' For any subset  {j1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=', js} of {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=', n}, let  dn−s  j1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=',js¯z =  n�  j=1,  j̸∈{j1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=',js}  d¯zj  Let  ˜Dj =  \uf8f1  \uf8f2  \uf8f3  D if j ̸∈ {j1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=', js}  ∂D otherwise  (1)  and define  Dj1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=',js =  n  �  j=1  ˜Dj  On the set {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=', n}, we work modulo n so that n + r = r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' In all that follows we let C be an arbitrary  constant depending only on n, and Ψ be an arbitrary linear operator depending only on n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' Both the symbols  C and Ψ will act as local variables, and the same symbols will be used for different constants and operators  to simplify the exposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' By the Bochner-Martinelli formula [Henkin],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' given uniformly continuous u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' g on  Dn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' ¯∂u = g in the distributional sense,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' we have  u(z) = C ·  �  ∂Dn  n  �  j=1  (−1)j−1u(ζ)  ¯ζj − ¯zj  |ζ − z|2n dn−1  j  ¯ζ ∧ dnζ + Ψ[g] =: C ·  n  �  j=1  uj(z) + Ψ[g]  Observe that  ¯ζj − ¯zj  |ζ − z|2n =  −1  n − 1  ∂  ∂ζj+1  ¯ζj − ¯zj  |ζ − z|2n−2(ζj+1 − zj+1)  Therefore  uj(z) = C · (−1)  �  Dj  u(ζ)  ∂  ∂ζj+1  ¯ζj − ¯zj  |ζ − z|2n−2(ζj+1 − zj+1)d¯ζj+1 ∧ dn−2  j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='j+1¯ζ ∧ dnζ + Ψ[g]  For fixed z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' our integrand is a uniformly continuous form in ζ with uniformly continuous differential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' Thus  3  by Stokes’ theorem, we obtain  uj(z) = C · (−1)  �  Dj,j+1  u(ζ)  ¯ζj − ¯zj  |ζ − z|2n−2(ζj+1 − zj+1)dn−2  j,j+1 ¯ζ ∧ dnζ + Ψ[g]  Here, the domain of integration is Dj,j+1 since the pullback of the form in the integral vanishes on the other  components of ∂Dj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' By repeating this procedure n − 2 more times, each time observing that  ¯ζj − ¯zj  |ζ − z|2n−2s �s  r=1(ζj+r − zj+r) = −  1  (n − s − 1)  ∂  ∂ζs+1  ¯ζj − ¯zj  |ζ − z|2n−2s−2 �s+1  r=1(ζj+r − zj+r)  We obtain  uj(z) = C · (−1)(j−1)(−1)(j−1)(n−j)(−1)(n−1)  �  (∂D)n  j  u(ζ)  ¯ζj − ¯zj  |ζ − z|2 �  r̸=j(ζr − zr)dnζ + Ψ[g]  Here, (∂D)n  j is the set (∂D)n with orientation induced by successive applications of the boundary operator be-  ginning with the j’th copy of D in the product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' It is clear from inspection that (∂D)n  j = (−1)(j−1)(n−j+1)(∂D)n  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='  Therefore,  uj(z) = C ·  �  T1  u(ζ)  ¯ζj − ¯zj  |ζ − z|2 �  r̸=j(ζr − zr)dnζ + Ψ[g]  Summing over j and noting that  n  �  j=1  ¯ζj − ¯zj  |ζ − z|2 �  r̸=j(ζr − zr) =  1  �n  r=1(ζr − zr)  We see that  u(z) = C · K[u](z) + Ψ[g]  Since this equation holds for any u, by considering for example u = 1, we see that C = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' By taking Φ = Ψ,  this concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='  Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' There is a canonical solution operator L : Z(0,1)(¯Dn) → C(¯Dn) to the ¯∂−equation such that  KL = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' Furthermore, L = H and for all non-negative integers k and 0 < α < 1 we have L|Zk+α  (0,1)(¯Dn) = T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' By Lemma 1, we need only show that KH = 0 and KT = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' To see the former, fix g ∈ Z(0,1)(¯Dn),  and let d2ζj = d¯ζj ∧ dζj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' We note that for |z1| = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' = |zn| = 1, H is the sum of terms, each with a region of  integration given by some  γJ(z) = {ζ ∈ Dn : ζj1 = zj1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=', ζjr = zjr, 1 ≥ |ζk1| = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' = |ζkn−r|}  4  Hence, K[H[g]] is given by K applied to the sum of terms of the form  �  |ζk1 |=.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='=|ζkn−r |≤1  gks(ζk1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='ζkn−r, zJ)  (ζk1 − zk1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='(ζkn−r − zkn−r)dζk1 ∧ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' ∧ d2ζks ∧ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' ∧ dζkn−r  =  �  |ζks |≤1  � �  |ζk1 |=.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='= �  |ζks|=.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='|ζkn−r|≤1  gks(ζk1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='ζkn−r, zJ)  (ζk1 − zk1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='  �  (ζks − zks).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='(ζkn−r − zkn−r)  dζk1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' �  dζks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='dζkn−r  �  d2ζks  (ζks − zks)  = Tks[G]  Here,  G(z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=', ζks, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='., zn) =  �  |ζk1 |=.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='= �  |ζks|=.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='|ζkn−r |≤1  gks(ζk1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='ζkn−r, zJ)  (ζk1 − zk1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='  �  (ζks − zks).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='(ζkn−r − zkn−r)  dζk1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' �  dζks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='dζkn−r  But since SjTj = 0 for all j = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='n and K = Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='S1, as these operators commute, KTks[G] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' Hence  KH[g] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' Since g was arbitrary, KH = 0 Likewise, since T [g] = �n  j=1 Tj ˜Sj[gj], we have KT = 0  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' In light of Corollary 1, we may view both H and T as different formulae for the same canonical  solution operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' As such, Pan and Zhang’s analysis of the formula T carry over to Henkin’s formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' Hence  we obtain the following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' For all k ≥ 0, k ∈ Z and 0 < α < 1, Henkin’s canonical solution operator H is a bounded  linear operator  H : Zk+α  (0,1)(Dn) → C(¯Dn)  3  Proof of Theorem 3  Fix g ∈ Z(0,1)(¯Dn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' In order to prove Theorem 3 we make a short digression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' Fix a permutation σ of {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=', n}  and let Dσ denote the open sectors {z ∈ Dn : |zσ(1)| > .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' > |zσ(n)|}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' For z ∈ Dσ, the only terms in H[g](z)  which do not vanish, take the form  �  |ζk1 |=.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='=|ζkn−r|≤|zjr |  gks(ζk1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='ζkn−r, zJ)  (ζk1 − zk1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='(ζkn−r − zkn−r)dζk1 ∧ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' ∧ d2ζks ∧ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' ∧ dζkn−r  Here, g = �n  i=1 gid¯ζj, J = (σ(j1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='., σ(jr)), 1 ≤ j1 < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' < jn ≤ n  We therefore consider the following linear operator  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' Let a, b ∈ D, ζ, z ∈ Dq, ζ = (ζ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=', ζq), z = (z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=', zq), h ∈ C(Dq × D × D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' We write d2ζj for  d¯ζj ∧ dζj and define  P[h](z, a, b) =  �  |ζ1|=.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='=|ζq|≤|a|  h(ζ, a, b)  (ζ1 − z1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='(ζq − zq)d2ζ1 ∧ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' ∧ dζq  We have the following Lemmas:  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' For each s ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=', q} let Ds = {(z, a, b) ∈ Dq+2 : |z1| < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' < |zs| < |a| < |zs+1| < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' < |zq|},  and t ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='q}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' Then on any open sector Ds, P[h] is H¨older−α uniformly in the variable zt, with coefficient  independent of z, a, b, and proportional to ∥h∥Cα(Dq+2)  5  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' We first consider the case t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' Observe that after ignoring sign changes,  P[h](z, a, b) =  �  |ζ1|=.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='=|ζq|≤|a|  h(ζ, a, b)  (ζ1 − z1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='(ζq − zq)d2ζ1 ∧ dζ2 ∧ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' ∧ dζq  =  �  |ζ1|≤|a|  � �  |ζ2|=.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='=|ζq|=|ζ1|  h(ζ, a, b)  (ζ2 − z2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='(ζq − zq)dζ2 ∧ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' ∧ dζq  �  d2ζ1  (ζ1 − z1)  =:  �  |ζ1|≤|a|  H(ζ1, z2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=', zq, a, b)  d2ζ1  (ζ1 − z1)  Thus the theorem will hold provided we can show H is uniformly bounded with coefficient independent of  z, a, b, and proportional to ∥h∥Cα(Dq+2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' But this is clear since  H(ζ1, z2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=', zq, a, b) =  �  |ζ2|=.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='=|ζq|=|ζ1|  h(ζ, a, b)  (ζ2 − z2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='(ζq − zq)dζ2 ∧ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' ∧ dζq  =  �  |ξ2|=.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='=|ξq|=1  h(ζ1, |ζ1|ξ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='|ζ1|ξq, a, b)  (ξ2 −  z2  |ζ1|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='(ξq −  zq  |ζ1|) dξ2 ∧ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' ∧ dξq  Since the function (ζ1, ξ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=', ξq, a, b) �→ h(ζ1, |ζ1|ξ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='|ζ1|ξq, a, b) is Cα on the region of integration, its Cauchy  torus integral is uniformly bounded, so H is uniformly bounded as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' Furthermore, ∥H∥L∞ ≲ ∥h∥Cα(Dp+q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='  Since P[h] is obtained by applying the Cauchy integral on a domain to H, for every 0 < ǫ < 1, P[h](z, a, b)  is C1−ǫ in Ds uniformly in z1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' In particular, it is Cα in Ds uniformly in z1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='  For t ̸= 1 we observe that ζt¯ζt = ζ1¯ζ1, Therefore d¯ζ1 = ζt  ζ1 d¯ζt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' Hence, after ignoring changes in sign,  P[h](z, a, b) =  �  |ζ1|=.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='=|ζq|≤|a|  ζtζ−1  1 h(ζ, a, b)  (ζ1 − z1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='(ζq − zq)dζ1 ∧ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' ∧ d2ζt ∧ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' ∧ dζq  The same analysis as before shows that P[h](z, a, b) is Cα in zt uniformly with coefficient independent of  z, a, b, and proportional to ∥h∥Cα(Dq+2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' On any open sector Ds, P[h] is H¨older−α uniformly in the parameters a, b with coefficient  proportional to ∥h∥Cα(Dq+2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='  6  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' We first show that P[h] is Cα uniformly in b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' Indeed, let ǫ ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='  |P[h](z, a, b + ǫ) − P[h](z, a, b)| ≤  �  |ζ1|=.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='=|ζq|≤|a|  ∥h∥Cα(Dq+2)|ǫ|α  |ζ1 − z1|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='|ζq − zq||d2ζ1|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='|dζq|  ≤ ∥h∥Cα(Dq+2)|ǫ|α  � 1  0  � 2π  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='  � 2π  0  rq · dθ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='dθqdr  |reiθ1 − z1|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='|reiθq − zq|  ≲ ∥h∥Cα(Dq+2)|ǫ|α  � 1  0  � 2π  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='  � 2π  0  rq · dθ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='dθqdr  |rθ1 + |r − |z1|||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='|rθq + |r − |zq|||  ≲ ∥h∥Cα(Dq+2)|ǫ|α  � 1  0  �  ln (rθ1 + |r − |z1||)  �2π  0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='  �  ln (rθq + |r − |zq||)  �2π  0 dr  ≲ ∥h∥Cα(Dq+2)|ǫ|α  � 1  0  | ln (|r − |z1||)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='.| ln (|r − |zq||)|dr  ≲ ∥h∥Cα(Dq+2)|ǫ|α  � 1  0  q  �  i=1  | ln (|r − |zi||)|qdr  ≲ ∥h∥Cα(Dq+2)|ǫ|α  � 1  0  | ln (r)|qdr  ≲ ∥h∥Cα(Dq+2)|ǫ|α  Similarly, we can show P[h] is Cα uniformly in a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='  P[h](z, a + ǫ, b) − P[h](z, a, b) =  �  |ζ1|=.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='=|ζq|≤|a+ǫ|  h(ζ, a + ǫ, b) − h(ζ, a, b)  (ζ1 − z1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='(ζq − zq)  d2ζ1 ∧ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' ∧ dζq  +  �  |a|≤|ζ1|=.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='=|ζq|≤|a+ǫ|  h(ζ, a, b)  (ζ1 − z1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='(ζq − zq)d2ζ1 ∧ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' ∧ dζq =: I1 + I2  We denote these two terms I1 and I2 respectively and control them separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' Repeating the same analysis  as we did for b, we see that |I1| ≲ ∥h∥Cα(Dq+2)|ǫ|α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' Thus we need only control I2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='  |I2| =  ����  � |a+ǫ|  |a|  � 2π  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='  � 2π  0  h(reiθ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=', eriθq, a, b)  (reiθ1 − z1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='(reiθq − zq)rdθ1(ireiθ2dθ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='(ireiθqdθq)dr  ����  ≲ |ǫ|  ����  � 2π  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='  � 2π  0  h(|a|eiθ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='|a|eiθq, a, b)  (|a|eiθ1 − z1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='(|a|eiθq − zq)|a|qdθ1(ieiθ2dθ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='(ieiθqdθq)  ����  ≲ |ǫ|  ����  �  |ξ1|=1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='  �  |ξ|=1  h(|a|ξ, a, b)iξ1  (ξ1 − z1  |a|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='(ξq − |zq|  |a| )  dξ1 ∧ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' ∧ dξq  ����  ≲ |ǫ| · ∥h∥Cα(Dq+2)  Here, the last inequality holds as the function (ζ, a, b) �→ h(|a|ξ, a, b)iξ1 is Cα, so that its Cauchy torus  integral is bounded, depending only on ∥h∥Cα(Dq+2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='  Therefore P preserves H¨older−α regularity in the  parameters a, b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='  These lemmas assemble into the following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' Let 0 < α < 1, then for any Ds, P : Cα(Dq × D × D) → Cα(Ds) is a bounded linear  operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' By the preceding two lemmas, P[h] is Cα in Ds uniformly in each variable zt and parameters a, b,  7  with coefficient proportional to ∥h∥Cα(Dq+2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' Hence ∥P[h]∥Cα(Ds) ≲ ∥h∥Cα(Dq+2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='  Theorem 3 immediately follows from proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' (Theorem 3) Observe that the closures of the finitely many open sectors Dσ cover the polydisc Dn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='  By Proposition 1, H[g] is uniformly continuous on the polydisc Dn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' Therefore to show that H : Zα  (0,1)(Dn) →  Cα(Dn) is a bounded linear operator, we need only show that given Cα datum g, the solution H[g] is Cα on  each of the open sectors Dσ, with coefficient proportional to ∥g∥Cα(Dn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' But this follows immediately from  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='  4  Proof of Theorem 1  Having shown Theorem 3, by Corollary 1, we obtain that for 0 < α < 1, H : Zα  (0,1)(Dn) → Cα(Dn) is a  bounded linear operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' To complete the proof of Theorem 1, we need only induct on k using Nijenhuis  and Wolf’s formula T = H : Zk+α  (0,1)(Dn) → Ck+α(Dn)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' (Theorem 1) As noted in [Vekua], we may differentiate the singular integral Tj with respect to zj and  ¯zj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' From which we obtain  ∂  ∂zj  Tj[g](z) = Πj[g](z) :=  1  2πi  �  D  g(z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=', ζj, zj+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=', zn)  (ζj − zj)2  dζj ∧ d¯ζj  and  ∂  ∂ ¯zj  Tj[g](z) = g  Suppose now T = H : Zk+α  (0,1)(Dn) → Ck+α(Dn) is a bounded linear operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' Fix g ∈ Zk+1+α  (0,1)  (Dn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='  Then, as T is a solution operator to the ¯∂−equation, we have that ¯∂T [g] = g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' In addition, for all j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=', n,  ∂  ∂zj Tj[g](z) = Πj[g](z) and  ∂  ∂zj Sj[g](z) = Sj[ ∂g  ∂zj ](z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' Therefore by commuting the differentiation symbol  with the operator when possible, we obtain the formula  ∂  ∂zj  T [g](z) =  �  i̸=j  Ti ˜Si[ ∂  ∂ζj  gi] + Πj ˜Sj[gj]  Since k + 1 ≥ 1, we may apply Stokes’ Theorem, to the term Πj ˜Sj[gj] as in [Vekua] to obtain  Πj ˜Sj[gj] = Tj ˜Sj[ ∂  ∂ζj  gj] −  1  2πi  �  ∂D  ˜Sj[gj](z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=', zj−1, ζj, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='zn)  ζj − zj  d¯ζj = Tj ˜Sj[ ∂  ∂ζj  gj] + Sj[ ˜Sj[gj] · ¯ζ2  j ]  From which it immediately follows that  ∂  ∂zj  T [g](z) =  �  i  Ti ˜Si[ ∂  ∂ζj  gi] + Sj[ ˜Sj[gj] · ¯ζ2  j ] = T [ ∂  ∂ζj  g] + Sj[ ˜Sj[gj] · ¯ζ2  j ]  By the induction hypothesis, ∥T [ ∂  ∂ζj g]∥Ck+α(Dn) ≲ ∥ ∂  ∂ζj g∥Ck+α(D) ≲ ∥g∥Ck+1+α(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' Furthermore, for all j,  gj ∈ Ck+1+α(Dn) we have ∥Sj[ ˜Sj[gj] · ¯ζ2  j ]∥Ck+α ≲ ∥gj∥Ck+1+α as the Cauchy integral loses arbitrarily small  amounts of H¨older regularity in parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' This completes the induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='  8  References  [Henkin]  Henkin, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' The method of integral representations in complex analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' (Russian) Current  problems in mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' Fundamental directions, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' 7, 23–124, 258, Itogi Nauki i Tekhniki,  Akad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' Nauk SSSR, Vsesoyuz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' Nauchn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' i Tekhn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' Inform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=', Moscow, (1985).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='  [HP1]  Henkin, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' and Polyakov, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' Prolongement des fonctions holomorphes born´e d’une  sous-vari´et´e du polydisue, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' Acad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
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+page_content=' H¨older estimates for the ¯∂ problem for (p, q) forms on product domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='  Internat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' 32 (2021), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
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+page_content=' 2150014, 20 pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='  [PZ2]  Pan, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
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+page_content=' Cauchy singular integral operator with parameters in log-H¨older spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='  (English summary) J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
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+page_content='  [Tumanov]  Tumanov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' On the propagation of extendibility of CR functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
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+page_content=', 173, Dekker, New  York, (1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='  [Vekua]  Vekua, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
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+page_content=' Pergamon Press, London-Paris-Frankfurt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='  Addison-Wesley Publishing Company, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=', Reading, Mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' (1962) xxix+668 pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='  [Zhang]  Zhang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' Optimal H¨older regularity for the ¯∂ problem on product domains in C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content=' To appear  at Proceedings of the AMS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
+page_content='  9' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfYAqA/content/2301.04484v1.pdf'}
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+arXiv:2301.00368v1  [math.AP]  1 Jan 2023
+SLOW TRAVELING-WAVE SOLUTIONS FOR THE GENERALIZED
+SURFACE QUASI-GEOSTROPHIC EQUATION
+DAOMIN CAO, SHANFA LAI, GUOLIN QIN
+Abstract. In this paper, we systematically study the existence, asymptotic behaviors,
+uniqueness, and nonlinear orbital stability of traveling-wave solutions with small prop-
+agation speeds for the generalized surface quasi-geostrophic (gSQG) equation.
+Firstly
+we obtain the existence of a new family of global solutions via the variational method.
+Secondly we show the uniqueness of maximizers under our variational setting. Thirdly
+by using the variational framework, the uniqueness of maximizers and a concentration-
+compactness principle we establish some stability theorems. Moreover, after a suitable
+transformation, these solutions constitute the desingularization of traveling point vortex
+pairs.
+Keywords: The gSQG equation; Traveling-wave solutions; Variational methods; Existence
+and uniqueness of maximizer; Nonlinear stability.
+2020 MSC Primary: 76B47; Secondary: 76B03, 35A02.
+1. Introduction and Main results
+In this paper, we are concerned with the following generalized surface quasi-geostrophic
+(gSQG) equation
+�
+∂tθ + u · ∇θ = 0
+in R2 × (0, T),
+u = ∇⊥(−∆)−sθ
+in R2 × (0, T),
+(1.1)
+where 0 < s < 1, θ(x, t) : R2 × (0, T) → R is the active scalar being transported by the
+velocity field u(x, t) : R2 × (0, T) → R2 generated by θ, and (a1, a2)⊥ = (a2, −a1). The
+operator (−∆)−s is defined by
+(−∆)−sω(x) = Gsω(x) =
+�
+R2 Gs(x − y)ω(y)dy,
+where Gs is the fundamental solution of (−∆)s in R2 given by
+Gs(z) =
+cs
+|z|2−2s,
+cs = Γ(1 − s)
+22sπΓ(s).
+When s = 1/2, (1.1) corresponds to the inviscid surface quasi-geostrophic (SQG) equa-
+tion, which models the evolution of the temperature from a general quasi-geostrophic
+system for atmospheric and atmospheric flows (see e.g. [26, 52]). The SQG equation has
+1
+
+2
+DAOMIN CAO, SHANFA LAI, GUOLIN QIN
+received extensive concern as a simplified model for the three-dimensional Euler equations
+since [26]. Formally at least, in the limit s ↑ 1 we obtain the well-known two-dimensional
+Euler equation in vorticity formulation [56]. For general 0 < s < 1, (1.1) was proposed
+by C´ordoba et al. in [29] as an interpolation between the Euler equation and the SQG
+equation.
+The global well-posedness for the Cauchy problem for two-dimensional incompressible
+Euler equation (i.e., s = 1 in (1.1)) has been well studied.
+Global well-posedness for
+Cauchy problems with initial data in L1 ∩ L∞ was established by Yudovich [66]. The L1
+assumption can be replaced by an appropriate symmetry condition thanks to the work of
+Elgindi and Jeong [36]. We refer to [36, 56] and references therein for more discussions.
+However, to the best of our knowledge, the problem of whether the gSQG system presents
+finite time singularities or there is global well-posedness of classical solutions is still open;
+see [20, 21, 46, 49, 50] and references therein for more details.
+For vortex patch type global solutions, the first non-trivial example was constructed
+in [44] for
+1
+2 < s < 1 by using the contour dynamics equation and bifurcation theory.
+Numerous results on the vortex patch type solutions for the gSQG equations were then
+obtained in different situations (see e.g. [18, 19, 28, 32, 39, 41, 45, 47]). In [20], Castro et al.
+established the first result of existence on global smooth solutions for the gSQG equation
+by developing a bifurcation argument from a specific radially symmetric function. In [43],
+Gravejat and Smets, for the first time, proved the existence of smooth translating vortex
+pairs for the SQG equation. This result was then generalized to the gSQG equation with
+s ∈ (0, 1) by Godard-Cadillac [40]. In [5], Ao et al. successfully constructed traveling and
+rotating smooth solutions to the gSQG equation with s ∈ (0, 1) by the Lyapunov-Schmidt
+reduction method.
+In this paper, we are interested in traveling-wave solutions for the gSQG equation. Up
+to a rotation, we may assume, without loss of generality, that these waves have a negative
+speed −W in the vertical direction, so that
+θ(x, t) = ω(x1, x2 + Wt).
+In this setting, the first equation in (1.1) is also reduced to a stationary equation
+∇⊥(Gsω − Wx1) · ∇ω = 0,
+(1.2)
+which has a weak form
+�
+R2 ω∇⊥(Gsω − Wx1) · ∇ϕdx = 0,
+∀ ϕ ∈ C∞
+0 (R2).
+(1.3)
+In the study of traveling-wave solutions for ideal incompressible fluids, translating vortex
+pairs is the main concern. The literature on vortex pairs can be traced back to the work
+of Pocklington [59] in 1895. In 1906, Lamb [51] founded an explicit solution for the Euler
+equation which is now generally referred to as the Lamb dipole or Chaplygin-Lamb dipole;
+see also [58]. Besides those exact solutions, the existence (and abundance) of translating
+vortex pairs for the Euler equation has been rigorously established in [2, 8, 61, 64, 65]
+and so on. As mentioned above, for the gSQG equation, some examples of traveling-wave
+solutions were constructed in [5, 15, 16, 40, 43, 45, 47].
+
+TRAVELING-WAVE SOLUTIONS FOR THE GSQG EQUATION
+3
+In this paper, we will obtain a new family of traveling-wave solutions for the gSQG
+equation and further investigate their asymptotic behaviors, uniqueness, and nonlinear
+orbital stability.
+1.1. Main results. As pointed out by Arnol’d [6], a natural way of obtaining solutions to
+the stationary problem (1.2) is to impose that ω and Gsω − Wx1 are (locally) functional
+dependent. That is, one may impose that
+ω = f(Gsω − Wx1),
+for some Borel measurable function f : R → R.
+Usually f is supposed to satisfy the following hypotheses
+(H1). f(0) = 0, f is nonnegative and strictly increasing for t > 0;
+(H2). limt→0+ t−
+1
+1−sf(t) = +∞ and limt→+∞ t−
+1
+1−sf(t) = 0.
+Our first main result concerns the existence of traveling-wave solutions with slow trav-
+eling speeds and the fine asymptotic behaviors of these solutions. For convenience, we will
+take W = Wε3−2s for some constant W > 0 and some small ε > 0. Let e2 = (0, 1) be the
+unit vector along the x2-direction. Let R2
++ := {x ∈ R2 | x1 > 0} be the right half plane
+and 1S represents the characteristic function of a set S. Denote by spt(ω) the support of
+a function ω. For fixed W > 0, κ > 0 denote
+d0 =
+�(1 − s)csκ
+22−2sW
+�
+1
+3−2s
+.
+(1.4)
+Our first result is as follows.
+Theorem 1.1. Let 0 < s < 1, W > 0 κ > 0 be given. Suppose that f is a measurable
+function satisfying (H1) and (H2) and f ∈ C1−2s if 0 < s < 1
+2. Then there is a number
+ε0 > 0 small such that for any ε ∈ (0, ε0), (1.1) has a traveling-wave solution of the form
+θε(x, t) = ωtr,ε(x + Wε3−2ste2) for some function ωtr,ε ∈ L∞(R2) in the sense that ωtr,ε
+solves (1.3) with W = Wε3−2s. Moreover, ωtr,ε has the following properties:
+(i) ωtr,ε is odd in x1 and even in x2. That is,
+ωtr,ε(−x1, x2) = −ωtr,ε(x1, x2),
+ωtr,ε(x1, −x2) = ωtr,ε(x1, x2),
+∀ x ∈ R2;
+(ii) It holds for some constant µε
+ωtr,ε = f(Gsωtr,ε − Wε3−2sx1 − µε),
+in R2
++;
+(iii) Let ωε := ωtr,ε1R2
++ and denote the center of mass of ωε by xε := κ−1 �
+xωε. Then,
+there hold
+�
+R2
++
+ωε = κ and εxε = (d0, 0) + o(1),
+as ε → 0,
+where d0 s given by (1.4). Furthermore, there is a constant R > 0 independent of
+ε such that spt(ωε) is contained in the disk with center xε and radius R.
+
+4
+DAOMIN CAO, SHANFA LAI, GUOLIN QIN
+Remark 1.2. The assumption f ∈ C1−2s in the case 0 < s <
+1
+2 is used to improve the
+regularity of Gsωε (see e.g. Propositions 2.8 and 2.9 in [60]) so that the integral in (1.3)
+makes sense.
+Remark 1.3. Typical examples of f satisfying the assumptions in Theorem 1.1 includes any
+C0,1 smooth bounded strictly increasing functions with f(0) = 0, such as f(t) = arctan(t+)
+as well as some unbounded functions, for example, f(t) = tp
++ with p ∈ (1,
+1
+1−s). Here t+
+means max{0, t}. We will obtain finer asymptotic behaviors and prove the uniqueness and
+stability in the later case.
+Remark 1.4. As we shall see in the proof of Lemma 2.21, up to some translation, ωε
+tends to a nontrivial function ω0 in L1 ∩ L2−s(R2), which is a maximizer of the limiting
+problem considered in subsection 2.1. Therefore, the amplitude of the solutions obtained
+in Theorem 1.1 does not vanish as ε → 0.
+Let G+
+s (x, y) :=
+cs
+|x−y|2−2s −
+cs
+|x−¯y|2−2s with ¯y = (−y1, y2) and define
+G+
+s ω :=
+�
+R2
++
+G+
+s (x, y)ω(y)dy.
+The proof of Theorem 1.1 is based on a constrained maximization method. More precisely,
+take J be defined by J(t) =
+� t
+0 f −1(τ)dτ and let B(x, r) stand for the disk with center x
+and radius r and κ > 0 be a constant. We are to consider the problem of maximizing the
+following functional
+Eε(ω) := 1
+2
+�
+R2
++
+�
+R2
++
+ω(x)G+
+s (x, y)ω(y)dxdy−Wε3−2s
+�
+R2
++
+x1ω(x)dx−
+�
+R2
++
+J(ω(x))dx, (1.5)
+over the constraint
+Aε :=
+�
+ω ∈ L1 ∩ L2−s(R2
++) | ω ≥ 0, spt(ω) ⊂ B
+�
+(d0
+ε , 0), d0
+2ε
+�
+,
+�
+R2
++
+ω(x)dx = κ
+�
+. (1.6)
+Consider the following maximization problem:
+eε := sup
+ω∈Aε
+Eε(ω).
+(1.7)
+For the above maximizing problem we have the following result.
+Theorem 1.5. Let 0 < s < 1 and W > 0. Suppose that f is a measurable function
+satisfying (H1) and (H2).
+Then there is a number ε0 > 0 small such that eε can be
+achieved for any ε ∈ (0, ε0), that is Eε admits a maximizer ωε in Aε. Moreover, ωε has the
+following properties:
+(i) ωε is Steiner symmetric with respect to some plane {x2 = const.};
+(ii) There is a constant µε such that
+ωε = f
+�
+G+
+s ωε − Wε3−2sx1 − µε
+�
+,
+in B(ε−1(d0, 0), ε−1d0/2);
+
+TRAVELING-WAVE SOLUTIONS FOR THE GSQG EQUATION
+5
+(iii) The energy satisfies
+I0 + O(ε2−2s) ≤ Eε(ωε) ≤ I0,
+where I0 is given by (2.8);
+(iv) There exists a constant 0 < C < +∞ independent of ε such that
+lim sup
+ε→0+
+∥ωε∥L∞ ≤ C.
+(v) Denote the center of mass of ωε by xε := κ−1 �
+xωε. Then,
+εxε = (d0, 0) + o(1),
+as ε → 0,
+where d0 is given by (1.4). Furthermore, there is a constant R > 0 independent of
+ε such that spt(ωε) is contained in the disk with center xε and radius R.
+As we will see that Theorem 1.1 can be derived from Theorem 1.5. Indeed, we will prove
+in Lemma 2.24 that if we further assume f ∈ C1−2s in the case 0 < s < 1
+2, then after a
+translation in x2, ωε(x) − ωε(¯x) is the desired function ωtr,ε in Theorem 1.1.
+Next, we shall investigate the problem of uniqueness, which is crucial in the study of
+stability. We focus our attention on the case f = tp
++ for some p ∈ (1,
+1
+1−s). It can be seen
+that such f satisfies all the assumptions in Theorem 1.1 for s ∈ (0, 1). Therefore, for every
+ε > 0 small, Theorem 1.1 ensures a traveling-wave solution with ωtr,ε(x) = ωε(x) − ωε(¯x)
+and ωε being a maximizer of Eε over constraint Aε. Inspired by the work on rotating stars
+[48], we will prove that ωε is the unique maximizer in the sense that any maximizer of Eε
+is a translation of ωε.
+For fixed ε0 as in Theorem 1.5 we denote by Σε the set of all maximizers of Eε over Aε
+for ε ∈ (0, ε0). Our second main result is as follows.
+Theorem 1.6. Suppose that f(t) = tp
++ for some p ∈ (1,
+1
+1−s). Let ε0 be as in Theorem 1.5
+and ε ∈ (0, ε0). Let ωε ∈ Σε be a maximizer as obtained in Theorem 1.5. Then there is a
+number ε1 ∈ (0, ε0] such that for all ε ∈ (0, ε1),
+Σε = {ωε(· + ce2) | c ∈ R}.
+For relative equilibria of fluids, there are fewer mathematical results available on the
+uniqueness. The first result was due to Amick and Fraenkel [3], who proved that Hill’s
+spherical vortex is the unique solution when viewed in a natural weak formulation by the
+method of moving planes. Later Amick and Fraenkel [4] also established local uniqueness
+for Norbury’s nearly spherical vortex. The uniqueness of the Chaplygin-Lamb dipole was
+shown by Burton [10] using a similar method as [3]. Recently, Jang and Seok [48] proved
+the uniqueness of maximizers of a variational problem related to rotating binary stars,
+which inspired our proof of Theorem 1.6. As for the gSQG equation, to the best of our
+knowledge, the only result on this issue is the recent work [15], where a very special case
+was considered in order to apply the method of moving planes. Our result Theorem 1.6
+provides uniqueness in a wide range of cases.
+Our last main result concerns the orbital stability of the traveling-wave solutions ob-
+tained in Theorem 1.1. We first prove a general stability theorem in a similar spirit as
+
+6
+DAOMIN CAO, SHANFA LAI, GUOLIN QIN
+[13], where the stability of vortex pairs for the 2D Euler equation was considered. To be
+precise, we will consider the maximization problem of the functional
+˜EW(ζ) := 1
+2
+�
+R2
++
+ζ(x)G+
+s ζ(x)dx − W
+�
+R2
++
+x1ζ(x)dx
+over the set R(ζ0)w, which means the weak closure of the rearrangement class of a given
+function ζ0. Using the concentrate compactness principle due to Lions [55], we establish
+the compactness of maximizing sequence and derive a general nonlinear stability theorem
+on the set of maximizers (see Theorem 4.8).
+In the case f = tp
++ for p ∈ (1,
+1
+1−s), Theorem 1.1 provides a traveling-wave solution with
+ωtr,ε(x) = ωε(x) − ωε(¯x) and ωε being a maximizer of Eε. To apply the stability theorem
+of the set of maximizers, we need to consider an auxiliary variational problem: maximize
+˜EW with W = Wε3−2s over the set R(ωε)w. By studying the asymptotic behaviors of
+maximizers and using the uniqueness result in Theorem 1.6, we are able to show that all
+the maximizers of the second variational problem are actually translations of ωε in the x2-
+direction. As a consequence, we obtain the orbital stability of ωε by applying the nonlinear
+stability theorem on the set of maximizers proved in Theorem 4.8.
+Roughly speaking, our stability result is as follows.
+Theorem 1.7. Suppose that f(t) = tp
++ for some p ∈ (1,
+1
+1−s). Let ωtr,ε be the traveling-
+wave solution obtained in Theorem 1.1. Then for ε fixed small, ωtr,ε is orbitally stable
+in the following sense: for arbitrary M > 0 and η > 0, there exists δ > 0 such that for
+non-negative function ξ0 ∈ L1 ∩ L∞(R2
++) with ||ξ0||∞ < M and
+infc∈R
+�
+∥ξ0 − ωtr,ε(· + ce2)∥L1(R2
++) + ∥ξ0 − ωtr,ε(· + ce2)∥L2(R2
++)
++∥x1(ξ0 − ωtr,ε(· + ce2))∥L1(R2
++)
+�
+≤ δ,
+(1.8)
+if there exists a L∞-regular solution ξ(t) with initial data ξ0(x) for t ∈ [0, T) with 0 < T ≤
+∞, then all t ∈ [0, T),
+infc∈R
+�
+∥ξ(t) − ωtr,ε(· + ce2)∥L1(R2
++) + ∥ξ(t) − ωtr,ε(· + ce2)∥L2(R2
++)
++∥x1(ξ(t) − ωtr,ε(· + ce2))∥L1(R2
++)
+�
+≤ η.
+(1.9)
+Remark 1.8. For the rigorous definition of L∞-regular solutions for the gSQG equation,
+please see Section 4. Once the uniqueness of other solutions was established, one may
+apply the general stability theorem and the framework in this paper to obtain their orbital
+stability. Compared with the result in [13], we admit perturbations with non-compact
+supports, which is achieved by bringing in the L1-norm in our theorem.
+Much work has been done on the stability of steady solutions to the Euler equations,
+for which we refer the interested reader to [1, 11, 12, 17, 23, 24, 25, 38, 63] and references
+therein.
+
+TRAVELING-WAVE SOLUTIONS FOR THE GSQG EQUATION
+7
+1.2. Desingularize the traveling point vortices. The result in Theorem 1.1 also pro-
+vides a family of solutions that desingularize the traveling point vortices for the gSQG
+equation. Indeed, taking the transformation ˆωtr,ε(x) = ε−2ωtr,ε(ε−1x), we conclude from
+Theorem 1.1 immediately that
+Corollary 1.9. Let 0 < s < 1. Suppose that f is a function satisfying (H1) and (H2) and
+f ∈ C1−2s if 0 < s < 1
+2. Then there is a constant ε0 > 0 small such that for any ε ∈ (0, ε0),
+(1.1) has a traveling-wave solution of the form θε(x, t) = ˆωtr,ε(x+Wte2) for some function
+ˆωtr,ε ∈ L∞(R2) in the sense that ˆωtr,ε solves (1.3) with W = W. Moreover, ˆωtr,ε has the
+following properties:
+(i) ˆωtr,ε is odd in x1 and even in x2. That is,
+ˆωtr,ε(−x1, x2) = −ˆωtr,ε(x1, x2),
+ˆωtr,ε(x1, −x2) = ˆωtr,ε(x1, x2),
+∀ x ∈ R2;
+(ii) It holds
+ˆωtr,ε = f(ε2−2s(Gsˆωtr,ε − Wx1) − µε),
+in R2
++,
+for some constant µε;
+(iii) There holds in the sense of measure
+ˆωtr,ε(x) ⇀ κδ(x − (d0, 0)) − κδ(x + (d0, 0)),
+where d0 =
+�
+(1−s)csκ
+22−2sW
+�
+1
+3−2s . Furthermore, there is a constant R > 0 independent of
+ε such that spt(ˆωε) is contained in B((d0, 0), Rε) ∪ B((−d0, 0), Rε).
+Corollary 1.9 (iii) implies that {ˆωtr,ε}ε∈(0,ε0) is a sequence of regular solutions approxi-
+mating the traveling point vortex pair for the gSQG equations.
+In [5], Ao et al. constructed a family of solutions closed to the points vortices of the gSQG
+equation with the profile function f(t) = tp
++ for p ∈ (1, 1+s
+1−s) by the Lyapunov-Schmidt
+reduction method. It can be seen that our result Corollary 1.9 covers the remaining case
+p ∈ (0, 1] for 1
+2 ≤ s < 1.
+In the recent paper [16], a family of traveling solutions for the gSQG equations with
+1
+2 ≤ s < 1 were constructed by the variational method. The solutions {˜ωtr,ε} obtained in
+[16] solve the integral equation
+˜ωtr,ε = g(Gs˜ωtr,ε − Wx1 − ˜µε),
+in R2
++,
+for some constant ˜µε and bounded non-decreasing function g. It is obvious that Corollary
+1.9 can not be deduced from the result in [16], since the profile function f(ε2−2s·) in
+Corollary 1.9 (ii) varies along with ε and is allowed to be unbounded. Therefore, Corollary
+1.9 provides a new family of traveling-wave solutions for the gSQG equations.
+The paper is organized as follows. In Section 2, for a large class of f, we construct
+traveling-wave solutions for the gSQG equation via a variational method. We first study
+the properties of maximizers of a limiting problem. Based on these properties, we are
+able to construct traveling-wave solutions with small traveling speeds by maximizing the
+energy functional Eε. Then, we study the asymptotic behavior of the maximizers carefully
+in several steps. With detailed asymptotic behaviors in hand, we prove the uniqueness of
+
+8
+DAOMIN CAO, SHANFA LAI, GUOLIN QIN
+maximizers in Section 3. Section 4 is devoted to investigating nonlinear stability. We first
+prove a general orbital stability theorem for the set of maximizers based on a combination
+of the variational method and the concentrated compactness lemma of Lions [55]. Then, we
+investigate the asymptotic behavior of maximizers in the rearrangement class and obtain
+the orbital stability Theorem 1.7 by using the uniqueness result in Theorem 1.6.
+2. Proofs of Theorem 1.1 and Theorem 1.5
+In this section, we first consider the maximization problem (1.7) and prove Theorem 1.5.
+Theorem 1.1 follows immediately.
+We assume that J : [0, +∞) → [0, +∞) satisfies
+(H′
+1). J is strictly convex and nonnegative;
+(H′
+2). limt→0+ J′(t)ts−1 = 0 and lim inft→+∞ J′(t)ts−1 ≥ K.
+Here K is a large constant, which will be determined later. Note that if J(t) =
+� t
+0 f −1(τ)dτ
+for some f satisfying (H1) and (H2), then one can check that J satisfies (H′
+1) and (H′
+2).
+Let Eε(ω) and Aε be defined as in (1.5) and (1.6). To obtain the existence of maximizers
+for (1.7) we need to consider its limiting problem first.
+2.1. The limiting problem. We start with definitions of the energy functional and set
+of constraints for the limiting problem corresponding to (1.7).
+The energy functional
+associated with Eε is
+E0(ω) := cs
+2
+�
+R2
+�
+R2
+ω(x)ω(y)
+|x − y|2−2sdxdy −
+�
+R2 J(ω(x))dx,
+and the constraint associated with Aε is
+A0 :=
+�
+ω ∈ L1 ∩ L2−s(R2) | ω ≥ 0,
+�
+R2 ω(x)dx = κ
+�
+.
+The limiting maximization problem associated with (1.7) is
+e0 := sup
+ω∈A0
+E0(ω).
+(2.1)
+In the classical paper [55], under a bit weaker assumption limt→0+ J(t)t−1 = 0, Lions
+showed the existence of maximizers of E0 over A0 (see Theorem II.2 and Corollary II.1 in
+[55]). As we shall see later, our assumption limt→0+ J′(t)ts−1 = 0 ensures that every max-
+imizer is compactly supported, which is an essential property used in the next subsection
+(for similar results on rotating stars, we refer to [7, 53, 57]).
+In what follows, we will investigate some essential properties of maximizers under our
+hypotheses (H′
+1) and (H′
+2).
+Recall that Gsω :=
+cs
+|x|2−2s ∗ ω. Denote ∥ · ∥p := ∥ · ∥Lp(R2) for simplicity. The following
+two lemmas concerning convolution inequalities are needed in our later discussion.
+
+TRAVELING-WAVE SOLUTIONS FOR THE GSQG EQUATION
+9
+Lemma 2.1. Assume that ω ∈ L1 ∩ Lp(R2) for some p > 1.
+If 1 < p ≤ s−1, then
+Gsω ∈ Lq(R2) for any
+1
+1−s < q <
+p
+1−sp and for some constants 0 < a, b < 1,
+∥Gsω∥q ≤ C
+�
+∥ω∥a
+1∥ω∥1−a
+p
++ ∥ω∥b
+1∥ω∥1−b
+p
+�
+.
+(2.2)
+If p > s−1, then (2.2) holds with q = ∞.
+Proof. We first consider the case 1 < p ≤ s−1. We split the function |x|2s−2 into two
+parts: |x|2s−2 = |x|2s−21{|x|<1} + |x|2s−21{|x|≥1}. It is easy to check that |x|2s−21{|x|<1} ∈
+Lr, ∀ 1 ≤ r <
+1
+1−s and |x|2s−21{|x|≥1} ∈ Lr, ∀ r >
+1
+1−s. Suppose
+1
+1−s < q <
+p
+1−sp, then
+there exist 1 ≤ r1 <
+1
+1−s, r2 >
+1
+1−s and 1 < p1, p2 < p such that 1 + q−1 = r−1
+1
++ p−1
+1
+and
+1 + q−1 = r−1
+2
++ p−1
+2 .
+Then it remains to apply the following Young inequality
+∥f ∗ g∥r ≤ ∥f∥u∥g∥v,
+∀f ∈ Lu, g ∈ Lv,
+for 1 ≤ r, u, v ≤ +∞ such that 1 + r−1 = u−1 + v−1, and the interpolation inequality
+∥f∥r ≤ ∥f∥a
+u∥f∥1−a
+v
+,
+∀f ∈ Lu ∩ Lv,
+(2.3)
+with 1 ≤ u < r < v ≤ +∞ and a = (r−1 − v−1)/(u−1 − v−1) ∈ (0, 1).
+For p > s−1, the proof is similar, so we will omit the detail and finish the proof.
+□
+Lemma 2.2. Suppose that ω ∈ L1 ∩ L2−s(R2), then it holds
+���
+�
+R2 ω(x)Gsω(x)dx
+��� ≤ C∥ω∥s
+1
+�
+R2 |ω|2−s.
+(2.4)
+Proof. The well-known Hardy-Littlewood-Sobolev inequality states that
+∥Gsω∥q ≤ C∥ω∥p,
+∀1 < p < q < +∞ with 1
+q = 1
+p − s.
+(2.5)
+Applying H¨older’s inequality, (2.5) with q = 2−s
+1−s and the interpolation inequality (2.3),
+for any ω1, ω2 ∈ L1 ∩ L2−s(R2) we obtain
+���
+�
+R2 ω1(x)Gsω2(x)dx
+��� ≤ ∥ω1∥2−s∥Gsω2∥ 2−s
+1−s ≤ C∥ω1∥2−s∥ω2∥1−s
+2−s∥ω2∥s
+1,
+(2.6)
+which implies (2.4) by taking ω1 = ω2 = ω and completes the proof.
+□
+We first show the radial symmetry and derive the Euler-Lagrange equation for a maxi-
+mizer of (2.1).
+Lemma 2.3. Let ω0 ∈ A0 be a maximizer of E0 over A0.
+Then ω0 must be radially
+symmetric and non-increasing with respect to some point. Moreover, there exists a constant
+µ0 ∈ R such that
+�
+Gsω0 − J′(ω0) ≤ µ0,
+on {ω0 = 0},
+Gsω0 − J′(ω0) = µ0,
+on {ω0 > 0}.
+(2.7)
+
+10
+DAOMIN CAO, SHANFA LAI, GUOLIN QIN
+Proof. The radial symmetry and monotonicity of ω0 are easy consequences of the strict
+rearrangement inequality (see Theorems 3.7 and 3.9 in [54]).
+Note that for δ > 0 small, the set {ω0 > δ} ̸= ∅ due to
+�
+R2 ω0 = κ > 0. We fix a δ > 0
+such that {ω0 > δ} ̸= ∅ and take a function φ0 so that
+�
+R2 φ0 = 1 and spt(φ0) ⊂ {ω0 > δ}.
+For any function φ bounded from below such that φ ≥ 0 on the set {ω0 ≤ δ}, we take a
+family of test functions as follows:
+ωt := ω0 + t(φ − φ0
+�
+R2 φ),
+which belong to A0 for |t| small. Since ω0 is a maximizer, we have
+0 = dE0(ωt)
+dt
+�����
+t=0
+=
+�
+R2(Gsω0 − J′(ω0) − µ0)φdx,
+where µ0 :=
+�
+R2(Gsω0 −J′(ω0))φ0. Then (2.7) follows from the arbitrariness of φ and hence
+the proof is complete.
+□
+Denote
+I0 := sup
+ω∈A0
+E0(ω).
+(2.8)
+Then I0 < +∞ by [55].
+Lemma 2.4. There is a constant c0 > 0 such that if ω0 ∈ A0 is a maximizer of E0 over
+A0, then one has
+∥ω0∥2−s ≤ c0.
+(2.9)
+Proof. Take the constant K in the hypothesis (H′
+2) as K = Cκs+2, where C is the constant
+in (2.4). Then there is a constant t0 > 0 such that J(t) > (Cκs + 1)t2−s for t > t0. On the
+one hand, by the definition of E0 and (2.4), we deduce
+�
+R2 J(ω0) ≤ −I0 + Cκs
+�
+R2(ω0)2−sdx.
+On the other hand, we infer from the choice of t0 that
+(Cκs + 1)
+�
+R2(ω0)2−sdx = (Cκs + 1)
+�
+{ω0≤t0}
+(ω0)2−sdx + (Cκs + 1)
+�
+{ω0>t0}
+(ω0)2−sdx
+≤ (Cκs + 1)t1−s
+0
+κ +
+�
+R2 J(ω0).
+Therefore, we arrive at (2.9) by taking c0 = (−I0 + (Cκs + 1)t1−s
+0
+κ)
+1
+2−s.
+□
+Lemma 2.5. Let ω0 ∈ A0 be a maximizer and µ0 be the constant defined in Lemma 2.3,
+then one has
+µ0 > 0.
+(2.10)
+
+TRAVELING-WAVE SOLUTIONS FOR THE GSQG EQUATION
+11
+Proof. Take r0 > 0 such that
+�
+B(0,r0) ω0 ≥ κ
+2. Since for any r ≥ 0,
+�
+B(0,r) ω0 ≤ κ, there is
+a point xr ∈ B(0, r) such that ω0(xr) ≤ π−1κr−2. Then we infer from the hypothesis (H′
+2)
+that
+J′(ω0(xr)) = o(r2s−2),
+as r → +∞.
+On the other hand, we have
+Gsω0(xr) ≥
+cs
+(2r)2−2s
+�
+B(0,r)
+ω0 ≥ 4s−2csκr2s−2,
+∀ r ≥ r0.
+Thus, we derive from (2.7) that
+µ0 ≥ Gsω0(xr) − J′(ω0(xr)) ≥ (4s−2csκ + o(1))r2s−2 > 0,
+for r sufficiently large and hence the proof is finished.
+□
+Lemma 2.5 will give a uniform bound for all maximizers.
+Corollary 2.6. There is a constant c1 > 0 such that if ω0 ∈ A0 is a maximizer, then
+∥ω0∥∞ ≤ c1.
+(2.11)
+Proof. By Lemmas 2.3 and 2.5, we have
+J′(ω0) = Gsω0 − µ0 ≤ Gsω0,
+on {ω0 > 0}.
+Let p1 = 2 − s. Then we derive from Lemmas 2.1 and 2.4 that ∥Gsω0∥q ≤ C for
+1
+1−s <
+q <
+p1
+1−sp1. Using the fact the J′(t) ≥ K
+2 t1−s for t > t1 due to the hypothesis (H′
+2) with
+t1 a large constant, we obtain ∥ω0∥r ≤ C for 1 ≤ r < p2 with p2 := (1−s)p1
+1−sp1 . Notice that
+p2 = p1 + sp1(p1−1)
+1−sp1
+and sp1(p1−1)
+1−sp1
+> 0 is increasing in p1 ∈ (1, s−1). So, using Lemma 2.1, a
+simple bootstrap argument will prove this lemma.
+□
+Lemma 2.7. It holds
+I0 > 0.
+Proof. We take a function ρ := 1B(0,√
+κ/π) and define ρr(x) := (r−2)ρ(r−1x). It can be seen
+that ρr ∈ A0.
+By the hypothesis (H′
+2), we find
+�
+R2 J(ρr) = o(1)
+�
+R2(ρr)2−s = o(r2s−2),
+as r → +∞.
+On the other hand, a change of variables gives
+�
+R2 ρrGsρr = r2s−2
+�
+R2 ρGsρ.
+Thus, we can take a constant r1 sufficiently large such that
+I0 ≥ E0(ρr1) ≥ c3r2s−2
+1
+> 0.
+□
+
+12
+DAOMIN CAO, SHANFA LAI, GUOLIN QIN
+Corollary 2.8. If ω0 ∈ A0 is a maximizer of E0 over A0, then it holds
+∥Gsω0∥∞ ≥ 2I0κ−1.
+(2.12)
+Proof. One has
+I0 = E0(ω0) ≤ 1
+2
+�
+R2 ω0Gsω0dx ≤ κ
+2∥Gsω0∥∞,
+which implies (2.12) and completes the proof.
+□
+Lemma 2.9. There is a constant η > 0 such that if ω0 ∈ A0 is a maximizer of E0 over
+A0, then we have
+sup
+x∈R2
+�
+|x−y|<1
+ω0(y)dy ≥ η > 0.
+(2.13)
+Proof. Denote η0 := supx∈R2
+�
+|x−y|<1 ω0(y)dy. Let r := (csI−1
+0 κ2)
+1
+2−2s . For any x ∈ R2, we
+calculate
+Gsω0(x) =
+�
+|x−y|<1
+csω0(y)
+|x − y|2−2sdy +
+�
+1≤|x−y|<r
+csω0(y)
+|x − y|2−2sdy +
+�
+|x−y|≥r
+csω0(y)
+|x − y|2−2sdy
+=: A1 + A2 + A3.
+For the first term A1, we use Lemma 2.1 and Corollary 2.6 to obtain
+A1 ≤ C(ηa
+0 + ηb
+0),
+for some a, b ∈ (0, 1).
+For the second term A2, noticing that the annulus {y | 1 < |x − y| < r} can be covered by
+≤ Cr2 disks with radius 1, we find
+A2 ≤ Ccsr2η0.
+Now, by the choice of r, the last integral A3 can be estimated by
+A3 ≤ csr2s−2κ = I0κ−1.
+So, by the arbitrariness of x and Corollary 2.8, we arrive at
+C(ηa
+0 + ηb
+0 + η0) ≥ I0κ−1,
+which implies (2.13) and completes the proof.
+□
+Now we show that the Lagrange multiplier µ0 is uniformly bounded from below.
+Lemma 2.10. There is a constant µ∗ > 0 such that if ω0 ∈ A0 is a maximizer of E0 over
+A0, then there holds
+µ0 ≥ µ∗.
+(2.14)
+Proof. By the previous lemma, one can find a point xη ∈ R2 such that
+�
+|xη−y|<1
+ω0(y)dy ≥ η
+2 > 0.
+
+TRAVELING-WAVE SOLUTIONS FOR THE GSQG EQUATION
+13
+In view of Lemma 2.3, we may assume that ω0 is radially symmetric with respect to the
+origin and non-increasing. Thus, we get
+�
+|y|<1
+ω0(y)dy ≥
+�
+|xη−y|<1
+ω0(y)dy ≥ η
+2.
+For any r > 1 large, there is a point xr ∈ B(0, r) such that ω0(xr) ≤ π−1κr−2. Then we
+infer from the hypothesis (H′
+2) that
+J′(ω0(xr)) = o(r2s−2),
+as r → +∞.
+On the other hand, we have
+Gsω0(xr) ≥
+cs
+(2r)2−2s
+�
+B(0,1)
+ω0 ≥ 4s−2csηr2s−2.
+Thus, by (2.7), we can take a large constant r0 such that
+µ0 ≥ Gsω0(xr0) − J′(ω0(xr0)) ≥ (4s−2csη0 + o(1))r2s−2
+0
+≥ 4s−3csη0r2s−2
+0
+=: µ∗ > 0.
+The proof is therefore finished.
+□
+Lemma 2.11. There exists a constant R∗ > 0 such that if ω0 ∈ A0 is a maximizer of E0
+over A0, then the diameter of the support of ω0 is less than 2R∗. That is,
+diam(spt(ω0)) ≤ 2R∗.
+(2.15)
+Proof. Without loss of generality, we may assume that ω0 is radially symmetric and non-
+increasing with respect to the origin.
+Since
+�
+ω0 = κ < +∞ and ω0(x) = ω0(|x|) is
+non-increasing in |x|, we have
+�
+|x−y|<1
+ω0(y) ≤ C|x|−2
+for |x| large. Then, through similar calculations as the proof of Lemma 2.9, for large |x|
+and any constant r, we get
+Gsω0(x) =
+�
+|x−y|<1
+csω0(y)
+|x − y|2−2sdy +
+�
+1≤|x−y|<r
+csω0(y)
+|x − y|2−2sdy +
+�
+|x−y|≥r
+csω0(y)
+|x − y|2−2sdy
+≤ C(|x|−2a + |x|−2b + r2|x|−2) + csκr2s−2,
+which will tend to 0 if we first take |x| → +∞ then r → +∞. Fixed r such that csκr2s−2 ≤
+µ∗
+2 . Then, one can find a constant R∗ such that Gsω0(x) < µ∗ whenever |x| > R∗. Take
+R4 = max{R3, c1, R∗ + 1}. We infer from Lemmas 2.3 and 2.10 that ω0(x) = 0 for any
+|x| > R∗. The proof of this lemma is hence completed.
+□
+Next, we further study the properties of the maximizers in the special case J(t) = Lt1+ 1
+p
+for some constants L > 0 and p ∈ (0,
+1
+1−s). We first determine the Lagrange multiplier µ0.
+Recall that we denote I0 = supω∈A0 E0(ω) ∈ (0, +∞) to be maximum value of E0.
+
+14
+DAOMIN CAO, SHANFA LAI, GUOLIN QIN
+Lemma 2.12. Suppose that J(t) = Lt1+ 1
+p for some constants L > 0 and p ∈ (0,
+1
+1−s). Let
+ω0 be a maximizer of E0 over A0 with (2.7) for some µ0. Then, we have
+µ0κ = Cs,pI0,
+(2.16)
+for some constant Cs,p depending only on s, p.
+Proof. Let γ := 1 + 1/p.
+We take a family of functions (ω0)t(x) := t−2ω0(t−1x).
+By
+changing of variables, we find
+E0((ω0)t) = t2s−2
+2
+�
+R2 ω0Gsω0 − t2−2γ
+�
+R2 J(ω0).
+Since ω0 is a maximizer, we have
+0 = d
+dt
+����
+t=1
+E0((ω0)t) = (s − 1)
+�
+R2 ω0Gsω0 − (2 − 2γ)
+�
+R2 J(ω0).
+(2.17)
+By (2.7), one has
+Lγωγ−1
+0
+= (Gsω0 − µ0)+.
+Multiplying the above equation by ω0 and integrating, we obtain
+κµ0 =
+�
+R2 ω0Gsω0 − γ
+�
+R2 J(ω0).
+(2.18)
+Then (2.16) follows from simply calculations by using the definition of E0, (2.17) and
+(2.18). The constant Cs,p = Aγ := 2−γ−sγ
+2−s−γ with γ = 1 + 1/p.
+□
+Lemma 2.12 states that the Lagrange multiplier is the same for all maximizers. Set
+ψ0 := Gsω0. Then, ψ0 is radially symmetric and satisfies the following equation by Lemma
+2.3.
+�
+(−∆)sψ0 =
+�
+p
+L(p+1)
+�p
+(ψ0 − µ0)p
++,
+in R2,
+ψ0(x) → 0,
+as |x| → +∞.
+(2.19)
+Thus, the uniqueness result in [22] is applicable for p ∈ (1,
+1
+1−s). Furthermore, Ao et
+al. [5] showed the non-degeneracy of the linearized equation for p ∈ (1, 1+s
+1−s). Note that
+for p ∈ (0, 1] the uniqueness of maximizers has been proved in [33]. Summarizing these
+results, we obtain
+Proposition 2.13. Suppose that J(t) = Lt1+ 1
+p for some constants L > 0 and p ∈ (0,
+1
+1−s).
+Then, up to translations E0 has a unique maximizer ω0 over A0. Moreover, the following
+properties hold:
+(i) ω0 is compact supported and radially symmetric and decreasing about some point;
+(ii) ω0 =
+�
+p
+L(p+1)
+�p
+(Gsω0 − µ0)p
++ for µ0 = κ−1Cs,pE0(ω0) > 0;
+(iii) If in addition p ∈ (1, 1+s
+1−s), then ω0 ∈ C1 and the kernal of the linearized operator
+ω �→ ω − p
+�
+p
+L(p + 1)
+�p
+(Gsω0 − µ0)p−1
++
+Gsω
+
+TRAVELING-WAVE SOLUTIONS FOR THE GSQG EQUATION
+15
+in the space L1 ∩ L∞(R2) is
+span{∂x1ω0, ∂x2ω0}.
+Proof. The existence of a maximizer ω0, (i) and (ii) follow from the above lemmas. For
+uniqueness, we refer to [22, 33]. The non-degeneracy was proved in Proposition 3.2 in [5]
+in terms of ψ0 = Gsω0, form which one can obtain (iii) easily.
+□
+2.2. Existence of traveling-wave solutions via maximization. In this subsection, we
+will obtain the existence of traveling-wave solutions by considering the maximization prob-
+lem (1.7), whose associated limiting problem has been studied in the preceding subsection.
+2.2.1. Existence of maximizers. To obtain the compactness of maximizing sequence, we
+first maximize Eε over a set smaller than Aε given by
+Aε,Γ :=
+�
+ω ∈ L1∩L∞(R2
++)
+��� 0 ≤ ω ≤ Γ,
+spt(ω) ⊂ B(ε−1(d0, 0), ε−1d0/2),
+�
+R2
++
+ω(x)dx = κ
+�
+,
+where Γ > 0 is a number that will be fixed later.
+Lemma 2.14. For given ε, Γ > 0, there exists a function ωε,Γ ∈ Aε,Γ such that
+Eε(ωε,Γ) = sup
+ω∈Aε,Γ
+Eε(ω).
+Proof. Let {ωj}∞
+j=1 ⊂ Aε,Γ be a maximizing sequence. By the definition of Aε,Γ, we know
+that {ωj}∞
+j=1 is uniformly bounded in L1 ∩ L∞(B(ε−1(d0, 0), ε−1d0/2)). Passing to a sub-
+sequence (still denoted by {ωj}∞
+j=1), we may assume ωj → ωε,Γ weakly star in L∞. By the
+weak star convergence, it holds
+0 ≤ ωε,Γ ≤ Γ,
+�
+R2
++
+ωε,Γ(x)dx = lim
+j→+∞
+�
+R2
++
+ωj(x)dx = κ.
+Note that Gs(x, y) ∈ Lr(B(ε−1(d0, 0), ε−1d0/2) × B(ε−1(d0, 0), ε−1d0/2)) for 1 ≤ r <
+1
+1−s.
+We have
+lim
+j→+∞
+cs
+2
+�
+R2
++
+�
+R2
++
+ωj(x)ωj(y)
+|x − y|2−2s dxdy = cs
+2
+�
+R2
++
+�
+R2
++
+ωε,Γ(x)ωε,Γ(y)
+|x − y|2−2s
+dxdy.
+On the other hand, since J is convex, by the lower semi-continuity, we find
+lim
+j→+∞
+�
+R2
++
+J(ωj(x))dx ≥
+�
+R2
++
+J(ωε,Γ(x))dx.
+Therefore,
+sup
+ω∈Aε,Γ
+Eε(ω) = lim
+j→+∞ Eε(ωj) ≤ Eε(ωε,Γ) ≤ sup
+ω∈Aε,Γ
+Eε(ω),
+which implies that ωε,Γ is a maximizer and completes the proof.
+□
+
+16
+DAOMIN CAO, SHANFA LAI, GUOLIN QIN
+For a non-negative function ζ, we shall say that ζ is Steiner symmetric in the x2-variable
+if for any fixed x1, ζ is the unique even function of x2 such that
+ζ(x1, x2) > τ
+if and only if
+|x2| < 1
+2 | {y2 ∈ R | ζ(x1, y2) > τ} |R,
+where | · |R denotes the Lebesgue measure on R.
+For a function 0 ≤ ζ ∈ L1(D) with D a domain symmetric with respect to x1-axis,
+we denote by ζ⋆ the Steiner symmetrization of ζ, which is the unique function in the
+rearrangement class that is Steiner symmetric in the x2-variable (see [54] for more details
+about rearrangement). A key fact about the Steiner symmetrization is the rearrangement
+inequality (see e.g. Theorems 3.7 and 3.9 in [54])
+�
+ζ⋆Gsζ⋆ ≥
+�
+ζGsζ,
+�
+ζ⋆G+
+s ζ⋆ ≥
+�
+ζG+
+s ζ,
+with strict inequality unless ζ(·) ≡ ζ⋆(· + (0, c)) for some c ∈ R.
+Lemma 2.15. For given ε, Γ > 0, let ωε,Γ ∈ Aε,Γ be a maximizer of Eε over Aε,Γ. Then
+ωε,Γ is symmetric non-increasing with respect to some line {x2 = const.}. Moreover, there
+exists a constant µε,Γ ∈ R such that
+
+
+
+
+
+G+
+s ωε,Γ − Wε3−2sx1 − J′(ωε,Γ) ≤ µε,Γ,
+on {ωε,Γ = 0},
+G+
+s ωε,Γ − Wε3−2sx1 − J′(ωε,Γ) = µε,Γ,
+on {0 < ωε,Γ < Γ},
+G+
+s ωε,Γ − Wε3−2sx1 − J′(ωε,Γ) ≥ µε,Γ,
+on {ωε,Γ = Γ}.
+(2.20)
+Proof. The symmetry and monotonicity of ωε,Γ with respect to some line {x2 = const.} is
+an easy consequence of the strict rearrangement inequality.
+For any ω ∈ Aε,Γ, we take a family of test functions as follows:
+ωt := ωε,Γ + t(ω − ωε,Γ),
+t ∈ [0, 1].
+Since ωε,Γ is a maximizer, we have
+0 ≥ dEε(ωt)
+dt
+�����
+t=0+
+=
+�
+R2
++
+(G+
+s ωε,Γ − Wε3−2sx1 − J′(ωε,Γ))(ω − ωε,Γ)dx.
+That is,
+�
+R2
++
+(G+
+s ωε,Γ − Wε3−2sx1 − J′(ωε,Γ))ωdx ≥
+�
+R2
++
+(G+
+s ωε,Γ − Wε3−2sx1 − J′(ωε,Γ))ωε,Γdx.
+Then (2.20) follows by applying an adaption of the bathtub principle (see section 1.14 in
+[54]). The proof is thus complete.
+□
+By the definition of I0 (see (2.8)) in the previous subsection we have
+Lemma 2.16. There are two constants ε0, Γ0 > 0 such that for any ε ∈ (0, ε0) and Γ > Γ0,
+it holds
+sup
+ω∈Aε,Γ
+Eε(ω) ≥ I0 + O(ε2−2s),
+as ε → 0.
+(2.21)
+
+TRAVELING-WAVE SOLUTIONS FOR THE GSQG EQUATION
+17
+Proof. Let ω0 be the maximizer of E0 over A0. We may assume that ω0 is symmetric and
+non-increasing with respect to the origin. Since ω0 has compact support and is bounded,
+we may take Γ0 > 0 large and ε0 > 0 small such that
+¯ω0(x) := ω0(x − ε−1(d0, 0)) ∈ Aε,Γ.
+Then direct computation shows
+sup
+ω∈Aε,Γ
+Eε(ω) ≥ Eε(¯ω0) = E0(ω0) + O(ε2−2s),
+which implies the desired estimate and finishes the proof.
+□
+Lemma 2.17. There are two constants C0, ε1 > 0 such that if ωε,Γ ∈ Aε,Γ is a maximizer
+of Eε over Aε,Γ for ε ∈ (0, ε1) and Γ > Γ0, then it holds
+∥ωε,Γ∥2−s ≤ C0.
+(2.22)
+Proof. Notice that by Lemma 2.16, there exists 0 < ε1 ≤ ε0 such that Eε(ωε,Γ) ≥ I0
+2 > 0
+for ε ∈ (0, ε1) and Γ > Γ0. Thus, we have
+�
+R2
++
+J(ωε,Γ) ≤ 1
+2
+�
+R2
++
+ωε,ΓGsωε,Γ − Eε(ωε,Γ) < 1
+2
+�
+R2
++
+ωε,ΓGsωε,Γ,
+from which, by Lemma 2.2 and a similar argument as the proof of Lemma 2.4, we obtain
+the estimate (2.22) and complete the proof.
+□
+Lemma 2.18. There is a constant ε2 > 0 such that if ωε,Γ ∈ Aε,Γ is a maximizer of Eε
+over Aε,Γ for ε ∈ (0, ε2) and Γ > Γ0, then it holds
+µε,Γ > 0.
+(2.23)
+Proof. For fixed Γ > Γ0, we see from Lemma 2.16 that {ωε,Γ}ε∈(0,ε0) is a maximizing
+sequence of E0. Then Theorem II.2 and Corollary II.1 in [55] give a subsequence (still
+denoted by {ωε,Γ} for convenience) converges to a maximizer ω0 of E0 in L1 ∩ L2−s(R2)
+after suitable translations. Thus, for ε small , we have
+�
+B(pε,R∗) ωε,Γ ≥ κ
+2 for some point
+pε. Here, R∗ is the constant in Lemma 2.11. Since
+�
+ωε,Γ = κ, for R∗ < r < d0
+2ε large, we
+can take a point xr ∈ B(pε, r) ∩ B(ε−1(d0, 0), ε−1d0/2) such that ωε,Γ(xr) ≤ Cr−2. Then,
+using (2.20) and the hypothesis (H′
+2), by similar calculations as the proof of Lemma 2.10,
+we have
+µε,Γ ≥ G+
+s ωε,Γ − Wε3−2sx1 − J′(ωε,Γ) ≥ (4s−2csκ + o(1))r2s−2 − O(ε2−2s),
+which implies (2.23) by taking r = ε− 1
+2 and ε sufficiently small.
+□
+As an immediate consequence of Lemmas 2.15, 2.17 and 2.18, we can obtain the following
+result through a similar argument as the proof of Corollary 2.6. We leave the details of
+the proof to readers.
+Corollary 2.19. There is a constant C1 > 0 such that if ωε,Γ ∈ Aε,Γ is a maximizer of Eε
+over Aε,Γ for ε ∈ (0, ε2) and Γ > Γ0, then
+∥ωε,Γ∥∞ ≤ C1.
+(2.24)
+
+18
+DAOMIN CAO, SHANFA LAI, GUOLIN QIN
+Moreover, if Γ > max{Γ0, C1}, then
+ωε,Γ = (J′)−1 �
+(G+
+s ωε,Γ − Wε3−2sx1 − µε,Γ)+
+�
+,
+in B(ε−1(d0, 0), ε−1d0/2).
+(2.25)
+Having made the necessary preparation, we next establish the existence and properties
+of maximizers for (1.7).
+2.2.2. Existence and properties of maximizers. We first state some basic properties of max-
+imizers.
+Lemma 2.20. For each ε ∈ (0, ε2), there exists a maximizer for (1.7).
+Let ωε be a
+maximizer of Eε over Aε. Then the following assertions hold:
+(i) ωε is Steiner symmetric with respect to some plane {x2 = const.};
+(ii) There is a constant µε such that
+ωε = (J′)−1 �
+(G+
+s ωε − Wε3−2sx1 − µε)+
+�
+,
+in B(ε−1(d0, 0), ε−1d0/2);
+(2.26)
+(iii) The energy satisfies
+I0 + O(ε2−2s) ≤ Eε(ωε) ≤ I0;
+(iv) There exists a constant 0 < C < +∞ independent of ε such that
+lim sup
+ε→0+
+∥ωε∥∞ ≤ C.
+(2.27)
+Proof. Fix a Γ > max{Γ0, C1}, then ωε,Γ ∈ Aε,Γ is a maximizer of Eε over Aε for ε ∈ (0, ε2).
+That is, we obtain a maximizer of Eε over Aε for each ε small, which we denote by ωε for
+simplicity. The properties of these maximizers can be derived by arguments quite similar
+to those in the preceding subsection, so we omit the details.
+□
+We note that (2.26) is not yet sufficient to provide a dynamically possible steady vortex
+flow for the gSQG equation. This is because of the presence of the truncation function
+1B(ε−1(d0,0),ε−1d0/2), which makes ωε and G+
+s ωε − Wε3−2sx1 − µε may be not functional
+dependent in the whole space R2. To get the desired solution, we need to prove that the
+support of ωε is away from the boundary of B(ε−1(d0, 0), ε−1d0/2). We will show that this
+is the case when ε is sufficiently small. It is based on the observation that in order to
+maximize energy, the diameter of the support of a maximizer can not be too large. We
+will reach this conclusion in several steps. We begin by giving a lower bound of µε.
+Lemma 2.21. For any δ > 0, there exist an εδ > 0 such that
+µε ≥ µ∗ − δ,
+∀ ε ∈ (0, εδ),
+(2.28)
+where µ∗ > 0 is the constant in Lemma 2.10
+Proof. Suppose on the contrary that there are constant δ0 > 0 and a sequence {εj}∞
+j=1 with
+εj → 0 as j → ∞ such that lim supj→+∞ µεj ≤ µ∗ − δ0. By Lemma 2.20, we know that
+{ωεj}∞
+j=1 is a maximizing sequence of E0. Then Theorem II.2 and Corollary II.1 in [55] give
+a subsequence (still denoted by {ωεj}∞
+j=1 for convenience) converges to a maximizer ω0 of
+E0 in L1∩L2−s(R2) after suitable translations. Since |x|2s−2 ∈ L
+1
+1−s ,∞ (see [42] for the weak
+
+TRAVELING-WAVE SOLUTIONS FOR THE GSQG EQUATION
+19
+Lp spaces), we deduce from the generalized Young’s inequality ( Theorem 1.4.25 in [42])
+that Gsωεj converges to Gsω0 strongly in L
+2−s
+(1−s)2 . Then, extracting another subsequence,
+one may assume that both ωεj and Gsωεj converge to ω0 and Gsω0 a.e. respectively.
+On the support of ωεj, the Euler-Lagrange equation (2.26) implies
+µεj = G+
+s ωεj − Wε3−2s
+j
+x1 − J′(ωεj) ≥ Gsωεj − J′(ωεj) + O(ε2−2s).
+Letting j → +∞, the a.e. convergence and the assumption on µεj imply that
+µ∗ − δ0 ≥ Gsω0 − J′(ω0).
+on the other hand, by the Euler-Lagrange equation for ω0 and Lemma 2.10, one has
+Gsω0 − J′(ω0) = µ0,
+for some µ0 ≥ µ∗, which is a contradiction. The proof of this lemma is thus finished.
+□
+Now, we determine the size of the support of ωε.
+Lemma 2.22. There exists a constant R > 0 such that for ε sufficiently small and for any
+maximizer ωε, the support of ωε is contained in a disk of radius R.
+Proof. By the previous lemma, we can take ε small such that µε ≥ µ∗
+2 . Then the Euler-
+Lagrange equation (2.26) implies
+J′(ωε) = (G+
+s ωε − Wε3−2sx1 − µε)+ ≤ (Gsωε − µ∗
+2 + O(ε2−2s))+.
+(2.29)
+Theorem II.2 and Corollary II.1 in [55] provide a subsequence (still denoted by {ωε}),
+which tends to a maximizer ω0 of E0 in L1 ∩ L2−s after translation. So, for any δ > 0, we
+can find a constant εδ > 0 such that
+∥ωε − ωε
+0∥2−s ≤ δ,
+∀ ε ∈ (0, εδ),
+where ωε
+0 is a translation of ω0. By Lemma 2.11, we conclude that spt(ωε
+0) ⊂ B(˜xε, R∗) for
+some point ˜xε. Thus, we get by using the H¨older inequality
+�
+B(˜xε,R∗)
+ωε ≥
+�
+B(˜xε,R∗)
+ωε
+0 − ∥ωε − ωε
+0∥L1(B(˜xε,R∗))
+≥ κ − C∥ωε − ωε
+0∥2−s ≥ κ − Cδ,
+which, combined with
+�
+ωε = κ, implies
+�
+B(˜xε,R∗)c ωε ≤ Cδ,
+∀ ε ∈ (0, εδ).
+(2.30)
+Take a large R ≥ R∗ such that
+csκ
+|R−R∗|2−2s ≤ µ∗
+6 . Then for any x ∈ B(ε−1(d0, 0), ε−1d0/2)\
+B(˜xε, R), using Lemma 2.1 and (2.27), we have
+Gsωε(x) =
+�
+B(˜xε,R∗)
+csωε(y)
+|x − y|2−sdy +
+�
+B(˜xε,R∗)c
+csωε(y)
+|x − y|2−sdy
+
+20
+DAOMIN CAO, SHANFA LAI, GUOLIN QIN
+≤
+csκ
+|R − R∗|2−2s + C
+���
+B(˜xε,R∗)c ωε
+�a
++
+��
+B(˜xε,R∗)c ωε
+�b�
+(2.31)
+≤ µ∗
+6 + C(δa + δb) ≤ µ∗
+3 ,
+by taking δ small. (2.29) implies that ωε(x) = 0 for arbitrary x ∈ B(ε−1(d0, 0), ε−1d0/2) \
+B(˜xε, R). Hence, we find the constant R > 0 such that the support of ωε is contained in a
+disk of radius R for ε small and finish the proof.
+□
+To find the location of ωε, let xε := κ−1 �
+xωε be the center of mass. Then xε = (ε−1dε, 0)
+for some dε > 0. By replacing R with 2R, we may assume that for ε small,
+spt(ωε) ⊂ B(xε, R).
+Lemma 2.23. There holds
+dε → d0,
+as ε → 0.
+(2.32)
+Proof. Since xε ∈ B(ε−1(d0, 0), ε−1d0/2), we deduce d0
+2 < dε < 3d0
+2 . Up to a subsequence,
+we may assume that
+dε → d∗ ∈
+�d0
+2 , 3d0
+2
+�
+,
+as ε → 0.
+We prove that d∗ = d0. Indeed, take
+¯ωε(x) := ωε(x + xε − (ε−1d0, 0)) ∈ Aε.
+Noticing that ωε is a maximizer, i.e. Eε(ωε) ≥ Eε(¯ωε), we find
+cs
+2
+�
+R2
++
+ωε(x)ωε(y)
+|x − ¯y|2−2s dxdy + Wε3−2s
+�
+R2
++
+x1ωε(x)dx
+≤ cs
+2
+�
+R2
++
+¯ωε(x)¯ωε(y)
+|x − ¯y|2−2s dxdy + Wε3−2s
+�
+R2
++
+x1¯ωε(x)dx.
+Letting ε → 0 in the above inequality, we obtain
+csκ
+23−2sd2−2s
+∗
++ Wd∗ ≤
+csκ
+23−2sd2−2s
+0
++ Wd0,
+which implies d∗ = d0 since d0 is the unique minimizer of the function h(τ) =
+csκ
+23−2sτ 2−2s +Wτ
+on (0, +∞).
+Noting that, by the above proof, one can see that each sequence in {dε} has a convergent
+subsequence that converges to the same limit d0. Then a simple contradiction argument
+shows that {dε} itself y converges to d0. The proof of this lemma is thus complete.
+□
+Now we are ready to prove Theorem 1.5.
+Proof of Theorem 1.5: Suppose J(t) =
+� t
+0 f −1(τ)dτ for some f satisfying (H1) and
+(H2). Combining Lemma 2.20–2.23, we can finish proof of Theorem 1.5.
+□
+In view of Lemmas 2.22 and 2.23, for sufficiently small ε, the support of ωε is far away
+from the boundary of B(ε−1(d0, 0), ε−1d0/2). We extend ωε to the half-space R2
++ by defining
+
+TRAVELING-WAVE SOLUTIONS FOR THE GSQG EQUATION
+21
+ωε to be 0 in R2
++ \ B(ε−1(d0, 0), ε−1d0/2). With this fact in hand, we can now show that
+ωtr,ε(x) := ωε(x) − ωε(¯x) is a steady solution in the sense of (1.3). More precisely, we have
+Lemma 2.24. Suppose that J(t) =
+� t
+0 f −1(τ)dτ for some f satisfying (H1) and (H2) with
+f ∈ C1−2s if 0 < s < 1
+2. Let ωε be a maximizer obtained above and ωtr,ε(x) := ωε(x)−ωε(¯x),
+then provided that ε is sufficiently small, it holds
+�
+R2 ωtr,ε∇⊥(Gsωtr,ε − Wε3−2sx1) · ∇ϕdx = 0,
+∀ ϕ ∈ C∞
+0 (R2).
+(2.33)
+Proof. Using the regularity theory for fractional Laplacians (Propositions 2.8 and 2.9 in
+[60]), for s ̸=
+1
+2, we can prove that Gsωtr,ε ∈ C0,1 by our assumptions on f, the fact
+ωε ∈ L1 ∩ L∞ and Gsωtr,ε ∈ L∞ due to Lemma 2.1. For s = 1
+2, we use the standard theory
+on potentials to deduce that Gsωtr,ε ∈ W 2s,p for any p > 1. Therefore, the integral in (2.33)
+makes sense for all s ∈ (0, 1).
+By the definition of ωtr,ε, it is sufficient to prove
+�
+R2
++
+ωε∇⊥(G+
+s ωε − Wε3−2sx1) · ∇ϕdx = 0,
+∀ ϕ ∈ C∞
+0 (R2
++).
+Recall that
+ωε = (J′)−1 �
+(G+
+s ωε − Wε3−2sx1 − µε)+
+�
+,
+∀ x ∈ R2
++.
+Let F(t) =
+� t
+0(J′)−1(τ)dτ. For any ϕ ∈ C∞
+0 (R2
++), we apply integrate by parts to obtain
+�
+R2
++
+ωε∇⊥(G+
+s ωε − Wε3−2sx1) · ∇ϕdx
+= −
+�
+B(ε−1(d0,0),ε−1d0/2)
+F((G+
+s ωε − Wε3−2sx1 − µε)+)(∂x2∂x1ϕ − ∂x1∂x2ϕ)dx = 0,
+where we have used the fact (G+
+s ωε − Wε3−2sx1 − µε)+ = 0 on ∂B(ε−1(d0, 0), ε−1d0/2).
+This completes the proof.
+□
+Now we are ready to prove Theorem 1.1.
+Proof of Theorem 1.1: Let ωtr,ε(x) := ωε(x) − ωε(¯x). Then the statements of Theorem
+1.1 follow from Theorem 1.5 and Lemma 2.24.
+□
+3. Uniqueness of maximizers
+In this section, we show the uniqueness of maximizers in the case J(t) = Lt1+ 1
+p for
+some L > 0 and p ∈ (0,
+1
+1−s). We first establish some finer estimates of the asymptotic
+behavior by careful analysis. Then, we obtain the uniqueness by using the non-degeneracy
+of linearized equations (i.e. Proposition 2.13 (iii)). In what follows, we sometimes leave
+out the domain in the integral symbol and abbreviate it to
+�
+when there is no risk of
+confusion.
+
+22
+DAOMIN CAO, SHANFA LAI, GUOLIN QIN
+3.1. Refined estimates on asymptotic behaviors. In the cases J(t) = Lt1+ 1
+p, we study
+in detail the asymptotic behaviors of the maximizers.
+Lemma 3.1. Suppose that J(t) = Lt1+ 1
+p for some L > 0 and p ∈ (0,
+1
+1−s). For each ε small,
+let ωε be a maximizer of Eε over Aε. Define ˜ωε(x) := ωε(x + xε), where xε = κ−1 �
+xωε is
+the center of mass of ωε. Let ω0 be the unique maximizer of E0 over A0 with
+�
+xω0 = 0.
+Then, we have
+lim
+ε→0 ∥˜ωε − ω0∥∞ = 0.
+Proof. We first prove the convergence of the energy. On the one hand, since ˜ωε ∈ A0 and
+ω0 is a maximizer of E0 over A0, we have
+E0(ω0) ≥ E0(˜ωε).
+On the other hand, noticing that ω0(· − ε−1(d0, 0)) ∈ Aε, we deduce
+Eε(ωε) ≥ Eε(ω0(· − ε−1(d0, 0))).
+It is easy to see that Eε(ωε) = E0(˜ωε) + O(ε2−2s) and Eε(ω0(· − ε−1(d0, 0))) = E0(ω0) +
+O(ε2−2s). So, we conclude
+E0(ω0) ≥ E0(˜ωε) ≥ E0(ω0) + O(ε2−2s),
+which implies
+lim
+ε→0 E0(˜ωε) = E0(ω0).
+(3.1)
+Note that for arbitrary ε small, one has ∥˜ωε∥∞ = ∥ωε∥∞ ≤ C for some C > 0 independent
+of ε and spt(˜ωε) ⊂ B(0, R) for some R independent of ε by previous lemmas. Therefore,
+we may assume that up to a subsequence ˜ωε → ˆω0 weakly star in L∞ for some ˆω0. Similar
+argument as the proof of Lemma 2.14 shows that E0(ˆω0) = limε→0 E0(˜ωε) = E0(ω0), so ˆω0
+is a maximizer of E0 over A0. Moreover, by the weakly star convergence, we find
+�
+xˆω0 = lim
+ε→0
+�
+x˜ωε = 0.
+This indicates that ˆω0 = ω0 by the uniqueness of maximizers of E0 due to Proposition 2.13.
+We have proved the weakly star convergence. Now, we show the strong convergence.
+Since ωε is uniformly bounded in L1 ∩ L∞, then by Lemma 2.1 and Proposition 2.9 in [60],
+we have Gsωε is uniformly bounded in Cα for some 0 < α < 1. Then, we deduce from the
+representation (2.26) that ωε is uniform bounded in Cα. So by the Arela-Ascoli theorem,
+we may assume that up to a subsequence, {˜ωε} strongly converges in L∞ to ω0.
+Since each sequence in {˜ωε} has a convergent subsequence that strongly converges in
+L∞ to the same limit ω0, a simple contradiction argument shows that {˜ωε} itself strongly
+converges in L∞ to ω0. The proof of this lemma is thus finished.
+□
+We derive the integral representation for the Lagrange multipliers µε in terms of ωε.
+
+TRAVELING-WAVE SOLUTIONS FOR THE GSQG EQUATION
+23
+Lemma 3.2. Suppose that J(t) = Lt1+ 1
+p for some L > 0 and p ∈ (0,
+1
+1−s). Let ωε be a
+maximizer of Eε over Aε satisfying (2.26) for some µε. Then we have
+µεκ = AγE0(ωε) + BγWε3−2s
+�
+R2
++
+x1ωε + Cγ
+�
+R2
++
+�
+R2
++
+csωε(x)ωε(y)
+|x − ¯y|2−2s dxdy,
+(3.2)
+where the constants Aγ = 2−γ−sγ
+2−s−γ , Bγ = 2−s
+s−1 −
+2−γ−γs
+2(s−1)(2−s−γ), Cγ = − 2−γ−γs
+2(2−s−γ) and γ = 1 + 1
+p.
+Proof. We may assume that ωε is symmetric non-increasing in x2. Then Lemmas 2.22 and
+2.23 shows that the support of ωε is far away from the boundary ∂B(ε−1(d0, 0), ε−1d0/2).
+So the function (ωε)t(x) := t−2ωε(t−1x) is supported in B(ε−1(d0, 0), ε−1d0/2) for t ≈ 1 and
+hence belongs to Aε. Since ωε is a maximizer, we deduce
+dEε((ωε)t)
+dt
+�����
+t=1
+= 0,
+which, by the definition of Eε and straightforward computations, gives
+(s − 1)
+�
+R2
++
+ωεG+
+s ωε − Wε3−2s
+�
+R2
++
+x1ωε − (2 − 2γ)
+�
+R2
++
+J(ωε) = 0.
+Then, we infer from the above identity that
+�
+R2
++
+ωεGsωε = 2 − 2γ
+s − 1
+�
+R2
++
+J(ωε)+ W
+s − 1ε3−2s
+�
+R2
++
+x1ωε +cs
+�
+R2
++
+�
+R2
++
+ωε(x)ωε(y)
+|x − ¯y|2−2s dxdy. (3.3)
+By the definition of E0 and (3.3), we deduce
+E0(ωε) = 1
+2
+�
+R2
++
+ωεGsωε −
+�
+R2
++
+J(ωε)
+= 2 − s − γ
+s − 1
+�
+R2
++
+J(ωε) +
+W
+2(s − 1)ε3−2s
+�
+R2
++
+x1ωε + cs
+2
+�
+R2
++
+�
+R2
++
+ωε(x)ωε(y)
+|x − ¯y|2−2s dxdy.
+(3.4)
+On the other hand, multiplying the equation (J′) (ωε) = (G+
+s ωε − Wε3−2sx1 − µε)+ by
+ωε and integrating, we have
+µεκ =
+�
+R2
++
+ωεG+
+s ωε − Wε3−2s
+�
+R2
++
+x1ωε − γ
+�
+R2
++
+J(ωε)
+= 2 − γ − γs
+s − 1
+�
+R2
++
+J(ωε) + 2 − s
+s − 1Wε3−2s
+�
+R2
++
+x1ωε
+= 2 − γ − γs
+2 − s − γ E0(ωε) +
+�2 − s
+s − 1 −
+2 − γ − γs
+2(s − 1)(2 − s − γ)
+�
+Wε3−2s
+�
+R2
++
+x1ωε
+− 2 − γ − γs
+2(2 − s − γ)
+�
+R2
++
+�
+R2
++
+csωε(x)ωε(y)
+|x − ¯y|2−2s dxdy,
+(3.5)
+
+24
+DAOMIN CAO, SHANFA LAI, GUOLIN QIN
+where we have used (3.3) and (3.4) in above calculations. So we obtain (3.2) and finish
+the proof of this lemma.
+□
+As an immediate consequence of Lemma 3.2, we obtain the following estimate for µε.
+Corollary 3.3. One has
+|µε − µ0| → 0,
+as ε → 0+.
+Next, we derive another integral identity for general J, which determines the location of
+ωε.
+Lemma 3.4. Suppose that J satisfies (H′
+1) and (H′
+2). Let ωε be a maximizer of Eε over
+Aε. Then the following identity is true.
+2(1 − s)cs
+�
+R2
++
+�
+R2
++
+ωε(x)(x1 + y1)ωε(y)
+|x − ¯y|4−2s
+dxdy = Wε3−2sκ.
+(3.6)
+Proof. We may assume that ωε is symmetric non-increasing in x2. Then Lemmas 2.22 and
+2.23 show that the support of ωε is far away from the boundary ∂B(ε−1(d0, 0), ε−1d0/2).
+So the function (ωε)t(x) := ωε(x + (t, 0)) is supported in B(ε−1(d0, 0), ε−1d0/2) for t ≈ 0
+and hence belongs to Aε. Since ωε is a maximizer, we deduce
+dEε((ωε)t)
+dt
+�����
+t=0
+= 0,
+which, by the definition of Eε and straightforward computations, implies (3.6). So the
+proof this lemma is completed.
+□
+Using the identity (3.6), we can sharpen the estimate of xε in Lemma 2.23 as follows.
+Lemma 3.5. Suppose that J satisfies (H′
+1) and (H′
+2). Let ωε be a maximizer of Eε over
+Aε. If xε = κ−1 �
+xωε = (ε−1dε, 0), then we have
+|dε − d0| = O(ε2).
+Proof. By Lemma 2.22, we know spt(ωε) ⊂ B(xε, R) for some R independent of ε and
+xε = (ε−1dε, 0) with dε → d0 as ε → 0+. We use the Taylor expansion to calculate the
+left-hand side of (3.6)
+� � ωε(x)(x1 + y1)ωε(y)
+|x − ¯y|4−2s
+dxdy = ε3−2s
+� � ˜ωε(x)(2dε + ε(x1 + y1))˜ωε(y)
+|(2dε, 0) + ε(x − ¯y)|4−2s
+dxdy
+=ε3−2s
+� �
+˜ωε(x)˜ωε(y)
+�
+1
+(2dε)3−2s + (2s − 3)ε(x1 + y1)
+(2dε)4−2s
++ O(ε2)
+�
+dxdy
+= ε3−2sκ2
+(2dε)3−2s + O(ε5−2s).
+(3.7)
+Here we have used that
+�
+x˜ωε(x)dx = 0, which follows from the definition of ˜ωε.
+
+TRAVELING-WAVE SOLUTIONS FOR THE GSQG EQUATION
+25
+Then using (3.6) and (3.7), we obtain
+2(1 − s)csκ2
+(2dε)3−2s
+− Wκ = O(ε2),
+which implies
+d2s−3
+ε
+=
+22−2sW
+(1 − s)csκ + O(ε2) = d2s−3
+0
++ O(ε2).
+(3.8)
+Here we have used the definition of d0. Since d(t2s−3)
+dt
+���
+t=d0 ̸= 0, we infer from (3.8) that
+|dε − d0| = O(ε2) and complete the proof.
+□
+3.2. Proof of the uniqueness. In this subsection, we prove the uniqueness of maximizers
+when J(t) = Lt1+ 1
+p for some L > 0 and p ∈ (1,
+1
+1−s). Let ωε be a maximizer of Eε over
+Aε. Recall that xε = κ−1 �
+xωε = (ε−1dε, 0) and ˜ωε = ωε(· + ε−1(dε, 0)). Set Lγ =
+�
+1
+Lγ
+�
+1
+γ−1
+with γ = 1 + 1
+p. Then, by (2.26), (3.2) and (3.6), we obtain the equations of ˜ωε and dε:
+�
+˜ωε(x) = Lγ (Gs˜ωε(x) − κ−1AγE0(˜ωε) + Sε(˜ωε, dε))p
++ ,
+2(1 − s)cs
+� � ˜ωε(x)(2dε+ε(x1+y1))˜ωε(y)
+|(2dε,0)+ε(x−¯y)|4−2s
+dxdy − Wκ = 0,
+(3.9)
+where the operator Sε(˜ωε, dε) is given by
+Sε(˜ωε, dε) = − Wε3−2s(x1 + ε−1dε) −
+�
+ε2−2scs˜ωε(y)
+|(2dε, 0) + ε(x − ¯y)|2−2sdy
+−κ−1BγWε2−2sdε − κ−1Cγ
+� �
+ε2−2scs˜ωε(x)˜ωε(y)
+|(2dε, 0) + ε(x − ¯y)|2−2sdxdy.
+(3.10)
+For simplicity of notations, we denote the operators Pε and Qε as follows.
+Pε(ω, d) := Lγ
+�
+Gsω(x) − κ−1AγE0(ω) + Sε(ω, d)
+�p
++ ,
+and
+Qε(ω, d) := 2(1 − s)cs
+� � ω(x)(2d + ε(x1 + y1))ω(y)
+|(2d, 0) + ε(x − ¯y)|4−2s
+dxdy − Wκ.
+Noting that for dε ∈ ( d0
+2 , 3d0
+2 ), x, y ∈ B(0, R) and ε small, we always have |(2dε, 0) +
+ε(x − ¯y)| is bounded from below by a positive constant. Therefore, we easily obtain the
+following lemma, whose proof we leave to readers.
+Lemma 3.6. For (ω, d) ∈ X 1 := L1(B(0, R)) × ( d0
+2 , 3d0
+2 ) and ε sufficiently small, there
+holds
+∥Sε(ω, d)∥∞ + ∥∇Sε(ω, d)∥X 1→L∞ + ∥∇2Sε(ω, d)∥X 1×X 1→L∞ = O(ε2−2s).
+Using the definition of E0 and (2.7), one can see that
+E′
+0(ω0)φ = µ0
+�
+φ.
+
+26
+DAOMIN CAO, SHANFA LAI, GUOLIN QIN
+If we define
+P0(ω, d) := Lγ
+�
+Gsω(x) − κ−1AγE0(ω)
+�p
++ ,
+and
+Q0(ω, d) := 2(1 − s)cs
+(2d)3−2s
+� �
+ω(x)ω(y)dxdy − Wκ.
+By direct calculations, one can verify that
+ω0 = P0(ω0, d0),
+∇P0(ω0, d0)(φ, l) = pLγ(Gsω0 − κ−1AγE0(ω0))p−1
++
+�
+Gsφ − κ−1Aγµ0
+�
+φ
+�
+and
+Q0(ω0, d0) = 0,
+∇Q0(ω0, d0)(φ, l) = (1 − s)csκ
+21−2sd3−2s
+0
+�
+φ − (1 − s)(3 − 2s)csκ2
+22−2sd4−2s
+0
+l.
+By Lemmas 2.23 and 3.1, ˜ωε and dε take the following form:
+˜ωε = ω0 + φε,
+dε = d0 + lε,
+where φε ∈ L∞(B(0, R)) satisfies
+�
+xφε =
+�
+φε = 0, ∥φε∥∞ = o(1) and lε is a real number
+with |lε| = o(1) as ε → 0.
+Then we deduce from (3.9) that (φ, l) = (φε, lε) solves the following system
+�
+φ − pLγ(Gsω0 − κ−1AγE0(ω0))p−1
++
+�
+Gsφ − κ−1Aγµ0
+�
+φ
+�
+= R1(φ, l),
+(1−s)csκ
+21−2sd3−2s
+0
+�
+φ − (1−s)(3−2s)csκ2
+22−2sd4−2s
+0
+l = R2(φ, l),
+(3.11)
+where the super-linear terms R1(φ, l) and R2(φ, l) are given by
+R1(φ, l) = Lγ
+�
+Gs(ω0 + φ) − κ−1AγE0(ω0 + φ) + Sε(ω0 + φ, d0 + l)
+�p
++
+−Lγ
+�
+Gsω0 − κ−1AγE0(ω0)
+�p
++ − pLγ(Gsω0 − κ−1AγE0(ω0))p−1
++
+�
+Gsφ − κ−1Aγµ0
+�
+φ
+�
+(3.12)
+and
+R2(φ, l) = 2(1 − s)cs
+� � (ω0 + φ)(x)(2d0 + 2lε + ε(x1 + y1))(ω0 + φ)(y)
+|(2d0 + 2lε, 0) + ε(x − ¯y)|4−2s
+dxdy − Wκ
+−(1 − s)csκ
+21−2sd3−2s
+0
+�
+φ + (1 − s)(3 − 2s)csκ2
+22−2sd4−2s
+0
+l
+(3.13)
+To study the linearized equation (3.11), we set
+L0φ := φ − pLγ(Gsω0 − κ−1AγE0(ω0))p−1
++
+�
+Gsφ − κ−1Aγµ0
+�
+φ
+�
+.
+Using the non-degeneracy Proposition 2.13 (iii), we have the following key estimate of L0.
+Lemma 3.7. There exists a constant C0 > 0 such that
+∥L0φ∥2 ≥ C0∥φ∥2,
+∀ φ ∈ L2(B(0, R)) with
+�
+xφ =
+�
+φ = 0.
+
+TRAVELING-WAVE SOLUTIONS FOR THE GSQG EQUATION
+27
+Proof. We normalize φ by replacing φ with φ/∥φ∥2 so that ∥φ∥2 = 1. Now suppose on the
+contrary that there is a sequence {φn}∞
+n=1 ⊂ L2(B(0, R)) such that
+∥φn∥ = 1,
+�
+xφn =
+�
+φn = 0,
+while
+∥L0φn∥ → 0,
+as n → +∞.
+That is
+φn = pLγ(Gsω0 − κ−1AγE0(ω0))p−1
++ Gsφn + o(1).
+(3.14)
+We first assume that up to a subsequence φn → φ∞ weakly in L2. Note that the support
+of (Gsω0 − κ−1AγE0(ω0))p−1
++
+is contained in B(0, R). Then by the regularity theory on
+potentials and the compact embedding theorem for fractional Sobolev spaces (see e.g.
+[34, 62]), we have up to a subsequence pLγ(Gsω0 − κ−1AγE0(ω0))p−1
++ Gsφn → pLγ(Gsω0 −
+κ−1AγE0(ω0))p−1
++ Gsφ∞ strongly in L2(B(0, R)) and hence by (3.14), we conclude φn → φ∞
+strongly in L2. Thus, we have φ∞ satisfies
+φ∞ = pLγ(Gsω0 − κ−1AγE0(ω0))p−1
++ Gsφ∞,
+∥φ∞∥2 = 1,
+�
+xφ∞ =
+�
+φ∞ = 0.
+By the regularity theory on potentials, Sobolev embedding and bootstrap argument, one
+can prove that φ∞ ∈ L∞(B(0, R)). Thus, we infer from Proposition 2.13 (iii) that φ∞ ∈
+span{∂x1ω0, ∂x2ω0}, which combined with
+�
+xφ∞ = 0 implies φ∞ ≡ 0 due to the radial
+symmetry of ω0. This is a contradiction with ∥φ∞∥2 = 1. Therefore, this lemma must hold
+true.
+□
+Now we study the right-hand side of (3.11).
+Lemma 3.8. For any ǫ > 0, there is a δ > 0 such that if φ1, φ2 ∈ L∞(B(0, R)) and
+l1, l2 ∈ R satisfy
+�
+φ1 =
+�
+φ2 = 0 and
+∥φ1∥∞ + ∥φ2∥∞ + |l1| + |l2| ≤ δ,
+then we have
+lim sup
+ε→0 (∥R1(φ1, l1) − R1(φ2, l2)∥2 + |R2(φ1, l1) − R2(φ2, l2)|) ≤ ǫ(∥φ1 − φ2∥2 + |l1 − l2|).
+Proof. By direct calculations, we find
+R1(φ1, l1) − R1(φ2, l2)
+=Lγ
+�
+Gs(ω0 + φ1) − κ−1AγE0(ω0 + φ1) + Sε(ω0 + φ1, d0 + l1)
+�p
++
+− Lγ
+�
+Gs(ω0 + φ2) − κ−1AγE0(ω0 + φ2) + Sε(ω0 + φ2, d0 + l2)
+�p
++
+− pLγ(Gsω0 − κ−1AγE0(ω0))p−1
++ Gs (φ1 − φ2)
+=pLγ
+� 1
+0
+��
+Gs(ω0 + φ(τ)) − κ−1AγE0(ω0 + φ(τ)) + Sε(ω0 + φ(τ), d0 + l(τ))
+�p−1
++
+−(Gsω0 − κ−1AγE0(ω0))p−1
++
+�
+dτGs(φ1 − φ2) + O(∥∇Sε(ω0, d0)∥)∥φ1 − φ2∥2,
+
+28
+DAOMIN CAO, SHANFA LAI, GUOLIN QIN
+where φ(τ) = τφ1 + (1 − τ)φ2, l(τ) = τl1 + (1 − τ)l2. To continue, we expand Gs(ω0 +
+φ(τ)) − κ−1AγE0(ω0 + φ(τ)) + Sε(ω0 + φ(τ), d0 + l(τ)) as follows.
+Gs(ω0 + φ(τ)) − κ−1AγE0(ω0 + φ(τ)) + Sε(ω0 + φ(τ), d0 + l(τ))
+=Gsω0 − κ−1AγE0(ω0)
++ Gsφ(τ) +
+�
+φ(τ)Gsω0 −
+�
+(J(ω0 + φ(τ)) − J(ω0)) + 1
+2
+�
+φ(τ)Gsφ(τ) + O(∥Sε∥∞)
+=Gsω0 − κ−1AγE0(ω0) + O(∥φ(τ)∥∞ + ∥φ(τ)∥2
+∞ + ∥Sε∥∞),
+where we have used H¨older’s inequality, Lemmas 2.1 and 2.2. Using h(t) = tp−1
++
+∈ C0,αp
+with αp = min{1, p − 1} ∈ (0, 1], we continue calculating R1(φ1, l1) − R1(φ2, l2).
+R1(φ1, l1) − R1(φ2, l2)
+=O((∥φ(τ)∥∞ + ∥φ(τ)∥2
+∞ + ∥Sε∥)αp)Gs(φ1 − φ2) + O(∥∇Sε(ω0, d0)∥)∥φ1 − φ2∥
+Thus, if ∥φ1∥, ∥φ2∥ ≤ δ, then we obtain
+∥R1(φ1, l1) − R1(φ2, l2)∥2 ≤ C(δαp + ε(2−2s)αp)∥φ1 − φ2∥2 = oδ,ε(1)∥φ1 − φ2∥2.
+(3.15)
+Since R2 is C2 smooth, it is easy to see that
+∥R2(φ1, l1)−R2(φ2, l2)∥ ≤ C(∥φ1 −φ2∥2 +|l1 −l2|2) = oδ,ε(1)(∥φ1 −φ2∥+|l1 −l2|). (3.16)
+The proof of this lemma is thus finished.
+□
+Now, we are ready to prove our uniqueness.
+Proof of Theorem 1.6:
+Suppose on the contrary that there are two different maximizers ω1,ε ̸= ω2,ε. Then,
+we obtain two pairs (φ1,ε, l1,ε) and (φ2,ε, l2,ε) such that ∥φ1,ε − φ2,ε∥2 + |l1,ε − l2,ε| ̸= 0,
+�
+xφ1,ε =
+�
+φ1,ε =
+�
+xφ2,ε =
+�
+φ2,ε = 0 and ∥φ1,ε∥∞ + ∥φ2,ε∥∞ + |l1,ε| + |l2,ε| = o(1) as
+ε → 0. More over, both (φ1,ε, l1,ε) and (φ2,ε, l2,ε) satisfy the system (3.11).
+On the one hand, by Lemma 3.7, the difference between the left-hand side satisfies
+∥L0(φ1,ε−φ2,ε)∥2+(1 − s)(3 − 2s)csκ2
+22−2sd4−2s
+0
+|l1,ε−l2,ε| ≥ C0∥φ1,ε−φ2,ε∥2+(1 − s)(3 − 2s)csκ2
+22−2sd4−2s
+0
+|l1,ε−l2,ε|.
+On the other hand, by Lemma 3.8, the difference between the right-hand side satisfies
+∥R1(φ1,ε, l1,ε)−R1(φ2,ε, l2,ε)∥2+|R2(φ1,ε, l1,ε)−R2((φ2,ε, l2,ε)| = o(1)(∥φ1,ε−φ2,ε∥2+|l1,ε−l2,ε|),
+which leads to a contradiction for ε small. Therefore, we have established the uniqueness
+of maximizers for ε small and completed the proof of Theorem 1.6.
+□
+4. Nonlinear orbital stability
+This section is devoted to investigating the nonlinear stability of traveling solutions
+obtained in Section 2. We first prove a general stability theorem in a similar spirit as [13],
+where the stability of vortex pairs for the 2D Euler equation was considered. Throughout
+this section, we always assume that D ⊂ R2
++ is the domain D = R2
++ if s > 1
+2 and D =
+{x1 ≥ 1} if 0 < s ≤ 1
+2.
+
+TRAVELING-WAVE SOLUTIONS FOR THE GSQG EQUATION
+29
+4.1. A general stability theorem on the set of maximizers. Let ξ be a non-negative
+Lebesgue integrable function on R2, we denote by R(ξ) the set of (equimeasurable) rear-
+rangements of ξ on D defined by
+R(ξ) =
+�
+0 ≤ ζ ∈ L1(D)
+���|{x : ζ(x) > τ}| = |{x : ξ(x) > τ}|, ∀ τ > 0
+�
+.
+Note that all functions in R(ξ) have the same Lq norm. Following [13], we also define
+R+(ξ) =
+�
+ζ1S
+��ζ ∈ R(ξ),
+S ⊂ D measurable
+�
+,
+and
+R(ξ)w =
+�
+ζ ≥ 0 measurable
+����
+�
+D
+(ζ − α)+dx ≤
+�
+D
+(ξ − α)+dx,
+∀α > 0
+�
+.
+It is easy to see that the inclusions R(ξ) ⊂ R+(ξ) ⊂ R(ξ)w hold. The key fact is that
+R(ξ)w is convex and is the weak closure of R(ξ) in Lp (see [13, 35]).
+We denote the kinetic energy as
+E(ζ) = 1
+2
+�
+D
+ζ(x)G+
+s ζ(x)dx,
+and the impulse
+I(ζ) =
+�
+D
+x1ζ(x)dx.
+For a constant W, set the energy functional as
+˜EW(ζ) := 1
+2
+�
+D
+ζ(x)G+
+s ζ(x)dx − W
+�
+D
+x1ζ(x)dx.
+For a function ζ0 and constant W > 0, we will consider the maximization problem
+sup
+ζ∈R(ζ0)w
+˜EW(ζ).
+The following two lemmas are needed.
+Lemma 4.1. Suppose ζ ∈ L1 ∩ Lr(D) for some s−1 < r ≤ +∞ if 0 < s ≤
+1
+2 and
+2
+2s−1 < r ≤ +∞ if 1
+2 < s < 1. Then, one has
+|G+
+s ζ(x)| ≤ C (∥ζ∥r + ∥ζ∥1) min{x1, x
+2s− 2
+r
+1
+},
+∀ x ∈ D.
+(4.1)
+Proof. We first consider the case x1 ≥ 1. By the mean value theorem, there holds
+G+
+s (x, y) ≤
+4csx1y1
+|x − y|4−2s,
+∀x, y ∈ R+.
+Therefore, using H¨older’s inequality and y1 < x1 + |x − y| ≤ 2|x − y| if |x − y| > x1, we
+have
+|G+
+s ζ(x)| ≤
+�
+R2
++
+G+
+s (x, y)|ζ(y)|dy
+
+30
+DAOMIN CAO, SHANFA LAI, GUOLIN QIN
+≤
+�
+|x−y|≤x1
+c2,s
+|x − y|2−2s|ζ(y)|dy +
+�
+{|x−y|>x1}
+4x1y1
+|x − y|4−2s|ζ(y)|dy
+≤ C
+�
+x
+2s− 2
+r
+1
+∥ζ∥r + x2s−2
+1
+∥ζ∥1
+�
+.
+This proves (4.1) in the case 0 < s < 1 and x1 > 1. Now, we turn to the remaining case
+1
+2 < s < 1 and 0 < x1 < 1. By H¨older’s inequality, we find
+|∇G+
+s ζ(x)| ≤ C
+�
+R+
+|ζ(y)|
+|x − y|3−2sdy
+≤
+�
+|x−y|≤1
+c2,s
+|x − y|3−2s|ζ(y)|dy +
+�
+{|x−y|>1}
+|ζ(y)|dy
+≤ C (∥ζ∥r + ∥ζ∥1) .
+Noticing that G+
+s ζ(x)
+���
+x1=0 ≡ 0, we conclude
+|G+
+s ζ(x)| ≤ x1∥∇G+
+s ζ∥∞ ≤ C (∥ζ∥q + ∥ζ∥1) x1.
+The proof is thus finished.
+□
+Lemma 4.2. Suppose x1ζ ∈ L1(R2
++) and ζ ∈ L1 ∩Lr(D) for some r with s−1 < r < +∞ if
+0 < s ≤ 1
+2 and
+2
+2s−1 < r < +∞ if 1
+2 < s < 1. If ζ is Steiner symmetric in the x2-variable,
+then for x ∈ D, there holds
+|G+
+s ζ(x)| ≤ C
+��
+|x2|− 1
+2r + |x2|− 1
+2
+�
+(∥ζ∥r + ∥ζ∥1) min{1, x
+2s− 2
+r −1
+1
+} + |x2|s−2∥x1ζ∥1
+�
+x1.
+(4.2)
+Proof. For x ∈ R2
++ fixed, let
+ζ1(y) =
+�
+ζ(y),
+if |y2 − x2| <
+�
+|x2|,
+0,
+if |y2 − x2| ≥
+�
+|x2|.
+Using equation (2.11) in [8] (see also (6) in [12]), it is easy to see that for any 1 ≤ q ≤ r
+∥ζ1∥q ≤
+�
+|x2|
+1
+2
+|x2|
+� 1
+q
+∥ζ∥q = |x2|− 1
+2q ∥ζ∥q.
+Hence, by (4.1), we have
+|G+
+s ζ1(x)| ≤ C
+�
+(∥ζ1∥r + ∥ζ1∥1) min{1, x
+2s− 2
+r −1
+1
+}
+�
+x1
+≤ C
+��
+|x2|− 1
+2r + |x2|− 1
+2
+�
+(∥ζ∥r + ∥ζ∥1) min{1, x
+2s− 2
+r −1
+1
+}
+�
+x1.
+(4.3)
+Letting ζ2 = ζ − ζ1, we have
+|G+
+s ζ2(x)| = cs
+�
+|x2−y2|>√
+|x2|
+�
+1
+|x − y|2−2s −
+1
+|x − ¯y|2−2s
+�
+|ζ(y)|dy
+
+TRAVELING-WAVE SOLUTIONS FOR THE GSQG EQUATION
+31
+≤ C
+�
+|x−y|>√
+|x2|
+x1y1
+|x − y|4−2sζ(y)dy
+(4.4)
+≤
+Cx1
+|x2|2−s∥x1ζ∥1,
+which, together with (4.3), gives (4.2) and completes the proof.
+□
+The following property enables us to control the supports of maximizers.
+Lemma 4.3. Suppose that ζ ∈ L1 ∩ Lq(D) with some s−1 < q < ∞ and W > 0 is a given
+constant. Let h = ζ1V for some set V ⊂ {G+
+s ζ − Wx1 ≤ 0}, then
+˜EW(ζ − h) ≥ ˜EW(ζ)
+with strict inequality unless h ≡ 0.
+Proof. It is easy to see that
+˜EW(ζ − h) = 1
+2
+�
+R2
++
+(ζ − h)(x)G+
+s (ζ − h)(x)dν − W
+�
+R2
++
+(ζ − h)(x)x1dx
+= ˜EW(ζ) + 1
+2
+�
+R2
++
+hG+
+s h −
+�
+R2
++
+h(x)
+�
+G+
+s ζ(x) − Wx1
+�
+dx
+≥ ˜EW(ζ) + 1
+2
+�
+R2
++
+hG+
+s h,
+which implies ˜EW(ζ − h) ≥ ˜EW(ζ) since 1
+2
+�
+hG+
+s h ≥ 0 and 1
+2
+�
+hG+
+s h = 0 if and only if
+h ≡ 0. The proof is thus complete.
+□
+Lemma 4.4. Let 0 ≤ ζ0 ∈ L1 ∩ Lq(D) with some q with s−1 < q ≤ +∞ if 0 < s ≤ 1
+2 and
+2
+2s−1 < q ≤ +∞ if 1
+2 < s < 1 and W > 0 is a given constant. Then
+sup
+ζ∈R(ζ0)w
+˜EW(ζ) < +∞,
+and any maximizer (if exists) is supported in [0, M0]×R, where M0 is a constant depending
+on ∥ζ0∥1 + ∥ζ0∥q and W.
+Proof. The upper bounded of ˜EW over R(ζ0)w follows from Lemma 2.2. By Lemma 4.1 and
+the fact that ζ0 ∈ Lr for any r ∈ [1, q], there is a constant M0 depending on ∥ζ0∥1 + ∥ζ0∥q
+and W such that G+
+s ζ(x) − Wx1 ≤ 0 for all x ∈ D with x1 ≥ M0 and for any ζ ∈ R(ζ0)w.
+Suppose that ζ0 ∈ R(ζ0)w is a maximizer, let h := ζ01(M0,∞)×R, then we infer from Lemma
+4.3 that h ≡ 0 since (M0, ∞) × R ⊂ {G+
+s ζ0 − Wx1 ≤ 0}. The proof of this lemma is thus
+finished.
+□
+To obtain the compactness of maximizing sequences, we need the following concentration
+compactness lemma due to Lions [55].
+
+32
+DAOMIN CAO, SHANFA LAI, GUOLIN QIN
+Lemma 4.5. Let {un}∞
+n=1 be a sequence of nonnegative functions in L1(D) satisfying
+lim sup
+n→∞
+�
+D
+undx → µ,
+for some 0 ≤ µ < ∞. Then, after passing to a subsequence, one of the following holds:
+(i) (Compactness) There exists a sequence {yn}∞
+n=1 in R2
++ such that for arbitrary ε > 0,
+there exists R > 0 satisfying
+�
+D∩BR(yn)
+undx ≥ µ − ε,
+∀n ≥ 1.
+(ii) (Vanishing) For each R > 0,
+lim
+n→∞ sup
+y∈D
+�
+D∩BR(yn)
+undx = 0.
+(iii) (Dichotomy) There exists a constant 0 < α < µ such that for any ε > 0, there exist
+N = N(ε) ≥ 1 and 0 ≤ ui,n ≤ un, i = 1, 2 satisfying
+�
+∥un − u1,n − u2,n∥L1(D) + |α −
+�
+D u1,ndx| + |µ − α −
+�
+D u2,ndx| < ε,
+for n ≥ N,
+dn := dist(spt(u1,n), spt(u2,n)) → ∞,
+as n → ∞.
+Moreover, if µ = 0 then only vanishing will occur.
+Proof. This lemma is a slight reformulation of Lemma 1.1 in [55], so we omit the proof.
+□
+For a function 0 ≤ ζ0 ∈ L1 ∩ Lq(D) and a constant W > 0, we denote Sζ0,W :=
+supζ∈R(ζ0)w ˜EW(ζ) as the maximum value and ˜Σζ0,W := {ζ ∈ R(ζ0)w | ˜EW(ζ) = Sζ0,W} as
+the set of all the maximizers. To continue, we first show the compactness of maximizing
+sequences by using Lemma 4.5.
+Proposition 4.6. For q with max{2, s−1} < q ≤ ∞ if 0 < s ≤ 1
+2 and
+2
+2s−1 < q ≤ +∞ if
+1
+2 < s < 1, let 0 ≤ ζ0 ∈ Lq(D) be a function with 0 < |spt(ζ0)| < ∞ and W > 0 be a given
+constant. Assume that
+∅ ̸= ˜Σζ0,W ⊂ R(ζ0).
+Suppose that {ζn}∞
+n=1 ⊂ R+(ζ0) is a maximizing sequence in the sense that
+˜EW(ζn) → Sζ0,W,
+as n → ∞.
+(4.5)
+Then, there exist ζ0 ∈ ˜Σζ0,W, a subsequence {ζnk}∞
+k=1 and a sequence of real numbers
+{ck}∞
+k=1 such that as k → ∞,
+∥ζnk(· + cke2) − ζ0∥2 → 0.
+(4.6)
+Proof. Note that since 0 ∈ R(ζ0)w \ R(ζ0), the condition ∅ ̸= ˜Σζ0,W ⊂ R(ζ0) implies that
+0 is not a maximizer and hence we have Sζ0,W > 0.
+
+TRAVELING-WAVE SOLUTIONS FOR THE GSQG EQUATION
+33
+Take un = ζ2
+n. Since 0 ≤
+�
+D undx ≤ ∥ζ0∥2
+2 < ∞, we may assume that, up to a subse-
+quence (still denoted by {un}∞
+n=1),
+�
+D
+undx → µ
+for some 0 ≤ µ ≤ ∥ζ0∥2
+2. Applying Lemma 4.5, we find that for a certain subsequence, still
+denoted by {ζn}∞
+n=1, one of the three cases in Lemma 4.5 should occur. In what follows,
+we divide the proof into three steps.
+Step 1. Vanishing excluded: Suppose that for each fixed R > 0,
+lim
+n→∞ sup
+y∈D
+�
+BR(y)∩D
+ζ2
+ndx = 0.
+(4.7)
+By the property of rearrangement and H¨older’s inequality, we have for any R > 0 and
+1 ≤ τ ≤ 2
+�
+BR(y)∩D
+ζτ
+ndx → 0
+as n → +∞ uniformly over y ∈ D. On the other hand, we have G+
+s (ζn(1 − 1BR(y)))(y) ≤
+C ∥ζn∥1
+R2−2s . Therefore, we get
+�
+ζnG+
+s ζn ≤
+C
+R2−2s + on(1),
+for any R > 0 and hence limn→∞ ˜EW(ζn) ≤ 0. This is a contradiction to Sζ0,W > 0. Thus,
+vanishing can not occur.
+Step 2. Dichotomy excluded: We may assume that ζn is supported in [0, M0] × R, where
+M0 is the constant obtained in Lemma 4.4. Suppose that there is a constant α ∈ (0, µ)
+such that for any ε > 0, there exist N(ε) ≥ 1 and 0 ≤ ζi,n ≤ ζn, i = 1, 2, 3 satisfying
+
+
+
+
+
+ζn = ζ1,n + ζ2,n + ζ3,n,
+�
+D ζ2
+3,ndx + |α − αn| + |µ − α − βn| < ε,
+for n ≥ N(ε),
+dn := dist(spt(ζ1,n), spt(ζ2,n)) → ∞,
+as n → ∞,
+where αn =
+�
+D ζ2
+1,ndx and βn =
+�
+D ζ2
+2,ndx. Using a diagonal argument, we obtain that there
+exists a subsequence, still denoted by {ζn}∞
+n=1, such that
+
+
+
+
+
+ζn = ζ1,n + ζ2,n + ζ3,n,
+0 ≤ ζi,n ≤ ζn, i = 1, 2, 3
+�
+D ζ2
+3,ndx + |α − αn| + |µ − α − βn| → 0,
+as n → ∞,
+dn = dist(spt(ζ1,n), spt(ζ2,n)) → ∞,
+as n → ∞.
+By direct calculations, one has
+�
+D
+ζnG+
+s ζn =
+�
+D
+(ζ1,n + ζ2,n + ζ3,n)G+
+s (ζ1,n + ζ2,n + ζ3,n)
+=
+�
+D
+ζ1,nG+
+s ζ1,n +
+�
+D
+ζ2,nG+
+s ζ2,n + 2
+�
+D
+ζ1,nG+
+s ζ2,n +
+�
+D
+ζ3,nG+
+s (2ζn − ζ3,n).
+
+34
+DAOMIN CAO, SHANFA LAI, GUOLIN QIN
+By (2.6) and H¨older’s inequality, we derive
+�
+D
+ζ3,nG+
+s (2ζn − ζ3,n) ≤ C∥ζ3,n∥2−s∥2ζn − ζ3,n∥1−s
+2−s∥2ζn − ζ3,n∥s
+1
+≤ C|spt(ζ0)|
+s
+2(2−s) ∥ζ3,n∥2(∥ζ0∥1 + ∥ζ0∥2) = on(1).
+It is obvious that �
+D
+�
+D
+ζ1,n(x)G+
+s (x, y)ζ2,n(y)dxdy ≤ C∥ζ0∥2
+1
+d2−2s
+n
+= on(1).
+Hence, we arrive at
+˜EW(ζn) = 1
+2
+�
+D
+ζnG+
+s ζn − W
+�
+D
+x1ζndx ≤ ˜EW(ζ1,n) + ˜EW(ζ2,n) + on(1).
+Taking Steiner symmetrization in the x2-variable ζ∗
+i,n of ζi,n for i = 1, 2, by the rearrange-
+ment inequality, we obtain
+˜EW(ζn) ≤ ˜EW(ζ∗
+1,n) + ˜EW(ζ∗
+2,n) + on(1).
+By Lemma 4.2, there exists a constant N0 > 0 depending on ∥ζ0∥1 + ∥ζ0∥q, W and M0
+such that for all ζ ∈ R+(ζ0) with spt(ζ) ⊂ ([0, M0] × R) ∩ D,
+G+
+s ζ(x) − Wx1 ≤ 0,
+∀x with |x2| > N0
+Let
+ζ∗∗
+i,n(x) = ζ∗
+i,n1[0,M0]×[−N0,N0](x + (−1)iN0e2),
+i = 1, 2.
+Then, we find
+˜EW(ζn) ≤ ˜EW(ζ∗∗
+1,n) + ˜EW(ζ∗∗
+2,n) + on(1).
+and
+supp(ζ∗∗
+1,n) ⊂ [0, M0] × [0, 2N0],
+supp(ζ∗∗
+2,n) ⊂ [0, M0] × [−2N0, 0].
+We may assume that ζ∗∗
+i,n → ζ∗∗
+i
+weakly in Lr(D) for some s−1 < r < q and i = 1, 2. Then,
+ζ∗∗ := ζ∗∗
+1 + ζ∗∗
+2 ∈ R(ζ0)w. Moreover, by the weak convergence, we get
+lim
+n→∞
+˜EW(ζ∗∗
+i,n) = ˜EW(ζ∗∗
+i ), for i = 1, 2,
+and therefore we arrive at
+˜EW(ζ∗∗
+1 ) + ˜EW(ζ∗∗
+2 ) ≥ lim sup
+n→∞
+˜EW(ζn) = Sζ0,W.
+It can be seen that
+Sζ0,W ≥ ˜EW(ζ∗∗) = ˜EW(ζ∗∗
+1 + ζ∗∗
+2 )
+= ˜EW(ζ∗∗
+1 ) + ˜EW(ζ∗∗
+2 ) +
+�
+D
+�
+D
+ζ∗∗
+1 (x)G+
+s (x, y)ζ∗∗
+2 (y)dxdy
+≥ Sζ0,W +
+�
+D
+�
+D
+ζ∗∗
+1 (x)G+
+s (x, y)ζ∗∗
+2 (y)dxdy,
+
+TRAVELING-WAVE SOLUTIONS FOR THE GSQG EQUATION
+35
+from which we must have
+˜EW(ζ∗∗) = Sζ0,W
+and
+�
+D
+�
+D
+ζ∗∗
+1 (x)G+
+s (x, y)ζ∗∗
+2 (y)dxdy = 0.
+(4.8)
+Since ˜Σζ0,W ⊂ R(ζ0) by the assumption, we deduce ζ∗∗ ∈ R(ζ0) and µ = ∥ζ∗∗∥2
+2 =
+∥ζ∗∗
+1 ∥2
+2 + ∥ζ∗∗
+2 ∥2
+2.
+On the other hand, since ∥ζ∗∗
+1 ∥2
+2 ≤ α and ∥ζ∗∗
+2 ∥2
+2 ≤ µ − α by the weak convergence, we
+find that ∥ζ∗∗
+1 ∥2
+2 = α > 0 and ∥ζ∗∗
+1 ∥2
+2 = µ − α > 0, which implies that both ζ∗∗
+1 and ζ∗∗
+2 are
+non-zero and hence
+�
+D
+�
+D ζ∗∗
+1 (x)G+
+s (x, y)ζ∗∗
+2 (y)dxdy > 0, which is a contradiction to (4.8).
+Step 3. Compactness: Assume that there is a sequence {yn}∞
+n=1 in D such that for
+arbitrary ε > 0, there exists R > 0 satisfying
+�
+D∩BR(yn)
+ζ2
+ndx ≥ µ − ε,
+∀ n ≥ 1.
+(4.9)
+We may assume that yn = (yn,1, 0) after a suitable translation in x2-variable.
+Define
+ζ0
+n := ζn1(0,M0)×R and ζR
+n := ζn1(0,M)×(−R,R). Then {ζ0
+n}∞
+n=1 is also a maximizing sequence
+in R+(ζ0) by Lemma 4.3. Moreover, we infer from (4.9) that for arbitrary ε > 0, there
+exists R > 0 such that
+∥ζ0
+n − ζR
+n ∥2
+2 ≤ ε,
+∀ n ≥ 1.
+That is,
+∥ζ0
+n − ζR
+n ∥2 → 0,
+as R → ∞,
+uniformly over n.
+(4.10)
+We may assume that ζ0
+n → ζ0 weakly in L2(D) ∩ Lr(D) for some s−1 < r < q and hence
+ζR
+n → ζ01(0,M0)×(−R,R) weakly in L2(D)∩Lr(D). By the weak convergence and ζn ∈ R+(ζ0),
+we find
+∥ζ0∥2 ≤ lim inf
+n→∞ ∥ζ0
+n∥2 ≤ ∥ζ0∥2.
+(4.11)
+Using Lemma 2.2 and H¨older’s inequality, we conclude |E(ζ0
+n)−E(ζR
+n )| = oR(1). That is,
+E(ζR
+n ) → E(ζ0
+n) as R → ∞ uniformly over n by (4.10). On the other hand E(ζR
+n ) → E(ζR)
+as n → ∞ for fixed R by weak continuity of E in functions supported on bounded domains
+and E(ζR) → E(ζ0) by the monotone convergence theorem. Therefore, we obtain
+E(ζ0
+n) → E(ζ0).
+As for the impulse, we split
+|I(ζ0
+n) − I(ζ0)| ≤ |I(ζ0
+n) − I(ζR
+n )| + |I(ζR
+n ) − I(ζR)| + |I(ζR) − I(ζ0)|.
+For the first term, by H¨older’s inequality, we deduce
+|I(ζ0
+n) − I(ζR
+n )| ≤ M0|spt(ζ0
+n)|
+1
+2∥ζ0
+n − ζR
+n ∥2 → 0,
+as R → ∞ uniformly over n. For fixed R, we have the second term |I(ζR
+n ) − I(ζR)| → 0 as
+n → ∞ by the weak convergence. Since the third term |I(ζR) − I(ζ0)| → 0 as R → ∞ by
+the monotone convergence theorem, we have |I(ζ0
+n) − I(ζ0)| → 0 by first letting R → ∞
+and then n → ∞.
+Therefore, we have proved ˜EW(ζ0
+n) → ˜EW(ζ0) and hence ˜EW(ζ0) = Sζ0,W and ζ0 ∈ R(ζ0)
+by our assumption Σζ0,W ⊂ R(ζ0). Then we deduce that ∥ζ0∥2 = ∥ζ0∥2 by the property of
+
+36
+DAOMIN CAO, SHANFA LAI, GUOLIN QIN
+rearrangement, which implies limn→∞ ∥ζ0
+n∥2 = ∥ζ0∥2. So, we obtain the strong convergence
+∥ζ0
+n − ζ0∥2 → 0 by the uniform convexity of L2(D).
+Now we want to show that ζn → ζ0 strongly. Indeed, since the supports of ζ0
+n and ζn −ζ0
+n
+are disjoint and ζn ∈ R(ζ0), we conclude
+∥ζn − ζ0
+n∥2
+2 = ∥ζn∥2
+2 − ∥ζ0
+n∥2
+2 ≤ ∥ζ0∥2
+2 − ∥ζ0
+n∥2
+2 → ∥ζ0∥2
+2 − ∥ζ0∥2
+2 = 0.
+Therefore, we obtain ∥ζn − ζ0∥2 → 0 and finish the proof.
+□
+To state our stability result, following [13], we need to introduce some definitions first.
+Definition 4.7. Let ξ0 ∈ L1 ∩ Lp(R2
++) be a given function. ξ ∈ L∞
+loc([0, T) , L1(R2
++)) ∩
+L∞
+loc([0, T) , Lp(R2
++)) is called a Lp-regular solution in (0, T) of (1.1) with initial data ξ0 if
+(i) ¯ξ(x, t) := ξ(x, t)−ξ(¯x, t) satisfies (1.1) in the sense of distributions with initial data
+¯ξ0(x) := ξ0(x) − ξ0(¯x);
+(ii) E(ξ(t, ·)) and I(ξ(t, ·)) are constant and ξ(t) ∈ R(ξ0) for t ∈ [0, T);
+(iii) ξ is non-negative for t ∈ (0, T) provided that ξ0 is non-negative;
+(iv) For 0 < s ≤ 1
+2, we require that ξ is supported in D for t ∈ (0, T) provided that ξ0
+is supported in D.
+Generally speaking, an Lp-regular solution is a weak solution to (1.1) such that its
+kinetic energy, impulse and distribution are conserved, which is true for sufficiently regular
+solutions; see [14] for some discussion about the conservation laws. The existence of Lp-
+regular solutions for the Euler equation was obtained in [13] using the transport nature
+of the Euler equation, see also [1].
+Note that the gSQG equations are also transport
+equations.
+So it is possible to modify the method in [1, 13] to show the existence of
+Lp-regular solutions defined above for the gSQG equation provided that the existence of
+sufficiently smooth solutions for the Cauchy problem of the gSQG equation is a priori
+known. Thus, the Lp-regular solutions for the gSQG equation may be proved to exist for
+small T due to the local well-posedness theory. For large T, though a general theory for the
+global well-posedness for the Cauchy problem of the gSQG equation remains a challenging
+open problem now, various global solutions known as relative equilibria are constructed
+as mentioned in the introduction, which are trivial examples of the Lp-regular solutions.
+Invariant measures and a large class of global solutions for the SQG equation are also
+obtained in [37]. Besides, certain blow-up scenarios have been ruled out analytically in
+[30, 31] and numerically in [27]. Therefore, we believe that the Lp-regular solutions are
+reasonable to exist for large T and for a large class of initial values.
+We say that an initial data ξ0 is admissible if ξ0 is nonnegative, supported in D if
+0 < s ≤ 1
+2 and there exists a L∞-regular solution with initial data ξ0 for T = +∞.
+Now, we are ready to establish the following stability theorem on the set of maximizers.
+Theorem 4.8. For q with max{2, s−1} < q ≤ ∞ if 0 < s ≤ 1
+2 and
+2
+2s−1 < q ≤ +∞ if
+1
+2 < s < 1, let 0 ≤ ζ0 ∈ Lq(D) be a function with 0 < |spt(ζ0)| < ∞ and W > 0 is a
+given constant. Suppose that ˜Σζ0,W, the set of maximizers of ˜EW := E − WI over R(ζ0)w,
+satisfies
+∅ ̸= ˜Σζ0,W ⊂ R(ζ0),
+
+TRAVELING-WAVE SOLUTIONS FOR THE GSQG EQUATION
+37
+and all elements of ˜Σζ0,W are admissible. Then ˜Σζ0,W is orbitally stable in the following
+sense:
+For arbitrary η > 0 and M > 0, there exists δ > 0 such that if there exist a L∞-regular
+solution ξ(t) in (0, T) with initial data ξ0, ∥ξ0∥∞ ≤ M and
+inf
+ζ∈˜Σζ0,W
+{∥ξ0 − ζ∥1 + ∥ξ0 − ζ∥2 + |I(ξ0 − ζ)|} ≤ δ,
+then for all t ∈ (0, T), we have
+inf
+ζ∈˜Σζ0,W
+{∥ξ(t) − ζ∥1 + ∥ξ(t) − ζ∥2} ≤ η.
+(4.12)
+If in addition
+I(ζ1) = I(ζ2),
+∀ ζ1, ζ2 ∈ ˜Σζ0,W,
+then for all t ∈ (0, T), we have
+inf
+ζ∈˜Σζ0,W
+{∥ξ(t) − ζ∥1 + ∥ξ(t) − ζ∥2 + I(|ξ(t) − ζ|)} ≤ η.
+(4.13)
+Proof. By Lemma 4.4, any maximizer (if exists) is supported in [0, M0]×R. So, we deduce
+that supζ1∈˜Σζ0,W I(ζ1) ≤ M0∥ζ0∥1 < ∞. By an argument similar to the proof of Lemma 4.4,
+for ζ with the following properties:
+ζ ≥ 0,
+∥ζ∥1 ≤ ∥ζ0∥1 + 1,
+∥ζ∥∞ ≤ M,
+there exists a constant M1 ≥ M0 independent of ζ such that
+G+
+s ζ(x) − Wx1 < 0,
+∀x with x1 > M1.
+Now we finish our proof by contradiction. Suppose on the contrary that there exist a
+sequence of non-negative functions {ωn
+0}∞
+n=1, each of which is admissible, ∥ωn
+0∥∞ ≤ M and
+as n → ∞,
+inf
+ζ∈˜Σζ0,W
+{∥ωn
+0 − ζ∥1 + ∥ωn
+0 − ζ∥2 + |I(ωn
+0 − ζ)|} → 0,
+while
+sup
+t∈(0,Tn)
+inf
+ζ∈˜Σζ0,W
+{∥ωn(t) − ζ∥1 + ∥ωn(t) − ζ∥2} ≥ c0 > 0,
+for some positive constant c0, where ωn(t) is the L∞-regular solution of (1.1) with initial
+data ωn
+0 in (0, Tn). By the choice of ωn
+0, we infer from Lemma 2.2 and the H¨older inequality
+that as n → ∞,
+˜EW(ωn
+0 ) → Sζ0,W.
+(4.14)
+We choose tn ∈ (0, Tn) such that for each n,
+inf
+ζ∈˜Σζ0,W
+{∥ωn(tn) − ζ∥1 + ∥ωn(tn) − ζ∥2} ≥ c0
+2 > 0.
+(4.15)
+Take a sequence of functions ζn
+0 ∈ ˜Σζ0,W ⊂ R(ζ0) such that as n → ∞,
+∥ωn
+0 − ζn
+0 ∥1 + ∥ωn
+0 − ζn
+0 ∥2 + |I(ωn
+0 − ζn
+0 )| → 0.
+(4.16)
+Then, {ζn
+0 }∞
+n=1 is a maximizing sequence due to Lemma 2.2.
+
+38
+DAOMIN CAO, SHANFA LAI, GUOLIN QIN
+For a function ω, we use ¯ω := ω1(0,M1)×R to denote the restriction of ω. By Lemma 4.3,
+the conservation of energy and impulse, we conclude
+˜EW(ωn(tn)) ≥ ˜EW(ωn(tn)) = ˜EW(ωn
+0) → Sζ0,W.
+(4.17)
+Since ωn(tn) is a rearrangement of ωn
+0 by Definition 4.7, one can find a rearrangement
+ζn
+1 ∈ R(ζn
+0 ) = R(ζ0) such that as n → +∞
+∥ωn(tn) − ζn
+1 ∥1 + ∥ωn(tn) − ζn
+1 ∥2 = ∥ωn
+0 − ζn
+0 ∥1 + ∥ωn
+0 − ζn
+0 ∥2 → 0.
+(4.18)
+Then, we infer from (2.6), (4.18) and H¨older’s inequality that
+|I(¯ζn
+1 ) − I(¯ωn(tn))| ≤ M1∥¯ζn
+1 − ¯ωn(tn)∥1
+≤ M1∥ζn
+1 − ωn(tn)∥1
+= M1∥ζn
+0 − ωn
+0∥1,
+(4.19)
+and
+|E(¯ζn
+1 ) − E(¯ωn(tn))| ≤ C∥¯ζn
+1 − ¯ωn(tn)∥1−s
+2−s ≤ C∥ζn
+0 − ωn
+0∥1−s
+2−s.
+(4.20)
+So, we deduce from (4.16)–(4.20) that {¯ζn
+1 } ⊂ R+(ζ0) is a maximizing sequence and hence
+by Proposition 4.6 after some translations, we have
+¯ζn
+1 → ζ∗∗ strongly in L2(D),
+(4.21)
+as n → ∞ for some function ζ∗∗ ∈ ˜Σζ0,W, which implies ζn
+1 → ζ∗∗ strongly. Indeed, since
+the supports of ¯ζn
+1 and ζn
+1 − ¯ζn
+1 are disjoint and ζn
+1 ∈ R(ζ0), we conclude
+∥ζn
+1 − ¯ζn
+1 ∥2
+2 = ∥ζn
+1 ∥2
+2 − ∥¯ζn
+1 ∥2
+2 ≤ ∥ζ0∥2
+2 − ∥¯ζn
+1 ∥2
+2 → ∥ζ0∥2
+2 − ∥ζ∗∗∥2
+2 = 0,
+from which we get
+∥ζn
+1 − ζ∗∗∥2 ≤ ∥ζn
+1 − ¯ζn
+1 ∥2 + ∥ζ∗∗ − ¯ζn
+1 ∥2 → 0.
+Hence, by (4.18) and (4.21), we deduce
+∥ωn(tn) − ζ∗∗∥2 ≤ ∥ζn
+1 − ωn(tn)∥2 + ∥ζn
+1 − ζ∗∗∥2
+= ∥ζn
+0 − ωn
+0∥2 + ∥ζn
+1 − ζ∗∗∥2 = on(1).
+Next, we will estimate ∥ωn(tn) − ζ∗∗∥1. By the conservation of the L1-norm and (4.21),
+we have
+∥ωn(tn) − ζ∗∗∥1 ≤
+�
+spt(ζ∗∗)
+|ωn(tn) − ζ∗∗|dx +
+�
+D\spt(ζ∗∗)
+ωn(tn)dx
+≤
+�
+spt(ζ∗∗)
+|ωn(tn) − ζ∗∗|dx +
+�
+D
+ωn(tn)dx −
+�
+D
+ζn
+0 dx +
+�
+D
+ζ∗∗dx −
+�
+spt(ζ∗∗)
+ωn(tn)dx
+≤ 2
+�
+spt(ζ∗∗)
+|ωn(tn) − ζ∗∗|dx +
+�
+D
+|ωn
+0 − ζn
+0 |dx
+≤ 2|supp(ζ0)|
+1
+2∥ωn(tn) − ζ∗∗∥2 +
+�
+D
+|ωn
+0 − ζn
+0 |dx = on(1),
+
+TRAVELING-WAVE SOLUTIONS FOR THE GSQG EQUATION
+39
+which contradicts the choice of tn in (4.15) and completes the proof of (4.12). Here we
+have used
+�
+D ζn
+0 dx =
+�
+D ζ∗∗dx =
+�
+D ζ0dx and |spt(ζn
+0 )| = |spt(ζ∗∗)| = |spt(ζ0)| due to the
+properties of rearrangement.
+If in addition
+I(ζ1) = I(ζ2),
+∀ ζ1, ζ2 ∈ ˜Σζ0,W,
+then by the conservation of the impulse, we find
+I(|ω(t) − ζ1|) ≤
+�
+spt(ζ1)
+x1|ω(t) − ζ1|dx +
+�
+D\spt(ζ1)
+x1ω(t)dx
+≤
+�
+spt(ζ1)
+x1|ζ1 − ω(t)|dx +
+�
+D
+x1ω(t)dx −
+�
+D
+x1ζ2dx +
+�
+D
+x1ζ1dx −
+�
+spt(ζ1)
+x1ω(t)dx
+≤ 2
+�
+spt(ζ1)
+x1|ζ1 − ω(t)|dx +
+�
+D
+x1ω0dx −
+�
+D
+x1ζ2dx
+≤ 2M0|spt(ζ0)|
+1
+2 ∥ω(t) − ζ1∥2 + |I(ω0 − ζ2)|,
+∀ ζ1, ζ2 ∈ ˜Σζ0,W,
+which implies (4.13) by (4.12) and taking δ smaller if necessary.
+□
+Remark 4.9. Our proof also works well for the case s = 1. Compared with Theorem 1 in
+[13], we admit perturbations with non-compact supports. This is achieved by bringing in
+the L1-norm in our theorem. Note that, if the perturbation ω0 has a compact support with
+measure less than A for some constant A as in Theorem 1 in [13] and is closed to ˜Σζ0,W in
+L2, then the H¨older inequality will imply that ω0 is closed to ˜Σζ0,W in L1. Thus, one can
+obtain a generalization of Theorem 1 in [13] by the argument in this paper.
+4.2. Maximizers in rearrangement class and the stability of traveling-wave so-
+lutions. In Sections 2 and 3, for J(t) = Lt1+ 1
+p for some L > 0 and p ∈ (1,
+1
+1−s), we have
+obtained existence and uniqueness of a traveling solution ωε. To apply our stability result
+Theorem 4.8 to obtain the stability of ωε, we consider the following maximizing problem
+sup
+ζ∈R(ωε)w
+˜Eε(ζ),
+(4.22)
+where (with abuse of notations for simplicity)
+˜Eε(ζ) := ˜EW ε3−2s(ζ) = 1
+2
+�
+D
+ζG+
+s ζdx − Wε3−2s
+�
+D
+x1ζdx.
+We denote ˜Σε as the set of maximizers of ˜Eε over the rearrangement class R(ωε)w. Our
+main result in this subsection is the following theorem.
+Theorem 4.10. For ε > 0 small, the set of maximizers ˜Σε satisfies
+∅ ̸= ˜Σε ⊂ R(ωε).
+Moreover, for each ζε ∈ ˜Σε, the following claims hold.
+(i). ζε = gε(G+
+s ζε − Wε3−2sx1 − ˜µε) for some non-negative and non-decreasing function
+gε : R → R satisfying gε(τ) > 0 if τ > 0 and gε(τ) = 0 if τ ≤ 0 and some constant
+˜µε.
+
+40
+DAOMIN CAO, SHANFA LAI, GUOLIN QIN
+(ii). diam (spt(ζε)) < ˜Rε for some constant 0 < ˜R < ∞ and up to a suitable translation
+in the x2 direction
+sup
+x∈spt(ζε)
+|εx − (d0, 0)| = o(1).
+In order to prove Theorem 4.10, a series of lemmas is needed.
+Lemma 4.11. The energy satisfies
+max
+ζ∈R(ωε)w
+˜Eε(ζ) ≥ I0 + O(ε2−2s).
+Proof. This is a simple consequence of the fact ωε ∈ R(ωε)w and Lemma 2.20.
+□
+Lemma 4.12. ˜Eε attains its maximum value over R(ωε)w at some ζε, which is Steiner
+symmetric in the x2-variable.
+Proof. By Lemma 4.1, there exists a constant 1 < Mε = O(ε−1) depending on ε, ωε and
+W such that
+G+
+s ζ(x) − Wε3−2sx1 < 0,
+∀x with x1 > Mε,
+∀ζ ∈ R(ωε)w.
+It is easy to check that ˜Eε is bounded from above over R(ωε)w by Lemma 2.2. Take a
+sequence {ζj} ⊂ R(ωε)w such that as j → +∞
+˜Eε(ζj) → sup
+R(ωε)w
+˜Eε.
+We may assume that ζj is supported in (0, Mε)×R by Lemma 4.3. We can also assume ζj is
+Steiner symmetric about the x1-axis by replacing ζj with its own Steiner symmetrization.
+Since
+�
+D x1ζjdx ≤ Rεκ, there exists a constant Nε > 0 such that G+
+s ζj(x) − Wε3−2sx1 <
+0,
+∀x ∈ D with |x2| > Nε,
+∀ j due to Lemma 4.2. That is, we assume that ζj is
+supported in [0, Mε] × [−Nε, Nε]. There is a subsequence (still denoted by {ζj}) such that
+as j → +∞, ζj → ζε ∈ R(ωε)w weakly star in L∞(D). Since G+
+s (·, ·) ∈ Lr
+loc(R2
++ × R2
++) for
+any 1 ≤ r <
+1
+1−s, we deduce that
+lim
+j→+∞
+˜Eε(ζj) = ˜Eε(ζε).
+This means that ζε is a maximizer and thus the proof is finished.
+□
+To study the properties of maximizers. We need the following lemma from [9].
+Lemma 4.13 (Lemmas 2.4 and 2.9 in [9]). Let (Ω, ν) be a finite positive measure space.
+Let ξ0 : Ω → R and ζ0 : Ω → R be ν-measurable functions, and suppose that every level
+set of ζ0 has zero measure. Then there is a non-decreasing function f such that f ◦ ζ0 is a
+rearrangement of ξ0. Moreover, if ξ0 ∈ Lq(Ω, ν) for some 1 ≤ q < +∞ and ζ0 ∈ Lq′(Ω, ν),
+then f ◦ ζ0 is the unique maximizer of linear functional
+M(ξ) :=
+�
+Ω
+ξ(x)ζ0(x)dν(x)
+relative to R(ξ0)w.
+
+TRAVELING-WAVE SOLUTIONS FOR THE GSQG EQUATION
+41
+Lemma 4.14. Let ζε ∈ ˜Σε be a maximizer. Then, up to a translation in the x2 direction,
+ζε must be Steiner symmetric in the x2-variable and supported in [0, Mε] × [−Nε, Nε].
+Moreover, for ε > 0 small, it holds
+0 ̸≡ ζε ∈ R(ζε)
+and there exists a non-negative and non-decreasing function gε : R → R satisfying gε(τ) > 0
+if τ > 0 and gε(τ) = 0 if τ ≤ 0, such that for some constant ˜µε,
+ζε(x) = gε(G1ζε(x) − Wε3−2sx1 − ˜µε),
+∀ x ∈ R2
++.
+(4.23)
+Proof. For ε > 0 small, we see from Lemma 4.11 that ˜Eε(ζε) > 0 and hence ζε ̸≡ 0.
+Since R(ωε)w is a convex set (see e.g. [8, 13] and references therein). Thus for each
+ζ ∈ R(ωε)w, it holds ζτ := ζε + τ(ζ − ζε) ∈ R(ωε)w for any τ ∈ [0, 1]. Noting that ζε is a
+maximizer of ˜Eε, we have
+d
+dτ
+����
+τ=0+
+˜Eε(ζτ) =
+�
+D
+(ζ − ζε)
+�
+G+
+s ζε − Wε3−2sx1
+�
+dx ≤ 0,
+which yields
+�
+D
+ζ
+�
+G+
+s ζε − Wε3−2sx1
+�
+dx ≤
+�
+D
+ζε �
+G+
+s ζε − Wε3−2sx1
+�
+dx,
+∀ ζ ∈ R(ωε)w.
+By the strict rearrangement inequality, we know that after a translation in the x2 direc-
+tion, ζε must be Steiner symmetric in the x2-variable. Then by Lemmas 4.1– 4.3, we find
+that ζε is supported in [0, Mε] × [−Nε, Nε]. Using the fact that ζε is Steiner symmetric in
+the x2-variable, one can verified that G+
+s ζε is even and strictly decreasing with respect to
+x2. It follows that every level set of G+
+s ζε − Wε3−2sx1 has measure zero. By Lemma 4.13,
+there exists a non-decreasing function ˜gε : R → R, such that,
+˜ζε(x) = ˜gε
+�
+G+
+s ζε(x) − Wε3−2sx1
+�
+.
+for some ˜ζε ∈ R(ωε). From the conclusion of Lemma 4.13, we also know ˜ζε(x) is the unique
+maximizer of the linear functional ζ �→
+�
+[0,Mε]×[−Nε,Nε] (G+
+s ζε − Wε3−2sx1) ζdx relative to
+R(ωε)w. Hence we must have ζε = ˜ζε ∈ R(ωε).
+Now, let
+˜µε := sup
+�
+G+
+s ζε(x) − Wε3−2sx1
+���� x ∈ R2
++ s.t. ζε(x) = 0
+�
+∈ R,
+and gε(·) = max{˜gε(· + ˜µε), 0}. We have
+ζε(x) = gε
+�
+G+
+s ζε(x) − Wε3−2sx1 − ˜µε
+�
+for any x ∈ D.
+The proof is thus complete.
+□
+Lemma 4.15. Let ζε ∈ ˜Σε ⊂ R(ωε) be a maximizer. Then there exists correspondingly
+zε ⊂ [−R, Mε + R] × {0} such that for ε > 0 small
+∥ζε − ˜ωε(· − zε)∥1 + ∥ζε − ˜ωε(· − zε)∥2−s = o(1).
+
+42
+DAOMIN CAO, SHANFA LAI, GUOLIN QIN
+Here, ˜ωε is the translation of ωε with
+�
+x˜ωε = 0 and R is the uniform constant such that
+spt(˜ωε) ⊂ B(0, R).
+Proof. Notice that ζε ∈ R(ωε) is a rearrangement of ωε. So, we have
+�
+ζε =
+�
+ωε = κ and
+�
+J(ζε) =
+�
+J(ωε). Then, we deduce
+Eε(ζε) = ˜Eε(ζε) −
+�
+J(ζε) ≥ ˜Eε(ωε) −
+�
+J(ωε) = Eε(ωε) ≥ I0 + O(ε2−2s),
+which implies that {ζε} is a maximizer sequence of E0 over A0. Therefore, Theorem II.2
+and Corollary II.1 in [55] provide a subsequence of {ζε} and a sequence of points {zε} such
+that
+∥ζε − ω0(· − zε)∥1 + ∥ζε − ω0(· − zε)∥2−s = o(1),
+as ε → 0.
+In view of Lemma 3.1, we obtain
+∥ζε − ˜ωε(· − zε)∥1 + ∥ζε − ˜ωε(· − zε)∥2−s = o(1),
+as ε → 0.
+Since ζε and ˜ωε are even and non-increasing in x2 and ζε is supported in [0, Mε]×[−Nε, Nε],
+we may take zε ∈ [−R, Mε + R] × {0}. This completes the proof of this lemma.
+□
+Lemma 4.16. Let zε = (zε,1, 0) be as in Lemma 4.15. Then zε,1 → +∞ as ε → 0.
+Proof. By the inequality ˜Eε(ζε) ≥ ˜Eε(ωε) and Mε = O(ε−1) due to the proof of Lemma
+4.12, we get
+�
+ζεG+
+s ζε ≥
+�
+ωεGsωε + o(1).
+(4.24)
+On the other hand, by Lemma 4.15, we have
+�
+ζεG+
+s ζε =
+� �
+˜ωε(x − zε)
+�
+cs
+|x − y|2−2s −
+cs
+|x − ¯y|2−2s
+�
+˜ωε(y − zε)dxdy + o(1)
+=
+�
+ωεGsωε −
+� �
+cs˜ωε(x)˜ωε(y)
+|x − ¯y + 2zε|2−2sdxdy + o(1),
+which, combined with (4.24), implies
+� �
+cs˜ωε(x)˜ωε(y)
+|x − ¯y + 2zε|2−2sdxdy = o(1).
+Thus, we must have zε = (zε,1, 0) with zε,1 → +∞ as ε → 0.
+□
+Denote ψε(x) = Gsζε and ψ0 = Gsω0. Since ∥ζε−ω0(·−zε)∥1+∥ζε−ω0(·−zε)∥2−s = o(1)
+and ∥ζε∥∞ ≤ ∥ωε∥∞ ≤ C, using Lemma 2.1 and a similar bootstrap argument as the proof
+of Corollary 2.6, we derive
+∥ψε − ψ0(· − zε)∥∞ = o(1).
+To continue our discussion, we next find a lower bound for the Lagrange multipliers.
+Lemma 4.17. One has for some constant ˜µ > 0,
+lim inf
+ε→0
+˜µε ≥ ˜µ > 0.
+
+Proof. Let R be the constant such that spt(˜ωε) ⊂ B(0, R) for every ε small. Since ζε is a
+rearrangement of ωε, there is a point ˜x ∈ B(zε, 2R) \ spt(ζε). At the point ˜x, by Lemmas
+4.14 and 4.15, we have
+˜µε ≥ Gsζε(˜x) −
+�
+csζε(y)
+|˜x − ¯y|2−2sdy − Wε3−2s˜x1
+≥ ψ0(˜x − zε) −
+�
+csω0(y)
+|˜x − ¯y + zε|2−2sdy + o(1)
+≥
+inf
+x∈B(0,2R) ψ0(x) + o(1),
+which proves this lemma by taking ˜µ = 1
+2 infx∈B(0,2R) ψ0(x).
+□
+Corollary 4.18. There is a constant ˜R > 0 such that spt(ζε) is contained in a disk of
+radius ˜R for arbitrary ε small.
+Proof. Using Lemmas 4.14 and 4.17, one can prove this lemma through an argument similar
+to Lemma 2.22, so we omit the details.
+□
+Let ˜xε := κ−1 �
+xζε be the center of mass of ζε. We have the following estimate of the
+location.
+Corollary 4.19. It holds
+|ε˜xε − (d0, 0)| = o(1).
+Proof. The proof is similar to Lemma 2.23, so we omit the details.
+□
+Now, we are able to prove Theorem 4.10.
+Proof of Theorem 4.10: The claims in Theorem 4.10 follow from the above lemmas.
+□
+Using Theorem 4.10 and the uniqueness result Theorem 1.6, we obtain the following
+conclusion about the set of maximizers.
+Proposition 4.20. For ε > 0 sufficiently small, one has
+˜Σε = {ωε(· + ce2) | c ∈ R}.
+Proof. Recall that Σε ⊂ Aε denotes the set of maximizers of Eε defined by (1.5) over Aε.
+The uniqueness result Theorem 1.6 states that Σε = {ωε(· + ce2) | c ∈ R} .
+Then, by Theorem 4.10, we have ∅ ̸= ˜Σε ⊂ R(ωε) and for arbitrary ζε ∈ ˜Σε, it
+holds spt(ζε) ⊂ B(ε−1(d0, 0), ε−1d0/2). This implies ˜Σε ⊂ Aε. Notice that
+�
+J(ζ)dx =
+�
+J(ωε)dx,
+∀ζ ∈ R(ωε) due to the property of rearrangement. Then one can see easily
+that
+˜Σε = Σε = {ωε(· + ce2) | c ∈ R} .
+□
+Having made all the preparation, we are now ready to give proof of Theorem 1.7 and
+end our paper.
+
+44
+DAOMIN CAO, SHANFA LAI, GUOLIN QIN
+Proof of Theorem 1.7: The orbital stability of ωε follows from a combination of Theorem
+4.8 and Proposition 4.20.
+□
+Acknowledgments: This work was supported by NNSF of China (Grant 11831009).
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+[66] V.I. Yudovich, Non-stationary flows of an ideal incompressible fluid, Zhurnal Vych Matematika, 3
+(1963), 1032–1106, English translation: U.S.S.R. Comput. Math. and Math. Phys., 3 (1963), 1407-
+1456.
+Institute of Applied Mathematics, Chinese Academy of Sciences, Beijing 100190, and
+University of Chinese Academy of Sciences, Beijing 100049, P.R. China
+Email address: dmcao@amt.ac.cn
+Institute of Applied Mathematics, Chinese Academy of Sciences, Beijing 100190, and
+University of Chinese Academy of Sciences, Beijing 100049, P.R. China
+Email address: laishanfa@amss.ac.cn
+Institute of Applied Mathematics, Chinese Academy of Sciences, Beijing 100190, and
+University of Chinese Academy of Sciences, Beijing 100049, P.R. China
+Email address: qinguolin18@mails.ucas.ac.cn
+
diff --git a/OtAyT4oBgHgl3EQfg_hE/content/tmp_files/load_file.txt b/OtAyT4oBgHgl3EQfg_hE/content/tmp_files/load_file.txt
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf,len=1975
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='00368v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='AP]  1 Jan 2023  SLOW TRAVELING-WAVE SOLUTIONS FOR THE GENERALIZED  SURFACE QUASI-GEOSTROPHIC EQUATION  DAOMIN CAO, SHANFA LAI, GUOLIN QIN  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' In this paper, we systematically study the existence, asymptotic behaviors,  uniqueness, and nonlinear orbital stability of traveling-wave solutions with small prop-  agation speeds for the generalized surface quasi-geostrophic (gSQG) equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Firstly  we obtain the existence of a new family of global solutions via the variational method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Secondly we show the uniqueness of maximizers under our variational setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Thirdly  by using the variational framework, the uniqueness of maximizers and a concentration-  compactness principle we establish some stability theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Moreover, after a suitable  transformation, these solutions constitute the desingularization of traveling point vortex  pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Keywords: The gSQG equation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Traveling-wave solutions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Variational methods;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Existence  and uniqueness of maximizer;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Nonlinear stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  2020 MSC Primary: 76B47;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Secondary: 76B03, 35A02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Introduction and Main results  In this paper, we are concerned with the following generalized surface quasi-geostrophic  (gSQG) equation  �  ∂tθ + u · ∇θ = 0  in R2 × (0, T),  u = ∇⊥(−∆)−sθ  in R2 × (0, T),  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='1)  where 0 < s < 1, θ(x, t) : R2 × (0, T) → R is the active scalar being transported by the  velocity field u(x, t) : R2 × (0, T) → R2 generated by θ, and (a1, a2)⊥ = (a2, −a1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' The  operator (−∆)−s is defined by  (−∆)−sω(x) = Gsω(x) =  �  R2 Gs(x − y)ω(y)dy,  where Gs is the fundamental solution of (−∆)s in R2 given by  Gs(z) =  cs  |z|2−2s,  cs = Γ(1 − s)  22sπΓ(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  When s = 1/2, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='1) corresponds to the inviscid surface quasi-geostrophic (SQG) equa-  tion, which models the evolution of the temperature from a general quasi-geostrophic  system for atmospheric and atmospheric flows (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' [26, 52]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' The SQG equation has  1  2  DAOMIN CAO, SHANFA LAI, GUOLIN QIN  received extensive concern as a simplified model for the three-dimensional Euler equations  since [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Formally at least, in the limit s ↑ 1 we obtain the well-known two-dimensional  Euler equation in vorticity formulation [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' For general 0 < s < 1, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='1) was proposed  by C´ordoba et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' in [29] as an interpolation between the Euler equation and the SQG  equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  The global well-posedness for the Cauchy problem for two-dimensional incompressible  Euler equation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=', s = 1 in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='1)) has been well studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Global well-posedness for  Cauchy problems with initial data in L1 ∩ L∞ was established by Yudovich [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' The L1  assumption can be replaced by an appropriate symmetry condition thanks to the work of  Elgindi and Jeong [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' We refer to [36, 56] and references therein for more discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  However, to the best of our knowledge, the problem of whether the gSQG system presents  finite time singularities or there is global well-posedness of classical solutions is still open;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  see [20, 21, 46, 49, 50] and references therein for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  For vortex patch type global solutions, the first non-trivial example was constructed  in [44] for  1  2 < s < 1 by using the contour dynamics equation and bifurcation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Numerous results on the vortex patch type solutions for the gSQG equations were then  obtained in different situations (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' [18, 19, 28, 32, 39, 41, 45, 47]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' In [20], Castro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  established the first result of existence on global smooth solutions for the gSQG equation  by developing a bifurcation argument from a specific radially symmetric function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' In [43],  Gravejat and Smets, for the first time, proved the existence of smooth translating vortex  pairs for the SQG equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' This result was then generalized to the gSQG equation with  s ∈ (0, 1) by Godard-Cadillac [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' In [5], Ao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' successfully constructed traveling and  rotating smooth solutions to the gSQG equation with s ∈ (0, 1) by the Lyapunov-Schmidt  reduction method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  In this paper, we are interested in traveling-wave solutions for the gSQG equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Up  to a rotation, we may assume, without loss of generality, that these waves have a negative  speed −W in the vertical direction, so that  θ(x, t) = ω(x1, x2 + Wt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  In this setting, the first equation in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='1) is also reduced to a stationary equation  ∇⊥(Gsω − Wx1) · ∇ω = 0,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='2)  which has a weak form  �  R2 ω∇⊥(Gsω − Wx1) · ∇ϕdx = 0,  ∀ ϕ ∈ C∞  0 (R2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='3)  In the study of traveling-wave solutions for ideal incompressible fluids, translating vortex  pairs is the main concern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' The literature on vortex pairs can be traced back to the work  of Pocklington [59] in 1895.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' In 1906, Lamb [51] founded an explicit solution for the Euler  equation which is now generally referred to as the Lamb dipole or Chaplygin-Lamb dipole;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  see also [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Besides those exact solutions, the existence (and abundance) of translating  vortex pairs for the Euler equation has been rigorously established in [2, 8, 61, 64, 65]  and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' As mentioned above, for the gSQG equation, some examples of traveling-wave  solutions were constructed in [5, 15, 16, 40, 43, 45, 47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  TRAVELING-WAVE SOLUTIONS FOR THE GSQG EQUATION  3  In this paper, we will obtain a new family of traveling-wave solutions for the gSQG  equation and further investigate their asymptotic behaviors, uniqueness, and nonlinear  orbital stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' As pointed out by Arnol’d [6], a natural way of obtaining solutions to  the stationary problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='2) is to impose that ω and Gsω − Wx1 are (locally) functional  dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' That is, one may impose that  ω = f(Gsω − Wx1),  for some Borel measurable function f : R → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Usually f is supposed to satisfy the following hypotheses  (H1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' f(0) = 0, f is nonnegative and strictly increasing for t > 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  (H2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' limt→0+ t−  1  1−sf(t) = +∞ and limt→+∞ t−  1  1−sf(t) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Our first main result concerns the existence of traveling-wave solutions with slow trav-  eling speeds and the fine asymptotic behaviors of these solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' For convenience, we will  take W = Wε3−2s for some constant W > 0 and some small ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Let e2 = (0, 1) be the  unit vector along the x2-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Let R2  + := {x ∈ R2 | x1 > 0} be the right half plane  and 1S represents the characteristic function of a set S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Denote by spt(ω) the support of  a function ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' For fixed W > 0, κ > 0 denote  d0 =  �(1 − s)csκ  22−2sW  �  1  3−2s  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='4)  Our first result is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Let 0 < s < 1, W > 0 κ > 0 be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Suppose that f is a measurable  function satisfying (H1) and (H2) and f ∈ C1−2s if 0 < s < 1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Then there is a number  ε0 > 0 small such that for any ε ∈ (0, ε0), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='1) has a traveling-wave solution of the form  θε(x, t) = ωtr,ε(x + Wε3−2ste2) for some function ωtr,ε ∈ L∞(R2) in the sense that ωtr,ε  solves (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='3) with W = Wε3−2s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Moreover, ωtr,ε has the following properties:  (i) ωtr,ε is odd in x1 and even in x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' That is,  ωtr,ε(−x1, x2) = −ωtr,ε(x1, x2),  ωtr,ε(x1, −x2) = ωtr,ε(x1, x2),  ∀ x ∈ R2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  (ii) It holds for some constant µε  ωtr,ε = f(Gsωtr,ε − Wε3−2sx1 − µε),  in R2  +;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  (iii) Let ωε := ωtr,ε1R2  + and denote the center of mass of ωε by xε := κ−1 �  xωε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Then,  there hold  �  R2  +  ωε = κ and εxε = (d0, 0) + o(1),  as ε → 0,  where d0 s given by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Furthermore, there is a constant R > 0 independent of  ε such that spt(ωε) is contained in the disk with center xε and radius R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  4  DAOMIN CAO, SHANFA LAI, GUOLIN QIN  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' The assumption f ∈ C1−2s in the case 0 < s <  1  2 is used to improve the  regularity of Gsωε (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Propositions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='8 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='9 in [60]) so that the integral in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='3)  makes sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Typical examples of f satisfying the assumptions in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='1 includes any  C0,1 smooth bounded strictly increasing functions with f(0) = 0, such as f(t) = arctan(t+)  as well as some unbounded functions, for example, f(t) = tp  + with p ∈ (1,  1  1−s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Here t+  means max{0, t}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' We will obtain finer asymptotic behaviors and prove the uniqueness and  stability in the later case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' As we shall see in the proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='21, up to some translation, ωε  tends to a nontrivial function ω0 in L1 ∩ L2−s(R2), which is a maximizer of the limiting  problem considered in subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Therefore, the amplitude of the solutions obtained  in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='1 does not vanish as ε → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Let G+  s (x, y) :=  cs  |x−y|2−2s −  cs  |x−¯y|2−2s with ¯y = (−y1, y2) and define  G+  s ω :=  �  R2  +  G+  s (x, y)ω(y)dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  The proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='1 is based on a constrained maximization method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' More precisely,  take J be defined by J(t) =  � t  0 f −1(τ)dτ and let B(x, r) stand for the disk with center x  and radius r and κ > 0 be a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' We are to consider the problem of maximizing the  following functional  Eε(ω) := 1  2  �  R2  +  �  R2  +  ω(x)G+  s (x, y)ω(y)dxdy−Wε3−2s  �  R2  +  x1ω(x)dx−  �  R2  +  J(ω(x))dx, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='5)  over the constraint  Aε :=  �  ω ∈ L1 ∩ L2−s(R2  +) | ω ≥ 0, spt(ω) ⊂ B  �  (d0  ε , 0), d0  2ε  �  ,  �  R2  +  ω(x)dx = κ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='6)  Consider the following maximization problem:  eε := sup  ω∈Aε  Eε(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='7)  For the above maximizing problem we have the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Let 0 < s < 1 and W > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Suppose that f is a measurable function  satisfying (H1) and (H2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Then there is a number ε0 > 0 small such that eε can be  achieved for any ε ∈ (0, ε0), that is Eε admits a maximizer ωε in Aε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Moreover, ωε has the  following properties:  (i) ωε is Steiner symmetric with respect to some plane {x2 = const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' };' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  (ii) There is a constant µε such that  ωε = f  �  G+  s ωε − Wε3−2sx1 − µε  �  ,  in B(ε−1(d0, 0), ε−1d0/2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  TRAVELING-WAVE SOLUTIONS FOR THE GSQG EQUATION  5  (iii) The energy satisfies  I0 + O(ε2−2s) ≤ Eε(ωε) ≤ I0,  where I0 is given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='8);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  (iv) There exists a constant 0 < C < +∞ independent of ε such that  lim sup  ε→0+  ∥ωε∥L∞ ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  (v) Denote the center of mass of ωε by xε := κ−1 �  xωε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Then,  εxε = (d0, 0) + o(1),  as ε → 0,  where d0 is given by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Furthermore, there is a constant R > 0 independent of  ε such that spt(ωε) is contained in the disk with center xε and radius R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  As we will see that Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='1 can be derived from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Indeed, we will prove  in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='24 that if we further assume f ∈ C1−2s in the case 0 < s < 1  2, then after a  translation in x2, ωε(x) − ωε(¯x) is the desired function ωtr,ε in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Next, we shall investigate the problem of uniqueness, which is crucial in the study of  stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' We focus our attention on the case f = tp  + for some p ∈ (1,  1  1−s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' It can be seen  that such f satisfies all the assumptions in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='1 for s ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Therefore, for every  ε > 0 small, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='1 ensures a traveling-wave solution with ωtr,ε(x) = ωε(x) − ωε(¯x)  and ωε being a maximizer of Eε over constraint Aε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Inspired by the work on rotating stars  [48], we will prove that ωε is the unique maximizer in the sense that any maximizer of Eε  is a translation of ωε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  For fixed ε0 as in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='5 we denote by Σε the set of all maximizers of Eε over Aε  for ε ∈ (0, ε0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Our second main result is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Suppose that f(t) = tp  + for some p ∈ (1,  1  1−s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Let ε0 be as in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='5  and ε ∈ (0, ε0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Let ωε ∈ Σε be a maximizer as obtained in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Then there is a  number ε1 ∈ (0, ε0] such that for all ε ∈ (0, ε1),  Σε = {ωε(· + ce2) | c ∈ R}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  For relative equilibria of fluids, there are fewer mathematical results available on the  uniqueness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' The first result was due to Amick and Fraenkel [3], who proved that Hill’s  spherical vortex is the unique solution when viewed in a natural weak formulation by the  method of moving planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Later Amick and Fraenkel [4] also established local uniqueness  for Norbury’s nearly spherical vortex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' The uniqueness of the Chaplygin-Lamb dipole was  shown by Burton [10] using a similar method as [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Recently, Jang and Seok [48] proved  the uniqueness of maximizers of a variational problem related to rotating binary stars,  which inspired our proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' As for the gSQG equation, to the best of our  knowledge, the only result on this issue is the recent work [15], where a very special case  was considered in order to apply the method of moving planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Our result Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='6  provides uniqueness in a wide range of cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Our last main result concerns the orbital stability of the traveling-wave solutions ob-  tained in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' We first prove a general stability theorem in a similar spirit as  6  DAOMIN CAO, SHANFA LAI, GUOLIN QIN  [13], where the stability of vortex pairs for the 2D Euler equation was considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' To be  precise, we will consider the maximization problem of the functional  ˜EW(ζ) := 1  2  �  R2  +  ζ(x)G+  s ζ(x)dx − W  �  R2  +  x1ζ(x)dx  over the set R(ζ0)w, which means the weak closure of the rearrangement class of a given  function ζ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Using the concentrate compactness principle due to Lions [55], we establish  the compactness of maximizing sequence and derive a general nonlinear stability theorem  on the set of maximizers (see Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  In the case f = tp  + for p ∈ (1,  1  1−s), Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='1 provides a traveling-wave solution with  ωtr,ε(x) = ωε(x) − ωε(¯x) and ωε being a maximizer of Eε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' To apply the stability theorem  of the set of maximizers, we need to consider an auxiliary variational problem: maximize  ˜EW with W = Wε3−2s over the set R(ωε)w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' By studying the asymptotic behaviors of  maximizers and using the uniqueness result in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='6, we are able to show that all  the maximizers of the second variational problem are actually translations of ωε in the x2-  direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' As a consequence, we obtain the orbital stability of ωε by applying the nonlinear  stability theorem on the set of maximizers proved in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Roughly speaking, our stability result is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Suppose that f(t) = tp  + for some p ∈ (1,  1  1−s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Let ωtr,ε be the traveling-  wave solution obtained in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Then for ε fixed small, ωtr,ε is orbitally stable  in the following sense: for arbitrary M > 0 and η > 0, there exists δ > 0 such that for  non-negative function ξ0 ∈ L1 ∩ L∞(R2  +) with ||ξ0||∞ < M and  infc∈R  �  ∥ξ0 − ωtr,ε(· + ce2)∥L1(R2  +) + ∥ξ0 − ωtr,ε(· + ce2)∥L2(R2  +)  +∥x1(ξ0 − ωtr,ε(· + ce2))∥L1(R2  +)  �  ≤ δ,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='8)  if there exists a L∞-regular solution ξ(t) with initial data ξ0(x) for t ∈ [0, T) with 0 < T ≤  ∞, then all t ∈ [0, T),  infc∈R  �  ∥ξ(t) − ωtr,ε(· + ce2)∥L1(R2  +) + ∥ξ(t) − ωtr,ε(· + ce2)∥L2(R2  +)  +∥x1(ξ(t) − ωtr,ε(· + ce2))∥L1(R2  +)  �  ≤ η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='9)  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' For the rigorous definition of L∞-regular solutions for the gSQG equation,  please see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Once the uniqueness of other solutions was established, one may  apply the general stability theorem and the framework in this paper to obtain their orbital  stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Compared with the result in [13], we admit perturbations with non-compact  supports, which is achieved by bringing in the L1-norm in our theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Much work has been done on the stability of steady solutions to the Euler equations,  for which we refer the interested reader to [1, 11, 12, 17, 23, 24, 25, 38, 63] and references  therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  TRAVELING-WAVE SOLUTIONS FOR THE GSQG EQUATION  7  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Desingularize the traveling point vortices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' The result in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='1 also pro-  vides a family of solutions that desingularize the traveling point vortices for the gSQG  equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Indeed, taking the transformation ˆωtr,ε(x) = ε−2ωtr,ε(ε−1x), we conclude from  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='1 immediately that  Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Let 0 < s < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Suppose that f is a function satisfying (H1) and (H2) and  f ∈ C1−2s if 0 < s < 1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Then there is a constant ε0 > 0 small such that for any ε ∈ (0, ε0),  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='1) has a traveling-wave solution of the form θε(x, t) = ˆωtr,ε(x+Wte2) for some function  ˆωtr,ε ∈ L∞(R2) in the sense that ˆωtr,ε solves (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='3) with W = W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Moreover, ˆωtr,ε has the  following properties:  (i) ˆωtr,ε is odd in x1 and even in x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' That is,  ˆωtr,ε(−x1, x2) = −ˆωtr,ε(x1, x2),  ˆωtr,ε(x1, −x2) = ˆωtr,ε(x1, x2),  ∀ x ∈ R2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  (ii) It holds  ˆωtr,ε = f(ε2−2s(Gsˆωtr,ε − Wx1) − µε),  in R2  +,  for some constant µε;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  (iii) There holds in the sense of measure  ˆωtr,ε(x) ⇀ κδ(x − (d0, 0)) − κδ(x + (d0, 0)),  where d0 =  �  (1−s)csκ  22−2sW  �  1  3−2s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Furthermore, there is a constant R > 0 independent of  ε such that spt(ˆωε) is contained in B((d0, 0), Rε) ∪ B((−d0, 0), Rε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='9 (iii) implies that {ˆωtr,ε}ε∈(0,ε0) is a sequence of regular solutions approxi-  mating the traveling point vortex pair for the gSQG equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  In [5], Ao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' constructed a family of solutions closed to the points vortices of the gSQG  equation with the profile function f(t) = tp  + for p ∈ (1, 1+s  1−s) by the Lyapunov-Schmidt  reduction method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' It can be seen that our result Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='9 covers the remaining case  p ∈ (0, 1] for 1  2 ≤ s < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  In the recent paper [16], a family of traveling solutions for the gSQG equations with  1  2 ≤ s < 1 were constructed by the variational method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' The solutions {˜ωtr,ε} obtained in  [16] solve the integral equation  ˜ωtr,ε = g(Gs˜ωtr,ε − Wx1 − ˜µε),  in R2  +,  for some constant ˜µε and bounded non-decreasing function g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' It is obvious that Corollary  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='9 can not be deduced from the result in [16], since the profile function f(ε2−2s·) in  Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='9 (ii) varies along with ε and is allowed to be unbounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Therefore, Corollary  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='9 provides a new family of traveling-wave solutions for the gSQG equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' In Section 2, for a large class of f, we construct  traveling-wave solutions for the gSQG equation via a variational method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' We first study  the properties of maximizers of a limiting problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Based on these properties, we are  able to construct traveling-wave solutions with small traveling speeds by maximizing the  energy functional Eε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Then, we study the asymptotic behavior of the maximizers carefully  in several steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' With detailed asymptotic behaviors in hand, we prove the uniqueness of  8  DAOMIN CAO, SHANFA LAI, GUOLIN QIN  maximizers in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Section 4 is devoted to investigating nonlinear stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' We first  prove a general orbital stability theorem for the set of maximizers based on a combination  of the variational method and the concentrated compactness lemma of Lions [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Then, we  investigate the asymptotic behavior of maximizers in the rearrangement class and obtain  the orbital stability Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='7 by using the uniqueness result in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Proofs of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='1 and Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='5  In this section, we first consider the maximization problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='7) and prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='1 follows immediately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  We assume that J : [0, +∞) → [0, +∞) satisfies  (H′  1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' J is strictly convex and nonnegative;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  (H′  2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' limt→0+ J′(t)ts−1 = 0 and lim inft→+∞ J′(t)ts−1 ≥ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Here K is a large constant, which will be determined later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Note that if J(t) =  � t  0 f −1(τ)dτ  for some f satisfying (H1) and (H2), then one can check that J satisfies (H′  1) and (H′  2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Let Eε(ω) and Aε be defined as in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='5) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' To obtain the existence of maximizers  for (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='7) we need to consider its limiting problem first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' The limiting problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' We start with definitions of the energy functional and set  of constraints for the limiting problem corresponding to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  The energy functional  associated with Eε is  E0(ω) := cs  2  �  R2  �  R2  ω(x)ω(y)  |x − y|2−2sdxdy −  �  R2 J(ω(x))dx,  and the constraint associated with Aε is  A0 :=  �  ω ∈ L1 ∩ L2−s(R2) | ω ≥ 0,  �  R2 ω(x)dx = κ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  The limiting maximization problem associated with (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='7) is  e0 := sup  ω∈A0  E0(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='1)  In the classical paper [55], under a bit weaker assumption limt→0+ J(t)t−1 = 0, Lions  showed the existence of maximizers of E0 over A0 (see Theorem II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='2 and Corollary II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='1 in  [55]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' As we shall see later, our assumption limt→0+ J′(t)ts−1 = 0 ensures that every max-  imizer is compactly supported, which is an essential property used in the next subsection  (for similar results on rotating stars, we refer to [7, 53, 57]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  In what follows, we will investigate some essential properties of maximizers under our  hypotheses (H′  1) and (H′  2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Recall that Gsω :=  cs  |x|2−2s ∗ ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Denote ∥ · ∥p := ∥ · ∥Lp(R2) for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' The following  two lemmas concerning convolution inequalities are needed in our later discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  TRAVELING-WAVE SOLUTIONS FOR THE GSQG EQUATION  9  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Assume that ω ∈ L1 ∩ Lp(R2) for some p > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  If 1 < p ≤ s−1, then  Gsω ∈ Lq(R2) for any  1  1−s < q <  p  1−sp and for some constants 0 < a, b < 1,  ∥Gsω∥q ≤ C  �  ∥ω∥a  1∥ω∥1−a  p  + ∥ω∥b  1∥ω∥1−b  p  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='2)  If p > s−1, then (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='2) holds with q = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' We first consider the case 1 < p ≤ s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' We split the function |x|2s−2 into two  parts: |x|2s−2 = |x|2s−21{|x|<1} + |x|2s−21{|x|≥1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' It is easy to check that |x|2s−21{|x|<1} ∈  Lr, ∀ 1 ≤ r <  1  1−s and |x|2s−21{|x|≥1} ∈ Lr, ∀ r >  1  1−s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Suppose  1  1−s < q <  p  1−sp, then  there exist 1 ≤ r1 <  1  1−s, r2 >  1  1−s and 1 < p1, p2 < p such that 1 + q−1 = r−1  1  + p−1  1  and  1 + q−1 = r−1  2  + p−1  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Then it remains to apply the following Young inequality  ∥f ∗ g∥r ≤ ∥f∥u∥g∥v,  ∀f ∈ Lu, g ∈ Lv,  for 1 ≤ r, u, v ≤ +∞ such that 1 + r−1 = u−1 + v−1, and the interpolation inequality  ∥f∥r ≤ ∥f∥a  u∥f∥1−a  v  ,  ∀f ∈ Lu ∩ Lv,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='3)  with 1 ≤ u < r < v ≤ +∞ and a = (r−1 − v−1)/(u−1 − v−1) ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  For p > s−1, the proof is similar, so we will omit the detail and finish the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  □  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Suppose that ω ∈ L1 ∩ L2−s(R2), then it holds  ���  �  R2 ω(x)Gsω(x)dx  ��� ≤ C∥ω∥s  1  �  R2 |ω|2−s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='4)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' The well-known Hardy-Littlewood-Sobolev inequality states that  ∥Gsω∥q ≤ C∥ω∥p,  ∀1 < p < q < +∞ with 1  q = 1  p − s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='5)  Applying H¨older’s inequality, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='5) with q = 2−s  1−s and the interpolation inequality (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='3),  for any ω1, ω2 ∈ L1 ∩ L2−s(R2) we obtain  ���  �  R2 ω1(x)Gsω2(x)dx  ��� ≤ ∥ω1∥2−s∥Gsω2∥ 2−s  1−s ≤ C∥ω1∥2−s∥ω2∥1−s  2−s∥ω2∥s  1,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='6)  which implies (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='4) by taking ω1 = ω2 = ω and completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  □  We first show the radial symmetry and derive the Euler-Lagrange equation for a maxi-  mizer of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Let ω0 ∈ A0 be a maximizer of E0 over A0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Then ω0 must be radially  symmetric and non-increasing with respect to some point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Moreover, there exists a constant  µ0 ∈ R such that  �  Gsω0 − J′(ω0) ≤ µ0,  on {ω0 = 0},  Gsω0 − J′(ω0) = µ0,  on {ω0 > 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='7)  10  DAOMIN CAO, SHANFA LAI, GUOLIN QIN  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' The radial symmetry and monotonicity of ω0 are easy consequences of the strict  rearrangement inequality (see Theorems 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='7 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='9 in [54]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Note that for δ > 0 small, the set {ω0 > δ} ̸= ∅ due to  �  R2 ω0 = κ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' We fix a δ > 0  such that {ω0 > δ} ̸= ∅ and take a function φ0 so that  �  R2 φ0 = 1 and spt(φ0) ⊂ {ω0 > δ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  For any function φ bounded from below such that φ ≥ 0 on the set {ω0 ≤ δ}, we take a  family of test functions as follows:  ωt := ω0 + t(φ − φ0  �  R2 φ),  which belong to A0 for |t| small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Since ω0 is a maximizer, we have  0 = dE0(ωt)  dt  �����  t=0  =  �  R2(Gsω0 − J′(ω0) − µ0)φdx,  where µ0 :=  �  R2(Gsω0 −J′(ω0))φ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Then (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='7) follows from the arbitrariness of φ and hence  the proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  □  Denote  I0 := sup  ω∈A0  E0(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='8)  Then I0 < +∞ by [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' There is a constant c0 > 0 such that if ω0 ∈ A0 is a maximizer of E0 over  A0, then one has  ∥ω0∥2−s ≤ c0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='9)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Take the constant K in the hypothesis (H′  2) as K = Cκs+2, where C is the constant  in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Then there is a constant t0 > 0 such that J(t) > (Cκs + 1)t2−s for t > t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' On the  one hand, by the definition of E0 and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='4), we deduce  �  R2 J(ω0) ≤ −I0 + Cκs  �  R2(ω0)2−sdx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  On the other hand, we infer from the choice of t0 that  (Cκs + 1)  �  R2(ω0)2−sdx = (Cκs + 1)  �  {ω0≤t0}  (ω0)2−sdx + (Cκs + 1)  �  {ω0>t0}  (ω0)2−sdx  ≤ (Cκs + 1)t1−s  0  κ +  �  R2 J(ω0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Therefore, we arrive at (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='9) by taking c0 = (−I0 + (Cκs + 1)t1−s  0  κ)  1  2−s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  □  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Let ω0 ∈ A0 be a maximizer and µ0 be the constant defined in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='3,  then one has  µ0 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='10)  TRAVELING-WAVE SOLUTIONS FOR THE GSQG EQUATION  11  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Take r0 > 0 such that  �  B(0,r0) ω0 ≥ κ  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Since for any r ≥ 0,  �  B(0,r) ω0 ≤ κ, there is  a point xr ∈ B(0, r) such that ω0(xr) ≤ π−1κr−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Then we infer from the hypothesis (H′  2)  that  J′(ω0(xr)) = o(r2s−2),  as r → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  On the other hand, we have  Gsω0(xr) ≥  cs  (2r)2−2s  �  B(0,r)  ω0 ≥ 4s−2csκr2s−2,  ∀ r ≥ r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Thus, we derive from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='7) that  µ0 ≥ Gsω0(xr) − J′(ω0(xr)) ≥ (4s−2csκ + o(1))r2s−2 > 0,  for r sufficiently large and hence the proof is finished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  □  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='5 will give a uniform bound for all maximizers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' There is a constant c1 > 0 such that if ω0 ∈ A0 is a maximizer, then  ∥ω0∥∞ ≤ c1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='11)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' By Lemmas 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='3 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='5, we have  J′(ω0) = Gsω0 − µ0 ≤ Gsω0,  on {ω0 > 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Let p1 = 2 − s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Then we derive from Lemmas 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='4 that ∥Gsω0∥q ≤ C for  1  1−s <  q <  p1  1−sp1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Using the fact the J′(t) ≥ K  2 t1−s for t > t1 due to the hypothesis (H′  2) with  t1 a large constant, we obtain ∥ω0∥r ≤ C for 1 ≤ r < p2 with p2 := (1−s)p1  1−sp1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Notice that  p2 = p1 + sp1(p1−1)  1−sp1  and sp1(p1−1)  1−sp1  > 0 is increasing in p1 ∈ (1, s−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' So, using Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='1, a  simple bootstrap argument will prove this lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  □  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' It holds  I0 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' We take a function ρ := 1B(0,√  κ/π) and define ρr(x) := (r−2)ρ(r−1x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' It can be seen  that ρr ∈ A0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  By the hypothesis (H′  2), we find  �  R2 J(ρr) = o(1)  �  R2(ρr)2−s = o(r2s−2),  as r → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  On the other hand, a change of variables gives  �  R2 ρrGsρr = r2s−2  �  R2 ρGsρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Thus, we can take a constant r1 sufficiently large such that  I0 ≥ E0(ρr1) ≥ c3r2s−2  1  > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  □  12  DAOMIN CAO, SHANFA LAI, GUOLIN QIN  Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' If ω0 ∈ A0 is a maximizer of E0 over A0, then it holds  ∥Gsω0∥∞ ≥ 2I0κ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='12)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' One has  I0 = E0(ω0) ≤ 1  2  �  R2 ω0Gsω0dx ≤ κ  2∥Gsω0∥∞,  which implies (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='12) and completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  □  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' There is a constant η > 0 such that if ω0 ∈ A0 is a maximizer of E0 over  A0, then we have  sup  x∈R2  �  |x−y|<1  ω0(y)dy ≥ η > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='13)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Denote η0 := supx∈R2  �  |x−y|<1 ω0(y)dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Let r := (csI−1  0 κ2)  1  2−2s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' For any x ∈ R2, we  calculate  Gsω0(x) =  �  |x−y|<1  csω0(y)  |x − y|2−2sdy +  �  1≤|x−y|<r  csω0(y)  |x − y|2−2sdy +  �  |x−y|≥r  csω0(y)  |x − y|2−2sdy  =: A1 + A2 + A3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  For the first term A1, we use Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='1 and Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='6 to obtain  A1 ≤ C(ηa  0 + ηb  0),  for some a, b ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  For the second term A2, noticing that the annulus {y | 1 < |x − y| < r} can be covered by  ≤ Cr2 disks with radius 1, we find  A2 ≤ Ccsr2η0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Now, by the choice of r, the last integral A3 can be estimated by  A3 ≤ csr2s−2κ = I0κ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  So, by the arbitrariness of x and Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='8, we arrive at  C(ηa  0 + ηb  0 + η0) ≥ I0κ−1,  which implies (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='13) and completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  □  Now we show that the Lagrange multiplier µ0 is uniformly bounded from below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' There is a constant µ∗ > 0 such that if ω0 ∈ A0 is a maximizer of E0 over  A0, then there holds  µ0 ≥ µ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='14)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' By the previous lemma, one can find a point xη ∈ R2 such that  �  |xη−y|<1  ω0(y)dy ≥ η  2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  TRAVELING-WAVE SOLUTIONS FOR THE GSQG EQUATION  13  In view of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='3, we may assume that ω0 is radially symmetric with respect to the  origin and non-increasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Thus, we get  �  |y|<1  ω0(y)dy ≥  �  |xη−y|<1  ω0(y)dy ≥ η  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  For any r > 1 large, there is a point xr ∈ B(0, r) such that ω0(xr) ≤ π−1κr−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Then we  infer from the hypothesis (H′  2) that  J′(ω0(xr)) = o(r2s−2),  as r → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  On the other hand, we have  Gsω0(xr) ≥  cs  (2r)2−2s  �  B(0,1)  ω0 ≥ 4s−2csηr2s−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Thus, by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='7), we can take a large constant r0 such that  µ0 ≥ Gsω0(xr0) − J′(ω0(xr0)) ≥ (4s−2csη0 + o(1))r2s−2  0  ≥ 4s−3csη0r2s−2  0  =: µ∗ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  The proof is therefore finished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  □  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' There exists a constant R∗ > 0 such that if ω0 ∈ A0 is a maximizer of E0  over A0, then the diameter of the support of ω0 is less than 2R∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' That is,  diam(spt(ω0)) ≤ 2R∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='15)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Without loss of generality, we may assume that ω0 is radially symmetric and non-  increasing with respect to the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Since  �  ω0 = κ < +∞ and ω0(x) = ω0(|x|) is  non-increasing in |x|, we have  �  |x−y|<1  ω0(y) ≤ C|x|−2  for |x| large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Then, through similar calculations as the proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='9, for large |x|  and any constant r, we get  Gsω0(x) =  �  |x−y|<1  csω0(y)  |x − y|2−2sdy +  �  1≤|x−y|<r  csω0(y)  |x − y|2−2sdy +  �  |x−y|≥r  csω0(y)  |x − y|2−2sdy  ≤ C(|x|−2a + |x|−2b + r2|x|−2) + csκr2s−2,  which will tend to 0 if we first take |x| → +∞ then r → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Fixed r such that csκr2s−2 ≤  µ∗  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Then, one can find a constant R∗ such that Gsω0(x) < µ∗ whenever |x| > R∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Take  R4 = max{R3, c1, R∗ + 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' We infer from Lemmas 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='3 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='10 that ω0(x) = 0 for any  |x| > R∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' The proof of this lemma is hence completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  □  Next, we further study the properties of the maximizers in the special case J(t) = Lt1+ 1  p  for some constants L > 0 and p ∈ (0,  1  1−s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' We first determine the Lagrange multiplier µ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Recall that we denote I0 = supω∈A0 E0(ω) ∈ (0, +∞) to be maximum value of E0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  14  DAOMIN CAO, SHANFA LAI, GUOLIN QIN  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Suppose that J(t) = Lt1+ 1  p for some constants L > 0 and p ∈ (0,  1  1−s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Let  ω0 be a maximizer of E0 over A0 with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='7) for some µ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Then, we have  µ0κ = Cs,pI0,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='16)  for some constant Cs,p depending only on s, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Let γ := 1 + 1/p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  We take a family of functions (ω0)t(x) := t−2ω0(t−1x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  By  changing of variables, we find  E0((ω0)t) = t2s−2  2  �  R2 ω0Gsω0 − t2−2γ  �  R2 J(ω0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Since ω0 is a maximizer, we have  0 = d  dt  ����  t=1  E0((ω0)t) = (s − 1)  �  R2 ω0Gsω0 − (2 − 2γ)  �  R2 J(ω0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='17)  By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='7), one has  Lγωγ−1  0  = (Gsω0 − µ0)+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Multiplying the above equation by ω0 and integrating, we obtain  κµ0 =  �  R2 ω0Gsω0 − γ  �  R2 J(ω0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='18)  Then (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='16) follows from simply calculations by using the definition of E0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='17) and  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' The constant Cs,p = Aγ := 2−γ−sγ  2−s−γ with γ = 1 + 1/p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  □  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='12 states that the Lagrange multiplier is the same for all maximizers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Set  ψ0 := Gsω0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Then, ψ0 is radially symmetric and satisfies the following equation by Lemma  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  �  (−∆)sψ0 =  �  p  L(p+1)  �p  (ψ0 − µ0)p  +,  in R2,  ψ0(x) → 0,  as |x| → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='19)  Thus, the uniqueness result in [22] is applicable for p ∈ (1,  1  1−s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Furthermore, Ao et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' [5] showed the non-degeneracy of the linearized equation for p ∈ (1, 1+s  1−s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Note that  for p ∈ (0, 1] the uniqueness of maximizers has been proved in [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Summarizing these  results, we obtain  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Suppose that J(t) = Lt1+ 1  p for some constants L > 0 and p ∈ (0,  1  1−s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Then, up to translations E0 has a unique maximizer ω0 over A0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Moreover, the following  properties hold:  (i) ω0 is compact supported and radially symmetric and decreasing about some point;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  (ii) ω0 =  �  p  L(p+1)  �p  (Gsω0 − µ0)p  + for µ0 = κ−1Cs,pE0(ω0) > 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  (iii) If in addition p ∈ (1, 1+s  1−s), then ω0 ∈ C1 and the kernal of the linearized operator  ω �→ ω − p  �  p  L(p + 1)  �p  (Gsω0 − µ0)p−1  +  Gsω  TRAVELING-WAVE SOLUTIONS FOR THE GSQG EQUATION  15  in the space L1 ∩ L∞(R2) is  span{∂x1ω0, ∂x2ω0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' The existence of a maximizer ω0, (i) and (ii) follow from the above lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' For  uniqueness, we refer to [22, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' The non-degeneracy was proved in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='2 in [5]  in terms of ψ0 = Gsω0, form which one can obtain (iii) easily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  □  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Existence of traveling-wave solutions via maximization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' In this subsection, we  will obtain the existence of traveling-wave solutions by considering the maximization prob-  lem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='7), whose associated limiting problem has been studied in the preceding subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Existence of maximizers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' To obtain the compactness of maximizing sequence, we  first maximize Eε over a set smaller than Aε given by  Aε,Γ :=  �  ω ∈ L1∩L∞(R2  +)  ��� 0 ≤ ω ≤ Γ,  spt(ω) ⊂ B(ε−1(d0, 0), ε−1d0/2),  �  R2  +  ω(x)dx = κ  �  ,  where Γ > 0 is a number that will be fixed later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' For given ε, Γ > 0, there exists a function ωε,Γ ∈ Aε,Γ such that  Eε(ωε,Γ) = sup  ω∈Aε,Γ  Eε(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Let {ωj}∞  j=1 ⊂ Aε,Γ be a maximizing sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' By the definition of Aε,Γ, we know  that {ωj}∞  j=1 is uniformly bounded in L1 ∩ L∞(B(ε−1(d0, 0), ε−1d0/2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Passing to a sub-  sequence (still denoted by {ωj}∞  j=1), we may assume ωj → ωε,Γ weakly star in L∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' By the  weak star convergence, it holds  0 ≤ ωε,Γ ≤ Γ,  �  R2  +  ωε,Γ(x)dx = lim  j→+∞  �  R2  +  ωj(x)dx = κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Note that Gs(x, y) ∈ Lr(B(ε−1(d0, 0), ε−1d0/2) × B(ε−1(d0, 0), ε−1d0/2)) for 1 ≤ r <  1  1−s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  We have  lim  j→+∞  cs  2  �  R2  +  �  R2  +  ωj(x)ωj(y)  |x − y|2−2s dxdy = cs  2  �  R2  +  �  R2  +  ωε,Γ(x)ωε,Γ(y)  |x − y|2−2s  dxdy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  On the other hand, since J is convex, by the lower semi-continuity, we find  lim  j→+∞  �  R2  +  J(ωj(x))dx ≥  �  R2  +  J(ωε,Γ(x))dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Therefore,  sup  ω∈Aε,Γ  Eε(ω) = lim  j→+∞ Eε(ωj) ≤ Eε(ωε,Γ) ≤ sup  ω∈Aε,Γ  Eε(ω),  which implies that ωε,Γ is a maximizer and completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  □  16  DAOMIN CAO, SHANFA LAI, GUOLIN QIN  For a non-negative function ζ, we shall say that ζ is Steiner symmetric in the x2-variable  if for any fixed x1, ζ is the unique even function of x2 such that  ζ(x1, x2) > τ  if and only if  |x2| < 1  2 | {y2 ∈ R | ζ(x1, y2) > τ} |R,  where | · |R denotes the Lebesgue measure on R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  For a function 0 ≤ ζ ∈ L1(D) with D a domain symmetric with respect to x1-axis,  we denote by ζ⋆ the Steiner symmetrization of ζ, which is the unique function in the  rearrangement class that is Steiner symmetric in the x2-variable (see [54] for more details  about rearrangement).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' A key fact about the Steiner symmetrization is the rearrangement  inequality (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Theorems 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='7 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='9 in [54])  �  ζ⋆Gsζ⋆ ≥  �  ζGsζ,  �  ζ⋆G+  s ζ⋆ ≥  �  ζG+  s ζ,  with strict inequality unless ζ(·) ≡ ζ⋆(· + (0, c)) for some c ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' For given ε, Γ > 0, let ωε,Γ ∈ Aε,Γ be a maximizer of Eε over Aε,Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Then  ωε,Γ is symmetric non-increasing with respect to some line {x2 = const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Moreover, there  exists a constant µε,Γ ∈ R such that  \uf8f1  \uf8f4  \uf8f2  \uf8f4  \uf8f3  G+  s ωε,Γ − Wε3−2sx1 − J′(ωε,Γ) ≤ µε,Γ,  on {ωε,Γ = 0},  G+  s ωε,Γ − Wε3−2sx1 − J′(ωε,Γ) = µε,Γ,  on {0 < ωε,Γ < Γ},  G+  s ωε,Γ − Wε3−2sx1 − J′(ωε,Γ) ≥ µε,Γ,  on {ωε,Γ = Γ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='20)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' The symmetry and monotonicity of ωε,Γ with respect to some line {x2 = const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='} is  an easy consequence of the strict rearrangement inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  For any ω ∈ Aε,Γ, we take a family of test functions as follows:  ωt := ωε,Γ + t(ω − ωε,Γ),  t ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Since ωε,Γ is a maximizer, we have  0 ≥ dEε(ωt)  dt  �����  t=0+  =  �  R2  +  (G+  s ωε,Γ − Wε3−2sx1 − J′(ωε,Γ))(ω − ωε,Γ)dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  That is,  �  R2  +  (G+  s ωε,Γ − Wε3−2sx1 − J′(ωε,Γ))ωdx ≥  �  R2  +  (G+  s ωε,Γ − Wε3−2sx1 − J′(ωε,Γ))ωε,Γdx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Then (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='20) follows by applying an adaption of the bathtub principle (see section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='14 in  [54]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' The proof is thus complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  □  By the definition of I0 (see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='8)) in the previous subsection we have  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' There are two constants ε0, Γ0 > 0 such that for any ε ∈ (0, ε0) and Γ > Γ0,  it holds  sup  ω∈Aε,Γ  Eε(ω) ≥ I0 + O(ε2−2s),  as ε → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='21)  TRAVELING-WAVE SOLUTIONS FOR THE GSQG EQUATION  17  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Let ω0 be the maximizer of E0 over A0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' We may assume that ω0 is symmetric and  non-increasing with respect to the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Since ω0 has compact support and is bounded,  we may take Γ0 > 0 large and ε0 > 0 small such that  ¯ω0(x) := ω0(x − ε−1(d0, 0)) ∈ Aε,Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Then direct computation shows  sup  ω∈Aε,Γ  Eε(ω) ≥ Eε(¯ω0) = E0(ω0) + O(ε2−2s),  which implies the desired estimate and finishes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  □  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' There are two constants C0, ε1 > 0 such that if ωε,Γ ∈ Aε,Γ is a maximizer  of Eε over Aε,Γ for ε ∈ (0, ε1) and Γ > Γ0, then it holds  ∥ωε,Γ∥2−s ≤ C0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='22)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Notice that by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='16, there exists 0 < ε1 ≤ ε0 such that Eε(ωε,Γ) ≥ I0  2 > 0  for ε ∈ (0, ε1) and Γ > Γ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Thus, we have  �  R2  +  J(ωε,Γ) ≤ 1  2  �  R2  +  ωε,ΓGsωε,Γ − Eε(ωε,Γ) < 1  2  �  R2  +  ωε,ΓGsωε,Γ,  from which, by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='2 and a similar argument as the proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='4, we obtain  the estimate (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='22) and complete the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  □  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' There is a constant ε2 > 0 such that if ωε,Γ ∈ Aε,Γ is a maximizer of Eε  over Aε,Γ for ε ∈ (0, ε2) and Γ > Γ0, then it holds  µε,Γ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='23)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' For fixed Γ > Γ0, we see from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='16 that {ωε,Γ}ε∈(0,ε0) is a maximizing  sequence of E0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Then Theorem II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='2 and Corollary II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='1 in [55] give a subsequence (still  denoted by {ωε,Γ} for convenience) converges to a maximizer ω0 of E0 in L1 ∩ L2−s(R2)  after suitable translations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Thus, for ε small , we have  �  B(pε,R∗) ωε,Γ ≥ κ  2 for some point  pε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Here, R∗ is the constant in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Since  �  ωε,Γ = κ, for R∗ < r < d0  2ε large, we  can take a point xr ∈ B(pε, r) ∩ B(ε−1(d0, 0), ε−1d0/2) such that ωε,Γ(xr) ≤ Cr−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Then,  using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='20) and the hypothesis (H′  2), by similar calculations as the proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='10,  we have  µε,Γ ≥ G+  s ωε,Γ − Wε3−2sx1 − J′(ωε,Γ) ≥ (4s−2csκ + o(1))r2s−2 − O(ε2−2s),  which implies (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='23) by taking r = ε− 1  2 and ε sufficiently small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  □  As an immediate consequence of Lemmas 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='15, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='17 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='18, we can obtain the following  result through a similar argument as the proof of Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' We leave the details of  the proof to readers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' There is a constant C1 > 0 such that if ωε,Γ ∈ Aε,Γ is a maximizer of Eε  over Aε,Γ for ε ∈ (0, ε2) and Γ > Γ0, then  ∥ωε,Γ∥∞ ≤ C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='24)  18  DAOMIN CAO, SHANFA LAI, GUOLIN QIN  Moreover, if Γ > max{Γ0, C1}, then  ωε,Γ = (J′)−1 �  (G+  s ωε,Γ − Wε3−2sx1 − µε,Γ)+  �  ,  in B(ε−1(d0, 0), ε−1d0/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='25)  Having made the necessary preparation, we next establish the existence and properties  of maximizers for (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Existence and properties of maximizers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' We first state some basic properties of max-  imizers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' For each ε ∈ (0, ε2), there exists a maximizer for (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Let ωε be a  maximizer of Eε over Aε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Then the following assertions hold:  (i) ωε is Steiner symmetric with respect to some plane {x2 = const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' };' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  (ii) There is a constant µε such that  ωε = (J′)−1 �  (G+  s ωε − Wε3−2sx1 − µε)+  �  ,  in B(ε−1(d0, 0), ε−1d0/2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='26)  (iii) The energy satisfies  I0 + O(ε2−2s) ≤ Eε(ωε) ≤ I0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  (iv) There exists a constant 0 < C < +∞ independent of ε such that  lim sup  ε→0+  ∥ωε∥∞ ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='27)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Fix a Γ > max{Γ0, C1}, then ωε,Γ ∈ Aε,Γ is a maximizer of Eε over Aε for ε ∈ (0, ε2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  That is, we obtain a maximizer of Eε over Aε for each ε small, which we denote by ωε for  simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' The properties of these maximizers can be derived by arguments quite similar  to those in the preceding subsection, so we omit the details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  □  We note that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='26) is not yet sufficient to provide a dynamically possible steady vortex  flow for the gSQG equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' This is because of the presence of the truncation function  1B(ε−1(d0,0),ε−1d0/2), which makes ωε and G+  s ωε − Wε3−2sx1 − µε may be not functional  dependent in the whole space R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' To get the desired solution, we need to prove that the  support of ωε is away from the boundary of B(ε−1(d0, 0), ε−1d0/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' We will show that this  is the case when ε is sufficiently small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' It is based on the observation that in order to  maximize energy, the diameter of the support of a maximizer can not be too large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' We  will reach this conclusion in several steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' We begin by giving a lower bound of µε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' For any δ > 0, there exist an εδ > 0 such that  µε ≥ µ∗ − δ,  ∀ ε ∈ (0, εδ),  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='28)  where µ∗ > 0 is the constant in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='10  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Suppose on the contrary that there are constant δ0 > 0 and a sequence {εj}∞  j=1 with  εj → 0 as j → ∞ such that lim supj→+∞ µεj ≤ µ∗ − δ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='20, we know that  {ωεj}∞  j=1 is a maximizing sequence of E0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Then Theorem II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='2 and Corollary II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='1 in [55] give  a subsequence (still denoted by {ωεj}∞  j=1 for convenience) converges to a maximizer ω0 of  E0 in L1∩L2−s(R2) after suitable translations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Since |x|2s−2 ∈ L  1  1−s ,∞ (see [42] for the weak  TRAVELING-WAVE SOLUTIONS FOR THE GSQG EQUATION  19  Lp spaces), we deduce from the generalized Young’s inequality ( Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='25 in [42])  that Gsωεj converges to Gsω0 strongly in L  2−s  (1−s)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Then, extracting another subsequence,  one may assume that both ωεj and Gsωεj converge to ω0 and Gsω0 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  On the support of ωεj, the Euler-Lagrange equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='26) implies  µεj = G+  s ωεj − Wε3−2s  j  x1 − J′(ωεj) ≥ Gsωεj − J′(ωεj) + O(ε2−2s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Letting j → +∞, the a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' convergence and the assumption on µεj imply that  µ∗ − δ0 ≥ Gsω0 − J′(ω0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  on the other hand, by the Euler-Lagrange equation for ω0 and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='10, one has  Gsω0 − J′(ω0) = µ0,  for some µ0 ≥ µ∗, which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' The proof of this lemma is thus finished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  □  Now, we determine the size of the support of ωε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' There exists a constant R > 0 such that for ε sufficiently small and for any  maximizer ωε, the support of ωε is contained in a disk of radius R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' By the previous lemma, we can take ε small such that µε ≥ µ∗  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Then the Euler-  Lagrange equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='26) implies  J′(ωε) = (G+  s ωε − Wε3−2sx1 − µε)+ ≤ (Gsωε − µ∗  2 + O(ε2−2s))+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='29)  Theorem II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='2 and Corollary II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='1 in [55] provide a subsequence (still denoted by {ωε}),  which tends to a maximizer ω0 of E0 in L1 ∩ L2−s after translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' So, for any δ > 0, we  can find a constant εδ > 0 such that  ∥ωε − ωε  0∥2−s ≤ δ,  ∀ ε ∈ (0, εδ),  where ωε  0 is a translation of ω0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='11, we conclude that spt(ωε  0) ⊂ B(˜xε, R∗) for  some point ˜xε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Thus, we get by using the H¨older inequality  �  B(˜xε,R∗)  ωε ≥  �  B(˜xε,R∗)  ωε  0 − ∥ωε − ωε  0∥L1(B(˜xε,R∗))  ≥ κ − C∥ωε − ωε  0∥2−s ≥ κ − Cδ,  which, combined with  �  ωε = κ, implies  �  B(˜xε,R∗)c ωε ≤ Cδ,  ∀ ε ∈ (0, εδ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='30)  Take a large R ≥ R∗ such that  csκ  |R−R∗|2−2s ≤ µ∗  6 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Then for any x ∈ B(ε−1(d0, 0), ε−1d0/2)\\  B(˜xε, R), using Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='1 and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='27), we have  Gsωε(x) =  �  B(˜xε,R∗)  csωε(y)  |x − y|2−sdy +  �  B(˜xε,R∗)c  csωε(y)  |x − y|2−sdy  20  DAOMIN CAO, SHANFA LAI, GUOLIN QIN  ≤  csκ  |R − R∗|2−2s + C  ���  B(˜xε,R∗)c ωε  �a  +  ��  B(˜xε,R∗)c ωε  �b�  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='31)  ≤ µ∗  6 + C(δa + δb) ≤ µ∗  3 ,  by taking δ small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='29) implies that ωε(x) = 0 for arbitrary x ∈ B(ε−1(d0, 0), ε−1d0/2) \\  B(˜xε, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Hence, we find the constant R > 0 such that the support of ωε is contained in a  disk of radius R for ε small and finish the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  □  To find the location of ωε, let xε := κ−1 �  xωε be the center of mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Then xε = (ε−1dε, 0)  for some dε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' By replacing R with 2R, we may assume that for ε small,  spt(ωε) ⊂ B(xε, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' There holds  dε → d0,  as ε → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='32)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Since xε ∈ B(ε−1(d0, 0), ε−1d0/2), we deduce d0  2 < dε < 3d0  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Up to a subsequence,  we may assume that  dε → d∗ ∈  �d0  2 , 3d0  2  �  ,  as ε → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  We prove that d∗ = d0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Indeed, take  ¯ωε(x) := ωε(x + xε − (ε−1d0, 0)) ∈ Aε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Noticing that ωε is a maximizer, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Eε(ωε) ≥ Eε(¯ωε), we find  cs  2  �  R2  +  ωε(x)ωε(y)  |x − ¯y|2−2s dxdy + Wε3−2s  �  R2  +  x1ωε(x)dx  ≤ cs  2  �  R2  +  ¯ωε(x)¯ωε(y)  |x − ¯y|2−2s dxdy + Wε3−2s  �  R2  +  x1¯ωε(x)dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Letting ε → 0 in the above inequality, we obtain  csκ  23−2sd2−2s  ∗  + Wd∗ ≤  csκ  23−2sd2−2s  0  + Wd0,  which implies d∗ = d0 since d0 is the unique minimizer of the function h(τ) =  csκ  23−2sτ 2−2s +Wτ  on (0, +∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Noting that, by the above proof, one can see that each sequence in {dε} has a convergent  subsequence that converges to the same limit d0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Then a simple contradiction argument  shows that {dε} itself y converges to d0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' The proof of this lemma is thus complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  □  Now we are ready to prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='5: Suppose J(t) =  � t  0 f −1(τ)dτ for some f satisfying (H1) and  (H2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Combining Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='20–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='23, we can finish proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  □  In view of Lemmas 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='22 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='23, for sufficiently small ε, the support of ωε is far away  from the boundary of B(ε−1(d0, 0), ε−1d0/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' We extend ωε to the half-space R2  + by defining  TRAVELING-WAVE SOLUTIONS FOR THE GSQG EQUATION  21  ωε to be 0 in R2  + \\ B(ε−1(d0, 0), ε−1d0/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' With this fact in hand, we can now show that  ωtr,ε(x) := ωε(x) − ωε(¯x) is a steady solution in the sense of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' More precisely, we have  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Suppose that J(t) =  � t  0 f −1(τ)dτ for some f satisfying (H1) and (H2) with  f ∈ C1−2s if 0 < s < 1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Let ωε be a maximizer obtained above and ωtr,ε(x) := ωε(x)−ωε(¯x),  then provided that ε is sufficiently small, it holds  �  R2 ωtr,ε∇⊥(Gsωtr,ε − Wε3−2sx1) · ∇ϕdx = 0,  ∀ ϕ ∈ C∞  0 (R2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='33)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Using the regularity theory for fractional Laplacians (Propositions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='8 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='9 in  [60]), for s ̸=  1  2, we can prove that Gsωtr,ε ∈ C0,1 by our assumptions on f, the fact  ωε ∈ L1 ∩ L∞ and Gsωtr,ε ∈ L∞ due to Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' For s = 1  2, we use the standard theory  on potentials to deduce that Gsωtr,ε ∈ W 2s,p for any p > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Therefore, the integral in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='33)  makes sense for all s ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  By the definition of ωtr,ε, it is sufficient to prove  �  R2  +  ωε∇⊥(G+  s ωε − Wε3−2sx1) · ∇ϕdx = 0,  ∀ ϕ ∈ C∞  0 (R2  +).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Recall that  ωε = (J′)−1 �  (G+  s ωε − Wε3−2sx1 − µε)+  �  ,  ∀ x ∈ R2  +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Let F(t) =  � t  0(J′)−1(τ)dτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' For any ϕ ∈ C∞  0 (R2  +), we apply integrate by parts to obtain  �  R2  +  ωε∇⊥(G+  s ωε − Wε3−2sx1) · ∇ϕdx  = −  �  B(ε−1(d0,0),ε−1d0/2)  F((G+  s ωε − Wε3−2sx1 − µε)+)(∂x2∂x1ϕ − ∂x1∂x2ϕ)dx = 0,  where we have used the fact (G+  s ωε − Wε3−2sx1 − µε)+ = 0 on ∂B(ε−1(d0, 0), ε−1d0/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  □  Now we are ready to prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='1: Let ωtr,ε(x) := ωε(x) − ωε(¯x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Then the statements of Theorem  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='1 follow from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='5 and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  □  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Uniqueness of maximizers  In this section, we show the uniqueness of maximizers in the case J(t) = Lt1+ 1  p for  some L > 0 and p ∈ (0,  1  1−s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' We first establish some finer estimates of the asymptotic  behavior by careful analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Then, we obtain the uniqueness by using the non-degeneracy  of linearized equations (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='13 (iii)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' In what follows, we sometimes leave  out the domain in the integral symbol and abbreviate it to  �  when there is no risk of  confusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  22  DAOMIN CAO, SHANFA LAI, GUOLIN QIN  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Refined estimates on asymptotic behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' In the cases J(t) = Lt1+ 1  p, we study  in detail the asymptotic behaviors of the maximizers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Suppose that J(t) = Lt1+ 1  p for some L > 0 and p ∈ (0,  1  1−s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' For each ε small,  let ωε be a maximizer of Eε over Aε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Define ˜ωε(x) := ωε(x + xε), where xε = κ−1 �  xωε is  the center of mass of ωε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Let ω0 be the unique maximizer of E0 over A0 with  �  xω0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Then, we have  lim  ε→0 ∥˜ωε − ω0∥∞ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' We first prove the convergence of the energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' On the one hand, since ˜ωε ∈ A0 and  ω0 is a maximizer of E0 over A0, we have  E0(ω0) ≥ E0(˜ωε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  On the other hand, noticing that ω0(· − ε−1(d0, 0)) ∈ Aε, we deduce  Eε(ωε) ≥ Eε(ω0(· − ε−1(d0, 0))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  It is easy to see that Eε(ωε) = E0(˜ωε) + O(ε2−2s) and Eε(ω0(· − ε−1(d0, 0))) = E0(ω0) +  O(ε2−2s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' So, we conclude  E0(ω0) ≥ E0(˜ωε) ≥ E0(ω0) + O(ε2−2s),  which implies  lim  ε→0 E0(˜ωε) = E0(ω0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='1)  Note that for arbitrary ε small, one has ∥˜ωε∥∞ = ∥ωε∥∞ ≤ C for some C > 0 independent  of ε and spt(˜ωε) ⊂ B(0, R) for some R independent of ε by previous lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Therefore,  we may assume that up to a subsequence ˜ωε → ˆω0 weakly star in L∞ for some ˆω0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Similar  argument as the proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='14 shows that E0(ˆω0) = limε→0 E0(˜ωε) = E0(ω0), so ˆω0  is a maximizer of E0 over A0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Moreover, by the weakly star convergence, we find  �  xˆω0 = lim  ε→0  �  x˜ωε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  This indicates that ˆω0 = ω0 by the uniqueness of maximizers of E0 due to Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  We have proved the weakly star convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Now, we show the strong convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Since ωε is uniformly bounded in L1 ∩ L∞, then by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='1 and Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='9 in [60],  we have Gsωε is uniformly bounded in Cα for some 0 < α < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Then, we deduce from the  representation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='26) that ωε is uniform bounded in Cα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' So by the Arela-Ascoli theorem,  we may assume that up to a subsequence, {˜ωε} strongly converges in L∞ to ω0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Since each sequence in {˜ωε} has a convergent subsequence that strongly converges in  L∞ to the same limit ω0, a simple contradiction argument shows that {˜ωε} itself strongly  converges in L∞ to ω0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' The proof of this lemma is thus finished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  □  We derive the integral representation for the Lagrange multipliers µε in terms of ωε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  TRAVELING-WAVE SOLUTIONS FOR THE GSQG EQUATION  23  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Suppose that J(t) = Lt1+ 1  p for some L > 0 and p ∈ (0,  1  1−s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Let ωε be a  maximizer of Eε over Aε satisfying (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='26) for some µε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Then we have  µεκ = AγE0(ωε) + BγWε3−2s  �  R2  +  x1ωε + Cγ  �  R2  +  �  R2  +  csωε(x)ωε(y)  |x − ¯y|2−2s dxdy,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='2)  where the constants Aγ = 2−γ−sγ  2−s−γ , Bγ = 2−s  s−1 −  2−γ−γs  2(s−1)(2−s−γ), Cγ = − 2−γ−γs  2(2−s−γ) and γ = 1 + 1  p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' We may assume that ωε is symmetric non-increasing in x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Then Lemmas 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='22 and  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='23 shows that the support of ωε is far away from the boundary ∂B(ε−1(d0, 0), ε−1d0/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  So the function (ωε)t(x) := t−2ωε(t−1x) is supported in B(ε−1(d0, 0), ε−1d0/2) for t ≈ 1 and  hence belongs to Aε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Since ωε is a maximizer, we deduce  dEε((ωε)t)  dt  �����  t=1  = 0,  which, by the definition of Eε and straightforward computations, gives  (s − 1)  �  R2  +  ωεG+  s ωε − Wε3−2s  �  R2  +  x1ωε − (2 − 2γ)  �  R2  +  J(ωε) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Then, we infer from the above identity that  �  R2  +  ωεGsωε = 2 − 2γ  s − 1  �  R2  +  J(ωε)+ W  s − 1ε3−2s  �  R2  +  x1ωε +cs  �  R2  +  �  R2  +  ωε(x)ωε(y)  |x − ¯y|2−2s dxdy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='3)  By the definition of E0 and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='3), we deduce  E0(ωε) = 1  2  �  R2  +  ωεGsωε −  �  R2  +  J(ωε)  = 2 − s − γ  s − 1  �  R2  +  J(ωε) +  W  2(s − 1)ε3−2s  �  R2  +  x1ωε + cs  2  �  R2  +  �  R2  +  ωε(x)ωε(y)  |x − ¯y|2−2s dxdy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='4)  On the other hand, multiplying the equation (J′) (ωε) = (G+  s ωε − Wε3−2sx1 − µε)+ by  ωε and integrating, we have  µεκ =  �  R2  +  ωεG+  s ωε − Wε3−2s  �  R2  +  x1ωε − γ  �  R2  +  J(ωε)  = 2 − γ − γs  s − 1  �  R2  +  J(ωε) + 2 − s  s − 1Wε3−2s  �  R2  +  x1ωε  = 2 − γ − γs  2 − s − γ E0(ωε) +  �2 − s  s − 1 −  2 − γ − γs  2(s − 1)(2 − s − γ)  �  Wε3−2s  �  R2  +  x1ωε  − 2 − γ − γs  2(2 − s − γ)  �  R2  +  �  R2  +  csωε(x)ωε(y)  |x − ¯y|2−2s dxdy,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='5)  24  DAOMIN CAO, SHANFA LAI, GUOLIN QIN  where we have used (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='3) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='4) in above calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' So we obtain (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='2) and finish  the proof of this lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  □  As an immediate consequence of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='2, we obtain the following estimate for µε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' One has  |µε − µ0| → 0,  as ε → 0+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Next, we derive another integral identity for general J, which determines the location of  ωε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Suppose that J satisfies (H′  1) and (H′  2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Let ωε be a maximizer of Eε over  Aε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Then the following identity is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  2(1 − s)cs  �  R2  +  �  R2  +  ωε(x)(x1 + y1)ωε(y)  |x − ¯y|4−2s  dxdy = Wε3−2sκ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='6)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' We may assume that ωε is symmetric non-increasing in x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Then Lemmas 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='22 and  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='23 show that the support of ωε is far away from the boundary ∂B(ε−1(d0, 0), ε−1d0/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  So the function (ωε)t(x) := ωε(x + (t, 0)) is supported in B(ε−1(d0, 0), ε−1d0/2) for t ≈ 0  and hence belongs to Aε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Since ωε is a maximizer, we deduce  dEε((ωε)t)  dt  �����  t=0  = 0,  which, by the definition of Eε and straightforward computations, implies (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' So the  proof this lemma is completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  □  Using the identity (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='6), we can sharpen the estimate of xε in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='23 as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Suppose that J satisfies (H′  1) and (H′  2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Let ωε be a maximizer of Eε over  Aε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' If xε = κ−1 �  xωε = (ε−1dε, 0), then we have  |dε − d0| = O(ε2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='22, we know spt(ωε) ⊂ B(xε, R) for some R independent of ε and  xε = (ε−1dε, 0) with dε → d0 as ε → 0+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' We use the Taylor expansion to calculate the  left-hand side of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='6)  � � ωε(x)(x1 + y1)ωε(y)  |x − ¯y|4−2s  dxdy = ε3−2s  � � ˜ωε(x)(2dε + ε(x1 + y1))˜ωε(y)  |(2dε, 0) + ε(x − ¯y)|4−2s  dxdy  =ε3−2s  � �  ˜ωε(x)˜ωε(y)  �  1  (2dε)3−2s + (2s − 3)ε(x1 + y1)  (2dε)4−2s  + O(ε2)  �  dxdy  = ε3−2sκ2  (2dε)3−2s + O(ε5−2s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='7)  Here we have used that  �  x˜ωε(x)dx = 0, which follows from the definition of ˜ωε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  TRAVELING-WAVE SOLUTIONS FOR THE GSQG EQUATION  25  Then using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='6) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='7), we obtain  2(1 − s)csκ2  (2dε)3−2s  − Wκ = O(ε2),  which implies  d2s−3  ε  =  22−2sW  (1 − s)csκ + O(ε2) = d2s−3  0  + O(ε2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='8)  Here we have used the definition of d0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Since d(t2s−3)  dt  ���  t=d0 ̸= 0, we infer from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='8) that  |dε − d0| = O(ε2) and complete the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  □  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Proof of the uniqueness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' In this subsection, we prove the uniqueness of maximizers  when J(t) = Lt1+ 1  p for some L > 0 and p ∈ (1,  1  1−s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Let ωε be a maximizer of Eε over  Aε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Recall that xε = κ−1 �  xωε = (ε−1dε, 0) and ˜ωε = ωε(· + ε−1(dε, 0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Set Lγ =  �  1  Lγ  �  1  γ−1  with γ = 1 + 1  p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Then, by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='26), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='2) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='6), we obtain the equations of ˜ωε and dε:  �  ˜ωε(x) = Lγ (Gs˜ωε(x) − κ−1AγE0(˜ωε) + Sε(˜ωε, dε))p  + ,  2(1 − s)cs  � � ˜ωε(x)(2dε+ε(x1+y1))˜ωε(y)  |(2dε,0)+ε(x−¯y)|4−2s  dxdy − Wκ = 0,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='9)  where the operator Sε(˜ωε, dε) is given by  Sε(˜ωε, dε) = − Wε3−2s(x1 + ε−1dε) −  �  ε2−2scs˜ωε(y)  |(2dε, 0) + ε(x − ¯y)|2−2sdy  −κ−1BγWε2−2sdε − κ−1Cγ  � �  ε2−2scs˜ωε(x)˜ωε(y)  |(2dε, 0) + ε(x − ¯y)|2−2sdxdy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='10)  For simplicity of notations, we denote the operators Pε and Qε as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Pε(ω, d) := Lγ  �  Gsω(x) − κ−1AγE0(ω) + Sε(ω, d)  �p  + ,  and  Qε(ω, d) := 2(1 − s)cs  � � ω(x)(2d + ε(x1 + y1))ω(y)  |(2d, 0) + ε(x − ¯y)|4−2s  dxdy − Wκ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Noting that for dε ∈ ( d0  2 , 3d0  2 ), x, y ∈ B(0, R) and ε small, we always have |(2dε, 0) +  ε(x − ¯y)| is bounded from below by a positive constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Therefore, we easily obtain the  following lemma, whose proof we leave to readers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' For (ω, d) ∈ X 1 := L1(B(0, R)) × ( d0  2 , 3d0  2 ) and ε sufficiently small, there  holds  ∥Sε(ω, d)∥∞ + ∥∇Sε(ω, d)∥X 1→L∞ + ∥∇2Sε(ω, d)∥X 1×X 1→L∞ = O(ε2−2s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Using the definition of E0 and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='7), one can see that  E′  0(ω0)φ = µ0  �  φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  26  DAOMIN CAO, SHANFA LAI, GUOLIN QIN  If we define  P0(ω, d) := Lγ  �  Gsω(x) − κ−1AγE0(ω)  �p  + ,  and  Q0(ω, d) := 2(1 − s)cs  (2d)3−2s  � �  ω(x)ω(y)dxdy − Wκ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  By direct calculations, one can verify that  ω0 = P0(ω0, d0),  ∇P0(ω0, d0)(φ, l) = pLγ(Gsω0 − κ−1AγE0(ω0))p−1  +  �  Gsφ − κ−1Aγµ0  �  φ  �  and  Q0(ω0, d0) = 0,  ∇Q0(ω0, d0)(φ, l) = (1 − s)csκ  21−2sd3−2s  0  �  φ − (1 − s)(3 − 2s)csκ2  22−2sd4−2s  0  l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  By Lemmas 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='23 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='1, ˜ωε and dε take the following form:  ˜ωε = ω0 + φε,  dε = d0 + lε,  where φε ∈ L∞(B(0, R)) satisfies  �  xφε =  �  φε = 0, ∥φε∥∞ = o(1) and lε is a real number  with |lε| = o(1) as ε → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Then we deduce from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='9) that (φ, l) = (φε, lε) solves the following system  �  φ − pLγ(Gsω0 − κ−1AγE0(ω0))p−1  +  �  Gsφ − κ−1Aγµ0  �  φ  �  = R1(φ, l),  (1−s)csκ  21−2sd3−2s  0  �  φ − (1−s)(3−2s)csκ2  22−2sd4−2s  0  l = R2(φ, l),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='11)  where the super-linear terms R1(φ, l) and R2(φ, l) are given by  R1(φ, l) = Lγ  �  Gs(ω0 + φ) − κ−1AγE0(ω0 + φ) + Sε(ω0 + φ, d0 + l)  �p  +  −Lγ  �  Gsω0 − κ−1AγE0(ω0)  �p  + − pLγ(Gsω0 − κ−1AγE0(ω0))p−1  +  �  Gsφ − κ−1Aγµ0  �  φ  �  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='12)  and  R2(φ, l) = 2(1 − s)cs  � � (ω0 + φ)(x)(2d0 + 2lε + ε(x1 + y1))(ω0 + φ)(y)  |(2d0 + 2lε, 0) + ε(x − ¯y)|4−2s  dxdy − Wκ  −(1 − s)csκ  21−2sd3−2s  0  �  φ + (1 − s)(3 − 2s)csκ2  22−2sd4−2s  0  l  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='13)  To study the linearized equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='11), we set  L0φ := φ − pLγ(Gsω0 − κ−1AγE0(ω0))p−1  +  �  Gsφ − κ−1Aγµ0  �  φ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Using the non-degeneracy Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='13 (iii), we have the following key estimate of L0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' There exists a constant C0 > 0 such that  ∥L0φ∥2 ≥ C0∥φ∥2,  ∀ φ ∈ L2(B(0, R)) with  �  xφ =  �  φ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  TRAVELING-WAVE SOLUTIONS FOR THE GSQG EQUATION  27  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' We normalize φ by replacing φ with φ/∥φ∥2 so that ∥φ∥2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Now suppose on the  contrary that there is a sequence {φn}∞  n=1 ⊂ L2(B(0, R)) such that  ∥φn∥ = 1,  �  xφn =  �  φn = 0,  while  ∥L0φn∥ → 0,  as n → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  That is  φn = pLγ(Gsω0 − κ−1AγE0(ω0))p−1  + Gsφn + o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='14)  We first assume that up to a subsequence φn → φ∞ weakly in L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Note that the support  of (Gsω0 − κ−1AγE0(ω0))p−1  +  is contained in B(0, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Then by the regularity theory on  potentials and the compact embedding theorem for fractional Sobolev spaces (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  [34, 62]), we have up to a subsequence pLγ(Gsω0 − κ−1AγE0(ω0))p−1  + Gsφn → pLγ(Gsω0 −  κ−1AγE0(ω0))p−1  + Gsφ∞ strongly in L2(B(0, R)) and hence by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='14), we conclude φn → φ∞  strongly in L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Thus, we have φ∞ satisfies  φ∞ = pLγ(Gsω0 − κ−1AγE0(ω0))p−1  + Gsφ∞,  ∥φ∞∥2 = 1,  �  xφ∞ =  �  φ∞ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  By the regularity theory on potentials, Sobolev embedding and bootstrap argument, one  can prove that φ∞ ∈ L∞(B(0, R)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Thus, we infer from Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='13 (iii) that φ∞ ∈  span{∂x1ω0, ∂x2ω0}, which combined with  �  xφ∞ = 0 implies φ∞ ≡ 0 due to the radial  symmetry of ω0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' This is a contradiction with ∥φ∞∥2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Therefore, this lemma must hold  true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  □  Now we study the right-hand side of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' For any ǫ > 0, there is a δ > 0 such that if φ1, φ2 ∈ L∞(B(0, R)) and  l1, l2 ∈ R satisfy  �  φ1 =  �  φ2 = 0 and  ∥φ1∥∞ + ∥φ2∥∞ + |l1| + |l2| ≤ δ,  then we have  lim sup  ε→0 (∥R1(φ1, l1) − R1(φ2, l2)∥2 + |R2(φ1, l1) − R2(φ2, l2)|) ≤ ǫ(∥φ1 − φ2∥2 + |l1 − l2|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' By direct calculations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' we find  R1(φ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' l1) − R1(φ2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' l2)  =Lγ  �  Gs(ω0 + φ1) − κ−1AγE0(ω0 + φ1) + Sε(ω0 + φ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' d0 + l1)  �p  +  − Lγ  �  Gs(ω0 + φ2) − κ−1AγE0(ω0 + φ2) + Sε(ω0 + φ2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' d0 + l2)  �p  +  − pLγ(Gsω0 − κ−1AγE0(ω0))p−1  + Gs (φ1 − φ2)  =pLγ  � 1  0  ��  Gs(ω0 + φ(τ)) − κ−1AγE0(ω0 + φ(τ)) + Sε(ω0 + φ(τ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' d0 + l(τ))  �p−1  +  −(Gsω0 − κ−1AγE0(ω0))p−1  +  �  dτGs(φ1 − φ2) + O(∥∇Sε(ω0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' d0)∥)∥φ1 − φ2∥2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  28  DAOMIN CAO,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' SHANFA LAI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' GUOLIN QIN  where φ(τ) = τφ1 + (1 − τ)φ2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' l(τ) = τl1 + (1 − τ)l2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' To continue, we expand Gs(ω0 +  φ(τ)) − κ−1AγE0(ω0 + φ(τ)) + Sε(ω0 + φ(τ), d0 + l(τ)) as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Gs(ω0 + φ(τ)) − κ−1AγE0(ω0 + φ(τ)) + Sε(ω0 + φ(τ), d0 + l(τ))  =Gsω0 − κ−1AγE0(ω0)  + Gsφ(τ) +  �  φ(τ)Gsω0 −  �  (J(ω0 + φ(τ)) − J(ω0)) + 1  2  �  φ(τ)Gsφ(τ) + O(∥Sε∥∞)  =Gsω0 − κ−1AγE0(ω0) + O(∥φ(τ)∥∞ + ∥φ(τ)∥2  ∞ + ∥Sε∥∞),  where we have used H¨older’s inequality, Lemmas 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Using h(t) = tp−1  +  ∈ C0,αp  with αp = min{1, p − 1} ∈ (0, 1], we continue calculating R1(φ1, l1) − R1(φ2, l2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  R1(φ1, l1) − R1(φ2, l2)  =O((∥φ(τ)∥∞ + ∥φ(τ)∥2  ∞ + ∥Sε∥)αp)Gs(φ1 − φ2) + O(∥∇Sε(ω0, d0)∥)∥φ1 − φ2∥  Thus, if ∥φ1∥, ∥φ2∥ ≤ δ, then we obtain  ∥R1(φ1, l1) − R1(φ2, l2)∥2 ≤ C(δαp + ε(2−2s)αp)∥φ1 − φ2∥2 = oδ,ε(1)∥φ1 − φ2∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='15)  Since R2 is C2 smooth, it is easy to see that  ∥R2(φ1, l1)−R2(φ2, l2)∥ ≤ C(∥φ1 −φ2∥2 +|l1 −l2|2) = oδ,ε(1)(∥φ1 −φ2∥+|l1 −l2|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='16)  The proof of this lemma is thus finished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  □  Now, we are ready to prove our uniqueness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='6:  Suppose on the contrary that there are two different maximizers ω1,ε ̸= ω2,ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Then,  we obtain two pairs (φ1,ε, l1,ε) and (φ2,ε, l2,ε) such that ∥φ1,ε − φ2,ε∥2 + |l1,ε − l2,ε| ̸= 0,  �  xφ1,ε =  �  φ1,ε =  �  xφ2,ε =  �  φ2,ε = 0 and ∥φ1,ε∥∞ + ∥φ2,ε∥∞ + |l1,ε| + |l2,ε| = o(1) as  ε → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' More over, both (φ1,ε, l1,ε) and (φ2,ε, l2,ε) satisfy the system (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  On the one hand, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='7, the difference between the left-hand side satisfies  ∥L0(φ1,ε−φ2,ε)∥2+(1 − s)(3 − 2s)csκ2  22−2sd4−2s  0  |l1,ε−l2,ε| ≥ C0∥φ1,ε−φ2,ε∥2+(1 − s)(3 − 2s)csκ2  22−2sd4−2s  0  |l1,ε−l2,ε|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  On the other hand, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='8, the difference between the right-hand side satisfies  ∥R1(φ1,ε, l1,ε)−R1(φ2,ε, l2,ε)∥2+|R2(φ1,ε, l1,ε)−R2((φ2,ε, l2,ε)| = o(1)(∥φ1,ε−φ2,ε∥2+|l1,ε−l2,ε|),  which leads to a contradiction for ε small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Therefore, we have established the uniqueness  of maximizers for ε small and completed the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  □  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Nonlinear orbital stability  This section is devoted to investigating the nonlinear stability of traveling solutions  obtained in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' We first prove a general stability theorem in a similar spirit as [13],  where the stability of vortex pairs for the 2D Euler equation was considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Throughout  this section, we always assume that D ⊂ R2  + is the domain D = R2  + if s > 1  2 and D =  {x1 ≥ 1} if 0 < s ≤ 1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  TRAVELING-WAVE SOLUTIONS FOR THE GSQG EQUATION  29  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' A general stability theorem on the set of maximizers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Let ξ be a non-negative  Lebesgue integrable function on R2, we denote by R(ξ) the set of (equimeasurable) rear-  rangements of ξ on D defined by  R(ξ) =  �  0 ≤ ζ ∈ L1(D)  ���|{x : ζ(x) > τ}| = |{x : ξ(x) > τ}|, ∀ τ > 0  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Note that all functions in R(ξ) have the same Lq norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Following [13], we also define  R+(ξ) =  �  ζ1S  ��ζ ∈ R(ξ),  S ⊂ D measurable  �  ,  and  R(ξ)w =  �  ζ ≥ 0 measurable  ����  �  D  (ζ − α)+dx ≤  �  D  (ξ − α)+dx,  ∀α > 0  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  It is easy to see that the inclusions R(ξ) ⊂ R+(ξ) ⊂ R(ξ)w hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' The key fact is that  R(ξ)w is convex and is the weak closure of R(ξ) in Lp (see [13, 35]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  We denote the kinetic energy as  E(ζ) = 1  2  �  D  ζ(x)G+  s ζ(x)dx,  and the impulse  I(ζ) =  �  D  x1ζ(x)dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  For a constant W, set the energy functional as  ˜EW(ζ) := 1  2  �  D  ζ(x)G+  s ζ(x)dx − W  �  D  x1ζ(x)dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  For a function ζ0 and constant W > 0, we will consider the maximization problem  sup  ζ∈R(ζ0)w  ˜EW(ζ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  The following two lemmas are needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Suppose ζ ∈ L1 ∩ Lr(D) for some s−1 < r ≤ +∞ if 0 < s ≤  1  2 and  2  2s−1 < r ≤ +∞ if 1  2 < s < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Then, one has  |G+  s ζ(x)| ≤ C (∥ζ∥r + ∥ζ∥1) min{x1, x  2s− 2  r  1  },  ∀ x ∈ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='1)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' We first consider the case x1 ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' By the mean value theorem, there holds  G+  s (x, y) ≤  4csx1y1  |x − y|4−2s,  ∀x, y ∈ R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Therefore, using H¨older’s inequality and y1 < x1 + |x − y| ≤ 2|x − y| if |x − y| > x1, we  have  |G+  s ζ(x)| ≤  �  R2  +  G+  s (x, y)|ζ(y)|dy  30  DAOMIN CAO, SHANFA LAI, GUOLIN QIN  ≤  �  |x−y|≤x1  c2,s  |x − y|2−2s|ζ(y)|dy +  �  {|x−y|>x1}  4x1y1  |x − y|4−2s|ζ(y)|dy  ≤ C  �  x  2s− 2  r  1  ∥ζ∥r + x2s−2  1  ∥ζ∥1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  This proves (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='1) in the case 0 < s < 1 and x1 > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Now, we turn to the remaining case  1  2 < s < 1 and 0 < x1 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' By H¨older’s inequality, we find  |∇G+  s ζ(x)| ≤ C  �  R+  |ζ(y)|  |x − y|3−2sdy  ≤  �  |x−y|≤1  c2,s  |x − y|3−2s|ζ(y)|dy +  �  {|x−y|>1}  |ζ(y)|dy  ≤ C (∥ζ∥r + ∥ζ∥1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Noticing that G+  s ζ(x)  ���  x1=0 ≡ 0, we conclude  |G+  s ζ(x)| ≤ x1∥∇G+  s ζ∥∞ ≤ C (∥ζ∥q + ∥ζ∥1) x1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  The proof is thus finished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  □  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Suppose x1ζ ∈ L1(R2  +) and ζ ∈ L1 ∩Lr(D) for some r with s−1 < r < +∞ if  0 < s ≤ 1  2 and  2  2s−1 < r < +∞ if 1  2 < s < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' If ζ is Steiner symmetric in the x2-variable,  then for x ∈ D, there holds  |G+  s ζ(x)| ≤ C  ��  |x2|− 1  2r + |x2|− 1  2  �  (∥ζ∥r + ∥ζ∥1) min{1, x  2s− 2  r −1  1  } + |x2|s−2∥x1ζ∥1  �  x1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='2)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' For x ∈ R2  + fixed, let  ζ1(y) =  �  ζ(y),  if |y2 − x2| <  �  |x2|,  0,  if |y2 − x2| ≥  �  |x2|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Using equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='11) in [8] (see also (6) in [12]), it is easy to see that for any 1 ≤ q ≤ r  ∥ζ1∥q ≤  �  |x2|  1  2  |x2|  � 1  q  ∥ζ∥q = |x2|− 1  2q ∥ζ∥q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Hence, by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='1), we have  |G+  s ζ1(x)| ≤ C  �  (∥ζ1∥r + ∥ζ1∥1) min{1, x  2s− 2  r −1  1  }  �  x1  ≤ C  ��  |x2|− 1  2r + |x2|− 1  2  �  (∥ζ∥r + ∥ζ∥1) min{1, x  2s− 2  r −1  1  }  �  x1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='3)  Letting ζ2 = ζ − ζ1, we have  |G+  s ζ2(x)| = cs  �  |x2−y2|>√  |x2|  �  1  |x − y|2−2s −  1  |x − ¯y|2−2s  �  |ζ(y)|dy  TRAVELING-WAVE SOLUTIONS FOR THE GSQG EQUATION  31  ≤ C  �  |x−y|>√  |x2|  x1y1  |x − y|4−2sζ(y)dy  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='4)  ≤  Cx1  |x2|2−s∥x1ζ∥1,  which, together with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='3), gives (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='2) and completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  □  The following property enables us to control the supports of maximizers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Suppose that ζ ∈ L1 ∩ Lq(D) with some s−1 < q < ∞ and W > 0 is a given  constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Let h = ζ1V for some set V ⊂ {G+  s ζ − Wx1 ≤ 0}, then  ˜EW(ζ − h) ≥ ˜EW(ζ)  with strict inequality unless h ≡ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' It is easy to see that  ˜EW(ζ − h) = 1  2  �  R2  +  (ζ − h)(x)G+  s (ζ − h)(x)dν − W  �  R2  +  (ζ − h)(x)x1dx  = ˜EW(ζ) + 1  2  �  R2  +  hG+  s h −  �  R2  +  h(x)  �  G+  s ζ(x) − Wx1  �  dx  ≥ ˜EW(ζ) + 1  2  �  R2  +  hG+  s h,  which implies ˜EW(ζ − h) ≥ ˜EW(ζ) since 1  2  �  hG+  s h ≥ 0 and 1  2  �  hG+  s h = 0 if and only if  h ≡ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' The proof is thus complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  □  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Let 0 ≤ ζ0 ∈ L1 ∩ Lq(D) with some q with s−1 < q ≤ +∞ if 0 < s ≤ 1  2 and  2  2s−1 < q ≤ +∞ if 1  2 < s < 1 and W > 0 is a given constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Then  sup  ζ∈R(ζ0)w  ˜EW(ζ) < +∞,  and any maximizer (if exists) is supported in [0, M0]×R, where M0 is a constant depending  on ∥ζ0∥1 + ∥ζ0∥q and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' The upper bounded of ˜EW over R(ζ0)w follows from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='1 and  the fact that ζ0 ∈ Lr for any r ∈ [1, q], there is a constant M0 depending on ∥ζ0∥1 + ∥ζ0∥q  and W such that G+  s ζ(x) − Wx1 ≤ 0 for all x ∈ D with x1 ≥ M0 and for any ζ ∈ R(ζ0)w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Suppose that ζ0 ∈ R(ζ0)w is a maximizer, let h := ζ01(M0,∞)×R, then we infer from Lemma  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='3 that h ≡ 0 since (M0, ∞) × R ⊂ {G+  s ζ0 − Wx1 ≤ 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' The proof of this lemma is thus  finished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  □  To obtain the compactness of maximizing sequences, we need the following concentration  compactness lemma due to Lions [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  32  DAOMIN CAO, SHANFA LAI, GUOLIN QIN  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Let {un}∞  n=1 be a sequence of nonnegative functions in L1(D) satisfying  lim sup  n→∞  �  D  undx → µ,  for some 0 ≤ µ < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Then, after passing to a subsequence, one of the following holds:  (i) (Compactness) There exists a sequence {yn}∞  n=1 in R2  + such that for arbitrary ε > 0,  there exists R > 0 satisfying  �  D∩BR(yn)  undx ≥ µ − ε,  ∀n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  (ii) (Vanishing) For each R > 0,  lim  n→∞ sup  y∈D  �  D∩BR(yn)  undx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  (iii) (Dichotomy) There exists a constant 0 < α < µ such that for any ε > 0, there exist  N = N(ε) ≥ 1 and 0 ≤ ui,n ≤ un, i = 1, 2 satisfying  �  ∥un − u1,n − u2,n∥L1(D) + |α −  �  D u1,ndx| + |µ − α −  �  D u2,ndx| < ε,  for n ≥ N,  dn := dist(spt(u1,n), spt(u2,n)) → ∞,  as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Moreover, if µ = 0 then only vanishing will occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' This lemma is a slight reformulation of Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='1 in [55], so we omit the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  □  For a function 0 ≤ ζ0 ∈ L1 ∩ Lq(D) and a constant W > 0, we denote Sζ0,W :=  supζ∈R(ζ0)w ˜EW(ζ) as the maximum value and ˜Σζ0,W := {ζ ∈ R(ζ0)w | ˜EW(ζ) = Sζ0,W} as  the set of all the maximizers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' To continue, we first show the compactness of maximizing  sequences by using Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' For q with max{2, s−1} < q ≤ ∞ if 0 < s ≤ 1  2 and  2  2s−1 < q ≤ +∞ if  1  2 < s < 1, let 0 ≤ ζ0 ∈ Lq(D) be a function with 0 < |spt(ζ0)| < ∞ and W > 0 be a given  constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Assume that  ∅ ̸= ˜Σζ0,W ⊂ R(ζ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Suppose that {ζn}∞  n=1 ⊂ R+(ζ0) is a maximizing sequence in the sense that  ˜EW(ζn) → Sζ0,W,  as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='5)  Then, there exist ζ0 ∈ ˜Σζ0,W, a subsequence {ζnk}∞  k=1 and a sequence of real numbers  {ck}∞  k=1 such that as k → ∞,  ∥ζnk(· + cke2) − ζ0∥2 → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='6)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Note that since 0 ∈ R(ζ0)w \\ R(ζ0), the condition ∅ ̸= ˜Σζ0,W ⊂ R(ζ0) implies that  0 is not a maximizer and hence we have Sζ0,W > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  TRAVELING-WAVE SOLUTIONS FOR THE GSQG EQUATION  33  Take un = ζ2  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Since 0 ≤  �  D undx ≤ ∥ζ0∥2  2 < ∞, we may assume that, up to a subse-  quence (still denoted by {un}∞  n=1),  �  D  undx → µ  for some 0 ≤ µ ≤ ∥ζ0∥2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Applying Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='5, we find that for a certain subsequence, still  denoted by {ζn}∞  n=1, one of the three cases in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='5 should occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' In what follows,  we divide the proof into three steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Vanishing excluded: Suppose that for each fixed R > 0,  lim  n→∞ sup  y∈D  �  BR(y)∩D  ζ2  ndx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='7)  By the property of rearrangement and H¨older’s inequality, we have for any R > 0 and  1 ≤ τ ≤ 2  �  BR(y)∩D  ζτ  ndx → 0  as n → +∞ uniformly over y ∈ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' On the other hand, we have G+  s (ζn(1 − 1BR(y)))(y) ≤  C ∥ζn∥1  R2−2s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Therefore, we get  �  ζnG+  s ζn ≤  C  R2−2s + on(1),  for any R > 0 and hence limn→∞ ˜EW(ζn) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' This is a contradiction to Sζ0,W > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Thus,  vanishing can not occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Dichotomy excluded: We may assume that ζn is supported in [0, M0] × R, where  M0 is the constant obtained in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Suppose that there is a constant α ∈ (0, µ)  such that for any ε > 0, there exist N(ε) ≥ 1 and 0 ≤ ζi,n ≤ ζn, i = 1, 2, 3 satisfying  \uf8f1  \uf8f4  \uf8f2  \uf8f4  \uf8f3  ζn = ζ1,n + ζ2,n + ζ3,n,  �  D ζ2  3,ndx + |α − αn| + |µ − α − βn| < ε,  for n ≥ N(ε),  dn := dist(spt(ζ1,n), spt(ζ2,n)) → ∞,  as n → ∞,  where αn =  �  D ζ2  1,ndx and βn =  �  D ζ2  2,ndx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Using a diagonal argument, we obtain that there  exists a subsequence, still denoted by {ζn}∞  n=1, such that  \uf8f1  \uf8f4  \uf8f2  \uf8f4  \uf8f3  ζn = ζ1,n + ζ2,n + ζ3,n,  0 ≤ ζi,n ≤ ζn, i = 1, 2, 3  �  D ζ2  3,ndx + |α − αn| + |µ − α − βn| → 0,  as n → ∞,  dn = dist(spt(ζ1,n), spt(ζ2,n)) → ∞,  as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  By direct calculations, one has  �  D  ζnG+  s ζn =  �  D  (ζ1,n + ζ2,n + ζ3,n)G+  s (ζ1,n + ζ2,n + ζ3,n)  =  �  D  ζ1,nG+  s ζ1,n +  �  D  ζ2,nG+  s ζ2,n + 2  �  D  ζ1,nG+  s ζ2,n +  �  D  ζ3,nG+  s (2ζn − ζ3,n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  34  DAOMIN CAO, SHANFA LAI, GUOLIN QIN  By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='6) and H¨older’s inequality, we derive  �  D  ζ3,nG+  s (2ζn − ζ3,n) ≤ C∥ζ3,n∥2−s∥2ζn − ζ3,n∥1−s  2−s∥2ζn − ζ3,n∥s  1  ≤ C|spt(ζ0)|  s  2(2−s) ∥ζ3,n∥2(∥ζ0∥1 + ∥ζ0∥2) = on(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  It is obvious that �  D  �  D  ζ1,n(x)G+  s (x, y)ζ2,n(y)dxdy ≤ C∥ζ0∥2  1  d2−2s  n  = on(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Hence, we arrive at  ˜EW(ζn) = 1  2  �  D  ζnG+  s ζn − W  �  D  x1ζndx ≤ ˜EW(ζ1,n) + ˜EW(ζ2,n) + on(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Taking Steiner symmetrization in the x2-variable ζ∗  i,n of ζi,n for i = 1, 2, by the rearrange-  ment inequality, we obtain  ˜EW(ζn) ≤ ˜EW(ζ∗  1,n) + ˜EW(ζ∗  2,n) + on(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='2, there exists a constant N0 > 0 depending on ∥ζ0∥1 + ∥ζ0∥q, W and M0  such that for all ζ ∈ R+(ζ0) with spt(ζ) ⊂ ([0, M0] × R) ∩ D,  G+  s ζ(x) − Wx1 ≤ 0,  ∀x with |x2| > N0  Let  ζ∗∗  i,n(x) = ζ∗  i,n1[0,M0]×[−N0,N0](x + (−1)iN0e2),  i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Then, we find  ˜EW(ζn) ≤ ˜EW(ζ∗∗  1,n) + ˜EW(ζ∗∗  2,n) + on(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  and  supp(ζ∗∗  1,n) ⊂ [0, M0] × [0, 2N0],  supp(ζ∗∗  2,n) ⊂ [0, M0] × [−2N0, 0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  We may assume that ζ∗∗  i,n → ζ∗∗  i  weakly in Lr(D) for some s−1 < r < q and i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Then,  ζ∗∗ := ζ∗∗  1 + ζ∗∗  2 ∈ R(ζ0)w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Moreover, by the weak convergence, we get  lim  n→∞  ˜EW(ζ∗∗  i,n) = ˜EW(ζ∗∗  i ), for i = 1, 2,  and therefore we arrive at  ˜EW(ζ∗∗  1 ) + ˜EW(ζ∗∗  2 ) ≥ lim sup  n→∞  ˜EW(ζn) = Sζ0,W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  It can be seen that  Sζ0,W ≥ ˜EW(ζ∗∗) = ˜EW(ζ∗∗  1 + ζ∗∗  2 )  = ˜EW(ζ∗∗  1 ) + ˜EW(ζ∗∗  2 ) +  �  D  �  D  ζ∗∗  1 (x)G+  s (x, y)ζ∗∗  2 (y)dxdy  ≥ Sζ0,W +  �  D  �  D  ζ∗∗  1 (x)G+  s (x, y)ζ∗∗  2 (y)dxdy,  TRAVELING-WAVE SOLUTIONS FOR THE GSQG EQUATION  35  from which we must have  ˜EW(ζ∗∗) = Sζ0,W  and  �  D  �  D  ζ∗∗  1 (x)G+  s (x, y)ζ∗∗  2 (y)dxdy = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='8)  Since ˜Σζ0,W ⊂ R(ζ0) by the assumption, we deduce ζ∗∗ ∈ R(ζ0) and µ = ∥ζ∗∗∥2  2 =  ∥ζ∗∗  1 ∥2  2 + ∥ζ∗∗  2 ∥2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  On the other hand, since ∥ζ∗∗  1 ∥2  2 ≤ α and ∥ζ∗∗  2 ∥2  2 ≤ µ − α by the weak convergence, we  find that ∥ζ∗∗  1 ∥2  2 = α > 0 and ∥ζ∗∗  1 ∥2  2 = µ − α > 0, which implies that both ζ∗∗  1 and ζ∗∗  2 are  non-zero and hence  �  D  �  D ζ∗∗  1 (x)G+  s (x, y)ζ∗∗  2 (y)dxdy > 0, which is a contradiction to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Compactness: Assume that there is a sequence {yn}∞  n=1 in D such that for  arbitrary ε > 0, there exists R > 0 satisfying  �  D∩BR(yn)  ζ2  ndx ≥ µ − ε,  ∀ n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='9)  We may assume that yn = (yn,1, 0) after a suitable translation in x2-variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Define  ζ0  n := ζn1(0,M0)×R and ζR  n := ζn1(0,M)×(−R,R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Then {ζ0  n}∞  n=1 is also a maximizing sequence  in R+(ζ0) by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Moreover, we infer from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='9) that for arbitrary ε > 0, there  exists R > 0 such that  ∥ζ0  n − ζR  n ∥2  2 ≤ ε,  ∀ n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  That is,  ∥ζ0  n − ζR  n ∥2 → 0,  as R → ∞,  uniformly over n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='10)  We may assume that ζ0  n → ζ0 weakly in L2(D) ∩ Lr(D) for some s−1 < r < q and hence  ζR  n → ζ01(0,M0)×(−R,R) weakly in L2(D)∩Lr(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' By the weak convergence and ζn ∈ R+(ζ0),  we find  ∥ζ0∥2 ≤ lim inf  n→∞ ∥ζ0  n∥2 ≤ ∥ζ0∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='11)  Using Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='2 and H¨older’s inequality, we conclude |E(ζ0  n)−E(ζR  n )| = oR(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' That is,  E(ζR  n ) → E(ζ0  n) as R → ∞ uniformly over n by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' On the other hand E(ζR  n ) → E(ζR)  as n → ∞ for fixed R by weak continuity of E in functions supported on bounded domains  and E(ζR) → E(ζ0) by the monotone convergence theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Therefore, we obtain  E(ζ0  n) → E(ζ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  As for the impulse, we split  |I(ζ0  n) − I(ζ0)| ≤ |I(ζ0  n) − I(ζR  n )| + |I(ζR  n ) − I(ζR)| + |I(ζR) − I(ζ0)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  For the first term, by H¨older’s inequality, we deduce  |I(ζ0  n) − I(ζR  n )| ≤ M0|spt(ζ0  n)|  1  2∥ζ0  n − ζR  n ∥2 → 0,  as R → ∞ uniformly over n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' For fixed R, we have the second term |I(ζR  n ) − I(ζR)| → 0 as  n → ∞ by the weak convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Since the third term |I(ζR) − I(ζ0)| → 0 as R → ∞ by  the monotone convergence theorem, we have |I(ζ0  n) − I(ζ0)| → 0 by first letting R → ∞  and then n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Therefore, we have proved ˜EW(ζ0  n) → ˜EW(ζ0) and hence ˜EW(ζ0) = Sζ0,W and ζ0 ∈ R(ζ0)  by our assumption Σζ0,W ⊂ R(ζ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Then we deduce that ∥ζ0∥2 = ∥ζ0∥2 by the property of  36  DAOMIN CAO, SHANFA LAI, GUOLIN QIN  rearrangement, which implies limn→∞ ∥ζ0  n∥2 = ∥ζ0∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' So, we obtain the strong convergence  ∥ζ0  n − ζ0∥2 → 0 by the uniform convexity of L2(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Now we want to show that ζn → ζ0 strongly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Indeed, since the supports of ζ0  n and ζn −ζ0  n  are disjoint and ζn ∈ R(ζ0), we conclude  ∥ζn − ζ0  n∥2  2 = ∥ζn∥2  2 − ∥ζ0  n∥2  2 ≤ ∥ζ0∥2  2 − ∥ζ0  n∥2  2 → ∥ζ0∥2  2 − ∥ζ0∥2  2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Therefore, we obtain ∥ζn − ζ0∥2 → 0 and finish the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  □  To state our stability result, following [13], we need to introduce some definitions first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Let ξ0 ∈ L1 ∩ Lp(R2  +) be a given function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' ξ ∈ L∞  loc([0, T) , L1(R2  +)) ∩  L∞  loc([0, T) , Lp(R2  +)) is called a Lp-regular solution in (0, T) of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='1) with initial data ξ0 if  (i) ¯ξ(x, t) := ξ(x, t)−ξ(¯x, t) satisfies (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='1) in the sense of distributions with initial data  ¯ξ0(x) := ξ0(x) − ξ0(¯x);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  (ii) E(ξ(t, ·)) and I(ξ(t, ·)) are constant and ξ(t) ∈ R(ξ0) for t ∈ [0, T);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  (iii) ξ is non-negative for t ∈ (0, T) provided that ξ0 is non-negative;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  (iv) For 0 < s ≤ 1  2, we require that ξ is supported in D for t ∈ (0, T) provided that ξ0  is supported in D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Generally speaking, an Lp-regular solution is a weak solution to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='1) such that its  kinetic energy, impulse and distribution are conserved, which is true for sufficiently regular  solutions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' see [14] for some discussion about the conservation laws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' The existence of Lp-  regular solutions for the Euler equation was obtained in [13] using the transport nature  of the Euler equation, see also [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Note that the gSQG equations are also transport  equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  So it is possible to modify the method in [1, 13] to show the existence of  Lp-regular solutions defined above for the gSQG equation provided that the existence of  sufficiently smooth solutions for the Cauchy problem of the gSQG equation is a priori  known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Thus, the Lp-regular solutions for the gSQG equation may be proved to exist for  small T due to the local well-posedness theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' For large T, though a general theory for the  global well-posedness for the Cauchy problem of the gSQG equation remains a challenging  open problem now, various global solutions known as relative equilibria are constructed  as mentioned in the introduction, which are trivial examples of the Lp-regular solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Invariant measures and a large class of global solutions for the SQG equation are also  obtained in [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Besides, certain blow-up scenarios have been ruled out analytically in  [30, 31] and numerically in [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Therefore, we believe that the Lp-regular solutions are  reasonable to exist for large T and for a large class of initial values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  We say that an initial data ξ0 is admissible if ξ0 is nonnegative, supported in D if  0 < s ≤ 1  2 and there exists a L∞-regular solution with initial data ξ0 for T = +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Now, we are ready to establish the following stability theorem on the set of maximizers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' For q with max{2, s−1} < q ≤ ∞ if 0 < s ≤ 1  2 and  2  2s−1 < q ≤ +∞ if  1  2 < s < 1, let 0 ≤ ζ0 ∈ Lq(D) be a function with 0 < |spt(ζ0)| < ∞ and W > 0 is a  given constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Suppose that ˜Σζ0,W, the set of maximizers of ˜EW := E − WI over R(ζ0)w,  satisfies  ∅ ̸= ˜Σζ0,W ⊂ R(ζ0),  TRAVELING-WAVE SOLUTIONS FOR THE GSQG EQUATION  37  and all elements of ˜Σζ0,W are admissible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Then ˜Σζ0,W is orbitally stable in the following  sense:  For arbitrary η > 0 and M > 0, there exists δ > 0 such that if there exist a L∞-regular  solution ξ(t) in (0, T) with initial data ξ0, ∥ξ0∥∞ ≤ M and  inf  ζ∈˜Σζ0,W  {∥ξ0 − ζ∥1 + ∥ξ0 − ζ∥2 + |I(ξ0 − ζ)|} ≤ δ,  then for all t ∈ (0, T), we have  inf  ζ∈˜Σζ0,W  {∥ξ(t) − ζ∥1 + ∥ξ(t) − ζ∥2} ≤ η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='12)  If in addition  I(ζ1) = I(ζ2),  ∀ ζ1, ζ2 ∈ ˜Σζ0,W,  then for all t ∈ (0, T), we have  inf  ζ∈˜Σζ0,W  {∥ξ(t) − ζ∥1 + ∥ξ(t) − ζ∥2 + I(|ξ(t) − ζ|)} ≤ η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='13)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='4, any maximizer (if exists) is supported in [0, M0]×R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' So, we deduce  that supζ1∈˜Σζ0,W I(ζ1) ≤ M0∥ζ0∥1 < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' By an argument similar to the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='4,  for ζ with the following properties:  ζ ≥ 0,  ∥ζ∥1 ≤ ∥ζ0∥1 + 1,  ∥ζ∥∞ ≤ M,  there exists a constant M1 ≥ M0 independent of ζ such that  G+  s ζ(x) − Wx1 < 0,  ∀x with x1 > M1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Now we finish our proof by contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Suppose on the contrary that there exist a  sequence of non-negative functions {ωn  0}∞  n=1, each of which is admissible, ∥ωn  0∥∞ ≤ M and  as n → ∞,  inf  ζ∈˜Σζ0,W  {∥ωn  0 − ζ∥1 + ∥ωn  0 − ζ∥2 + |I(ωn  0 − ζ)|} → 0,  while  sup  t∈(0,Tn)  inf  ζ∈˜Σζ0,W  {∥ωn(t) − ζ∥1 + ∥ωn(t) − ζ∥2} ≥ c0 > 0,  for some positive constant c0, where ωn(t) is the L∞-regular solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='1) with initial  data ωn  0 in (0, Tn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' By the choice of ωn  0, we infer from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='2 and the H¨older inequality  that as n → ∞,  ˜EW(ωn  0 ) → Sζ0,W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='14)  We choose tn ∈ (0, Tn) such that for each n,  inf  ζ∈˜Σζ0,W  {∥ωn(tn) − ζ∥1 + ∥ωn(tn) − ζ∥2} ≥ c0  2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='15)  Take a sequence of functions ζn  0 ∈ ˜Σζ0,W ⊂ R(ζ0) such that as n → ∞,  ∥ωn  0 − ζn  0 ∥1 + ∥ωn  0 − ζn  0 ∥2 + |I(ωn  0 − ζn  0 )| → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='16)  Then, {ζn  0 }∞  n=1 is a maximizing sequence due to Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  38  DAOMIN CAO, SHANFA LAI, GUOLIN QIN  For a function ω, we use ¯ω := ω1(0,M1)×R to denote the restriction of ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='3,  the conservation of energy and impulse, we conclude  ˜EW(ωn(tn)) ≥ ˜EW(ωn(tn)) = ˜EW(ωn  0) → Sζ0,W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='17)  Since ωn(tn) is a rearrangement of ωn  0 by Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='7, one can find a rearrangement  ζn  1 ∈ R(ζn  0 ) = R(ζ0) such that as n → +∞  ∥ωn(tn) − ζn  1 ∥1 + ∥ωn(tn) − ζn  1 ∥2 = ∥ωn  0 − ζn  0 ∥1 + ∥ωn  0 − ζn  0 ∥2 → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='18)  Then, we infer from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='6), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='18) and H¨older’s inequality that  |I(¯ζn  1 ) − I(¯ωn(tn))| ≤ M1∥¯ζn  1 − ¯ωn(tn)∥1  ≤ M1∥ζn  1 − ωn(tn)∥1  = M1∥ζn  0 − ωn  0∥1,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='19)  and  |E(¯ζn  1 ) − E(¯ωn(tn))| ≤ C∥¯ζn  1 − ¯ωn(tn)∥1−s  2−s ≤ C∥ζn  0 − ωn  0∥1−s  2−s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='20)  So, we deduce from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='16)–(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='20) that {¯ζn  1 } ⊂ R+(ζ0) is a maximizing sequence and hence  by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='6 after some translations, we have  ¯ζn  1 → ζ∗∗ strongly in L2(D),  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='21)  as n → ∞ for some function ζ∗∗ ∈ ˜Σζ0,W, which implies ζn  1 → ζ∗∗ strongly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Indeed, since  the supports of ¯ζn  1 and ζn  1 − ¯ζn  1 are disjoint and ζn  1 ∈ R(ζ0), we conclude  ∥ζn  1 − ¯ζn  1 ∥2  2 = ∥ζn  1 ∥2  2 − ∥¯ζn  1 ∥2  2 ≤ ∥ζ0∥2  2 − ∥¯ζn  1 ∥2  2 → ∥ζ0∥2  2 − ∥ζ∗∗∥2  2 = 0,  from which we get  ∥ζn  1 − ζ∗∗∥2 ≤ ∥ζn  1 − ¯ζn  1 ∥2 + ∥ζ∗∗ − ¯ζn  1 ∥2 → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Hence, by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='18) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='21), we deduce  ∥ωn(tn) − ζ∗∗∥2 ≤ ∥ζn  1 − ωn(tn)∥2 + ∥ζn  1 − ζ∗∗∥2  = ∥ζn  0 − ωn  0∥2 + ∥ζn  1 − ζ∗∗∥2 = on(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Next, we will estimate ∥ωn(tn) − ζ∗∗∥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' By the conservation of the L1-norm and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='21),  we have  ∥ωn(tn) − ζ∗∗∥1 ≤  �  spt(ζ∗∗)  |ωn(tn) − ζ∗∗|dx +  �  D\\spt(ζ∗∗)  ωn(tn)dx  ≤  �  spt(ζ∗∗)  |ωn(tn) − ζ∗∗|dx +  �  D  ωn(tn)dx −  �  D  ζn  0 dx +  �  D  ζ∗∗dx −  �  spt(ζ∗∗)  ωn(tn)dx  ≤ 2  �  spt(ζ∗∗)  |ωn(tn) − ζ∗∗|dx +  �  D  |ωn  0 − ζn  0 |dx  ≤ 2|supp(ζ0)|  1  2∥ωn(tn) − ζ∗∗∥2 +  �  D  |ωn  0 − ζn  0 |dx = on(1),  TRAVELING-WAVE SOLUTIONS FOR THE GSQG EQUATION  39  which contradicts the choice of tn in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='15) and completes the proof of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Here we  have used  �  D ζn  0 dx =  �  D ζ∗∗dx =  �  D ζ0dx and |spt(ζn  0 )| = |spt(ζ∗∗)| = |spt(ζ0)| due to the  properties of rearrangement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  If in addition  I(ζ1) = I(ζ2),  ∀ ζ1, ζ2 ∈ ˜Σζ0,W,  then by the conservation of the impulse, we find  I(|ω(t) − ζ1|) ≤  �  spt(ζ1)  x1|ω(t) − ζ1|dx +  �  D\\spt(ζ1)  x1ω(t)dx  ≤  �  spt(ζ1)  x1|ζ1 − ω(t)|dx +  �  D  x1ω(t)dx −  �  D  x1ζ2dx +  �  D  x1ζ1dx −  �  spt(ζ1)  x1ω(t)dx  ≤ 2  �  spt(ζ1)  x1|ζ1 − ω(t)|dx +  �  D  x1ω0dx −  �  D  x1ζ2dx  ≤ 2M0|spt(ζ0)|  1  2 ∥ω(t) − ζ1∥2 + |I(ω0 − ζ2)|,  ∀ ζ1, ζ2 ∈ ˜Σζ0,W,  which implies (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='13) by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='12) and taking δ smaller if necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  □  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Our proof also works well for the case s = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Compared with Theorem 1 in  [13], we admit perturbations with non-compact supports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' This is achieved by bringing in  the L1-norm in our theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Note that, if the perturbation ω0 has a compact support with  measure less than A for some constant A as in Theorem 1 in [13] and is closed to ˜Σζ0,W in  L2, then the H¨older inequality will imply that ω0 is closed to ˜Σζ0,W in L1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Thus, one can  obtain a generalization of Theorem 1 in [13] by the argument in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Maximizers in rearrangement class and the stability of traveling-wave so-  lutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' In Sections 2 and 3, for J(t) = Lt1+ 1  p for some L > 0 and p ∈ (1,  1  1−s), we have  obtained existence and uniqueness of a traveling solution ωε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' To apply our stability result  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='8 to obtain the stability of ωε, we consider the following maximizing problem  sup  ζ∈R(ωε)w  ˜Eε(ζ),  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='22)  where (with abuse of notations for simplicity)  ˜Eε(ζ) := ˜EW ε3−2s(ζ) = 1  2  �  D  ζG+  s ζdx − Wε3−2s  �  D  x1ζdx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  We denote ˜Σε as the set of maximizers of ˜Eε over the rearrangement class R(ωε)w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Our  main result in this subsection is the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' For ε > 0 small, the set of maximizers ˜Σε satisfies  ∅ ̸= ˜Σε ⊂ R(ωε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Moreover, for each ζε ∈ ˜Σε, the following claims hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' ζε = gε(G+  s ζε − Wε3−2sx1 − ˜µε) for some non-negative and non-decreasing function  gε : R → R satisfying gε(τ) > 0 if τ > 0 and gε(τ) = 0 if τ ≤ 0 and some constant  ˜µε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  40  DAOMIN CAO, SHANFA LAI, GUOLIN QIN  (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' diam (spt(ζε)) < ˜Rε for some constant 0 < ˜R < ∞ and up to a suitable translation  in the x2 direction  sup  x∈spt(ζε)  |εx − (d0, 0)| = o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  In order to prove Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='10, a series of lemmas is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' The energy satisfies  max  ζ∈R(ωε)w  ˜Eε(ζ) ≥ I0 + O(ε2−2s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' This is a simple consequence of the fact ωε ∈ R(ωε)w and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  □  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' ˜Eε attains its maximum value over R(ωε)w at some ζε, which is Steiner  symmetric in the x2-variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='1, there exists a constant 1 < Mε = O(ε−1) depending on ε, ωε and  W such that  G+  s ζ(x) − Wε3−2sx1 < 0,  ∀x with x1 > Mε,  ∀ζ ∈ R(ωε)w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  It is easy to check that ˜Eε is bounded from above over R(ωε)w by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Take a  sequence {ζj} ⊂ R(ωε)w such that as j → +∞  ˜Eε(ζj) → sup  R(ωε)w  ˜Eε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  We may assume that ζj is supported in (0, Mε)×R by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' We can also assume ζj is  Steiner symmetric about the x1-axis by replacing ζj with its own Steiner symmetrization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Since  �  D x1ζjdx ≤ Rεκ, there exists a constant Nε > 0 such that G+  s ζj(x) − Wε3−2sx1 <  0,  ∀x ∈ D with |x2| > Nε,  ∀ j due to Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' That is, we assume that ζj is  supported in [0, Mε] × [−Nε, Nε].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' There is a subsequence (still denoted by {ζj}) such that  as j → +∞, ζj → ζε ∈ R(ωε)w weakly star in L∞(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Since G+  s (·, ·) ∈ Lr  loc(R2  + × R2  +) for  any 1 ≤ r <  1  1−s, we deduce that  lim  j→+∞  ˜Eε(ζj) = ˜Eε(ζε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  This means that ζε is a maximizer and thus the proof is finished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  □  To study the properties of maximizers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' We need the following lemma from [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='13 (Lemmas 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='4 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='9 in [9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Let (Ω, ν) be a finite positive measure space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Let ξ0 : Ω → R and ζ0 : Ω → R be ν-measurable functions, and suppose that every level  set of ζ0 has zero measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Then there is a non-decreasing function f such that f ◦ ζ0 is a  rearrangement of ξ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Moreover, if ξ0 ∈ Lq(Ω, ν) for some 1 ≤ q < +∞ and ζ0 ∈ Lq′(Ω, ν),  then f ◦ ζ0 is the unique maximizer of linear functional  M(ξ) :=  �  Ω  ξ(x)ζ0(x)dν(x)  relative to R(ξ0)w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  TRAVELING-WAVE SOLUTIONS FOR THE GSQG EQUATION  41  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Let ζε ∈ ˜Σε be a maximizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Then, up to a translation in the x2 direction,  ζε must be Steiner symmetric in the x2-variable and supported in [0, Mε] × [−Nε, Nε].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Moreover, for ε > 0 small, it holds  0 ̸≡ ζε ∈ R(ζε)  and there exists a non-negative and non-decreasing function gε : R → R satisfying gε(τ) > 0  if τ > 0 and gε(τ) = 0 if τ ≤ 0, such that for some constant ˜µε,  ζε(x) = gε(G1ζε(x) − Wε3−2sx1 − ˜µε),  ∀ x ∈ R2  +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='23)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' For ε > 0 small, we see from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='11 that ˜Eε(ζε) > 0 and hence ζε ̸≡ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Since R(ωε)w is a convex set (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' [8, 13] and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Thus for each  ζ ∈ R(ωε)w, it holds ζτ := ζε + τ(ζ − ζε) ∈ R(ωε)w for any τ ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Noting that ζε is a  maximizer of ˜Eε, we have  d  dτ  ����  τ=0+  ˜Eε(ζτ) =  �  D  (ζ − ζε)  �  G+  s ζε − Wε3−2sx1  �  dx ≤ 0,  which yields  �  D  ζ  �  G+  s ζε − Wε3−2sx1  �  dx ≤  �  D  ζε �  G+  s ζε − Wε3−2sx1  �  dx,  ∀ ζ ∈ R(ωε)w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  By the strict rearrangement inequality, we know that after a translation in the x2 direc-  tion, ζε must be Steiner symmetric in the x2-variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Then by Lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='1– 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='3, we find  that ζε is supported in [0, Mε] × [−Nε, Nε].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Using the fact that ζε is Steiner symmetric in  the x2-variable, one can verified that G+  s ζε is even and strictly decreasing with respect to  x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' It follows that every level set of G+  s ζε − Wε3−2sx1 has measure zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='13,  there exists a non-decreasing function ˜gε : R → R, such that,  ˜ζε(x) = ˜gε  �  G+  s ζε(x) − Wε3−2sx1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  for some ˜ζε ∈ R(ωε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' From the conclusion of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='13, we also know ˜ζε(x) is the unique  maximizer of the linear functional ζ �→  �  [0,Mε]×[−Nε,Nε] (G+  s ζε − Wε3−2sx1) ζdx relative to  R(ωε)w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Hence we must have ζε = ˜ζε ∈ R(ωε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Now, let  ˜µε := sup  �  G+  s ζε(x) − Wε3−2sx1  ���� x ∈ R2  + s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' ζε(x) = 0  �  ∈ R,  and gε(·) = max{˜gε(· + ˜µε), 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' We have  ζε(x) = gε  �  G+  s ζε(x) − Wε3−2sx1 − ˜µε  �  for any x ∈ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  The proof is thus complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  □  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Let ζε ∈ ˜Σε ⊂ R(ωε) be a maximizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Then there exists correspondingly  zε ⊂ [−R, Mε + R] × {0} such that for ε > 0 small  ∥ζε − ˜ωε(· − zε)∥1 + ∥ζε − ˜ωε(· − zε)∥2−s = o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  42  DAOMIN CAO, SHANFA LAI, GUOLIN QIN  Here, ˜ωε is the translation of ωε with  �  x˜ωε = 0 and R is the uniform constant such that  spt(˜ωε) ⊂ B(0, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Notice that ζε ∈ R(ωε) is a rearrangement of ωε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' So, we have  �  ζε =  �  ωε = κ and  �  J(ζε) =  �  J(ωε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Then, we deduce  Eε(ζε) = ˜Eε(ζε) −  �  J(ζε) ≥ ˜Eε(ωε) −  �  J(ωε) = Eε(ωε) ≥ I0 + O(ε2−2s),  which implies that {ζε} is a maximizer sequence of E0 over A0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Therefore, Theorem II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='2  and Corollary II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='1 in [55] provide a subsequence of {ζε} and a sequence of points {zε} such  that  ∥ζε − ω0(· − zε)∥1 + ∥ζε − ω0(· − zε)∥2−s = o(1),  as ε → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  In view of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='1, we obtain  ∥ζε − ˜ωε(· − zε)∥1 + ∥ζε − ˜ωε(· − zε)∥2−s = o(1),  as ε → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Since ζε and ˜ωε are even and non-increasing in x2 and ζε is supported in [0, Mε]×[−Nε, Nε],  we may take zε ∈ [−R, Mε + R] × {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' This completes the proof of this lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  □  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Let zε = (zε,1, 0) be as in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Then zε,1 → +∞ as ε → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' By the inequality ˜Eε(ζε) ≥ ˜Eε(ωε) and Mε = O(ε−1) due to the proof of Lemma  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='12, we get  �  ζεG+  s ζε ≥  �  ωεGsωε + o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='24)  On the other hand, by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='15, we have  �  ζεG+  s ζε =  � �  ˜ωε(x − zε)  �  cs  |x − y|2−2s −  cs  |x − ¯y|2−2s  �  ˜ωε(y − zε)dxdy + o(1)  =  �  ωεGsωε −  � �  cs˜ωε(x)˜ωε(y)  |x − ¯y + 2zε|2−2sdxdy + o(1),  which, combined with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='24), implies  � �  cs˜ωε(x)˜ωε(y)  |x − ¯y + 2zε|2−2sdxdy = o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Thus, we must have zε = (zε,1, 0) with zε,1 → +∞ as ε → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  □  Denote ψε(x) = Gsζε and ψ0 = Gsω0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Since ∥ζε−ω0(·−zε)∥1+∥ζε−ω0(·−zε)∥2−s = o(1)  and ∥ζε∥∞ ≤ ∥ωε∥∞ ≤ C, using Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='1 and a similar bootstrap argument as the proof  of Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='6, we derive  ∥ψε − ψ0(· − zε)∥∞ = o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  To continue our discussion, we next find a lower bound for the Lagrange multipliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' One has for some constant ˜µ > 0,  lim inf  ε→0  ˜µε ≥ ˜µ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Let R be the constant such that spt(˜ωε) ⊂ B(0, R) for every ε small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Since ζε is a  rearrangement of ωε, there is a point ˜x ∈ B(zε, 2R) \\ spt(ζε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' At the point ˜x, by Lemmas  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='14 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='15, we have  ˜µε ≥ Gsζε(˜x) −  �  csζε(y)  |˜x − ¯y|2−2sdy − Wε3−2s˜x1  ≥ ψ0(˜x − zε) −  �  csω0(y)  |˜x − ¯y + zε|2−2sdy + o(1)  ≥  inf  x∈B(0,2R) ψ0(x) + o(1),  which proves this lemma by taking ˜µ = 1  2 infx∈B(0,2R) ψ0(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  □  Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' There is a constant ˜R > 0 such that spt(ζε) is contained in a disk of  radius ˜R for arbitrary ε small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Using Lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='14 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='17, one can prove this lemma through an argument similar  to Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='22, so we omit the details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  □  Let ˜xε := κ−1 �  xζε be the center of mass of ζε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' We have the following estimate of the  location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' It holds  |ε˜xε − (d0, 0)| = o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' The proof is similar to Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='23, so we omit the details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  □  Now, we are able to prove Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='10: The claims in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='10 follow from the above lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  □  Using Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='10 and the uniqueness result Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='6, we obtain the following  conclusion about the set of maximizers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' For ε > 0 sufficiently small, one has  ˜Σε = {ωε(· + ce2) | c ∈ R}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Recall that Σε ⊂ Aε denotes the set of maximizers of Eε defined by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='5) over Aε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  The uniqueness result Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='6 states that Σε = {ωε(· + ce2) | c ∈ R} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Then, by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='10, we have ∅ ̸= ˜Σε ⊂ R(ωε) and for arbitrary ζε ∈ ˜Σε, it  holds spt(ζε) ⊂ B(ε−1(d0, 0), ε−1d0/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' This implies ˜Σε ⊂ Aε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Notice that  �  J(ζ)dx =  �  J(ωε)dx,  ∀ζ ∈ R(ωε) due to the property of rearrangement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' Then one can see easily  that  ˜Σε = Σε = {ωε(· + ce2) | c ∈ R} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  □  Having made all the preparation, we are now ready to give proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='7 and  end our paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  44  DAOMIN CAO, SHANFA LAI, GUOLIN QIN  Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='7: The orbital stability of ωε follows from a combination of Theorem  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='8 and Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  □  Acknowledgments: This work was supported by NNSF of China (Grant 11831009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
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+page_content='  195, 42 pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
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+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
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+page_content='  https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
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+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
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+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content=' and Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
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+page_content=', 3 (1963), 1407-  1456.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
+page_content='  Institute of Applied Mathematics, Chinese Academy of Sciences, Beijing 100190, and  University of Chinese Academy of Sciences, Beijing 100049, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAyT4oBgHgl3EQfg_hE/content/2301.00368v1.pdf'}
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+Springer Nature 2021 LATEX template
+Discrete Morse Functions and Watersheds
+Gilles Bertrand1, Nicolas Boutry2 and Laurent Najman1
+1Univ Gustave Eiffel, CNRS, LIGM, F-77454 Marne-la-Vall´ee, France.
+2EPITA, Research and Development Laboratory (LRDE), France.
+Contributing authors: gilles.bertrand@esiee.fr; nicolas.boutry@lrde.epita.fr;
+laurent.najman@esiee.fr;
+Abstract
+Any watershed, when defined on a stack on a normal pseudomanifold of dimension d, is a
+pure (d − 1)-subcomplex that satisfies a drop-of-water principle. In this paper, we introduce
+Morse stacks, a class of functions that are equivalent to discrete Morse functions. We show
+that the watershed of a Morse stack on a normal pseudomanifold is uniquely defined, and
+can be obtained with a linear-time algorithm relying on a sequence of collapses. Last, but
+not the least, we prove that such a watershed is the cut of the unique minimum spanning
+forest, rooted in the minima of the Morse stack, of the facet graph of the pseudomanifold.
+Keywords: Topological Data Analysis, Mathematical Morphology, Discrete Morse Theory, Simplicial Stacks,
+Minimum Spanning Forest.
+1 Introduction
+Watershed is a fundamental tool in computer
+vision, since its inception as an algorithm by the
+school of mathematical morphology [1, 2]. It is
+still true in this era of deep learning, where it is
+used as a post-processing tool [3]. From a discrete,
+theoretical point of view, the first topologically-
+sound approach was proposed in [4, 5]. Building
+on those results, in [6–8], it is demonstrated that
+watersheds are included in skeletons on pseudo-
+manifolds of arbitrary dimension.
+In this paper, we continue exploring the link
+between watershed and topology, in the frame-
+work of discrete Morse theory [9, 10]. Indeed,
+mathematical morphology [11] and discrete Morse
+theory, although they pursue different objec-
+tives, share many similar ideas. In particular, as
+demonstrated in [12–14], filtering minima using
+morphological dynamics [15] in watershed-based
+image-segmentation, is equivalent to filtering the
+minima by persistence, a fundamental tool from
+Persistent Homology [16] used for topological data
+analysis [17, 18].
+Although the main ideas of the present paper
+originate in [19], we have worked towards a
+simpler, unifying framework for exposing these
+ideas. This leads us to introduce Morse stacks:
+these functions correspond to the inverse of flat
+Witten-Morse functions that, according to R. For-
+man [20], seem to have shown themselves to be
+the appropriate combinatorial analogue of smooth
+non-degenerate Morse functions. We also propose
+a new definition for normal pseudomanifolds, a
+class of manifolds on which path-connectivities are
+all equivalent. Relying on these notions, we prove
+that a watershed, a pure (d − 1)-subcomplex of
+a normal pseudomanifold, has several interesting
+properties when defined on a Morse stack F. In
+particular, in this setting, a watershed is uniquely
+1
+arXiv:2301.03840v1  [cs.DM]  10 Jan 2023
+
+Springer Nature 2021 LATEX template
+2
+Discrete Morse Functions and Watersheds
+defined, and can be obtained thanks to a linear-
+time algorithm, relying on a sequence of collapses.
+Furthermore, a watershed is the cut of the unique
+minimum spanning forest of the facet graph of
+the normal pseudomanifold weighted by F, rooted
+in the minima of F. Relations between water-
+sheds and Morse theory have long been informally
+known [21], but this is the first time that a link is
+presented in the discrete setting, relying on a pre-
+cise definition of the watershed. Furthermore, as
+far as we know, this is the first time that a concept
+from Discrete Morse Theory is linked to a classical
+combinatorial optimization problem.
+The plan of this paper is the following. Section
+2 provides some basic definitions of simplicial com-
+plexes. We introduce here the notion of a covering
+pair, that is fundamental for the definition of
+Morse stacks. Section 3 recalls some definitions
+of simplicial stacks, which are a class of weighted
+simplicial complexes whose upper threshold sets
+are also complexes. Section 4 proposes a new
+definition for normal pseudomanifolds. Section 5
+provides the necessary definitions for watersheds
+on stacks defined on normal pseudomanifolds.
+We propose here an algorithm for computing
+watershed relying on the collapse operation. In
+section 6, we introduce Morse stacks. Section 7
+studies the properties of watersheds on Morse
+stacks. Section 8 links watershed and the mini-
+mum spanning forest. We conclude the paper with
+a discussion in section 9, in which we highlight the
+importance of our results, both from a theoreti-
+cal and a practical point of views, and we propose
+some perspective for future work.
+Finally, appendix A shows that our defini-
+tion of normal pseudomanifold is equivalent to the
+classical one, and appendix B demonstrates that
+Morse stacks are equivalent to classical discrete
+Morse functions.
+2 Simplicial complexes
+A simplex x is a non-empty finite set; the dimen-
+sion of x, written dim(x), is the number of its
+elements minus one. We also say that x is a
+p-simplex if dim(x) = p.
+Let S be a finite set of simplexes. A p-simplex
+in S is a (p-)face of S. A (p-)facet of S is a p-face
+of S that is maximal for inclusion. If x and y are
+two distinct faces of S such that x ⊆ y, we say
+that x is a face of y (in S). The simplicial closure
+of S is the set S− = {y ⊆ x | y ̸= ∅ and x ∈ S}.
+The set S is a (simplicial) complex if S = S−.
+Let X be a complex. The dimension of X,
+written dim(X), is the largest dimension of its
+simplices, the dimension of ∅ being defined to be
+−1.
+A 0-face of X is a vertex of X and a 1-face of X
+is an edge of X.
+A complex X is a graph if the dimension of X is
+at most 1.
+Let X be a complex and let S ⊆ X. If S is a
+complex, we say that S is closed for X or that S
+is a subcomplex of X. We say that S is open for X
+or that S is an open subset of X if, for any x ∈ S,
+we have y ∈ S whenever x ⊆ y and y ∈ X. If X is
+a complex and S ⊆ X, we note that S is closed for
+X if and only if X \S is open for X. In particular,
+∅ and X are both closed and open for X.
+Let S be a finite set of simplexes. Let π =
+⟨x0, . . . , xk⟩ be a sequence of elements of S. The
+sequence π is a path in S (from x0 to xk) if, for
+any i ∈ [0, k−1], either xi ⊆ xi+1 or xi+1 ⊆ xi. We
+say that S is connected if, for any x, y ∈ S, there
+exists a path from x to y in S. We say that T ⊆ S
+is a connected component of S if T is connected
+and maximal, with respect to set inclusion, for this
+property.
+Remark 1 We observe that:
+– If X is a complex, then X is connected if and
+only if, for any vertices x, y ∈ X, there exists a
+sequence ⟨x = x0, . . . , xk = y⟩ of vertices of X
+such that, for any i ∈ [0, k − 1], xi ̸= xi+1 and
+xi ∪ xi+1 is an edge of X.
+– If S is an open subset of a complex X, then S
+is connected if and only if, for any facets x, y of
+S, there exists a sequence ⟨x = x0, . . . , xk = y⟩
+of facets of S such that, for any i ∈ [0, k − 1],
+xi ∩ xi+1 is in S.
+The following simple definition of a covering
+pair (or a p-pair) will play an important role in
+the sequel of the paper:
+– It will first allow us to define a free pair (Def-
+inition 3). This corresponds to the operation
+of collapse of a simplicial complex introduced
+by J.H.C. Whitehead [22], which is a discrete
+analogue of a retraction, that is, a continuous
+(homotopic) deformation of an object onto one
+
+Springer Nature 2021 LATEX template
+Discrete Morse Functions and Watersheds
+3
+of its subsets. In Section 3, free pairs for a sim-
+plicial complex will be extended to free pairs on
+stacks, which are maps on simplicial complexes
+(Definition 4).
+– In Section 6, we introduce the notion of a flat
+pair, which is a special case of a covering pair.
+This permits us to have a very simple and con-
+cise presentation of a Morse stack (Definition
+16). Indeed, a basic link between Morse stacks
+and the collapse operation is straightforward
+(Proposition 17). Furthermore, the notions of a
+gradient vector field and a gradient path follow
+immediately from covering and flat pairs.
+Definition 2 (Covering pair) Let X be a complex and
+x, y ∈ X, with dim(y) = p. We say that (x, y) is a
+covering pair of X or a p-pair of X if x is a face of y
+and dim(x) = p − 1.
+Definition 3 (Free pair) Let X be a complex and let
+(x, y) be a p-pair of X. We say that (x, y) is a free
+(p-)pair of X if y is the only face of X that contains x.
+Thus, if (x, y) is a free pair of X, we have nec-
+essarily dim(x) = dim(y) − 1. Furthermore, we
+observe that y is necessarily a facet of X.
+If (x, y) is a free p-pair of a complex X, then
+Y = X \{x, y} is an elementary (p-)collapse of X.
+We say that X collapses (resp. p-collapses) onto
+Y , if there exists a sequence ⟨X = X0, ..., Xk =
+Y ⟩ such that Xi is an elementary collapse (resp.
+elementary p-collapse) of Xi−1, i ∈ [1, k]. If, fur-
+thermore, Y has no free pair (resp. free p-pair),
+then Y is an ultimate collapse (resp. ultimate p-
+collapse) of X. A complex X is collapsible if X
+collapses onto a single vertex.
+3 Simplicial stacks
+Let X be a simplicial complex, and let F be a map
+from X to Z. If x is a face of X, the value F(x)
+is called the altitude of F at x. For any λ ∈ Z, we
+write F[λ] = {x ∈ X | F(x) ≥ λ}, F[λ] is the λ-
+section of F. We say that F is a (simplicial) stack
+on X if any λ-section of F is a simplicial complex.
+In other words, any λ-section of F is a closed set
+for X.
+Let F be a map from a complex X to Z. It may
+be easily seen that F is a simplicial stack if and
+only if, for any x, y ∈ X such that x ⊆ y, we have
+F(x) ≥ F(y). Also, a map F is a simplicial stack
+if and only if, for any covering pair (x, y) in X, we
+have F(x) ≥ F(y).
+Now, we extend the notion of free pairs of sim-
+plicial complexes to simplicial stacks. This exten-
+sion allows us to introduce some fundamental
+discrete homotopic transforms of these maps.
+Definition 4 Let F be a simplicial stack on X. We
+set λm = min{F(x) | x ∈ X}. Let (x, y) be a p-pair
+of X. We say that (x, y) is a free (p-)pair of F if (x, y)
+is a free (p-)pair of F[λ], with λ = F(x) and λ > λm.
+If (x, y) is a free pair of F, then both x and y
+are in F[λ], with λ = F(x). Thus F(y) ≥ F(x).
+Also, we have x ⊆ y. Then, since F is a stack, we
+have F(y) ≤ F(x). Thus, we have F(x) = F(y)
+whenever (x, y) is a free pair of F.
+Let F be a simplicial stack on a complex X.
+1. Let (x, y) be a free (p-)pair of F. Let G be the
+map such that G(z) = F(z) − 1 if z = x or z =
+y, and G(z) = F(z) if z ∈ X\{x, y}. We can see
+that G is a simplicial stack on X, the map G is
+called an elementary (p-)collapse of F through
+(x, y) or simply an elementary (p-)collapse of
+F.
+2. If G is the result of a sequence of elementary
+collapses (resp. p-collapses) of F, then we say
+that F collapses (resp. p-collapses) onto G.
+3. If F collapses (resp. p-collapses) onto a stack
+G that has no free pair (resp. no free p-pair),
+then G is an ultimate collapse (resp. ultimate
+p-collapse) of F.
+We conclude this section by giving a definition
+of a (regional) minimum of a stack, which plays a
+crucial role for the notion of a watershed.
+Let F be a simplicial stack on a complex X
+and let λ ∈ Z. A subset A of X is a minimum of
+F (at altitude λ) if A is a connected component
+of X \ F[λ + 1] and A ∩ (X \ F[λ]) = ∅. The divide
+of F is the set composed of all faces of X that are
+not in a minimum of F. Note that any minimum
+of F is an open set for X, and the divide of F is
+a simplicial complex.
+4 Normal pseudomanifolds
+The results of this paper hold true in a large fam-
+ily of n-dimensional discrete spaces, namely the
+
+Springer Nature 2021 LATEX template
+4
+Discrete Morse Functions and Watersheds
+normal pseudomanifolds. This section provides a
+presentation of these spaces.
+Let S be a finite set of simplexes. A
+strong
+p-path in S (from x0 to xk) is a path ⟨x0, ..., xk⟩
+such that, for each i ∈ [0, k−1], either (xi, xi+1) is
+a p-pair, or (xi+1, xi) is a p-pair. The set S is (d-
+)pure if all facets of S have the same dimension d.
+If S is d-pure, we say that a strong d-path in S is
+a strong path in S. Also, we say that S is strongly
+connected if, for any two facets x, y in S, there
+exists a strong path in S from x to y. A subset T
+of S is a strong connected component of S if T is
+strongly connected and maximal, with respect to
+set inclusion, for this property.
+Definition 5 (Normal pseudomanifold) A connected
+and d-pure complex X, with d ≥ 1, is a normal
+pseudomanifold (or a normal d-pseudomanifold) if:
+1. The complex X is non-branching, that is, each
+(d − 1)−face of X is included in exactly two
+d-faces of X.
+2. The complex X is strictly connected, that is,
+each connected open subset of X is strongly
+connected.
+Recall that a pure complex X is a pseudoman-
+ifold if it is non-branching and strongly connected
+[23]. Since the very set X is open for a complex X,
+we see that any normal pseudomanifold is a pseu-
+domanifold.
+In fact, the above definition is a new definition
+for a normal pseudomanifold. In Appendix A, we
+show that it is equivalent to the classical definition
+[24], [25], [26], which consists of a local condition
+together with the conditions that must be satisfied
+by a pseudomanifold.
+Let us consider Fig. 1. The triangulated torus
+(a) is a normal pseudomanifold. The triangulated
+pinched torus (b) is a pseudomanifold that is not
+normal: the set of all faces containing the pinch
+vertex is a connected open subset of the com-
+plex, but this set is not strongly connected. The
+triangulated pinched torus (c) is not a pseudo-
+manifold: the pinch segment does not satisfy the
+non-branching condition.
+Let X
+be a proper subcomplex of a d-
+pseudomanifold M. We can see that, if dim(X) =
+d, the complex X has necessarily a boundary, that
+is, there exists a free d-pair for X. By induction,
+it means that the dimension of an ultimate d-
+collapse of X is necessarily d − 1. See [7] for a
+formal proof.
+Important notations. In the sequel of the
+paper:
+– We denote by S the collection of all simplicial
+complexes.
+– If X ∈ S, we write Y ⪯ X whenever Y ⊆ X and
+Y ∈ S, that is, whenever Y is a subcomplex of
+X.
+– If X ∈ S, and S ⊆ X, we write S ⊑ X whenever
+S is an open subset of X.
+– We denote by M (resp. Md) the collection of
+all normal pseudomanifolds (resp. all normal d-
+pseudomanifolds).
+– If F is a stack on M ∈ M, the notation min(F)
+stands for the union of all minima of F, and
+we write div(F) for the divide of F. Thus, we
+have div(F) = M \ min(F), div(F) ⪯ M, and
+min(F) ⊑ M.
+We are now ready to introduce the notion of a
+watershed in the context of simplicial complexes.
+We will consider a normal pseudomanifold M ∈ M
+and a simplicial stack F on M, the map F may be
+seen as a “topographical relief” on the space M.
+A simplicial complex W ⪯ M may be a watershed
+of F if W “separates the minima of F”. It means
+that if A ⊑ M is a connected component of M\W,
+then A contains one and only one set B ⊑ M
+that is a minimum of F. Furthermore, the com-
+plex W must satisfy a “drop of water principle”:
+from each face of W, we may reach two distinct
+minima of F by following a descending path. Each
+connected component A of M \W will correspond
+to a “catchment basin” of the map F.
+5 Watersheds
+Let X ∈ S, and let A ⊑ X, with A ̸= ∅. We say
+that B ⊑ X is an extension of A if A ⊆ B, and if
+each connected component of B includes exactly
+one connected component of A. We also say that
+B is an extension of A if A = B = ∅.
+Proposition 6 Let M ∈ M and let A ⊑ M.
+1. A subset S of A is a connected component of A
+if and only if S is a strong connected component
+of A.
+
+Springer Nature 2021 LATEX template
+Discrete Morse Functions and Watersheds
+5
+(a)
+(b)
+(c)
+Fig. 1
+(a): A normal pseudomanifold, which is a torus, (b): A pseudomanifold, which is a pinched torus, and where the
+pinch face is a vertex, (c): A pinched torus where the pinch face is a segment. This is not a pseudomanifold.
+2. Let B ⊑ M, with A ⊆ B. The set B is an
+extension of A if and only if each strong con-
+nected component of B includes exactly one
+strong connected component of A.
+Proof 1) It may be seen that, if S is a connected com-
+ponent of the open set A, then S is necessarily an
+open set for M. Since M is a normal pseudomanifold,
+we deduce that S is strongly connected. Furthermore,
+S is a strong connected component of A, otherwise S
+would not be a maximal connected subset of A. Now,
+if S is a strong connected component of A, then S is
+a connected subset of A. Again, we see that S is a
+connected component of A. Otherwise, S would be a
+proper subset of a connected open subset T of A. Since
+M is a normal pseudomanifold, this subset T would
+be strongly connected, and S would not be a maximal
+strongly connected subset of A.
+2) is a direct consequence of 1).
+□
+Let X ∈ S and Y ⪯ X. Let A ⊑ X, with
+A ̸= ∅. We say that Y is a cut for A, if X \ Y
+is an extension of A, and if Y is minimal for this
+property. That is, if Z ⪯ Y , and if X \ Z is an
+extension of A, then we have necessarily Z = Y .
+Proposition 7 (from [7]) Let M ∈ M, A ⊑ M and
+X ⪯ M, with A ̸= ∅. If X is a cut for A, then the
+complex X is either empty or a pure (n − 1)-complex.
+Remark 8 It could be seen that the previous result
+no longer holds if we consider arbitrary pseudomani-
+folds instead of normal pseudomanifolds. For example,
+the pinched vertex of the pinched torus of Figure 1.(b)
+could be in a cut.
+In fact, it is possible to bypass this situation by con-
+sidering only strong paths between faces, as it is done
+in [7]. In this paper, in order to handle general con-
+nectedness and arbitrary paths, we have made the
+choice to settle our results in normal pseudomanifolds.
+Let F be a stack on M ∈ M. If π = ⟨x0, . . . , xk⟩
+is a path in M, we say that π is ascending for F
+(resp. descending for F) if, for any i ∈ [0, k], we
+have F(xi) ≤ F(xi+1) (resp. F(xi) ≥ F(xi+1)).
+Definition 9 (Cut) Let F be a stack on M ∈ M and
+let X ⪯ M be a cut for min(F). We say that X is
+a watershed of F if, for each x ∈ X, there exist two
+strong paths π1 = ⟨x0, . . . , xk⟩ and π2 = ⟨y0, . . . , yl⟩
+in M \ X, such that:
+– x ⊆ x0 and x ⊆ y0;
+– π1 and π2 are descending paths for F; and
+– xk and yl are simplices of two distinct minima
+of F.
+Let M ∈ M, F be a stack on M, and let W be
+a watershed for F. We say that B ⊑ M is a (catch-
+ment) basin of W if B is a connected component
+of W = M \ W. Since W is a cut for min(F),
+– any catchment basin B of W contains a unique
+minimum A of F, we say that A is the minimum
+of B;
+– any minimum A of F is included in a unique
+basin B of W, we say that B is the catchment
+basin of A.
+Proposition 10 (from [7]) Let M ∈ Md, F be a stack
+on M, and W be a watershed of F. Then, for any d-
+face x in M, there exists a strong path in M \ W from
+x to a d-face of a minimum of F, that is descending
+for F.
+From the previous result, we easily derive the
+following proposition.
+Proposition 11 Let M ∈ M, F be a stack on M, and
+W be a watershed of F. Let B be the catchment basin
+of a minimum A of F. Then, for any x ∈ B, there
+exists a descending path in B from x to a face of A.
+The two following results are crucial for linking
+a watershed of a stack F and the homotopy of F.
+
+Springer Nature 2021 LATEX template
+6
+Discrete Morse Functions and Watersheds
+Proposition 12 (from [7]) Let M ∈ Md. If F is a
+stack on M and H is a collapse of F, then:
+1. min(H) is an extension of min(F).
+2. div(H) is a collapse of div(F).
+It should be noted that the previous prop-
+erty is no longer true if we consider a stack on
+an arbitrary complex X ∈ S rather than a com-
+plex M ∈ M. See Fig 2 which provides a simple
+counter-example.
+Proposition 13 (from [7]) Let M ∈ Md and F be a
+stack on M.
+1. F contains a free d-pair if and only if div(F)
+contains a free d-pair.
+2. If dim(div(F)) = d, then there exists a free d-
+pair for F.
+Let F be a stack on M ∈ Md and x be a (d−1)-
+face of M. Let y, z be the two d-faces containing
+x. We say that x is (locally) separating for F if
+F(y) < F(x) and F(z) < F(x). We say that x
+is biconnected for F if y and z belong to distinct
+minima of F.
+Definition 14 Let F be a stack on M ∈ Md. Let
+X ⪯ M. We say that X is a cut by collapse of F,
+or a C-watershed of F, if there exists an ultimate d-
+collapse H of F such that X is the simplicial closure
+of the set of all faces of M that are biconnected for H.
+Theorem 15 (from [7]) Let M ∈ M and let F be a
+stack on M. A complex X ⪯ M is a watershed of F if
+and only if X is a C-watershed of F.
+Theorem 15 is illustrated in Fig. 3.
+From Theorem 15 we may derive the procedure
+WatershedCollapse(F, M) (Fig. 4) for obtaining
+a watershed of a stack F on M ∈ Md.
+The result W depends on the choices of the
+free pairs that are made at step 1. In any case,
+any watershed of F may be obtained by this
+procedure.
+As explained hereafter, a direct implementa-
+tion of the algorithm WatershedCollapse(F, M)
+can be slow.
+– Step 1 is the more complex one. A naive imple-
+mentation of this step is in the order of n2 ∗ h,
+where n is the number of d-faces, and h is the
+number of different altitudes of F. However, this
+step can be done in quasi-linear time, relying
+on a straightforward adaptation to simplicial
+complexes of the algorithm presented in [27].
+This algorithm relies on a tree structure, where
+the nodes of the tree are the connected com-
+ponents of all the level sets of F, and where
+the edges of the tree correspond to the par-
+enthood relationships between those connected
+components.
+– Step 2 is a simple labelling, and may be done
+in linear time with respect to the number of d-
+faces. By using such a labelling, checking if a
+d-face is biconnected can be done in constant
+time.
+– Finally, step 3 may be implemented in linear
+time, with respect to the number of incidence
+relations of M, that is the cardinal of the set
+{(x, y) | x, y ∈ M and x ⊊ y}.
+6 Morse stacks
+In this section, we transpose some basic notions
+of discrete Morse theory to stacks. We proceed by
+defining a Morse stack, which is the counterpart of
+a classical discrete Morse function. Morse stacks
+simply correspond to the inverse of flat discrete
+Morse functions. See Appendix B, which provides
+the few facts linking these two notions.
+Let F be a map from a complex X to Z. We
+say that a covering pair (x, y) of X is a flat pair
+of F whenever we have F(x) = F(y).
+Definition 16 (Morse stack) Let F be a simplicial
+stack on a complex X. We say that F is a Morse stack
+(on X) if any face of X is in at most one flat pair
+of F.
+Let F be an arbitrary simplicial stack, and let
+(x, y) be a covering pair of F. We have seen that, if
+(x, y) is a free pair of F, then necessarily (x, y) is a
+flat pair of F. Suppose now that (x, y) is a flat pair
+of F. Then, there may exist another covering pair
+(x, z) that is also a flat pair of F. In this case, we
+see that (x, y) is not a free pair of F. By the very
+definition of a Morse stack, this situation cannot
+occur. In fact, we have the following result.
+
+Springer Nature 2021 LATEX template
+Discrete Morse Functions and Watersheds
+7
+F
+H
+Fig. 2
+Two stacks F and H with two levels: altitude 0 (faces in light grey) and altitude 1 (faces in black). The stack H
+is an elementary collapse of F (at altitude 1). But F has three minima whereas H has only two. Thus, min(H) is not an
+extension of min(F).
+(a)
+(b)
+(c)
+Fig. 3
+(a) A simplicial stack F on a subset of a normal 2-pseudomanifold. (b) An ultimate 2-collapse of F. (c) A watershed
+of F. The pair (y, x) in (a) is a free-pair for F.
+Proposition 17 Let F be a Morse stack on a complex
+X. A covering pair (x, y) of X is a free pair of F if
+and only if (x, y) is a flat pair of F.
+Definition 18 (Regular and critical simplex) Let F
+be a Morse stack on a complex X and let x ∈ X with
+dim(x) = p.
+– We say that x is regular or p-regular for F if x
+is in a flat pair of F.
+– We say that x is critical or p-critical for F if x
+is not regular for F.
+A Morse stack defined on a normal pseudo-
+manifold, together with its critical and regular
+simplexes, can be seen on Fig. 5.
+Let F be a Morse stack on a complex X.
+The gradient vector field of F, written −−→
+grad(F), is
+the set of all flat pairs of F.
+If (x, y) is a covering pair of F such that F(x) >
+F(y), we say that (y, x) is a differential pair of F.
+We write −→
+diffF for the set of all differential pairs
+for F.
+We also set:
+
+y
+X
+20
+0-03203303
+03
+33
+03
+3-3
+03-0-03-3
+303202203
+000030033
+-033Springer Nature 2021 LATEX template
+8
+Discrete Morse Functions and Watersheds
+WatershedCollapse(F, M)
+1. Set H = F. Until H has no free d-pair, select arbitrarily a free d-pair (x, y) of H
+and replace H by the elementary collapse of H through (x, y);
+2. Label all d-faces of distinct minima of H with distinct labels;
+3. Extract from H the complex W that is the simplicial closure of the set of all
+d-faces of M which are biconnected for H.
+Fig. 4 The procedure WatershedCollapse(F, M) computes a watershed W of a stack F defined on a normal
+pseudomanifold M.
+– −−→
+gradp(F) = {(x, y) ∈ −−→
+grad(F) | dim(y) = p},
+and
+– −→
+diffp(F) = {(y, x) ∈ −→
+diff(F) | dim(y) = p}.
+A Λp-path in F (from x0 to xk) is a sequence
+π = ⟨x0, x1, ..., xk⟩ composed of faces of X such
+that, for all i ∈ [0, k − 1], the pair (xi, xi+1) is
+either in −−→
+gradp(F) or in −→
+diffp(F). A sequence π is
+a gradient path for F if π is a Λp-path for F for
+some p.
+Let π = ⟨x0, x1, ..., xk⟩ be a Λp-path in F. We
+observe that:
+– For any i ∈ [1, k − 1], the pair (xi, xi+1) is in
+−−→
+gradp(F) (resp. −→
+diffp(F)) whenever (xi−1, xi) is
+in −→
+diffp(F) (resp. −−→
+gradp(F)).
+– Each face of π is either a p-face or a (p−1)-face.
+For any i ∈ [0, k − 1], if xi is a p-face, then xi+1
+is a (p − 1)-face, and if xi is a (p − 1)-face, then
+xi+1 is a p-face.
+– The path π is a strong p-path.
+– If π is not trivial, then π cannot be closed, that
+is, we have necessarily k = 0 whenever xk = x0.
+– The path π is an ascending path, that is, we
+have F(xi) ≤ F(xi+1) for any i ∈ [0, k − 1].
+Furthermore, we have F(xi) < F(xi+2) for any
+i ∈ [0, k − 2].
+The following result is a basic fact about Morse
+functions.
+Proposition 19 Let F be a Morse stack on a complex
+X ∈ S, and let S be a subset of X. If S is a minimum
+of F, then S is composed of a single facet of X.
+In the sequel, we will say that a face x ∈ X
+is a minimum (of F) whenever the set {x} is a
+minimum of F.
+7 Morse stacks and
+watersheds
+Let F be a Morse stack on a complex X ∈ S. Let
+x, y be two faces of X. We say that x is Λp-linked
+to y if there is a Λp-path in F from x to y. Let
+π = ⟨x = x0, ..., xk = y⟩ be a Λp-path in F from x
+to y. We write ˜π = ⟨y = xk, ..., x0 = x⟩ and we say
+that ˜π is a ˜Λp-path in F from y to x. We say that a
+face z ∈ X is an extension of π if ⟨x = x0, ..., xk =
+y, z⟩ is a Λp-path in F from x to z. We say that z
+is an extension of ˜π if ⟨y = xk, ..., x0 = x, z⟩ is a
+˜Λp-path in F from y to z.
+Proposition 20 Let F be a Morse stack on M ∈ Md,
+and let x be a facet of X. Let ˜π be a ˜Λd-path in F from
+the facet x to a face y ∈ X. Then one and only one of
+the following statements is true:
+1. The face y is a minimum.
+2. There exists a unique face that is an extension
+of ˜π.
+Proof We set ˜π = ⟨x = x0, ..., xk = y⟩.
+i) Suppose y is a (d − 1)-face. In this case, y cannot
+be a minimum. Furthermore, we have k ≥ 1 (since x
+is a d-face), and the face t = xk−1 is necessarily a d-
+face. By the definition of a ˜Λp-path, the pair (y, t) is a
+flat pair. Since X is a pseudomanifold, there exists a
+unique d-face z ∈ X such that t∩z = y. We must have
+F(z) < F(y), otherwise y would belong to more than
+one flat pair. Therefore, the pair (z, y) is a differential
+pair and z is an extension of ˜π. Since (y, t) and (y, z)
+are the only covering pairs that contain y, the face z
+is the unique extension of ˜π.
+ii) Suppose y is a d-face. By the definition of a ˜Λp-
+path, a face z is an extension of ˜π if and only if (z, y)
+is a flat pair. Now we observe that y is not a minimum
+if and only if there is a face z such that (z, y) is a flat
+pair. But y belongs to at most one flat pair. Thus, ˜π
+has a unique extension whenever y is not a minimum.
+□
+
+Springer Nature 2021 LATEX template
+Discrete Morse Functions and Watersheds
+9
+Let F be a Morse stack on M ∈ Md. It should
+be noted that, if π is a Λd-path in F from a facet
+x to a face y, then π may have more than one
+extension. Nevertheless, by induction, we obtain
+the following result from Prop. 20.
+Proposition 21 Let F be a Morse stack on M ∈ Md,
+and let x be a facet of X. There exists a unique
+minimum m of F such that m is Λd-linked to x. Fur-
+thermore, there exists a unique Λd-path in F from m
+to x.
+We now consider the case where a Λd-path has
+no extension. Recall that a (d − 1)-face x is sepa-
+rating for F if the two d-faces y, z which contain
+x, are such that F(y) < F(x) and F(z) < F(x).
+Proposition 22 Let F be a Morse stack on M ∈ Md.
+Let x be a facet of X, and let π be a Λd-path in F
+from x to a face y ∈ M. If π has no extension, then
+the face y is necessarily separating for F.
+Proof Let π = ⟨x = x0, ..., xk = y⟩ be a Λd-path in F
+from x to y ∈ M.
+If dim(y) = d, then π has necessarily an extension.
+Now suppose dim(y) = d − 1, thus k ≥ 1. The face y
+is a face of two d-faces, the face z = xk−1 and another
+face t. Since π has no extension, we have F(t) < F(y).
+Furthermore, by the very definition of a Λd-path, the
+pair (z, y) is a differential pair, thus we have F(z) <
+F(y). Therefore, y is separating for F.
+□
+Let F be a Morse stack on a complex X ∈ S.
+Let π be a Λp-path in F. We say that π is maximal
+if neither π nor ˜π has an extension. The following
+result is a direct consequence of Prop. 20 and 22.
+Corollary 23 Let F be a Morse stack on M ∈ Md.
+Let π be a Λd-path in F from x to y. If π is maximal,
+then x is a minimum of F and y is a separating face
+for F.
+Let F be a Morse stack on M ∈ Md. Let x be
+a (d − 1)-face of X, and let y, z be the two dis-
+tinct d-faces containing x. According to Prop. 21,
+each of these faces is Λd-linked to a single mini-
+mum. We say that the face x is Λ-biconnected (for
+F) if these two minima are distinct. Observe that
+a face is necessarily separating whenever it is
+Λ-biconnected.
+Definition 24 (Morse watershed) Let F be a Morse
+stack on M
+∈ M. The Morse watershed of F is
+the complex that is the simplicial closure of the set
+composed of all faces that are Λ-biconnected for F.
+Theorem 25 Let F be a Morse stack on M ∈ M.
+The Morse watershed of F is a watershed of F. Fur-
+thermore, the Morse watershed of F is the unique
+watershed of F.
+Th. 25 is illustrated in Fig. 5.
+Proof Let W be the Morse watershed of F.
+1. Let A be a connected component of M \W. By
+Prop. 6, the set A is a strong connected com-
+ponent of M \ W. Let f be a facet of A. By
+Prop. 21, there exists a Λd-path π from a min-
+imum m of F to f. By the very definition of a
+Λd-path, it may be seen that π does not con-
+tain any separating face. Thus π is included in
+M \ W, and m is in A. Therefore, A contains a
+minimum m.
+Now let x be an arbitrary facet of A. Since A is
+strongly connected, there exists a strong path
+π = ⟨m = x0, ..., xk = x⟩ in A from m to x. By
+Prop. 21, for each facet of π, there is a unique
+minimum of F that is Λd-linked to this facet.
+Let xi be a facet of π, with i ≤ k − 2. Thus
+xi+1 is a (d−1)-face and xi+2 is a facet. Let mi
+(resp. mi+2) be the unique minimum of F that
+is Λd-linked to xi (resp. to xi+2). Since xi+1
+is not Λ-biconnected for F, it may be checked
+that we have necessarily mi = mi+2. By induc-
+tion, it follows that m is Λd-linked to the facet
+x. Since this result holds for any facet of A, this
+clearly implies that m is the unique minimum
+of F which is in A. Thus, any connected com-
+ponent of M \W contains exactly one minimum
+of F. But any minimum of F is included in
+M \W (since a minimum is a d-face). It follows
+that M \W is an extension of min(F). Further-
+more, by the definition of a Λ-biconnected face,
+W is minimal for this last property. Therefore,
+W is a cut for min(F). Since any ˜Λd-path is a
+descending strong path, it may be checked that
+W fulfills all the conditions of Definition 9: W
+is a watershed.
+2. Let W ′ be a watershed of F. Let x be a (d−1)-
+face of M that is Λ-biconnected for F, and let
+
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+5
+8
+8
+8
+9
+9
+10
+10
+27
+V
+0
+V
+0
+2
+2
+0
+3
+3
+V
+3
+4
+4
+12
+12
+11
+11
+1
+6
+6
+7
+7
+V
+2
+V
+2
+V
+1
+V
+1
+18
+18
+16
+16
+22
+25
+24
+24
+25
+30
+27
+19
+28
+23
+20
+28
+29
+29
+26
+30
+26
+21
+22
+27
+23
+21
+26
+30
+26
+25
+25
+30
+Fig. 5
+The Morse watershed (in red) on a Morse stack defined on a normal pseudomanifold, a 2d-torus. In black, critical
+simplexes: those of dimension 2 are minima, those of dimension 1 are saddle, and those of dimension 0 are maxima. Arrows
+represent gradient/collapse of dimension 2. We observe that the critical simplex 20 do not belong to the watershed.
+y, z be the two distinct d-faces containing x. By
+Prop. 10 there exist a descending strong path in
+M \W ′ from y to a minimum m and a descend-
+ing strong path in M \W ′ from z to a minimum
+m′. We observe that any descending strong
+path in M is also a ˜Λd-path in M. Thus, by the
+very definition of a Λ-biconnected face, we must
+have m ̸= m′. Therefore, the face x must be in
+W ′, otherwise y, z, m, and m′, would belong to
+the same connected component of M\W ′. Thus
+W ⊆ W ′. Since M \ W ′ ⊆ M \ W, each con-
+nected component of M \W ′ is included in one
+connected component of M \ W. But M \ W ′
+must be maximal for this last property, oth-
+erwise W ′ would not be a cut for min(F). It
+follows that we have W ′ = W.
+□
+By
+Theorem
+15,
+the
+Morse
+water-
+shed
+may
+be
+obtained
+by
+the
+algorithm
+WatershedCollapse(F, M). In this case we have
+a greedy procedure since the result W does not
+depend on the choice of the free pair of H that is
+made at each iteration.
+In
+fact,
+since
+F
+is
+a
+Morse
+stack,
+we
+can
+simplify
+this
+procedure.
+The
+algorithm
+MorseWatershed(F, M) (Fig. 6) extracts the
+Morse watershed W of a Morse stack F on M ∈
+Md. Also, it gives the catchment basin B of each
+minimum of F.
+The soundness of this algorithm is a direct
+consequence of the above results. It may be imple-
+mented in linear time, with respect to the number
+of incidence relations of M, that is the cardinal of
+the set {(x, y) | x, y ∈ M and x ⊊ y}.
+
+Springer Nature 2021 LATEX template
+Discrete Morse Functions and Watersheds
+11
+MorseWatershed(F, M)
+1) Label all d-faces x ∈ M with the label B(x) = 0;
+Label all faces x ∈ M with the label W(x) = False;
+Label all d-faces x ∈ M of distinct minima of F with distinct labels B(x) > 0;
+2) Insert all d-faces x such that B(x) > 0 in a list L;
+3) Until L is empty, do:
+3.1) Extract a face x from L;
+3.2) For all y such that z = x ∩ y is a (d − 1)-face do:
+3.2.1) If F(y) = F(z) insert y in L and do B(y) := B(x);
+3.2.2) If B(y) > 0 and B(y) ̸= B(x) do W(z) := True;
+4) For all (d − 1)-faces x ∈ M with W(x) = True and all y ⊊ x, do W(y) := True;
+5) For all d-faces x ∈ M and all y ⊊ x with W(y) = False, do B(y) = B(x).
+Fig. 6
+The MorseWatershed(F, M) algorithm computes the Morse watershed W of a Morse stack F defined on a
+normal pseudomanifold M.
+8 Morse watersheds and
+minimum spanning forests
+In [7], an equivalence result which links the notion
+of a watershed in an arbitrary stack with the one
+of a minimum spanning forest is given. In this
+section, we refine this result in the case of Morse
+stacks.
+Recall that we have defined a graph as a com-
+plex X ∈ S such that the dimension of X is at
+most 1.
+Let X ∈ S with dim(X) = 0, that is X is a
+non-empty set of vertices. Let Y be a graph such
+that X ⪯ Y . We say that Y is a forest rooted by
+X if:
+– we have X = Y , or
+– there exists a free pair {x, y} of Y such that
+Y \ {x, y} is a forest rooted by X. If {x, y} is a
+free pair for Y , we say that x is a leaf for Y .
+If X is made of a single vertex, then it may be
+seen that the previous definition is an inductive
+definition, which is equivalent to the notion of a
+rooted tree in the sense of graph theory. If X is
+made of k vertices, then Y has k connected com-
+ponents. Each of these connected components is a
+rooted tree for some vertex of X.
+Let M ∈ Md. The facet graph of M is the
+graph, denoted by ΥM, such that:
+– A vertex {x} is in ΥM if and only if x is a d-face
+of M;
+– An edge {x, y} is in ΥM if and only if x ∩ y is a
+(d − 1)-face of M.
+Let F be a Morse stack on M ∈ M, and let
+X = {{x}
+|
+x ∈ min(F)}. By Prop. 19, each
+{x} ∈ X is a vertex of ΥM. Let Y ⪯ ΥM be a
+forest rooted by X. We say that Y is a spanning
+forest for min(F) if all vertices of ΥM are in Y . We
+define the weight of Y as the sum of all numbers
+F(x ∩ y), where {x, y} is an edge of Y . We say
+that Y is a minimum spanning forest for min(F),
+if Y is a spanning forest for min(F) whose weight
+is minimum.
+Let F be a Morse stack on M ∈ Md. We denote
+by S the set of all couples of d-faces (x, y) in M
+such that (x, x ∩ y) is a differential pair of F, and
+(x∩y, y) is a flat pair of F. Thus, π = ⟨x, x∩y, y⟩
+is a Λd-path in F from x to y.
+The watershed forest of F is the graph G ⪯ ΥM
+such that:
+– All vertices of ΥM are in G;
+– An edge {x, y} is in G if and only if (x, y) or
+(y, x) is a couple in S.
+From Proposition 21, we can check that the
+watershed forest is indeed a forest. More precisely,
+we can derive the following result.
+Proposition 26 If F is a Morse stack on M ∈ Md,
+then the watershed forest of F is a spanning forest for
+min(F).
+Theorem 27 Let F be a Morse stack on M ∈ M. The
+watershed forest of F is a minimum spanning forest
+for min(F). Furthermore, the watershed forest of F is
+the unique minimum spanning forest for min(F).
+
+Springer Nature 2021 LATEX template
+12
+Discrete Morse Functions and Watersheds
+13
+13
+13
+13
+13
+14
+22
+V
+3
+V
+0
+V
+0
+V
+4
+V
+4
+17
+17
+17
+17
+17
+5
+5
+8
+8
+8
+9
+9
+10
+10
+10
+27
+V
+0
+V
+0
+2
+0
+3
+3
+V
+3
+4
+4
+12
+12
+12
+11
+11
+11
+1
+6
+6
+7
+7
+V
+2
+V
+2
+V
+1
+V
+1
+18
+18
+18
+16
+16
+16
+22
+25
+24
+24
+25
+30
+27
+19
+28
+23
+20
+28
+29
+29
+26
+30
+26
+21
+22
+27
+23
+21
+26
+30
+26
+25
+25
+30
+2
+14
+14
+Fig. 7
+The Morse watershed (in red) on a Morse stack F defined on a 2d-torus. In blue, the watershed forest, which is
+the minimum spanning forest for min(F).
+This theorem is illustrated in Fig. 7.
+Proof Let G be a minimum spanning forest for
+min(F). By a minimum spanning tree lemma [28], [29],
+if {x} is a vertex of G, then G must contain an edge
+{x, y} that is a minimum weighted edge containing
+{x}. Now let W be the watershed forest of F and let
+{x, y} be an edge of W. By the very definition of W,
+either (x ∩ y, x) or (x ∩ y, y) is a flat pair. By the defi-
+nition of a Morse stack, if (x ∩ y, x) is a flat pair, then
+{x, y} is the only minimum weighted edge containing
+{x}. Similarly, if (x ∩ y, y) is a flat pair, then {x, y}
+is the only minimum weighted edge containing {y}. It
+follows that, if e is an edge of W, then e is the only
+minimum weighted edge containing some vertex v of
+W. Since v is necessarily in G, we deduce by the above
+lemma that e ∈ G. Therefore, we have W ⊆ G. Since
+both G and W are spanning forests, we must have
+G = W. This shows that W is the unique minimum
+spanning forest for min(F).
+□
+9 Discussion, future work and
+conclusion
+In this paper, we propose, for the first time, a def-
+inition of watersheds for discrete Morse functions,
+and we study its properties. We are working in
+this paper on normal pseudomanifolds. This allows
+us in particular to exhibit a link between water-
+shed and minimum spanning tree, that relies on
+gradient paths.
+While a watershed definition have been pro-
+posed in the continuous Morse setting [30], the
+watershed notion was only a source of inspiration
+in discrete Morse theory. We mention in particular
+the following.
+– A watershed algorithm was used as a prepro-
+cessing in [31] for computing a gradient vector
+
+Springer Nature 2021 LATEX template
+Discrete Morse Functions and Watersheds
+13
+field, but without proof that watershed basins
+were related to any topological notion. Our
+approach directly provides watershed basins
+that are defined with gradient paths.
+– In [32], watershed ideas were used as a moti-
+vation for obtaining Morse cells similar to
+catchment basins, with application to image
+segmentation. Our framework allows clarifying
+the difference between Morse cells and water-
+shed basins. For example, in Fig. 5, the critical
+1-simplex 20 is not part of the watershed cut,
+while it is part of the boundary of the Morse
+cell. We intend to explore in more details those
+differences in future work. We also envision
+studying the Morse-Smale decomposition.
+One
+important
+difference
+between
+Morse
+stacks and discrete Morse functions is that min-
+ima are d-dimensional simplices in our framework,
+while they are 0-dimensional ones in Morse the-
+ory. Although this might appear minor, such a
+difference has important consequences. In partic-
+ular, the watershed is a pure (d − 1)-subcomplex,
+while a similar property is not possible with the
+boundary of Morse cells as classically defined (see
+[32] for example). Our approach allows for easily
+extracting topological features linking two regions,
+following the seminal paper [33]: indeed, we can
+for instance weight any simplex of the water-
+shed cut with the persistence/dynamics [14] at
+which it disappears in a filtering. Such a repre-
+sentation, illustrated in Fig. 8, is called a geodesic
+saliency map in mathematical morphology, and is
+widely used (under the name ultrametric contour
+map [3]) as a post-processing behind deep-learning
+approaches. See [34, 35] for theoretical studies of
+this notion, and [36] for a toolbox implementing
+many variations around it.
+Data analysis heavily relies on data simplifica-
+tion and data visualization. We advocate that the
+watershed, together with filtering operators such
+as morphological dynamics, is a cornerstone for
+data analysis [37]. We aim at controlling the topo-
+logical simplification, and understanding what is
+discarded in the simplification. The results of this
+paper is a first step in this direction. We envi-
+sion using skeleton algorithms such as [38], and
+tools from cross-section topology [39], that, up to
+now, have been used mainly for image analysis.
+Indeed, these tools can be applied to general data.
+In this regard, an important perspective of the
+current paper is to bring together the persistence
+homology framework with the morphological one,
+reaching an audience as large as possible.
+Acknowledgements
+The authors would like to thank both Julien
+Tierny and Thierry G´eraud, for many insightful
+discussions.
+Appendix A
+Normal pseudo-
+manifolds
+A normal pseudomanifold is usually defined as a
+pseudomanifold that satisfies a certain link condi-
+tion, which corresponds to a local property [24],
+[25], [26]. In this section, we show that this defi-
+nition is equivalent to the one given in Definition
+5.
+Let S be a finite set of simplexes. If x and y
+are facets of S, a p-chain (in S) from x to y is a
+sequence ⟨x = x0, ..., xk = y⟩ of facets of S such
+that, for each i ∈ [0, k − 1], xi ∩ xi+1 is a q-face of
+S, with q ≥ p. The set S is p-connected if, for any
+two facets x, y in S, there is a p-chain in S from x
+to y.
+We observe that:
+– A complex is connected if and only if it is 0-
+connected.
+– A d-pure complex is strongly connected if and
+only if it is (d − 1)-connected.
+Let X be a complex. Two faces x, y ∈ X are
+adjacent if x ∪ y ∈ X. The link of x ∈ X in X is
+the complex lk(x, X) = {y ∈ X | x ∩ y = ∅ and
+x ∪ y ∈ X}.
+The star of x ∈ X in X is the set st(x, X) = {y ∈
+X | x ⊆ y}.
+Let X be a d-pseudomanifold. We say that X
+satisfies the link condition if lk(x, X) is connected
+whenever x is a p-face of X and p ≤ d − 2.
+Let X be a complex and x be a p-face of X. Let
+st∗(x, X) = st(x, X) \ {x}. We have lk(x, X) =
+{y \ x | y ∈ st∗(x, X)} and st∗(x, X) = {z ∪ x |
+z ∈ lk(x, X)}.
+We note that there is a set isomorphism between
+lk(x, X) and st∗(x, X), which preserves set inclu-
+sion. If y ∈ st∗(x, X), the corresponding face y \ x
+of lk(x, X) is such that dim(y \ x) = dim(y) −
+
+Springer Nature 2021 LATEX template
+14
+Discrete Morse Functions and Watersheds
+Fig. 8
+Top: an image (left), together with a geodesic saliency map (right) where the darker a contour is, the more
+persistent it is. Bottom: two different views of a triangular mesh, superimposed with a geodesic saliency map where the
+whiter a contour is, the more persistent it is.
+(p + 1). Thus, dim(y \ x) = dim(y) − p+, where
+p+ = p + 1 is the number of elements in x.
+Let X be a d-pseudomanifold and x be a p-face
+of X. Let p+ = p + 1 and d′ = d − p+. The fol-
+lowing facts are a direct consequence of the above
+isomorphism:
+– The complex lk(x, X) is d′-pure.
+– The complex lk(x, X) is non-branching.
+– The set st∗(x, X) is q-connected if and only if
+lk(x, X) is q′-connected, with q′ = q − p+.
+Proposition 28 A pseudomanifold is normal if and
+only if it satisfies the link condition.
+Proof Let X be a d-pseudomanifold.
+1. Suppose X satisfies the link condition and let
+S be a connected open subset of X.
+Let x and y be two d-faces of S. By Remark 1,
+there exists a p-chain π in S from x to y. Thus
+π = ⟨x = x0, ..., xk = y⟩ is a sequence of facets
+of S such that, for each i ∈ [0, k − 1], xi ∩ xi+1
+is a q-face of S, with q ≥ p. We choose π such
+that p is maximal and, if p is maximal, such
+that the number K(π) of p-faces xi∩xi+1, with
+i ∈ [0, k − 1], is minimal. If p = d − 1, it means
+that S is strongly connected; then we are done.
+Suppose p < d − 1 and let xi, xi+1 such that
+z = xi ∩ xi+1 is a p-face.
+Since X satisfies the link condition, lk(z, X)
+is connected. By the isomorphism between
+lk(z, X) and st∗(z, X), it follows there is a q-
+chain ⟨xi = w0, ..., wl = xi+1⟩ in st∗(z, X)
+with q > p. Therefore, π′ = ⟨x = x0, ..., xi =
+w0, ..., wl = xi+1, ..., xk = y⟩ is a p-chain in S
+from x to y. But we have K(π′) < K(π), a con-
+tradiction. Thus, each connected open subset
+of X is strongly connected.
+2. Suppose X
+is strictly connected. That is,
+any connected open subset of X is (d − 1)-
+connected. Let x be a p-face of X with p ≤ d−2.
+The set st(x, X) is a connected open subset of
+X, thus it is (d−1)-connected. Since p < d−1,
+it means that st∗(x, X) is (d − 1)-connected.
+By the isomorphism between lk(x, X) and
+st∗(x, X), it follows that lk(x, X) is strongly
+connected. Thus lk(x, X) is connected.
+□
+In the second part of the proof of Prop. 28, we
+showed that lk(x, X) is strongly connected. Con-
+sequently, we have the following characterization
+of a normal pseudomanifold.
+Proposition 29 A pseudomanifold X is normal if
+and only if, for each p-face x of X, with p ≤ d−2, the
+complex lk(x, X) is a pseudomanifold.
+
+Springer Nature 2021 LATEX template
+Discrete Morse Functions and Watersheds
+15
+Appendix B
+discrete Morse
+fonctions
+Let us consider the following definition of a dis-
+crete Morse function:
+Definition 30 (Morse function) Let X be a complex
+and let F be a map from X to Z. We say that F is
+a discrete Morse function on X if any face of X is
+in at most one covering pair (x, y) in X such that
+F(x) ≥ F(y). If F is a discrete Morse function, we
+say that such a pair is a regular pair of F.
+It may be checked that this definition is equiv-
+alent to the classical one given by Forman (See
+Def. 2.1 and Lemma 2.5 of [40]).
+In this way, the gradient vector field of a dis-
+crete Morse function F, written −−→
+grad(F), is the
+set composed of all regular pairs of F.
+The following restriction of a discrete Morse
+function will lead us to Morse stacks.
+We say that a discrete Morse function F on X
+is flat if we have F(x) = F(y) whenever (x, y) is
+a regular pair of F, that is, if each regular pair of
+F is a flat pair of F.
+We can check that a map F from X to Z is a
+flat discrete Morse function if and only if:
+1. Each covering pair (x, y) in X is such that
+F(x) ≤ F(y);
+2. Each face of X is in at most one flat pair of F.
+Therefore, if we consider the function −F, we
+obtain the following:
+Proposition 31 Let X be a complex and let F be a
+map from X to Z. The map F is a Morse stack on
+X if and only if the map −F is a flat discrete Morse
+function on X.
+The following proposition claims that, up to
+an equivalence, we may assume that any discrete
+Morse function is flat (see Def. 2.27 and Prop. 4.16
+of [10]).
+Proposition 32 (from [10]) If F is a discrete Morse
+function on X, then there exists a flat discrete Morse
+function G on X such that, for every covering pair
+(x, y) in X, we have F(x) ≥ F(y) if and only if
+G(x) ≥ G(y). In other words, the function G is such
+that −−→
+grad(G) = −−→
+grad(F).
+References
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+rithms for the topological watershed. In:
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+Discrete Geometry for Computer Imagery,
+pp. 172–182. Springer, Berlin, Heidelberg
+(2005)
+[28] Cormen, T.H., Leiserson, C.E., Rivest, R.L.:
+Introduction to Algorithms. 23rd printing.
+The MIT Press and McGraw-Hill (1999)
+[29] Motwani, R., Raghavan, P.: Randomized
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+Springer Nature 2021 LATEX template
+Discrete Morse Functions and Watersheds
+17
+Algorithms.
+Cambridge
+university
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+[30] Najman, L., Schmitt, M.: Watershed of a
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+99–112 (1994)
+[31] ˇComi´c, L., De Floriani, L., Iuricich, F., Mag-
+illo, P.: Computing a discrete Morse gradient
+from a watershed decomposition. Computers
+& Graphics 58, 43–52 (2016)
+[32] Delgado-Friedrichs, O., Robins, V., Shep-
+pard, A.: Skeletonization and partitioning of
+digital images using discrete morse theory.
+IEEE transactions on pattern analysis and
+machine intelligence 37(3), 654–666 (2014)
+[33] Najman, L., Schmitt, M.: Geodesic saliency
+of watershed contours and hierarchical seg-
+mentation. IEEE Transactions on Pattern
+Analysis and Machine Intelligence 18(12),
+1163–1173 (1996)
+[34] Najman, L.: On the equivalence between
+hierarchical segmentations and ultrametric
+watersheds. Journal of Mathematical Imag-
+ing and Vision 40(3), 231–247 (2011)
+[35] Cousty,
+J.,
+Najman,
+L.,
+Kenmochi,
+Y.,
+Guimar˜aes, S.: Hierarchical segmentations
+with graphs: quasi-flat zones, minimum span-
+ning trees, and saliency maps. Journal of
+Mathematical Imaging and Vision 60(4),
+479–502 (2018)
+[36] Perret,
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+L.: Higra: Hierarchical graph analysis. Soft-
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+on topology. Journal of Electronic Imaging
+6(4), 395–405 (1997)
+[40] Forman, R.: Morse Theory for cell com-
+plexes. Advances in Mathematics 134, 90–
+145 (1998)
+
diff --git a/TtE2T4oBgHgl3EQfXAej/content/tmp_files/load_file.txt b/TtE2T4oBgHgl3EQfXAej/content/tmp_files/load_file.txt
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@@ -0,0 +1,866 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf,len=865
+page_content='Springer Nature 2021 LATEX template  Discrete Morse Functions and Watersheds  Gilles Bertrand1, Nicolas Boutry2 and Laurent Najman1  1Univ Gustave Eiffel, CNRS, LIGM, F-77454 Marne-la-Vall´ee, France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  2EPITA, Research and Development Laboratory (LRDE), France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  Contributing authors: gilles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='bertrand@esiee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='fr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' nicolas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='boutry@lrde.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='epita.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='fr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  laurent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='najman@esiee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='fr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  Abstract  Any watershed, when defined on a stack on a normal pseudomanifold of dimension d, is a  pure (d − 1)-subcomplex that satisfies a drop-of-water principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' In this paper, we introduce  Morse stacks, a class of functions that are equivalent to discrete Morse functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' We show  that the watershed of a Morse stack on a normal pseudomanifold is uniquely defined, and  can be obtained with a linear-time algorithm relying on a sequence of collapses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Last, but  not the least, we prove that such a watershed is the cut of the unique minimum spanning  forest, rooted in the minima of the Morse stack, of the facet graph of the pseudomanifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  Keywords: Topological Data Analysis, Mathematical Morphology, Discrete Morse Theory, Simplicial Stacks,  Minimum Spanning Forest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  1 Introduction  Watershed is a fundamental tool in computer  vision, since its inception as an algorithm by the  school of mathematical morphology [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' It is  still true in this era of deep learning, where it is  used as a post-processing tool [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' From a discrete,  theoretical point of view, the first topologically-  sound approach was proposed in [4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Building  on those results, in [6–8], it is demonstrated that  watersheds are included in skeletons on pseudo-  manifolds of arbitrary dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  In this paper, we continue exploring the link  between watershed and topology, in the frame-  work of discrete Morse theory [9, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Indeed,  mathematical morphology [11] and discrete Morse  theory, although they pursue different objec-  tives, share many similar ideas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' In particular, as  demonstrated in [12–14], filtering minima using  morphological dynamics [15] in watershed-based  image-segmentation, is equivalent to filtering the  minima by persistence, a fundamental tool from  Persistent Homology [16] used for topological data  analysis [17, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  Although the main ideas of the present paper  originate in [19], we have worked towards a  simpler, unifying framework for exposing these  ideas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' This leads us to introduce Morse stacks:  these functions correspond to the inverse of flat  Witten-Morse functions that, according to R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' For-  man [20], seem to have shown themselves to be  the appropriate combinatorial analogue of smooth  non-degenerate Morse functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' We also propose  a new definition for normal pseudomanifolds, a  class of manifolds on which path-connectivities are  all equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Relying on these notions, we prove  that a watershed, a pure (d − 1)-subcomplex of  a normal pseudomanifold, has several interesting  properties when defined on a Morse stack F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' In  particular, in this setting, a watershed is uniquely  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='03840v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='DM]  10 Jan 2023  Springer Nature 2021 LATEX template  2  Discrete Morse Functions and Watersheds  defined, and can be obtained thanks to a linear-  time algorithm, relying on a sequence of collapses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  Furthermore, a watershed is the cut of the unique  minimum spanning forest of the facet graph of  the normal pseudomanifold weighted by F, rooted  in the minima of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Relations between water-  sheds and Morse theory have long been informally  known [21], but this is the first time that a link is  presented in the discrete setting, relying on a pre-  cise definition of the watershed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Furthermore, as  far as we know, this is the first time that a concept  from Discrete Morse Theory is linked to a classical  combinatorial optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  The plan of this paper is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Section  2 provides some basic definitions of simplicial com-  plexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' We introduce here the notion of a covering  pair, that is fundamental for the definition of  Morse stacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Section 3 recalls some definitions  of simplicial stacks, which are a class of weighted  simplicial complexes whose upper threshold sets  are also complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Section 4 proposes a new  definition for normal pseudomanifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Section 5  provides the necessary definitions for watersheds  on stacks defined on normal pseudomanifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  We propose here an algorithm for computing  watershed relying on the collapse operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' In  section 6, we introduce Morse stacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Section 7  studies the properties of watersheds on Morse  stacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Section 8 links watershed and the mini-  mum spanning forest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' We conclude the paper with  a discussion in section 9, in which we highlight the  importance of our results, both from a theoreti-  cal and a practical point of views, and we propose  some perspective for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  Finally, appendix A shows that our defini-  tion of normal pseudomanifold is equivalent to the  classical one, and appendix B demonstrates that  Morse stacks are equivalent to classical discrete  Morse functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  2 Simplicial complexes  A simplex x is a non-empty finite set;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' the dimen-  sion of x, written dim(x), is the number of its  elements minus one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' We also say that x is a  p-simplex if dim(x) = p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  Let S be a finite set of simplexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' A p-simplex  in S is a (p-)face of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' A (p-)facet of S is a p-face  of S that is maximal for inclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' If x and y are  two distinct faces of S such that x ⊆ y, we say  that x is a face of y (in S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' The simplicial closure  of S is the set S− = {y ⊆ x | y ̸= ∅ and x ∈ S}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  The set S is a (simplicial) complex if S = S−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  Let X be a complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' The dimension of X,  written dim(X), is the largest dimension of its  simplices, the dimension of ∅ being defined to be  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  A 0-face of X is a vertex of X and a 1-face of X  is an edge of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  A complex X is a graph if the dimension of X is  at most 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  Let X be a complex and let S ⊆ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' If S is a  complex, we say that S is closed for X or that S  is a subcomplex of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' We say that S is open for X  or that S is an open subset of X if, for any x ∈ S,  we have y ∈ S whenever x ⊆ y and y ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' If X is  a complex and S ⊆ X, we note that S is closed for  X if and only if X \\S is open for X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' In particular,  ∅ and X are both closed and open for X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  Let S be a finite set of simplexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Let π =  ⟨x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' , xk⟩ be a sequence of elements of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' The  sequence π is a path in S (from x0 to xk) if, for  any i ∈ [0, k−1], either xi ⊆ xi+1 or xi+1 ⊆ xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' We  say that S is connected if, for any x, y ∈ S, there  exists a path from x to y in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' We say that T ⊆ S  is a connected component of S if T is connected  and maximal, with respect to set inclusion, for this  property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  Remark 1 We observe that:  – If X is a complex, then X is connected if and  only if, for any vertices x, y ∈ X, there exists a  sequence ⟨x = x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' , xk = y⟩ of vertices of X  such that, for any i ∈ [0, k − 1], xi ̸= xi+1 and  xi ∪ xi+1 is an edge of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  – If S is an open subset of a complex X, then S  is connected if and only if, for any facets x, y of  S, there exists a sequence ⟨x = x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' , xk = y⟩  of facets of S such that, for any i ∈ [0, k − 1],  xi ∩ xi+1 is in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  The following simple definition of a covering  pair (or a p-pair) will play an important role in  the sequel of the paper:  – It will first allow us to define a free pair (Def-  inition 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' This corresponds to the operation  of collapse of a simplicial complex introduced  by J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Whitehead [22], which is a discrete  analogue of a retraction, that is, a continuous  (homotopic) deformation of an object onto one  Springer Nature 2021 LATEX template  Discrete Morse Functions and Watersheds  3  of its subsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' In Section 3, free pairs for a sim-  plicial complex will be extended to free pairs on  stacks, which are maps on simplicial complexes  (Definition 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  – In Section 6, we introduce the notion of a flat  pair, which is a special case of a covering pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  This permits us to have a very simple and con-  cise presentation of a Morse stack (Definition  16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Indeed, a basic link between Morse stacks  and the collapse operation is straightforward  (Proposition 17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Furthermore, the notions of a  gradient vector field and a gradient path follow  immediately from covering and flat pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  Definition 2 (Covering pair) Let X be a complex and  x, y ∈ X, with dim(y) = p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' We say that (x, y) is a  covering pair of X or a p-pair of X if x is a face of y  and dim(x) = p − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  Definition 3 (Free pair) Let X be a complex and let  (x, y) be a p-pair of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' We say that (x, y) is a free  (p-)pair of X if y is the only face of X that contains x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  Thus, if (x, y) is a free pair of X, we have nec-  essarily dim(x) = dim(y) − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Furthermore, we  observe that y is necessarily a facet of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  If (x, y) is a free p-pair of a complex X, then  Y = X \\{x, y} is an elementary (p-)collapse of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  We say that X collapses (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' p-collapses) onto  Y , if there exists a sequence ⟨X = X0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=', Xk =  Y ⟩ such that Xi is an elementary collapse (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  elementary p-collapse) of Xi−1, i ∈ [1, k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' If, fur-  thermore, Y has no free pair (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' free p-pair),  then Y is an ultimate collapse (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' ultimate p-  collapse) of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' A complex X is collapsible if X  collapses onto a single vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  3 Simplicial stacks  Let X be a simplicial complex, and let F be a map  from X to Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' If x is a face of X, the value F(x)  is called the altitude of F at x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' For any λ ∈ Z, we  write F[λ] = {x ∈ X | F(x) ≥ λ}, F[λ] is the λ-  section of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' We say that F is a (simplicial) stack  on X if any λ-section of F is a simplicial complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  In other words, any λ-section of F is a closed set  for X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  Let F be a map from a complex X to Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' It may  be easily seen that F is a simplicial stack if and  only if, for any x, y ∈ X such that x ⊆ y, we have  F(x) ≥ F(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Also, a map F is a simplicial stack  if and only if, for any covering pair (x, y) in X, we  have F(x) ≥ F(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  Now, we extend the notion of free pairs of sim-  plicial complexes to simplicial stacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' This exten-  sion allows us to introduce some fundamental  discrete homotopic transforms of these maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  Definition 4 Let F be a simplicial stack on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' We  set λm = min{F(x) | x ∈ X}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Let (x, y) be a p-pair  of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' We say that (x, y) is a free (p-)pair of F if (x, y)  is a free (p-)pair of F[λ], with λ = F(x) and λ > λm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  If (x, y) is a free pair of F, then both x and y  are in F[λ], with λ = F(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Thus F(y) ≥ F(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  Also, we have x ⊆ y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Then, since F is a stack, we  have F(y) ≤ F(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Thus, we have F(x) = F(y)  whenever (x, y) is a free pair of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  Let F be a simplicial stack on a complex X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Let (x, y) be a free (p-)pair of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Let G be the  map such that G(z) = F(z) − 1 if z = x or z =  y, and G(z) = F(z) if z ∈ X\\{x, y}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' We can see  that G is a simplicial stack on X, the map G is  called an elementary (p-)collapse of F through  (x, y) or simply an elementary (p-)collapse of  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' If G is the result of a sequence of elementary  collapses (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' p-collapses) of F, then we say  that F collapses (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' p-collapses) onto G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' If F collapses (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' p-collapses) onto a stack  G that has no free pair (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' no free p-pair),  then G is an ultimate collapse (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' ultimate  p-collapse) of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  We conclude this section by giving a definition  of a (regional) minimum of a stack, which plays a  crucial role for the notion of a watershed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  Let F be a simplicial stack on a complex X  and let λ ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' A subset A of X is a minimum of  F (at altitude λ) if A is a connected component  of X \\ F[λ + 1] and A ∩ (X \\ F[λ]) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' The divide  of F is the set composed of all faces of X that are  not in a minimum of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Note that any minimum  of F is an open set for X, and the divide of F is  a simplicial complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  4 Normal pseudomanifolds  The results of this paper hold true in a large fam-  ily of n-dimensional discrete spaces, namely the  Springer Nature 2021 LATEX template  4  Discrete Morse Functions and Watersheds  normal pseudomanifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' This section provides a  presentation of these spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  Let S be a finite set of simplexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' A  strong  p-path in S (from x0 to xk) is a path ⟨x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=', xk⟩  such that, for each i ∈ [0, k−1], either (xi, xi+1) is  a p-pair, or (xi+1, xi) is a p-pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' The set S is (d-  )pure if all facets of S have the same dimension d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  If S is d-pure, we say that a strong d-path in S is  a strong path in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Also, we say that S is strongly  connected if, for any two facets x, y in S, there  exists a strong path in S from x to y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' A subset T  of S is a strong connected component of S if T is  strongly connected and maximal, with respect to  set inclusion, for this property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  Definition 5 (Normal pseudomanifold) A connected  and d-pure complex X, with d ≥ 1, is a normal  pseudomanifold (or a normal d-pseudomanifold) if:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' The complex X is non-branching, that is, each  (d − 1)−face of X is included in exactly two  d-faces of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' The complex X is strictly connected, that is,  each connected open subset of X is strongly  connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  Recall that a pure complex X is a pseudoman-  ifold if it is non-branching and strongly connected  [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Since the very set X is open for a complex X,  we see that any normal pseudomanifold is a pseu-  domanifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  In fact, the above definition is a new definition  for a normal pseudomanifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' In Appendix A, we  show that it is equivalent to the classical definition  [24], [25], [26], which consists of a local condition  together with the conditions that must be satisfied  by a pseudomanifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  Let us consider Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' The triangulated torus  (a) is a normal pseudomanifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' The triangulated  pinched torus (b) is a pseudomanifold that is not  normal: the set of all faces containing the pinch  vertex is a connected open subset of the com-  plex, but this set is not strongly connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' The  triangulated pinched torus (c) is not a pseudo-  manifold: the pinch segment does not satisfy the  non-branching condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  Let X  be a proper subcomplex of a d-  pseudomanifold M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' We can see that, if dim(X) =  d, the complex X has necessarily a boundary, that  is, there exists a free d-pair for X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' By induction,  it means that the dimension of an ultimate d-  collapse of X is necessarily d − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' See [7] for a  formal proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  Important notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' In the sequel of the  paper:  – We denote by S the collection of all simplicial  complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  – If X ∈ S, we write Y ⪯ X whenever Y ⊆ X and  Y ∈ S, that is, whenever Y is a subcomplex of  X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  – If X ∈ S, and S ⊆ X, we write S ⊑ X whenever  S is an open subset of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  – We denote by M (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Md) the collection of  all normal pseudomanifolds (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' all normal d-  pseudomanifolds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  – If F is a stack on M ∈ M, the notation min(F)  stands for the union of all minima of F, and  we write div(F) for the divide of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Thus, we  have div(F) = M \\ min(F), div(F) ⪯ M, and  min(F) ⊑ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  We are now ready to introduce the notion of a  watershed in the context of simplicial complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  We will consider a normal pseudomanifold M ∈ M  and a simplicial stack F on M, the map F may be  seen as a “topographical relief” on the space M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  A simplicial complex W ⪯ M may be a watershed  of F if W “separates the minima of F”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' It means  that if A ⊑ M is a connected component of M\\W,  then A contains one and only one set B ⊑ M  that is a minimum of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Furthermore, the com-  plex W must satisfy a “drop of water principle”:  from each face of W, we may reach two distinct  minima of F by following a descending path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Each  connected component A of M \\W will correspond  to a “catchment basin” of the map F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  5 Watersheds  Let X ∈ S, and let A ⊑ X, with A ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' We say  that B ⊑ X is an extension of A if A ⊆ B, and if  each connected component of B includes exactly  one connected component of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' We also say that  B is an extension of A if A = B = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  Proposition 6 Let M ∈ M and let A ⊑ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' A subset S of A is a connected component of A  if and only if S is a strong connected component  of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  Springer Nature 2021 LATEX template  Discrete Morse Functions and Watersheds  5  (a)  (b)  (c)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' 1  (a): A normal pseudomanifold, which is a torus, (b): A pseudomanifold, which is a pinched torus, and where the  pinch face is a vertex, (c): A pinched torus where the pinch face is a segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' This is not a pseudomanifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Let B ⊑ M, with A ⊆ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' The set B is an  extension of A if and only if each strong con-  nected component of B includes exactly one  strong connected component of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  Proof 1) It may be seen that, if S is a connected com-  ponent of the open set A, then S is necessarily an  open set for M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Since M is a normal pseudomanifold,  we deduce that S is strongly connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Furthermore,  S is a strong connected component of A, otherwise S  would not be a maximal connected subset of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Now,  if S is a strong connected component of A, then S is  a connected subset of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Again, we see that S is a  connected component of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Otherwise, S would be a  proper subset of a connected open subset T of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Since  M is a normal pseudomanifold, this subset T would  be strongly connected, and S would not be a maximal  strongly connected subset of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  2) is a direct consequence of 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  □  Let X ∈ S and Y ⪯ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Let A ⊑ X, with  A ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' We say that Y is a cut for A, if X \\ Y  is an extension of A, and if Y is minimal for this  property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' That is, if Z ⪯ Y , and if X \\ Z is an  extension of A, then we have necessarily Z = Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  Proposition 7 (from [7]) Let M ∈ M, A ⊑ M and  X ⪯ M, with A ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' If X is a cut for A, then the  complex X is either empty or a pure (n − 1)-complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  Remark 8 It could be seen that the previous result  no longer holds if we consider arbitrary pseudomani-  folds instead of normal pseudomanifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' For example,  the pinched vertex of the pinched torus of Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' (b)  could be in a cut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  In fact, it is possible to bypass this situation by con-  sidering only strong paths between faces, as it is done  in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' In this paper, in order to handle general con-  nectedness and arbitrary paths, we have made the  choice to settle our results in normal pseudomanifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  Let F be a stack on M ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' If π = ⟨x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' , xk⟩  is a path in M, we say that π is ascending for F  (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' descending for F) if, for any i ∈ [0, k], we  have F(xi) ≤ F(xi+1) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' F(xi) ≥ F(xi+1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  Definition 9 (Cut) Let F be a stack on M ∈ M and  let X ⪯ M be a cut for min(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' We say that X is  a watershed of F if, for each x ∈ X, there exist two  strong paths π1 = ⟨x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' , xk⟩ and π2 = ⟨y0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' , yl⟩  in M \\ X, such that:  – x ⊆ x0 and x ⊆ y0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  – π1 and π2 are descending paths for F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' and  – xk and yl are simplices of two distinct minima  of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  Let M ∈ M, F be a stack on M, and let W be  a watershed for F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' We say that B ⊑ M is a (catch-  ment) basin of W if B is a connected component  of W = M \\ W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Since W is a cut for min(F),  – any catchment basin B of W contains a unique  minimum A of F, we say that A is the minimum  of B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  – any minimum A of F is included in a unique  basin B of W, we say that B is the catchment  basin of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  Proposition 10 (from [7]) Let M ∈ Md, F be a stack  on M, and W be a watershed of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Then, for any d-  face x in M, there exists a strong path in M \\ W from  x to a d-face of a minimum of F, that is descending  for F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  From the previous result, we easily derive the  following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  Proposition 11 Let M ∈ M, F be a stack on M, and  W be a watershed of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Let B be the catchment basin  of a minimum A of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Then, for any x ∈ B, there  exists a descending path in B from x to a face of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  The two following results are crucial for linking  a watershed of a stack F and the homotopy of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  Springer Nature 2021 LATEX template  6  Discrete Morse Functions and Watersheds  Proposition 12 (from [7]) Let M ∈ Md.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' If F is a  stack on M and H is a collapse of F, then:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' min(H) is an extension of min(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' div(H) is a collapse of div(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  It should be noted that the previous prop-  erty is no longer true if we consider a stack on  an arbitrary complex X ∈ S rather than a com-  plex M ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' See Fig 2 which provides a simple  counter-example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  Proposition 13 (from [7]) Let M ∈ Md and F be a  stack on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' F contains a free d-pair if and only if div(F)  contains a free d-pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' If dim(div(F)) = d, then there exists a free d-  pair for F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  Let F be a stack on M ∈ Md and x be a (d−1)-  face of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Let y, z be the two d-faces containing  x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' We say that x is (locally) separating for F if  F(y) < F(x) and F(z) < F(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' We say that x  is biconnected for F if y and z belong to distinct  minima of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  Definition 14 Let F be a stack on M ∈ Md.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Let  X ⪯ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' We say that X is a cut by collapse of F,  or a C-watershed of F, if there exists an ultimate d-  collapse H of F such that X is the simplicial closure  of the set of all faces of M that are biconnected for H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  Theorem 15 (from [7]) Let M ∈ M and let F be a  stack on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' A complex X ⪯ M is a watershed of F if  and only if X is a C-watershed of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  Theorem 15 is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  From Theorem 15 we may derive the procedure  WatershedCollapse(F, M) (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' 4) for obtaining  a watershed of a stack F on M ∈ Md.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  The result W depends on the choices of the  free pairs that are made at step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' In any case,  any watershed of F may be obtained by this  procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  As explained hereafter, a direct implementa-  tion of the algorithm WatershedCollapse(F, M)  can be slow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  – Step 1 is the more complex one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' A naive imple-  mentation of this step is in the order of n2 ∗ h,  where n is the number of d-faces, and h is the  number of different altitudes of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' However, this  step can be done in quasi-linear time, relying  on a straightforward adaptation to simplicial  complexes of the algorithm presented in [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  This algorithm relies on a tree structure, where  the nodes of the tree are the connected com-  ponents of all the level sets of F, and where  the edges of the tree correspond to the par-  enthood relationships between those connected  components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  – Step 2 is a simple labelling, and may be done  in linear time with respect to the number of d-  faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' By using such a labelling, checking if a  d-face is biconnected can be done in constant  time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  – Finally, step 3 may be implemented in linear  time, with respect to the number of incidence  relations of M, that is the cardinal of the set  {(x, y) | x, y ∈ M and x ⊊ y}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  6 Morse stacks  In this section, we transpose some basic notions  of discrete Morse theory to stacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' We proceed by  defining a Morse stack, which is the counterpart of  a classical discrete Morse function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Morse stacks  simply correspond to the inverse of flat discrete  Morse functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' See Appendix B, which provides  the few facts linking these two notions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  Let F be a map from a complex X to Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' We  say that a covering pair (x, y) of X is a flat pair  of F whenever we have F(x) = F(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  Definition 16 (Morse stack) Let F be a simplicial  stack on a complex X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' We say that F is a Morse stack  (on X) if any face of X is in at most one flat pair  of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  Let F be an arbitrary simplicial stack, and let  (x, y) be a covering pair of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' We have seen that, if  (x, y) is a free pair of F, then necessarily (x, y) is a  flat pair of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Suppose now that (x, y) is a flat pair  of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Then, there may exist another covering pair  (x, z) that is also a flat pair of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' In this case, we  see that (x, y) is not a free pair of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' By the very  definition of a Morse stack, this situation cannot  occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' In fact, we have the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  Springer Nature 2021 LATEX template  Discrete Morse Functions and Watersheds  7  F  H  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' 2  Two stacks F and H with two levels: altitude 0 (faces in light grey) and altitude 1 (faces in black).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' The stack H  is an elementary collapse of F (at altitude 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' But F has three minima whereas H has only two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Thus, min(H) is not an  extension of min(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  (a)  (b)  (c)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' 3  (a) A simplicial stack F on a subset of a normal 2-pseudomanifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' (b) An ultimate 2-collapse of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' (c) A watershed  of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' The pair (y, x) in (a) is a free-pair for F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  Proposition 17 Let F be a Morse stack on a complex  X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' A covering pair (x, y) of X is a free pair of F if  and only if (x, y) is a flat pair of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  Definition 18 (Regular and critical simplex) Let F  be a Morse stack on a complex X and let x ∈ X with  dim(x) = p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  – We say that x is regular or p-regular for F if x  is in a flat pair of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  – We say that x is critical or p-critical for F if x  is not regular for F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  A Morse stack defined on a normal pseudo-  manifold, together with its critical and regular  simplexes, can be seen on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  Let F be a Morse stack on a complex X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  The gradient vector field of F, written −−→  grad(F), is  the set of all flat pairs of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  If (x, y) is a covering pair of F such that F(x) >  F(y), we say that (y, x) is a differential pair of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  We write −→  diffF for the set of all differential pairs  for F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  We also set:  y  X  20  0-03203303  03  33  03  3-3  03-0-03-3  303202203  000030033  033Springer Nature 2021 LATEX template  8  Discrete Morse Functions and Watersheds  WatershedCollapse(F, M)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Set H = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Until H has no free d-pair, select arbitrarily a free d-pair (x, y) of H  and replace H by the elementary collapse of H through (x, y);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Label all d-faces of distinct minima of H with distinct labels;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Extract from H the complex W that is the simplicial closure of the set of all  d-faces of M which are biconnected for H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' 4 The procedure WatershedCollapse(F, M) computes a watershed W of a stack F defined on a normal  pseudomanifold M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  – −−→  gradp(F) = {(x, y) ∈ −−→  grad(F) | dim(y) = p},  and  – −→  diffp(F) = {(y, x) ∈ −→  diff(F) | dim(y) = p}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  A Λp-path in F (from x0 to xk) is a sequence  π = ⟨x0, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=', xk⟩ composed of faces of X such  that, for all i ∈ [0, k − 1], the pair (xi, xi+1) is  either in −−→  gradp(F) or in −→  diffp(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' A sequence π is  a gradient path for F if π is a Λp-path for F for  some p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  Let π = ⟨x0, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=', xk⟩ be a Λp-path in F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' We  observe that:  – For any i ∈ [1, k − 1], the pair (xi, xi+1) is in  −−→  gradp(F) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' −→  diffp(F)) whenever (xi−1, xi) is  in −→  diffp(F) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' −−→  gradp(F)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  – Each face of π is either a p-face or a (p−1)-face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  For any i ∈ [0, k − 1], if xi is a p-face, then xi+1  is a (p − 1)-face, and if xi is a (p − 1)-face, then  xi+1 is a p-face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  – The path π is a strong p-path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  – If π is not trivial, then π cannot be closed, that  is, we have necessarily k = 0 whenever xk = x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  – The path π is an ascending path, that is, we  have F(xi) ≤ F(xi+1) for any i ∈ [0, k − 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  Furthermore, we have F(xi) < F(xi+2) for any  i ∈ [0, k − 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  The following result is a basic fact about Morse  functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  Proposition 19 Let F be a Morse stack on a complex  X ∈ S, and let S be a subset of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' If S is a minimum  of F, then S is composed of a single facet of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  In the sequel, we will say that a face x ∈ X  is a minimum (of F) whenever the set {x} is a  minimum of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  7 Morse stacks and  watersheds  Let F be a Morse stack on a complex X ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Let  x, y be two faces of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' We say that x is Λp-linked  to y if there is a Λp-path in F from x to y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Let  π = ⟨x = x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=', xk = y⟩ be a Λp-path in F from x  to y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' We write ˜π = ⟨y = xk, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=', x0 = x⟩ and we say  that ˜π is a ˜Λp-path in F from y to x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' We say that a  face z ∈ X is an extension of π if ⟨x = x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=', xk =  y, z⟩ is a Λp-path in F from x to z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' We say that z  is an extension of ˜π if ⟨y = xk, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=', x0 = x, z⟩ is a  ˜Λp-path in F from y to z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  Proposition 20 Let F be a Morse stack on M ∈ Md,  and let x be a facet of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Let ˜π be a ˜Λd-path in F from  the facet x to a face y ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Then one and only one of  the following statements is true:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' The face y is a minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' There exists a unique face that is an extension  of ˜π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  Proof We set ˜π = ⟨x = x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=', xk = y⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  i) Suppose y is a (d − 1)-face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' In this case, y cannot  be a minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Furthermore, we have k ≥ 1 (since x  is a d-face), and the face t = xk−1 is necessarily a d-  face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' By the definition of a ˜Λp-path, the pair (y, t) is a  flat pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Since X is a pseudomanifold, there exists a  unique d-face z ∈ X such that t∩z = y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' We must have  F(z) < F(y), otherwise y would belong to more than  one flat pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Therefore, the pair (z, y) is a differential  pair and z is an extension of ˜π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Since (y, t) and (y, z)  are the only covering pairs that contain y, the face z  is the unique extension of ˜π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  ii) Suppose y is a d-face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' By the definition of a ˜Λp-  path, a face z is an extension of ˜π if and only if (z, y)  is a flat pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Now we observe that y is not a minimum  if and only if there is a face z such that (z, y) is a flat  pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' But y belongs to at most one flat pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Thus, ˜π  has a unique extension whenever y is not a minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  □  Springer Nature 2021 LATEX template  Discrete Morse Functions and Watersheds  9  Let F be a Morse stack on M ∈ Md.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' It should  be noted that, if π is a Λd-path in F from a facet  x to a face y, then π may have more than one  extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Nevertheless, by induction, we obtain  the following result from Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  Proposition 21 Let F be a Morse stack on M ∈ Md,  and let x be a facet of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' There exists a unique  minimum m of F such that m is Λd-linked to x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Fur-  thermore, there exists a unique Λd-path in F from m  to x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  We now consider the case where a Λd-path has  no extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Recall that a (d − 1)-face x is sepa-  rating for F if the two d-faces y, z which contain  x, are such that F(y) < F(x) and F(z) < F(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  Proposition 22 Let F be a Morse stack on M ∈ Md.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  Let x be a facet of X, and let π be a Λd-path in F  from x to a face y ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' If π has no extension, then  the face y is necessarily separating for F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  Proof Let π = ⟨x = x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=', xk = y⟩ be a Λd-path in F  from x to y ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  If dim(y) = d, then π has necessarily an extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  Now suppose dim(y) = d − 1, thus k ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' The face y  is a face of two d-faces, the face z = xk−1 and another  face t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Since π has no extension, we have F(t) < F(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  Furthermore, by the very definition of a Λd-path, the  pair (z, y) is a differential pair, thus we have F(z) <  F(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Therefore, y is separating for F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  □  Let F be a Morse stack on a complex X ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  Let π be a Λp-path in F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' We say that π is maximal  if neither π nor ˜π has an extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' The following  result is a direct consequence of Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' 20 and 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  Corollary 23 Let F be a Morse stack on M ∈ Md.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  Let π be a Λd-path in F from x to y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' If π is maximal,  then x is a minimum of F and y is a separating face  for F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  Let F be a Morse stack on M ∈ Md.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Let x be  a (d − 1)-face of X, and let y, z be the two dis-  tinct d-faces containing x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' According to Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' 21,  each of these faces is Λd-linked to a single mini-  mum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' We say that the face x is Λ-biconnected (for  F) if these two minima are distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Observe that  a face is necessarily separating whenever it is  Λ-biconnected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  Definition 24 (Morse watershed) Let F be a Morse  stack on M  ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' The Morse watershed of F is  the complex that is the simplicial closure of the set  composed of all faces that are Λ-biconnected for F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  Theorem 25 Let F be a Morse stack on M ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  The Morse watershed of F is a watershed of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Fur-  thermore, the Morse watershed of F is the unique  watershed of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' 25 is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  Proof Let W be the Morse watershed of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Let A be a connected component of M \\W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' By  Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' 6, the set A is a strong connected com-  ponent of M \\ W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Let f be a facet of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' By  Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' 21, there exists a Λd-path π from a min-  imum m of F to f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' By the very definition of a  Λd-path, it may be seen that π does not con-  tain any separating face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Thus π is included in  M \\ W, and m is in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Therefore, A contains a  minimum m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  Now let x be an arbitrary facet of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Since A is  strongly connected, there exists a strong path  π = ⟨m = x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=', xk = x⟩ in A from m to x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' By  Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' 21, for each facet of π, there is a unique  minimum of F that is Λd-linked to this facet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  Let xi be a facet of π, with i ≤ k − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Thus  xi+1 is a (d−1)-face and xi+2 is a facet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Let mi  (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' mi+2) be the unique minimum of F that  is Λd-linked to xi (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' to xi+2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Since xi+1  is not Λ-biconnected for F, it may be checked  that we have necessarily mi = mi+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' By induc-  tion, it follows that m is Λd-linked to the facet  x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Since this result holds for any facet of A, this  clearly implies that m is the unique minimum  of F which is in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Thus, any connected com-  ponent of M \\W contains exactly one minimum  of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' But any minimum of F is included in  M \\W (since a minimum is a d-face).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' It follows  that M \\W is an extension of min(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Further-  more, by the definition of a Λ-biconnected face,  W is minimal for this last property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Therefore,  W is a cut for min(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Since any ˜Λd-path is a  descending strong path, it may be checked that  W fulfills all the conditions of Definition 9: W  is a watershed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Let W ′ be a watershed of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Let x be a (d−1)-  face of M that is Λ-biconnected for F, and let  Springer Nature 2021 LATEX template  10  Discrete Morse Functions and Watersheds  13  13  13  14  14  22  V  3  V  0  V  0  V  4  V  4  17  17  17  5  5  8  8  8  9  9  10  10  27  V  0  V  0  2  2  0  3  3  V  3  4  4  12  12  11  11  1  6  6  7  7  V  2  V  2  V  1  V  1  18  18  16  16  22  25  24  24  25  30  27  19  28  23  20  28  29  29  26  30  26  21  22  27  23  21  26  30  26  25  25  30  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' 5  The Morse watershed (in red) on a Morse stack defined on a normal pseudomanifold, a 2d-torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' In black, critical  simplexes: those of dimension 2 are minima, those of dimension 1 are saddle, and those of dimension 0 are maxima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Arrows  represent gradient/collapse of dimension 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' We observe that the critical simplex 20 do not belong to the watershed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  y, z be the two distinct d-faces containing x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' By  Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' 10 there exist a descending strong path in  M \\W ′ from y to a minimum m and a descend-  ing strong path in M \\W ′ from z to a minimum  m′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' We observe that any descending strong  path in M is also a ˜Λd-path in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Thus, by the  very definition of a Λ-biconnected face, we must  have m ̸= m′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Therefore, the face x must be in  W ′, otherwise y, z, m, and m′, would belong to  the same connected component of M\\W ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Thus  W ⊆ W ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Since M \\ W ′ ⊆ M \\ W, each con-  nected component of M \\W ′ is included in one  connected component of M \\ W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' But M \\ W ′  must be maximal for this last property, oth-  erwise W ′ would not be a cut for min(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' It  follows that we have W ′ = W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  □  By  Theorem  15,  the  Morse  water-  shed  may  be  obtained  by  the  algorithm  WatershedCollapse(F, M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' In this case we have  a greedy procedure since the result W does not  depend on the choice of the free pair of H that is  made at each iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  In  fact,  since  F  is  a  Morse  stack,  we  can  simplify  this  procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  The  algorithm  MorseWatershed(F, M) (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' 6) extracts the  Morse watershed W of a Morse stack F on M ∈  Md.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Also, it gives the catchment basin B of each  minimum of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  The soundness of this algorithm is a direct  consequence of the above results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' It may be imple-  mented in linear time, with respect to the number  of incidence relations of M, that is the cardinal of  the set {(x, y) | x, y ∈ M and x ⊊ y}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  Springer Nature 2021 LATEX template  Discrete Morse Functions and Watersheds  11  MorseWatershed(F, M)  1) Label all d-faces x ∈ M with the label B(x) = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  Label all faces x ∈ M with the label W(x) = False;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  Label all d-faces x ∈ M of distinct minima of F with distinct labels B(x) > 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  2) Insert all d-faces x such that B(x) > 0 in a list L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  3) Until L is empty, do:  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='1) Extract a face x from L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='2) For all y such that z = x ∩ y is a (d − 1)-face do:  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='1) If F(y) = F(z) insert y in L and do B(y) := B(x);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='2) If B(y) > 0 and B(y) ̸= B(x) do W(z) := True;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  4) For all (d − 1)-faces x ∈ M with W(x) = True and all y ⊊ x, do W(y) := True;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  5) For all d-faces x ∈ M and all y ⊊ x with W(y) = False, do B(y) = B(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' 6  The MorseWatershed(F, M) algorithm computes the Morse watershed W of a Morse stack F defined on a  normal pseudomanifold M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  8 Morse watersheds and  minimum spanning forests  In [7], an equivalence result which links the notion  of a watershed in an arbitrary stack with the one  of a minimum spanning forest is given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' In this  section, we refine this result in the case of Morse  stacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  Recall that we have defined a graph as a com-  plex X ∈ S such that the dimension of X is at  most 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  Let X ∈ S with dim(X) = 0, that is X is a  non-empty set of vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Let Y be a graph such  that X ⪯ Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' We say that Y is a forest rooted by  X if:  – we have X = Y , or  – there exists a free pair {x, y} of Y such that  Y \\ {x, y} is a forest rooted by X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' If {x, y} is a  free pair for Y , we say that x is a leaf for Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  If X is made of a single vertex, then it may be  seen that the previous definition is an inductive  definition, which is equivalent to the notion of a  rooted tree in the sense of graph theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' If X is  made of k vertices, then Y has k connected com-  ponents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Each of these connected components is a  rooted tree for some vertex of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  Let M ∈ Md.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' The facet graph of M is the  graph, denoted by ΥM, such that:  – A vertex {x} is in ΥM if and only if x is a d-face  of M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  – An edge {x, y} is in ΥM if and only if x ∩ y is a  (d − 1)-face of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  Let F be a Morse stack on M ∈ M, and let  X = {{x}  |  x ∈ min(F)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' By Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' 19, each  {x} ∈ X is a vertex of ΥM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Let Y ⪯ ΥM be a  forest rooted by X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' We say that Y is a spanning  forest for min(F) if all vertices of ΥM are in Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' We  define the weight of Y as the sum of all numbers  F(x ∩ y), where {x, y} is an edge of Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' We say  that Y is a minimum spanning forest for min(F),  if Y is a spanning forest for min(F) whose weight  is minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  Let F be a Morse stack on M ∈ Md.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' We denote  by S the set of all couples of d-faces (x, y) in M  such that (x, x ∩ y) is a differential pair of F, and  (x∩y, y) is a flat pair of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Thus, π = ⟨x, x∩y, y⟩  is a Λd-path in F from x to y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  The watershed forest of F is the graph G ⪯ ΥM  such that:  – All vertices of ΥM are in G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  – An edge {x, y} is in G if and only if (x, y) or  (y, x) is a couple in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  From Proposition 21, we can check that the  watershed forest is indeed a forest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' More precisely,  we can derive the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  Proposition 26 If F is a Morse stack on M ∈ Md,  then the watershed forest of F is a spanning forest for  min(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  Theorem 27 Let F be a Morse stack on M ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' The  watershed forest of F is a minimum spanning forest  for min(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Furthermore, the watershed forest of F is  the unique minimum spanning forest for min(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  Springer Nature 2021 LATEX template  12  Discrete Morse Functions and Watersheds  13  13  13  13  13  14  22  V  3  V  0  V  0  V  4  V  4  17  17  17  17  17  5  5  8  8  8  9  9  10  10  10  27  V  0  V  0  2  0  3  3  V  3  4  4  12  12  12  11  11  11  1  6  6  7  7  V  2  V  2  V  1  V  1  18  18  18  16  16  16  22  25  24  24  25  30  27  19  28  23  20  28  29  29  26  30  26  21  22  27  23  21  26  30  26  25  25  30  2  14  14  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' 7  The Morse watershed (in red) on a Morse stack F defined on a 2d-torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' In blue, the watershed forest, which is  the minimum spanning forest for min(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  This theorem is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  Proof Let G be a minimum spanning forest for  min(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' By a minimum spanning tree lemma [28], [29],  if {x} is a vertex of G, then G must contain an edge  {x, y} that is a minimum weighted edge containing  {x}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Now let W be the watershed forest of F and let  {x, y} be an edge of W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' By the very definition of W,  either (x ∩ y, x) or (x ∩ y, y) is a flat pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' By the defi-  nition of a Morse stack, if (x ∩ y, x) is a flat pair, then  {x, y} is the only minimum weighted edge containing  {x}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Similarly, if (x ∩ y, y) is a flat pair, then {x, y}  is the only minimum weighted edge containing {y}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' It  follows that, if e is an edge of W, then e is the only  minimum weighted edge containing some vertex v of  W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Since v is necessarily in G, we deduce by the above  lemma that e ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Therefore, we have W ⊆ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Since  both G and W are spanning forests, we must have  G = W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' This shows that W is the unique minimum  spanning forest for min(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  □  9 Discussion, future work and  conclusion  In this paper, we propose, for the first time, a def-  inition of watersheds for discrete Morse functions,  and we study its properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' We are working in  this paper on normal pseudomanifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' This allows  us in particular to exhibit a link between water-  shed and minimum spanning tree, that relies on  gradient paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  While a watershed definition have been pro-  posed in the continuous Morse setting [30], the  watershed notion was only a source of inspiration  in discrete Morse theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' We mention in particular  the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  – A watershed algorithm was used as a prepro-  cessing in [31] for computing a gradient vector  Springer Nature 2021 LATEX template  Discrete Morse Functions and Watersheds  13  field, but without proof that watershed basins  were related to any topological notion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Our  approach directly provides watershed basins  that are defined with gradient paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  – In [32], watershed ideas were used as a moti-  vation for obtaining Morse cells similar to  catchment basins, with application to image  segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Our framework allows clarifying  the difference between Morse cells and water-  shed basins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' For example, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' 5, the critical  1-simplex 20 is not part of the watershed cut,  while it is part of the boundary of the Morse  cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' We intend to explore in more details those  differences in future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' We also envision  studying the Morse-Smale decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  One  important  difference  between  Morse  stacks and discrete Morse functions is that min-  ima are d-dimensional simplices in our framework,  while they are 0-dimensional ones in Morse the-  ory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Although this might appear minor, such a  difference has important consequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' In partic-  ular, the watershed is a pure (d − 1)-subcomplex,  while a similar property is not possible with the  boundary of Morse cells as classically defined (see  [32] for example).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Our approach allows for easily  extracting topological features linking two regions,  following the seminal paper [33]: indeed, we can  for instance weight any simplex of the water-  shed cut with the persistence/dynamics [14] at  which it disappears in a filtering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Such a repre-  sentation, illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' 8, is called a geodesic  saliency map in mathematical morphology, and is  widely used (under the name ultrametric contour  map [3]) as a post-processing behind deep-learning  approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' See [34, 35] for theoretical studies of  this notion, and [36] for a toolbox implementing  many variations around it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  Data analysis heavily relies on data simplifica-  tion and data visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' We advocate that the  watershed, together with filtering operators such  as morphological dynamics, is a cornerstone for  data analysis [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' We aim at controlling the topo-  logical simplification, and understanding what is  discarded in the simplification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' The results of this  paper is a first step in this direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' We envi-  sion using skeleton algorithms such as [38], and  tools from cross-section topology [39], that, up to  now, have been used mainly for image analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  Indeed, these tools can be applied to general data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  In this regard, an important perspective of the  current paper is to bring together the persistence  homology framework with the morphological one,  reaching an audience as large as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  Acknowledgements  The authors would like to thank both Julien  Tierny and Thierry G´eraud, for many insightful  discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  Appendix A  Normal pseudo-  manifolds  A normal pseudomanifold is usually defined as a  pseudomanifold that satisfies a certain link condi-  tion, which corresponds to a local property [24],  [25], [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' In this section, we show that this defi-  nition is equivalent to the one given in Definition  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  Let S be a finite set of simplexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' If x and y  are facets of S, a p-chain (in S) from x to y is a  sequence ⟨x = x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=', xk = y⟩ of facets of S such  that, for each i ∈ [0, k − 1], xi ∩ xi+1 is a q-face of  S, with q ≥ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' The set S is p-connected if, for any  two facets x, y in S, there is a p-chain in S from x  to y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  We observe that:  – A complex is connected if and only if it is 0-  connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  – A d-pure complex is strongly connected if and  only if it is (d − 1)-connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  Let X be a complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Two faces x, y ∈ X are  adjacent if x ∪ y ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' The link of x ∈ X in X is  the complex lk(x, X) = {y ∈ X | x ∩ y = ∅ and  x ∪ y ∈ X}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  The star of x ∈ X in X is the set st(x, X) = {y ∈  X | x ⊆ y}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  Let X be a d-pseudomanifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' We say that X  satisfies the link condition if lk(x, X) is connected  whenever x is a p-face of X and p ≤ d − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  Let X be a complex and x be a p-face of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Let  st∗(x, X) = st(x, X) \\ {x}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' We have lk(x, X) =  {y \\ x | y ∈ st∗(x, X)} and st∗(x, X) = {z ∪ x |  z ∈ lk(x, X)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  We note that there is a set isomorphism between  lk(x, X) and st∗(x, X), which preserves set inclu-  sion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' If y ∈ st∗(x, X), the corresponding face y \\ x  of lk(x, X) is such that dim(y \\ x) = dim(y) −  Springer Nature 2021 LATEX template  14  Discrete Morse Functions and Watersheds  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' 8  Top: an image (left), together with a geodesic saliency map (right) where the darker a contour is, the more  persistent it is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Bottom: two different views of a triangular mesh, superimposed with a geodesic saliency map where the  whiter a contour is, the more persistent it is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  (p + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Thus, dim(y \\ x) = dim(y) − p+, where  p+ = p + 1 is the number of elements in x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  Let X be a d-pseudomanifold and x be a p-face  of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Let p+ = p + 1 and d′ = d − p+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' The fol-  lowing facts are a direct consequence of the above  isomorphism:  – The complex lk(x, X) is d′-pure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  – The complex lk(x, X) is non-branching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  – The set st∗(x, X) is q-connected if and only if  lk(x, X) is q′-connected, with q′ = q − p+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  Proposition 28 A pseudomanifold is normal if and  only if it satisfies the link condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  Proof Let X be a d-pseudomanifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Suppose X satisfies the link condition and let  S be a connected open subset of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  Let x and y be two d-faces of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' By Remark 1,  there exists a p-chain π in S from x to y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Thus  π = ⟨x = x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=', xk = y⟩ is a sequence of facets  of S such that, for each i ∈ [0, k − 1], xi ∩ xi+1  is a q-face of S, with q ≥ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' We choose π such  that p is maximal and, if p is maximal, such  that the number K(π) of p-faces xi∩xi+1, with  i ∈ [0, k − 1], is minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' If p = d − 1, it means  that S is strongly connected;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' then we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  Suppose p < d − 1 and let xi, xi+1 such that  z = xi ∩ xi+1 is a p-face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  Since X satisfies the link condition, lk(z, X)  is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' By the isomorphism between  lk(z, X) and st∗(z, X), it follows there is a q-  chain ⟨xi = w0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=', wl = xi+1⟩ in st∗(z, X)  with q > p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Therefore, π′ = ⟨x = x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=', xi =  w0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=', wl = xi+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=', xk = y⟩ is a p-chain in S  from x to y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' But we have K(π′) < K(π), a con-  tradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Thus, each connected open subset  of X is strongly connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Suppose X  is strictly connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' That is,  any connected open subset of X is (d − 1)-  connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Let x be a p-face of X with p ≤ d−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  The set st(x, X) is a connected open subset of  X, thus it is (d−1)-connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Since p < d−1,  it means that st∗(x, X) is (d − 1)-connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  By the isomorphism between lk(x, X) and  st∗(x, X), it follows that lk(x, X) is strongly  connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Thus lk(x, X) is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  □  In the second part of the proof of Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' 28, we  showed that lk(x, X) is strongly connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Con-  sequently, we have the following characterization  of a normal pseudomanifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  Proposition 29 A pseudomanifold X is normal if  and only if, for each p-face x of X, with p ≤ d−2, the  complex lk(x, X) is a pseudomanifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  Springer Nature 2021 LATEX template  Discrete Morse Functions and Watersheds  15  Appendix B  discrete Morse  fonctions  Let us consider the following definition of a dis-  crete Morse function:  Definition 30 (Morse function) Let X be a complex  and let F be a map from X to Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' We say that F is  a discrete Morse function on X if any face of X is  in at most one covering pair (x, y) in X such that  F(x) ≥ F(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' If F is a discrete Morse function, we  say that such a pair is a regular pair of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  It may be checked that this definition is equiv-  alent to the classical one given by Forman (See  Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='1 and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='5 of [40]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  In this way, the gradient vector field of a dis-  crete Morse function F, written −−→  grad(F), is the  set composed of all regular pairs of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  The following restriction of a discrete Morse  function will lead us to Morse stacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  We say that a discrete Morse function F on X  is flat if we have F(x) = F(y) whenever (x, y) is  a regular pair of F, that is, if each regular pair of  F is a flat pair of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  We can check that a map F from X to Z is a  flat discrete Morse function if and only if:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Each covering pair (x, y) in X is such that  F(x) ≤ F(y);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' Each face of X is in at most one flat pair of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  Therefore, if we consider the function −F, we  obtain the following:  Proposition 31 Let X be a complex and let F be a  map from X to Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' The map F is a Morse stack on  X if and only if the map −F is a flat discrete Morse  function on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  The following proposition claims that, up to  an equivalence, we may assume that any discrete  Morse function is flat (see Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='27 and Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='16  of [10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content='  Proposition 32 (from [10]) If F is a discrete Morse  function on X, then there exists a flat discrete Morse  function G on X such that, for every covering pair  (x, y) in X, we have F(x) ≥ F(y) if and only if  G(x) ≥ G(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
+page_content=' In other words, the function G is such  that −−→  grad(G) = −−→  grad(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfXAej/content/2301.03840v1.pdf'}
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diff --git a/TtFKT4oBgHgl3EQfky6p/content/tmp_files/2301.11851v1.pdf.txt b/TtFKT4oBgHgl3EQfky6p/content/tmp_files/2301.11851v1.pdf.txt
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@@ -0,0 +1,1655 @@
+Morphology and Magnetic vortex chiral symmetry of 2D arrays of magnetic trilayer
+disks with magnetostatic interlayer coupling determined by X ray resonant magnetic
+scattering∗
+J. D´ıaz1,2, L. M. ´Alvarez-Prado1,2, S. M. Valvidares3, I. Montoya4, C. Redondo4, R. Morales5,6 and M. V´elez1,2
+1Universidad de Oviedo, Calle Federico Garc´ıa Lorca 18, 33007, Oviedo
+2CINN (CSIC – Universidad de Oviedo), 33940 El Entrego, Spain
+3ALBA Synchrotron, 08290 Cerdanyola del Vall´es, Spain
+4Dept. of Physical Chemistry, Univ. of the Basque Country UPV/EHU, E-48940 Leioa, Spain
+5Dept. of Physical Chemistry, Univ. of the Basque Country UPV/EHU and BCMaterials, E-48940 Leioa, Spain. and
+6IKERBASQUE, Basque Foundation for Science, E-48011 Bilbao, Spain
+(Dated: January 30, 2023)
+X ray resonant magnetic scattering (XRMS) was used to characterize the magnetization of 2D
+arrays of trilayer submicron magnets. The interpretation of the data required the understanding of
+the morphology of the magnets which was also deduced from the scattered intensity. The magnets
+consisted of two magnetostatically coupled ferromagnetic layers separated by a non-magnetic spacer.
+The scattered intensity from the disks resulted to be dependent on the disks surface curvature. This
+made the collected intensity at each Bragg reflection (BR) to be correlated to the reflected light
+from locations of the disk with the same angle of curvature. Due to this, quantitative information
+was obtained, averaged over the disks illuminated by x rays, of the variations in thickness and
+magnetization across the entire area of the disks.
+This averaged magnetization mapping of the
+disks served to study their vortex configuration in each of their magnetic layers, determining the
+average location of the vortex, the chiral symmetry of its magnetic circulation, and the specific
+locations where the vortex nucleation starts within the disks. Chiral asymmetry appeared in the
+disks when the field was oriented at an oblique angle with respect to the easy axis of the array. The
+local magnetic sensitivity of the technique allowed to identify a non-centrosymmetric distribution
+of the magnetization of the disks that explains the observed chiral asymmetry. Unexpectedly, the
+magnetic circulation sense of the vortex was the same in both ferromagnetic layers. In addition,
+the magnetization of the buried layer was different in the descent branch than in the ascent branch
+of its hysteresis loops. This effect was also found in some of the hysteresis loops of both layers
+collected at different BRs in two different sample orientations, suggesting that the magnetization
+and demagnetization of the disks could be affected by collective stochastic process.
+I.
+INTRODUCTION
+Magnetic vortices formed in simple magnet forms have
+been the subject of investigation since the methods to
+create arrays of submicron size magnets are available [1–
+5].
+They are a stable magnetic configuration in disks
+and squares magnets, and they are relatively simply to
+describe. An interesting aspect of the vortex is that it can
+adopt 4 different configurations, attending to the handi-
+ness of the magnetic rotation and the sense of polariza-
+tion of its core. These configurations are robust and they
+have been intended to be used for information storage
+[6, 7], spin wave source [8, 9], magnetic sensors [10–12],
+and biotechnology [13, 14]. The difference in energy be-
+tween the configurations depends on the symmetries of
+the system in many cases, and its characterization and
+control has been an important subject of investigation
+for years.
+The behavior of stacking magnetic layers in the same
+shaped magnet is, however, less known. A double layer
+system allows more parameters to tune, like, for instance,
+the interaction between layers through the non-magnetic
+∗ jidiaz@uniovi.es
+spacer and the magnetization of the disks, increasing its
+functional capabilities [11, 15]. Such a double layer struc-
+ture is actually the chosen in sensors and memory units
+like in spin valves and magnetic tunnel junctions. How-
+ever, the studies where more than one magnetic layer in
+the disks are involved are scarce due to the more difficult
+characterization of the buried layers [8, 16, 17]. Most of
+the magnetic sensitive microscopies with submicron res-
+olution are surface sensitive, like MFM (Magentic Force
+Microscopy) [18, 19] and PEEM (Photoemission Electron
+Microscopy) [20]. Transmission electron microscopies are
+not layer sensitive [2, 15, 21]. They are also subjected to
+limitations in the substrates, which have to be transpar-
+ent to electrons, and the applied fields during measure-
+ments. These limitations are overcome in XMRS, making
+it the tool of choice for the characterization of these sys-
+tems due to its capability to peer into buried layers at
+relatively large thickness. The present experiment also
+shows that XRMS can have enough lateral resolution to
+register local changes in the magnetization of submicron
+size magnets, averaged over all the probed magnets.
+XMRS is a non-destructive photon-in photon-out tech-
+nique, which makes it compatible with the use of exter-
+nal fields, currents or temperature during measurements.
+The only restriction for the sample substrates is to be
+flat.
+The principles of magnetic scattering are similar
+arXiv:2301.11851v1  [cond-mat.mes-hall]  27 Jan 2023
+
+2
+to the observed using light in the visible spectrum [22–
+26]. Changing the polarization of the incident beam al-
+lows either to measure the longitudinal or the transverse
+component of the magnetization of the probed magnets
+[25, 27]. Using x rays permits the access to a wider range
+of moment transfer values, increasing the sensitivity to
+local changes. To distinguish the magnetic signal from
+each of the stacking layers, the energy and polarization
+of the x rays must be tuned, requiring a synchrotron
+radiation source. XRMS has been already used in this
+kind of systems before [16, 28, 29].
+In particular, the
+study done in reference [30] is the only one that recov-
+ers the magnetization of each layer in a bilayer square
+ring magnet at different subregions by deconvoluting the
+contribution of each of this subregions to the intensity
+of the diffracted spots. This method requires magnets
+with geometrically well differentiated regions and form
+factors. In all the cases, it is assumed that the magnets
+are flat, i.e., perfectly bidimensional.
+However, this is
+not always the case. Short wave length sources can be
+specially sensitive to this.
+The present study deepens
+in the interpretation of the intensity obtained at differ-
+ent x ray Bragg reflected angles for those cases in which
+the form of the magnets is not perfectly flat, finding a
+correlation between the angle at which the Bragg reflec-
+tion (BR) is collected and the region of the submicron
+magnet from where the light comes. For those case, this
+converts XRMS in a magnetic microscope capable to see
+the variations in the lateral component of the magneti-
+zation, averaged over all the submicron magnets illumi-
+nated by the x rays, at specific regions of the submicron
+magnets and at specific layers. It also evidenced the sen-
+sitivity of XRMS to the morphology of the disks in a
+quantitative way, making possible to determine changes
+in the thickness across the area of the submicron magnets
+with nanometer resolution. This is an aspect that it has
+been largely overlooked, since most of the studies done
+assumed flat surfaces in their magnetic forms.
+A direct consequence of this finding has to do with
+the sensitivity of the XRMS to the chiral asymmetry of
+the vortex. This was demonstrated by us in a previous
+experiment on an array of a single layer of Permalloy
+disks [31]. This sensitivity arises in XRMS from the ori-
+gin of the magnetic scattering intensity in these systems,
+which is due to the interference between the magnetic
+and charge scattering, and it has a similar origin than
+the observed using visible light [25, 26]. Although this
+interpretation is still perfectly valid in flat or nearly flat
+forms, the present experiment shows that chiral sensitiv-
+ity is enhanced by the curved surface of the submicron
+magnets.
+In the studied sample, chiral asymmetry was detected
+when the disks array was oriented at an oblique angle
+with respect to its easy axis, changing its vortex chiral
+sense with the direction of the initial magnetization in
+saturation. Thanks to the here reported microscopic sen-
+sitivity of XRMS, the location of the nucleation area in
+the disks was determined, demonstrating that the mag-
+netic chiral asymmetry of the disks is associated to a
+non-centrosymetric distribution of their magnetization.
+Chiral symmetry was also detected in each of the layers,
+resulting to be the same in both of them, what was un-
+expected specially due to the nature of the observed chi-
+ral asymmetry. The measured hysteresis loop branches
+in some locations of the disk, or even in all the disk,
+were not symmetric, suggesting a collective stochastic be-
+havior in the magnetization and demagnetization of the
+disks, probably induced by thermal fluctuations during
+measurements[29].
+II.
+SAMPLE PREPARATION AND
+CHARACTERIZATION
+The array of disks was produced by, first, creating an
+antidot array by interference laser lithography (ILL) on
+a negative resist that was spin wet on silicon substrates.
+The ILL procedure used a Lloyd’s mirror interferometer
+with a He-Cd laser (λ =325 nm) as the light source [32].
+ILL produces patterns of a constant period and similar
+disks shape over large areas of the order of cm2 in a sin-
+gle shot. In this way, the x ray beam had not restrictions
+in its size to probe the samples. The metallic layers were
+deposited on these antidot imprinted substrates by mag-
+netron sputtering.
+The measured disks array resulted
+after the resist was lift off. Evaporation of each layer was
+done at normal incidence in a vacuum chamber at a base
+pressure of 1 × 10−7 mbar and under an Ar pressure of
+3 × 10−3 mbar. The iron layer was 14.8 nm thick, and
+it was deposited directly on the substrate with no buffer
+layer, following by the deposited of the aluminum spacer
+layer (2.2 nm) and the cobalt layer (17.6 nm). A 3 nm
+aluminum capping layer was deposited on top to avoid
+contamination.
+A reference sample was deposited at the same time
+than the substrates with the imprinted pattern.
+Fig-
+ure 1 shows the hysteresis loops of the reference and
+the patterned samples measured by VSM. The reference
+sample was magnetically soft, with an in-plane magnetic
+anisotropy of about 40 Oe and a coercive field in the
+easy axis (EA) of about 15 Oe. The hysteresis loops of
+the array of magnetic disks are the expected in magnetic
+configurations that minimize their stray magnetic fields,
+with low remanence, coercivity and comparatively large
+saturation fields. Its relative remanence Mr/M is of the
+order of 30%, and its coercive field is of about 20 Oe.
+The saturation field of the sample changes from 550 Oe
+to more than 700 Oe at two orthogonal directions parallel
+to the symmetry axis of the square lattice of the array.
+Scanning electron microspe (SEM) images show that
+the lattice is perfectly squared with a lattice parameter,
+α, of 1.3 µm, confirmed by the x ray diffracted pattern.
+The magnets have a diamond shape with rounded cor-
+ners. The corners are aligned to the square lattice axis.
+The axis of the magnet related to these directions of the
+array have not the same length: their proportion ratio is
+
+3
+FIG. 1.
+(a) Hysteresis loops of a reference thin film prepared
+at the same time that the array of disks; (b) VSM hysteresis
+loops of the 2D array of disks along the (10) and (01) direction
+of the square lattice.
+FIG. 2.
+(a) SEM image of the array of disks; (b) MFM image
+of the array of disks.
+of about 0.9 (see figure 2). The size of the disks was of
+805 nm in the long axis. The EA of the sample was par-
+allel to this axis. MFM images confirmed the presence
+of a single vortex at the top layer of most of the disks in
+the remanent state of the samples (see figure 2).
+The thickness of the deposited layer and the quality
+of their interfaces was obtained by fitting the reflectivity
+curves of the reference sample taken at the Fe and Co res-
+onant energies using circularly polarized light. Figure 3
+displays the reflectivity curves and the magnetic dichro-
+ism asymmetry with their corresponding fitting curves
+obtained at the two resonant energies of Fe (706 eV)
+and Co (777 eV). The fitting of the curves was done
+by a home-made code using the methodology presented
+in reference [33]. The thickness of the cobalt and iron
+layers were 176 ˚A and 148 ˚A respectively. Their thick-
+ness ratio was chosen to be approximately the same that
+their magnetization ratio, which is 0.84, to have the same
+magnetization in the two layers. The thickness of the alu-
+minum spacer was of about 22 ˚A, which was large enough
+to consider that the magnetic interaction between layers
+was entirely dipolar.
+The interfaces of the cobalt and
+Fe layers with the nonmagnetic aluminum spacer had a
+roughness of about 9 ˚A. This value might include some
+possible intermixing between layers.
+Resonant magnetic reflectivity curves were also ob-
+tained from the disks, which are displayed in figure 4.
+Oscillations due to magnetic contrast were visible and
+they were used to determine the x ray incident angle un-
+der which magnetic contrast was the highest in the range
+FIG. 3.
+Fitted reflectivity and magnetic asymmetry curves
+of the reference thin film taken at the (a) Co (776 eV) and
+(b) Fe (706 eV) edges .
+FIG. 4.
+Reflectivity curves of the array of disks taken at
+the cobalt (776 eV) and iron (706 eV) edges. The position
+in qz(θi) chosen for obtained the diffraction patterns with
+magnetic contrast are marked in each curve with a vertical
+line.
+of large qx values. However, the curves cannot be fitted
+in the same way than the reference sample, likely due
+to the discrete nature of the probed surface which might
+introduce intensity unrelated to pure reflection from the
+disks.
+III.
+MAGNETIC SCATTERING
+The scattering of x rays is sensitive to the magnetic
+moment of the probed material by considering the terms
+of the photon scattering that are sensitive to the angular
+moment of the electrons. These terms are usually too
+small but they are enhanced when the photon energy is
+close to the absorption edge of the probed element where
+its magnetic properties are better manifested, the scat-
+tering becoming element specific. In this case, the chosen
+photon energies were 706 eV and 777 eV, the energies of
+the L3 edge for Fe and Co, respectively. Using circular
+polarized light and grazing incidence, the intensity ob-
+served is, to a good approximation, proportional to the
+scalar product between the scattered magnetic moment
+of the electron, ⃗m3d, and the wave vector of the incident
+beam, ⃗ki. In the present experiment, magnetic contrast
+
+1.0
+Sil/FelAl/Co
+0.5
+M/Ms
+0.0
+M/M
+0
+-0.5
+-A.00
+A.90°
+-1.0
+-1
+-10
+-5
+0
+5
+10
+-1000
+-500
+0
+500
+1000
+H (mT)
+H(Oe)(a
+(b)
+1.3 μm0
+Fe706ev
+log(Reflectivity)
+Co 777eV
+log(Reflectivity)
+-2
+2
+4
+.4
+-6
+c
+c
+c
+-6
+c
+-8
+10
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+q2(R-)
+qz(8-)
+1.0
+1.0
+Fe706eV
+Co777eV
+0.5
+0.5
+Asymmetry
+Asymmetry
+0.0
+0.0
+-0.5
+-0.5
+-1.0
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+q2(A-)
+q,(R-)Disks
+Co 777 eV
+c
+c
+Fe 706 e
+8
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+q2(R~)4
+was obtained at the BR directions only, indicating that
+this king of scattering corresponds to the term related
+to the interference between the charge scattering and the
+magnetic scattering. This scattered intensity, IMQi, has
+the following dependence on the scattering vectors and
+the magnetic configuration of the disks ([27]):
+IMQi = Re
+�
+F ∗
+0 ρ∗ (⃗q) F1
+�
+⃗ki · M (⃗q) + ⃗ksct · M (⃗q)
+�
+⃗ki · ⃗ksct
+���
+P3 ≈ 2Re
+�
+F ∗
+0 ρ∗ (⃗q) F1⃗ki · M (⃗q)
+�
+P3
+(1)
+M (⃗q) and ρ∗ (⃗q) are the Fourier transform of the mag-
+netic and charge configuration of the array of disks, F0
+and F1 are the scattering factors for charge and magnetic
+scattering, ⃗ki and ⃗ksct are the wavevectors of the incident
+and the scattered beams, respectively. P3 is the circular
+polarization degree and ⃗q = ⃗ksct−⃗ki is the moment trans-
+fer vector.
+The structure and magnetic characterization of the
+samples was done measuring the reflectivity curves over a
+relatively wide range of 2θ angles, from 0◦ to 50◦using a
+photodiode detector. The light scattered from the two di-
+mensional array was detected using a CCD camera placed
+at the same location than the photodiode.
+Magnetic
+fields were set at constant values for each measurement
+using a dedicated electromagnet [34]. Magnetic contrast
+at the BRs in the CCD images at each magnetic field
+intensity was obtained by calculating the circular dichro-
+ism: IM(⃗q) = IC+
+MQi − IC−
+MQi. Charge scattering was ex-
+tracted summing the intensity obtained at the two circu-
+lar polarization helicities, IM(⃗q) = IC+
+MQi + IC−
+MQi.
+The photon energy and incidence angle chosen were
+those in which the dichroism contrast occurred at a sim-
+ilar angle for the Co and Fe edges to insure that the
+probed areas of the sample were nearly the same. The
+chosen angle of incidence was θi = 7.7◦, which was high
+enough to include a large number of BRs in the qx di-
+rection, parallel to the direction of the beam. Figure 5
+shows the geometry and the axis orientation for the qx
+and qy moment transfer vectors used in the CCD images.
+Figure 4 indicates, in the qz scale (qz = 2k0 sin θi, k0 =
+2π/λ), the dichroism contrast for that angle of incidence
+at the iron and cobalt chosen photon energies.
+A to-
+tal of 10 snapshots with 0.1’ exposure were recorded to
+obtain the final Bragg intensity pattern covered by the
+CCD camera at each of the applied magnetic fields for
+each circular polarization helicity, C+ and C−, of the
+incident x-rays. This process was repeated at different
+applied fields to complete the two branches of an hys-
+teresis loop (HL). Each of the branches consisted of 30
+measurements at constant increments of the applied mag-
+netic field, where ∆H =67 Oe. HL started measuring at
+magnetic saturation fields of 1 kOe. The measurements
+were done in two different orientations of the disks arrays
+with respect to the magnetic field: 1) at the (11) orienta-
+tion of the array, oblique to its magnetic easy axis (EA),
+and 2) at the (10) orientation, parallel to the EA.
+FIG. 5.
+Geometry of the scattering experiment: orientation
+of the qx and qy axis with respect to the incident, ⃗ki, and
+scattered, ⃗ksct, beams.
+IV.
+BRAGG REFLEXIONS
+Figure 6 and 7 display the diffraction patterns recorded
+on the CCD at the Co and Fe resonant energies, 776 eV
+and 706 eV, respectively in the (11) array orientations
+with their corresponding profiles along the qy direction
+for all the qx values collected in Fe and Co at this ori-
+entation. The figures contain the diffraction pattern due
+to the scattering of the charge (IQ (⃗q)) and the magnetic
+dichroism (IM (⃗q)) obtained with the sample in magnetic
+saturation. The images have been scaled in the qx and qy
+components of the moment transfer. Due to the grazing
+incidence geometry, the range of qx values collected by
+the CCD camera is smaller than in the qy direction. The
+dynamic range in the CCD images have been reduced to
+enhance the intensity of Bragg reflections at angles far
+from the reflected beam, which is the most intense.
+The intensity of the diffraction pattern spreads from
+the center of the image, with more intensity at positive
+than at negative qx values. This intensity is also mod-
+ulated, i.e., its value oscillates from the center of the
+pattern, apparently forming parabolic curves with their
+vertex located at the positive side of the qx axis. Both
+(10) and (11) orientations have a similar intensity distri-
+bution pattern. There exists marked differences between
+the distributed intensity in Co and Fe: the intensity at
+
+qz
+k
+V
+X
+(10)
+(11)
+(01)
+7,7
+q=ksct- kin
+800 nm5
+FIG. 6.
+Diffraction pattern of the 2D array of disks taken
+at the Co (776 eV) and Fe (706 eV) edges at the (11) orien-
+tations due to charge scattering (Q) and magnetic scattering
+(M). The sample was magnetically saturated during image
+acquisition.
+FIG. 7. Charge (Q) and magnetic dichroism (M) scans along
+the qy direction for different values of qx in iron and cobalt in
+the (11) orientation
+the center looks broader in cobalt than in iron when mov-
+ing to negative qx values. In Fe, the BRs with the lowest
+intensity near the center of the diffraction pattern forms
+an incomplete ring. Cobalt magnetic contrast changes
+sign twice counting from the center to the border. Fe
+changes the sign of its magnetic contrast in few points
+just at the center of the image, but there is no change in
+contrast at higher qy values as in cobalt.
+The observed modulation in intensity of the BR peaks
+cannot be attributed to the form factor of the disks. Ac-
+tually, such a diffraction is missing from the pattern. If
+it existed, it should form concentric rings from the center
+of the pattern since its form factor depends only on the
+in-plane coordinates x and y, i.e., it is only a function of
+qx and qy. The regular spacing between the BR peaks,
+which depends on qx and qy only, avoids any possible de-
+formation of these rings due to a sample misalignment.
+The size of the observed rings does not correspond to the
+expected from the diameter of the disk. And the inten-
+sity outside the first zero in intensity is far higher than
+the expected from the diffraction of a disk, as deduced
+from figures 6 and 7. For instance, the intensity of the
+BRs at qx = 4q0 (q0 = 2π
+α , and α is the lattice parame-
+ter of the array) is comparable to the intensity near the
+region close to the reflected beam, BR [0, 0].
+The described intensity distribution in the diffracted
+patterns is better explained by assuming that the sur-
+face of the disks have a curvature due to a radial de-
+crease of their thickness. This is the only way to obtain
+the observed particular dispersion of the light along the
+qx and qy axis and the change in the sign of the magnetic
+contrast observed in the magnetic dichroism diffraction
+patterns (figures 6 and 7). This change in thickness prob-
+ably arises by shadowing effects during the deposition of
+the layers into the antidots. Therefore, the orientation of
+the surface at each point of the disk at a certain radial
+distance ξ from the center is characterized by an angle γ,
+formed between the normal to the surface in this point
+and the normal to the sample, and a layer thickness τ.
+The steepness of the surface should depend on the rate
+at which τ decreases, which is unknown. Figure 8 shows
+the angles and parameters that describe the proposed
+3-dimensional shape of the disks.
+In this model, the oscillations in intensity are due to
+the interference between the light scattered at each in-
+terface. Therefore, in the case of a single layer, the scat-
+tered light with the lowest intensity have the relation
+qzτ = 2πm + π, where m is an integer number.
+The
+thickness τ is a function of the position in the disk from
+where the scattered light comes, which depends on γ. qz
+is also a function of γ due to the way the light is expected
+to be scattered from the interfaces, which should be done
+mainly in the same direction than the reflected light. By
+rotation of the system of reference to align its z axis to
+the normal of the plane at each point of the disk, it is easy
+to show that the corresponding moment transfer vector,
+⃗q, of the scattered beam has the following variation with
+the angle γ (taking only the linear terms):
+
+qx(8-)
+x(A
+0.01.0mA
+-8x10-404
+5
+5
+CoQ(11)
+Co
+M (11)
+Fe 
+Q (11)
+Fe
+M (11)
+4:
+41
+4-
+4-
+3
+3
+?1
+3?-
+21
+F心
+2
+2
+-1
+qy(8")
+0
+0
+0
+N
+2
+2-
+2-
+2-
+3-
+3-
+3-
+3
+41
+41
+4-
+4Fe
+TT
+9x=-4
+Fe
+Q (11)
+9.-3
+M(11)
+-b
+9,=-2
+Scattering)
+q,=-2
+Dichroism)
+qx=-1
+9x=-1
+Intensity(Charge
+Intensity (Magnetic 
+0=*b
+0=*b
+q,=1
+q,=1
+q,=2
+9x=2
+9,=3
+q,=3
+q,=4
+4
+-4 x103
+4
+9y (A~)
+q (A-)
+9,=-4
+Co
+9,=-4
+Co
+Q (11)
+M (11)
+9,=-3
+E-="b
+Scattering)
+Intensity (Magnetic Dichroism)
+9,=-2
+9,=-2
+Intensity(Charge
+qx=-1
+qx=-1
+0=b
+0=*b
+qx=1
+a=2l
+qx=1
+9,=2
+=*b
+9=3
+9x=4
+9=4
+0
+4
+0
+4
+9y (A*)
+qy (A-)6
+FIG. 8. Model for the reflection of the x rays from a dome
+shaped disk. (a) Definition of γ and ϕ anges; (b) Transverse
+cut of the model propossed for the structure of the disks. H
+is the the thickness of the layer at the center of the disks,
+and τ is the thickness at a distance from the center. γ is the
+angle formed by the normal to the surface at that point with
+respect to the z axis; (c) and (d), dependence of qx, qx and qx
+with the angles γ in the longitudinal (x axis) and transverse
+(y axis) to the beam directions.
+qx ≈ qz0γ cos ϕ
+(2)
+qy ≈ qz0γ sin ϕ
+(3)
+qz ≈ qz0 − qx
+θi
+(4)
+qz0 = 2
+���⃗ki
+��� sin θi. The angle ϕ is formed by the posi-
+tion of the point in the plane of the disk with respect to
+the x axis (see figure 8). These equalities show that γ is
+related to the plane component of the moment transfer
+vector, q∥, by the dependence γ =
+q∥
+qz0 . Note that when
+qx = 0, qz = qz0 and, therefore, the variation along qy de-
+pends only on the variation of the thickness τ. Also, when
+qx > 0 (cos ϕ > 1), qz decreases, and increases when
+qx > 0. This means that the equality qzτ = 2π (m + 1/2)
+is reached at a lower value of γ (qx) than the correspond-
+ing in [0, qy] in the former case, but further away in the
+qx < 0. Moreover, due to the higher reflectivity coeffi-
+cients at grazing angles, it is expected that the scattered
+intensity decreases as the take off angle of the scattered
+beam increases, which occurs at qx < 0 (see figure 5). All
+of this agrees with what is experimentally observed.
+Thanks to this result, it is possible to have access to
+the morphology of the disks in a more quantitative way.
+Figure 9 shows the [qx = 0, qy] profiles of the magnetic
+dichroism and the charge scattering intensities measured
+in cobalt and iron, in the (11) orientation. This profile al-
+lows a better estimation of the change of the thickness of
+the layers across the disks since the change in qz is practi-
+cally negligible, all the variations observed are due to the
+thickness and the steepness of the disk surface. Also, the
+number of points available are much larger than in the
+qx direction. In cobalt, the magnetic contrast is perfectly
+coupled to the charge scattering: the magnetic contrast
+changes sign every time the charge scattering crosses a
+point of lowest intensity. This is what expected in the
+reflection from a single layer, but not from a trilayer sys-
+tem. Actually, the variation in intensity seems to follow
+a sin2 (qy) function. This suggest that the observed scat-
+tered intensity is mainly caused by diffuse scattering at
+the interfaces of cobalt [35]. This is probably the case
+since the large incidence angle used (θi = 7.7◦), which
+gives rise to a low reflectivity coefficient, the relatively
+large roughness of the interfaces and the resonant condi-
+tion.
+The profiles of iron have a similar oscillation period
+than the found in cobalt, but there is one oscillation less.
+In this case, its magnetic contrast does not change as
+in cobalt. In fact, it is more structured in the region of
+highest intensity, in the central region. There, magnetic
+contrast changes, but it remains constant in the rest of
+oscillations. This indicates that the exact understanding
+of the light scattered from the iron layer is apparently
+more complicated than in cobalt due to its buried condi-
+tion. But this complication affects to the magnetic con-
+trast mainly. The oscillation in intensity of the charge
+scattering is compatible with a single layer model, as in
+cobalt, which is the most expected behavior at the reso-
+nant photon energy.
+Then, the highest order of interference occurs at the
+point of highest intensity, where the angle γ = 0 and the
+thickness is the highest. This is m = 3 for cobalt and
+m = 2 for iron due to the lower thickness of the iron
+layer and the higher wavelength at the iron edge. This
+agrees with the one less intensity oscillation in iron than
+in cobalt and the different distribution of the intensity
+oscillation in the diffraction pattern at both absorption
+edges. Also, this confirms that the profile along [0, qy]
+covers scattered light from all the disk, from the highest
+thickness to the null thickness regions. The dependence
+of the thickness τ on qy (which is linear on γ) is correlated
+to the rate at which the thickness of the layer changes
+with the curvature of the surface, which is unknown. For
+instance, τ will have a quadratic dependence on qy if the
+radius of curvature of the surface of the two interfaces is
+constant. In that case, the zeros in intensity should occur
+at qy values proportional to the square root of the inter-
+ference order m. However, the experimentally observed
+relation is close to linear on m (see figure 9). This im-
+plies that the curvature of the surface of the disks needs
+to be stronger to have a change in the thickness of the
+layers. The adjustment of the zeros in the profiles of fig-
+ure 9 is done using the equality qzτ = 2π (m + 1/2) and
+taking the relation τ = H − σ
+2 γ = H − σ qy
+2qz0 , where H
+is the highest thickness of the disk, and σ a factor that
+indicates how fast the layer thickness changes with the
+curvature angle. This shows a slightly faster rate in iron
+than in cobalt, indicating a larger curvature in cobalt
+
+Z个
+(a)
+n
+(b)
+H
+y>
+Co
+Fe
+x
+(c)
+(d)
+z1
+0
+0,
+0+2
+H
+-x
+x
+y
+V
+qx=-qzo
+0×bz+0zb="b
+qy=-qzo
+Zb="b7
+FIG. 9.
+Intensity profile along the qy direction at qx = 0
+related to the charge scattering (blue) and the magnetic scat-
+tering (red) in cobalt (top) and iron (bottom).
+than in iron. This relation is the expected since cobalt is
+deposited on a curved surface, whereas iron is deposited
+in a flat surface. As a consequence of this, there is not a
+perfect one-to-one correspondence in the intensity of the
+BRs between cobalt and iron. Iron covers a larger area
+of the disk in a smaller range of q∥ than cobalt. This dif-
+ference is not excessively important. From the previous
+adjustment of the [qx = 0, qy] profile, it is estimated a
+expansion ratio in cobalt respect to iron of 1.4.
+The in-plane component of the moment transfer vec-
+tor, ⃗q∥, related to the scattered light from the disks will
+be distorted by a non-uniform curvature of the surface,
+i.e., by variations in the γ angle. This will give rise to
+a non-uniform distribution of the intensity besides the
+caused by the thickness. For instance, if the portion of
+the area of the disk that is flat is large, most of the inten-
+sity will be concentrated in a smaller [qx, qy] area. Then,
+although the BR peaks are correlated to the different re-
+gions of the disk, i.e., each [qx, qy] coordinate is related to
+a [x, y] position in the disk, this correlation is not com-
+pletely linear. A precise model of the scattered intensity
+is required for that, what will determine, therefore, the
+complete shape of the disks. Note also that the area of
+the disk covered by the CCD detector is constrained in
+the qx direction, equivalent to the x direction. This re-
+gion is delimited by the first interferential zero, meaning
+that the total decrease in thickness within it is of the
+order of 60 ˚A out of 350 ˚A.
+V.
+HYSTERESIS LOOPS
+The one-to-one correlation between the in-plane mo-
+ment transfer vector and the in-plane spatial coordinate
+of the disks eases the interpretation of the HLs collected
+at each BR. For instance, this explains why XRMS is
+specially sensitive to a chiral asymmetry in the disks:
+when the vortex is formed, the regions of the disks with
+magnetization parallel to the beam have their normal
+to their surface mainly pointing transverse to the field.
+When there is chiral asymmetry in the magnetic vortex
+circulation, each magnetic orientation points in a direc-
+tion opposite to the other one giving rise to the resulting
+asymmetric magnetic contrast in the qy axis. This expla-
+nation is different, but not contradictory, to the origin of
+the sensitivity of XRMS to chiral asymmetry proposed in
+[31], which still holds and it should be observed in perfect
+flat disks.
+In what follows, it will be assumed that each BR posi-
+tion [h, k] is related to a region around a position [x, y] in
+the disk. To describe the different regions of the disk, the
+direction of the incident beam is taken as the reference.
+This direction is the same than the positive direction of
+the applied field. Therefore, intensity at qx > 0 corre-
+sponds to the north (N) side of the disk, qx < 0 to the
+south (S) side, qy > 0 to the west (W) side and qy < 0
+to the east (E) side.
+The HLs presented here were normalized to 1. To im-
+prove their visualization, their noise was reduced using
+a binomial smoothing. The smoothing degree was the
+same for all the loops. This did not modify in essence
+the loops line-shape since the changes in magnetization
+should be smooth. However, the smoothing was unable
+to smearing out all the noise, leaving low frequency oscil-
+lations in the magnetization which were obviously more
+notorious in those loops with poorer signal to noise ra-
+tios. Although this did not impede to identify the general
+trends, it rested accuracy in the value of the onset fields
+obtained from them, whose highest accuracy is half the
+field step used to measure the HL, which is 33 Oe.
+Note that the HLs are averaged over hundred of disks.
+Therefore, the observed result will depend on the pos-
+sible number of magnetic configurations that the vortex
+can adopt. This number obviously decreases when the
+symmetry of the system decreases, like the one related
+to the sense of circulation of the magnetization in a vor-
+tex.
+Figure 10 displays the HLs observed at the E and W
+sides of the disk, and located relatively distant from the
+center, when the applied field was oriented parallel to the
+(11) orientation of the array. The differences between the
+two HLs are due to the broken chiral symmetry of the
+magnetic vortex circulation in this orientation. The HLs
+shows changes in the magnetic susceptibility at some crit-
+ical fields which define the onset for the creation, move-
+ment and annihilation of the vortex in the disks. The
+field at which the demagnetization of the disk is initiated
+is named H0. The location of the disk where this hap-
+
+Co 11
+nsity
+Inter
+-6 mA
+-4 
+-2
+0
+2
+4
+6
+Fe 11
+-6 mA-1
+-4
+-2
+0
+2
+4
+68
+pens is of interest since it sets the circulation sense of the
+vortex. Note that this means that, if the two branches of
+the loop are symmetrical, the nucleation occurs always
+in the same side making the sense of circulation in the
+vortex to invert in each branch.
+In the presented ex-
+ample, nucleation occurs in the E side, where H0 is the
+highest. Therefore, the circulation is clockwise (CW) in
+the downward branch and counter clockwise (CCW) in
+the upward branch. The creation of the vortex causes a
+fast reduction of the magnetization until a point where
+the magnetization reaches a value close to zero and the
+magnetic susceptibility changes again. The field where
+this occurs is named Hv0. Once the vortex is formed,
+the region of the disk with opposite magnetization to the
+initial one is mainly located in the E side if the circula-
+tion is CW. Therefore, its magnetization will be negative
+at Hv0, and positive in the W side, being Hv0 > 0. The
+opposite occurs in the upward branch because vortex cir-
+culation inverts. This makes the branches of the E side
+HL to cross each other twice near ±Hv0. As the field is
+increased to magnetized the disk in the opposite direc-
+tion, the core of the vortex moves transverse to the field.
+This movement is from the E toward the W in the down-
+ward branch. This movement starts at a critical field,
+named Hv1. This field has not to be symmetrical to Hv0.
+Also, the core movement to the edge might have a dif-
+ferent magnetic susceptibility than the changes produced
+during the creation of the vortex. This makes the HL to
+develop lobes near the magnetic saturation regions. The
+steepness of the magnetic susceptibility in the region be-
+tween Hv0 and Hv1 indicates how much the vortex moves
+in that range of fields. Therefore, this field region gives
+direct information of the regions of the disk where core
+vortex is stable. The HLs of the example shows that the
+vortex is relatively stable in the region from where the
+HLs are extracted, what is at the region of the disk far
+from the center. The killing of the vortex is usually pro-
+duced near saturation fields. The field at which magnetic
+saturation is produced is named Hs. This occurs first in
+the E side than in the W side. Therefore, the magneti-
+zation in the W side is always higher than in the E side
+in the downward branch, explaining the ”fat” shape of
+its HL, whose branches envelopes those of the E side HL.
+By contrast, the magnetization measured at any point in
+the center of the disk from the N to the S sides will be
+zero if the core of the vortex is in the center of the disk
+since their main magnetic component is transverse to the
+measured direction.
+Figures 11 and 12 shows the HLs of Fe and Co at dif-
+ferent BR positions in the (11) sample orientation, giving
+a more detailed description of the magnetization at dif-
+ferent locations of the disks.
+The BRs are ordered in
+the horizontal line from negative qx to positive qx values,
+which are related to the magnetization at the regions of
+the disk running from S to N. The first row are the HLs
+taken at qy = 0, i.e., the HLs located at the center of
+the disk. The other two rows have increasing qy values.
+They are related to regions of the disk which are increas-
+FIG. 10. On top, HLs collected at BRs at fixed qx but oppo-
+site qy in cobalt at the (11) orientation: in red, qy > 0; and
+in blue, qy < 0. In the middle, a schematics of the related
+regions of the disks probed at the corresponding BR orienta-
+tions: qy > 0 is the related to the W side of the disk, and
+qy < 0 to the E side of the disk.
+The probed regions are
+the overlaps between the disks and the related stripes, in the
+corresponding magnetic configuration of the magnetization of
+the disks at the fields H0, Hv0 and Hv1. The differences be-
+tween the HLs are due to a fixed magnetic vortex circulation
+which is inverted in each branch.
+In the drawing, N is on
+the right, S is on the left, W is on the top and E is on the
+bottom of the disk. The beam direction goes from S to the N
+direction. On the bottom, the hysteresis loops taken in iron
+in the (11) orientation and in similar regions than in cobalt.
+The values of H0, Hv0 and HS signaled in the figure are those
+of cobalt in the E side HL.
+ingly further from the center, either moving towards the
+E (HLs in blue) or to the W (HLs in red). The distant
+∆q between BRs along the qx and qy axis is
+√
+2 π
+α. Some
+of the [h, k] values displayed in the N side are not exactly
+the same than at the S side because the corresponding
+HLs were too distorted to be showed. They are at BRs
+where the magnetic contrast inverts its sign. The HLs
+displayed in the third row are taken at qy values that
+are further from the center than the allowed qx values
+
+Co (11)
+Hs
+H
+-----
+M/M
+W
+E
+1
+1
+--
+-
+Hs'
+1
+"H
+:Ho
+-40
+0
+40
+H (mT)
+W
+E
+Ho
+Hvo
+Hyi
+Fe (11)
+Hs:
+M/Ms
+H
+vo
+-40
+0
+40
+H (mT)9
+(h ≤ 4).
+At the cobalt layer, the HL in BR [0, 0] has a coercive
+field, indicating that the core of the vortex avoids the
+center of the disk. Note that this HL does not resemble
+the obtained by VSM (see figure 1). The magnetic be-
+havior in the N and S sides is not symmetric with respect
+to the center. The H0 field is higher in the S side than
+in the N side. Actually, the HL at the extreme S side
+does not seem to reach saturation. There, the coercive
+field is null, but it is significant in the N side. This asym-
+metry between the N and S sides occurs also in the HLs
+taken at qy ̸= 0. The H0 field is higher in the S side of
+the E side (HL in blue), decreasing as qy becomes more
+negative. The difference in the H0 and Hs fields between
+the E (red) and W (blue) side decreases going from the
+S side to the center. The same behavior occurs from the
+N side to the center with the important differences that,
+in this case, the H0 is significantly higher in the E side
+than in the W side. Moreover, the branches of the HL in
+the NE side never cross each other whereas this clearly
+happens in the SE side. Therefore, the N side of the disk
+has a lower probability of holding the core of the vortex
+than in the S side. The region where the vortex are more
+stable, i.e., where little changes in the magnetization oc-
+curs, is at the edges of the disk (large |qy| values) since it
+is where the susceptibility between Hv0 and Hv1 is flatter
+and the distance between both fields is increased. It is
+in the NE and NW sides where the distance between the
+Hv0 and Hv1 fields is the highest, confirming that most
+of the changes in the vortex in the cobalt layer occurs
+mainly in the S side.
+The HLs of the iron layer shown in figure 12 are of less
+quality than those of cobalt due to the lower scattered in-
+tensity. All of them have the downward branch different
+than the upward branch, they are not symmetric. Both
+branches cross each other once near H = 0. Figure 10 (at
+the bottom) shows the shape of these HLs more in detail.
+The start of the downward branch is similar to that of
+cobalt, which is related to the formation of vortices with
+the same magnetic circulation sense, CW. The magneti-
+zation of the downward branch falls down to lower values
+at H ≥-10 mT. The reduction of magnetization until sat-
+uration from there is done with a slow rate. The value of
+Hs is of about 55 mT, similar than the saturation fields
+found in cobalt. The upward branch starts at the same
+field H0 than the saturation field. This is a much higher
+H0 field than in cobalt. The increase in magnetization is
+also faster than in the downward branch, stabilizing the
+vortex at -20 mT. Again, the chirality of this vortex is the
+same as in cobalt for this orientation of the field, giving
+raise to a CCW chirality. As in cobalt, the onset field H0
+is higher in the S side than in the N side. The HL at [-4,0]
+has an slope as if magnetic saturation was not complete,
+something that does not happen at the conjugate BR in
+[4,0], with lower saturation fields. Therefore, nucleation
+starts at the same region than in cobalt.
+The annihilation of the vortex is obviously different for
+the two branches. The change in the magnetization at the
+field where the vortex in the downward branch was killed
+was more important in the SE side of the disk, indicating
+that the vortex was preferentially in this region, as in
+cobalt. Saturation occurs first in the NW side for this
+branch. For the upward branch, saturation field is higher
+than in cobalt. Saturation occurs at lower fields in the
+N side for this loop branch again. In general, saturation
+is produced first in the NW and at a much higher field
+in the SE side. The region of the disk where the change
+in the magnetization in the upward branch between Hv0
+and Hv1 is small, occurs as well in the edges. But the
+range of fields seems to be larger towards the S side. From
+these observations, it seems that the core of vortex in the
+cobalt and iron layers stays in similar regions of the disks,
+which could be the reason of the sudden anhinilation of
+the vortex in the iron layer and the resulting asymmetric
+HL branches.
+Figures 13 and 14 shows the Fe and Co HLs in the (10)
+orientation for comparison. In this case, there was not a
+clear asymmetry in the magnetic vortex circulation. The
+distance ∆q between BRs along the qx and qy axis is 2π
+α .
+The HL of the cobalt layer at BR [0, 0] is very similar to
+the obtained in the (11) orientation. The onset field H0
+is similar in the N and S sides and smaller than in the
+(11) orientation. Moreover, the highest H0 seems to be in
+the E and W sides, at [0, ±3] and [1, ±3]. There, the HLs
+have a different shape than at the other edges, indicating
+that it is in this region where the core of the vortex moves.
+There is still an asymmetry between the N and the S
+sides, as in the (11) orientation. At the S side, the HLs
+with qy ̸= 0 shows an imbalance in the chiral symmetry of
+the magnetic circulation. The magnetic circulation is, in
+this case, CCW in the downward branch and CW in the
+upward branch. Surprisingly, such a chiral asymmetry is
+not observed in the N side, remarking the asymmetry in
+the magnetic behavior between the two sides of the disk.
+In this side, the downward branch has an onset H0 field
+which is clearly different than in the downward branch.
+In the iron layer, its HL at BR [0, 0] has a coercive
+field which is larger than in cobalt. The onset field H0
+are similar in the N and S sides. It is also the highest
+of the registered for this orientation, but smaller than in
+the (11) orientation. The asymmetry between the N and
+S sides is not so clear as in cobalt. There is not either a
+clear imbalance in the chiral symmetry of the vortex.
+A.
+Discussion
+Having the same chiral asymmetry in both layers is not
+the expected behavior since the nucleation of the vortex
+in each layer creates magnetic poles of the same sign.
+This makes energetically more favorable for each disk to
+do the nucleation at opposite sides of the border of the
+disk, inducing an opposite chiral sense in each layer [5].
+This is in fact very critical in this case because the re-
+gion where the disk begins to demagnetize seems to be
+the same in both layers and for the two orientations of the
+
+10
+FIG. 11. Hysteresis loops of the cobalt layer in the (11) orientation, at chosen BRs. The way [h,k] numbers locate the BRs is
+described in the text. N side is at qx > 0, E side is at qy > 0 (red color HLs).
+FIG. 12. Hysteresis loops of the iron layer in the (11) orientation, at chosen BRs. The way [h,k] numbers locate the BRs is
+described in the text. N side is at qx > 0, E side is at qy > 0 (red color HLs).
+FIG. 13. Hysteresis loops of the cobalt layer in the (10) orientation, at chosen BRs, [qx,qy]=[h,k] and in conjugated sites ([h,±
+k]). N side is at qx > 0, E side is at qy > 0 (red color HLs).
+
+Co -4 0
+Co -2 0
+Co0 0
+Co 2 0
+Co40
+M/Ms
+Co -4 4
+Co -2 4
+Co0 4
+Co 2 6
+Co42
+Co -2-4
+Co 0-4
+M/Ms
+Co 4-2
+Co -4-4
+Co 2-6
+o-210
+Co -4 10
+Co'0 10
+Co 2 10
+Co 4 10
+-2-8
+Co 0 -24
+Co
+Co 4-10
+Co -4-10
+Co 2-10
+M/Ms
+L
+-40
+0
+40
+-40
+0
+40
+-40
+0
+40
+-40
+0
+40
+-40
+0
+40
+H (mT)
+H (mT)
+H (mT)
+H (mT)
+H (mT)Fe -4 0
+Fe -2 0
+Fe00
+Fe 2 0
+Fe4 0
+M/Ms
+Fe -4 4
+Fe06
+Fe 24
+Fe44
+M/Ms
+Fe 0-6
+Fe -4-4
+Fe 2-4
+Fe 4-4
+-2
+4
+Fe210
+Fe -4 10
+Fe -2 10
+Fe010
+Fe48
+Fe 2-10
+Fe -2-10
+Fe -4-10
+Fe 0-10
+M/Ms
+Fe 4-8
+-40
+0
+40
+-40
+0
+40
+-40
+0
+40
+-40
+0
+40
+-40
+0
+40
+H (mT)
+H (mT)
+H (mT)
+H (mT)
+H (mT)Co -3 0
+Co -1 0
+Co00
+Co1 0
+Co30
+M/Ms
+Co -3-3
+Co 0-3
+Co 3-3
+Co 1-3
+Co -1-3
+Co -1 3
+Co33
+Co -3 3
+Co03
+Co13
+40
+0
+-40
+0
+40
+-40
+0
+-40
+40
+-40
+0
+40
+-40
+0
+40
+H (mT)
+H (mT)
+H (mT)
+H (mT)
+H (mT)11
+FIG. 14. Hysteresis loops of the iron layer in the (10) orientation, at chosen BR, [qx,qy]=[h,k] and in conjugated sites ([h,±
+k]).. N side is at qx > 0, E side is at qy > 0 (red color HLs).
+FIG. 15. Model proposed to explain the chiral asymmetry in
+the array of disks.
+saturated magnetization in the disks. Therefore, the ori-
+gin of the asymmetry has to be related to either a higher
+energy process that should over compensate the polar
+repulsion between the two layers, or to an effective at-
+tractive interaction between layers, possibly induced by
+the roughness of the interfaces (N´eel peel orange effect
+[36]) and/or the observed thickness gradient.
+The observed type of chiral asymmetry, which changes
+of sign depending on the initial magnetization direc-
+tion, is induced in submicron magnets by making their
+shape non-centrosymetric:
+triangles, truncated disks,
+asymmetric rings, ”pac-man” shaped disks or asymmet-
+rical magnetic moment distribution[2, 15, 37–41]. Chiral
+asymmetry occurs in these systems when the applied field
+is at an angle with respect to the mirror symmetry axis
+of the system.
+To explain the origin of the chiral asymmetry in the
+studied system, we proposed the following model based
+on the existence of a non-centrosymmetry in the mag-
+nets of the array. In the (10) field orientation, the cobalt
+layer contains an asymmetry.Such an asymmetry might
+be related to the anisotropy energy which could be asym-
+metrically distributed across the area of the disk. This
+could happen if its shape is not fully symmetric. In this
+case, the S side has possibly a higher anisotropy than
+the N side, what is required to preserve mirror symme-
+try between the W and E sides, since chiral asymmetry
+is not fully manifested in this orientation [39].
+In the
+(11) orientation, the oblique angle direction of the ap-
+plied field breaks such a mirror symmetry causing the
+observed chiral asymmetry. Then, when the direction of
+the field is positive, the moments in the N side have a
+weaker anisotropy and align their moments with the ap-
+plied field at lower fields than in the S side. This creates
+an in-plane component of the magnetization perpendicu-
+lar to the direction of the field at that particular region,
+setting the sense of rotation of the magnetization in the
+vortex. When the direction of the field is the opposite,
+the orientation of this in-plane perpendicular component
+changes its direction as well, changing the sense of the
+magnetic circulation of the vortex to the opposite one.
+The model is depicted in figure 15. This process seems
+to be very solid in the cobalt layer since the branches of
+its HLs are symmetric at any point in the disk. It is also
+the layer that register an asymmetry between N a S sides
+in the (10) orientation as well.
+Figure 16 shows the Fast Fourier Transform (FFT) of
+the image obtained by SEM of the sample which included
+more than 103 disks. A visual inspection shows that the
+intensity of the peaks deviates from the expected sym-
+metric square shape resulting from the FFT of a diamond
+shape.
+A detailed analysis of the intensity of the BR
+peaks in this figure shows that the ideally diamond-shape
+disks are actually rhomboids.
+Their shape asymmetry
+arises because the distance between opposite sides of the
+diamond-like shape of the disks are not exactly the same.
+This makes the shape of the disks non-centrosymmetric,
+i.e., there is not perfect mirror symmetry across any of
+the symmetry axis oriented along the (10), (01), (11) or
+(-1-1) directions. This deviation from centrosymmetry in
+the shape of the magnets is much smaller than the non-
+centrosymmetry induced in magnets by purpose to fix
+their vortex chirality. This might indicate that a precise
+control of the ILL technique can be used to modulate the
+chiral properties of magnets arrays by inducing asymme-
+tries in their shape.
+Since the HL measurement was done in a single cycle,
+it is not possible to assert that the different magnetiza-
+
+Fe-2 0
+Fe00
+Fe10
+Fe 3 0
+M/Ms
+Fe 2 0
+-e
+Fe -2-2
+Fe 2-2
+Fe 3-2
+Fe 02
+Fe -2 2
+Fe 2 2
+M/Ms
+Fe32
+Fe.0-2
+Fe 1-2
+-40
+0
+40
+-40
+0
+40
+-40
+0
+40
+-40
+0
+40
+-40
+0
+40
+H (mT)
+H (mT)
+H (mT)
+H (mT)
+H (mT)12
+FIG. 16.
+Fourier transform of a SEM image of the array
+including more than 103 disks. On the bottom, the intensity
+of qx scans across the previous image in two opossite values
+of qy to evidence the deviation from perfect mirror symmetry
+with respect to the (10) and (01) axes. Scans have their qx
+zero shifted for better comparison. q0 = 2π/α where α is the
+lattice constant of the array
+tion paths taken in each of the branches of the HL in Fe
+in the (11) orientation, and the observed in some HLs in
+the (10) orientation, was a systematic and repetitive pro-
+cess. The HL measured by VSM, which covers a much
+larger area than the measured by XRMS, did not show
+any asymmetry between the branches. Therefore, these
+asymmetries are possibly due to a certain stochasticity in
+the magnetic inversion process of the disks, which have
+to be collective by the nature of the measurement.
+VI.
+CONCLUSIONS
+In conclusion, we observed a one-to-one correlation be-
+tween the in-plane moment vector transfer of the scat-
+tered light from an array of submicron magnets and the
+spatial location in the submicron magnet from where
+the light comes, converting the magnetic contrast of the
+diffraction pattern of the array in an image of the magne-
+tization of the disk, averaged over the illuminated disks.
+The conditions for this to happen are related to the mor-
+phology of the magnets and the configuration of the ex-
+periment. The surface of the magnets must have a certain
+curvature. In the present experiment, this is obtained us-
+ing magnetron sputtering deposition at normal incidence
+in patterned holes, which is one of the most used meth-
+ods to produce the kind of studied submicron magnet
+arrays. The grazing incidence angles employed were rel-
+atively large to allow the collection of scattered light at a
+wider range of in-plane moment transfer. At such large
+angles, the diffuse scattering due to the roughness and
+imperfections at the layer interfaces should be important
+what, joined to the resonant energies employed, allowed
+a better isolation of the targeted layers even if they were
+buried under 20 nm thick layers. This kind of magnetic
+microscope effect explains why XRMS is specially sen-
+sitive to the chiral asymmetry of the magnetization cir-
+culation of the vortex in this kind of samples. This was
+used to study the chiral asymmetry of the disks. Thanks
+to this effect, it was possible to identify the presence of a
+non-centro-symmetry in the magnetization of the sample
+that explained the apparition of the chiral asymmetry at
+the oblique angle orientation of the field with respect to
+the EA axis of the array. The physical origin of such a
+magnetic non-centro-symmetry was attributed to devia-
+tions from centro-symmetry in the shape of the magnets,
+giving an indication of the sensitivity of the studied sys-
+tem to such deviations. The presented experiment shows
+that XRMS can give a collective vision of the stability
+and symmetry breaking process in this kind of system
+which is complementary to the obtained by other micro-
+scopic techniques. There is plenty of room to increase
+the quality of the data and to increase the information
+extracted from the scattered light in the configuration
+used in this experiment, specially the related to the mor-
+phology of the magnets.
+ACKNOWLEDGMENTS
+This
+project
+has
+been
+supported
+by
+Spanish
+MINECO
+under
+grant
+PID2019-
+104604RB/AEI/10.13039/501100011033,
+and
+by
+Asturias
+FICYT
+under
+grant
+AYUD/2021/51185
+with the support of FEDER funds. R. M. acknowledge
+Basque Country grant No. IT1491-22.
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diff --git a/TtFKT4oBgHgl3EQfky6p/content/tmp_files/load_file.txt b/TtFKT4oBgHgl3EQfky6p/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..d32e97e981242ed825a2f46ca8f71b9fbb0b5e5f
--- /dev/null
+++ b/TtFKT4oBgHgl3EQfky6p/content/tmp_files/load_file.txt
@@ -0,0 +1,1052 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf,len=1051
+page_content='Morphology and Magnetic vortex chiral symmetry of 2D arrays of magnetic trilayer  disks with magnetostatic interlayer coupling determined by X ray resonant magnetic  scattering∗  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' D´ıaz1,2, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' ´Alvarez-Prado1,2, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' Valvidares3, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' Montoya4, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' Redondo4, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' Morales5,6 and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' V´elez1,2  1Universidad de Oviedo, Calle Federico Garc´ıa Lorca 18, 33007, Oviedo  2CINN (CSIC – Universidad de Oviedo), 33940 El Entrego, Spain  3ALBA Synchrotron, 08290 Cerdanyola del Vall´es, Spain  4Dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' of Physical Chemistry, Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' of the Basque Country UPV/EHU, E-48940 Leioa, Spain  5Dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' of Physical Chemistry, Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' of the Basque Country UPV/EHU and BCMaterials, E-48940 Leioa, Spain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' and  6IKERBASQUE, Basque Foundation for Science, E-48011 Bilbao, Spain  (Dated: January 30, 2023)  X ray resonant magnetic scattering (XRMS) was used to characterize the magnetization of 2D  arrays of trilayer submicron magnets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' The interpretation of the data required the understanding of  the morphology of the magnets which was also deduced from the scattered intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' The magnets  consisted of two magnetostatically coupled ferromagnetic layers separated by a non-magnetic spacer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='  The scattered intensity from the disks resulted to be dependent on the disks surface curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' This  made the collected intensity at each Bragg reflection (BR) to be correlated to the reflected light  from locations of the disk with the same angle of curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' Due to this, quantitative information  was obtained, averaged over the disks illuminated by x rays, of the variations in thickness and  magnetization across the entire area of the disks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='  This averaged magnetization mapping of the  disks served to study their vortex configuration in each of their magnetic layers, determining the  average location of the vortex, the chiral symmetry of its magnetic circulation, and the specific  locations where the vortex nucleation starts within the disks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' Chiral asymmetry appeared in the  disks when the field was oriented at an oblique angle with respect to the easy axis of the array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' The  local magnetic sensitivity of the technique allowed to identify a non-centrosymmetric distribution  of the magnetization of the disks that explains the observed chiral asymmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' Unexpectedly, the  magnetic circulation sense of the vortex was the same in both ferromagnetic layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' In addition,  the magnetization of the buried layer was different in the descent branch than in the ascent branch  of its hysteresis loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' This effect was also found in some of the hysteresis loops of both layers  collected at different BRs in two different sample orientations, suggesting that the magnetization  and demagnetization of the disks could be affected by collective stochastic process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='  INTRODUCTION  Magnetic vortices formed in simple magnet forms have  been the subject of investigation since the methods to  create arrays of submicron size magnets are available [1–  5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='  They are a stable magnetic configuration in disks  and squares magnets, and they are relatively simply to  describe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' An interesting aspect of the vortex is that it can  adopt 4 different configurations, attending to the handi-  ness of the magnetic rotation and the sense of polariza-  tion of its core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' These configurations are robust and they  have been intended to be used for information storage  [6, 7], spin wave source [8, 9], magnetic sensors [10–12],  and biotechnology [13, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' The difference in energy be-  tween the configurations depends on the symmetries of  the system in many cases, and its characterization and  control has been an important subject of investigation  for years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='  The behavior of stacking magnetic layers in the same  shaped magnet is, however, less known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' A double layer  system allows more parameters to tune, like, for instance,  the interaction between layers through the non-magnetic  ∗ jidiaz@uniovi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='es  spacer and the magnetization of the disks, increasing its  functional capabilities [11, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' Such a double layer struc-  ture is actually the chosen in sensors and memory units  like in spin valves and magnetic tunnel junctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' How-  ever, the studies where more than one magnetic layer in  the disks are involved are scarce due to the more difficult  characterization of the buried layers [8, 16, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' Most of  the magnetic sensitive microscopies with submicron res-  olution are surface sensitive, like MFM (Magentic Force  Microscopy) [18, 19] and PEEM (Photoemission Electron  Microscopy) [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' Transmission electron microscopies are  not layer sensitive [2, 15, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' They are also subjected to  limitations in the substrates, which have to be transpar-  ent to electrons, and the applied fields during measure-  ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' These limitations are overcome in XMRS, making  it the tool of choice for the characterization of these sys-  tems due to its capability to peer into buried layers at  relatively large thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' The present experiment also  shows that XRMS can have enough lateral resolution to  register local changes in the magnetization of submicron  size magnets, averaged over all the probed magnets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='  XMRS is a non-destructive photon-in photon-out tech-  nique, which makes it compatible with the use of exter-  nal fields, currents or temperature during measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='  The only restriction for the sample substrates is to be  flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='  The principles of magnetic scattering are similar  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='11851v1  [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='mes-hall]  27 Jan 2023  2  to the observed using light in the visible spectrum [22–  26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' Changing the polarization of the incident beam al-  lows either to measure the longitudinal or the transverse  component of the magnetization of the probed magnets  [25, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' Using x rays permits the access to a wider range  of moment transfer values, increasing the sensitivity to  local changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' To distinguish the magnetic signal from  each of the stacking layers, the energy and polarization  of the x rays must be tuned, requiring a synchrotron  radiation source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' XRMS has been already used in this  kind of systems before [16, 28, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='  In particular, the  study done in reference [30] is the only one that recov-  ers the magnetization of each layer in a bilayer square  ring magnet at different subregions by deconvoluting the  contribution of each of this subregions to the intensity  of the diffracted spots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' This method requires magnets  with geometrically well differentiated regions and form  factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' In all the cases, it is assumed that the magnets  are flat, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=', perfectly bidimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='  However, this is  not always the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' Short wave length sources can be  specially sensitive to this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='  The present study deepens  in the interpretation of the intensity obtained at differ-  ent x ray Bragg reflected angles for those cases in which  the form of the magnets is not perfectly flat, finding a  correlation between the angle at which the Bragg reflec-  tion (BR) is collected and the region of the submicron  magnet from where the light comes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' For those case, this  converts XRMS in a magnetic microscope capable to see  the variations in the lateral component of the magneti-  zation, averaged over all the submicron magnets illumi-  nated by the x rays, at specific regions of the submicron  magnets and at specific layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' It also evidenced the sen-  sitivity of XRMS to the morphology of the disks in a  quantitative way, making possible to determine changes  in the thickness across the area of the submicron magnets  with nanometer resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' This is an aspect that it has  been largely overlooked, since most of the studies done  assumed flat surfaces in their magnetic forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='  A direct consequence of this finding has to do with  the sensitivity of the XRMS to the chiral asymmetry of  the vortex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' This was demonstrated by us in a previous  experiment on an array of a single layer of Permalloy  disks [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' This sensitivity arises in XRMS from the ori-  gin of the magnetic scattering intensity in these systems,  which is due to the interference between the magnetic  and charge scattering, and it has a similar origin than  the observed using visible light [25, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' Although this  interpretation is still perfectly valid in flat or nearly flat  forms, the present experiment shows that chiral sensitiv-  ity is enhanced by the curved surface of the submicron  magnets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='  In the studied sample, chiral asymmetry was detected  when the disks array was oriented at an oblique angle  with respect to its easy axis, changing its vortex chiral  sense with the direction of the initial magnetization in  saturation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' Thanks to the here reported microscopic sen-  sitivity of XRMS, the location of the nucleation area in  the disks was determined, demonstrating that the mag-  netic chiral asymmetry of the disks is associated to a  non-centrosymetric distribution of their magnetization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='  Chiral symmetry was also detected in each of the layers,  resulting to be the same in both of them, what was un-  expected specially due to the nature of the observed chi-  ral asymmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' The measured hysteresis loop branches  in some locations of the disk, or even in all the disk,  were not symmetric, suggesting a collective stochastic be-  havior in the magnetization and demagnetization of the  disks, probably induced by thermal fluctuations during  measurements[29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='  SAMPLE PREPARATION AND  CHARACTERIZATION  The array of disks was produced by, first, creating an  antidot array by interference laser lithography (ILL) on  a negative resist that was spin wet on silicon substrates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='  The ILL procedure used a Lloyd’s mirror interferometer  with a He-Cd laser (λ =325 nm) as the light source [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='  ILL produces patterns of a constant period and similar  disks shape over large areas of the order of cm2 in a sin-  gle shot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' In this way, the x ray beam had not restrictions  in its size to probe the samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' The metallic layers were  deposited on these antidot imprinted substrates by mag-  netron sputtering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='  The measured disks array resulted  after the resist was lift off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' Evaporation of each layer was  done at normal incidence in a vacuum chamber at a base  pressure of 1 × 10−7 mbar and under an Ar pressure of  3 × 10−3 mbar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' The iron layer was 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='8 nm thick, and  it was deposited directly on the substrate with no buffer  layer, following by the deposited of the aluminum spacer  layer (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='2 nm) and the cobalt layer (17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='6 nm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' A 3 nm  aluminum capping layer was deposited on top to avoid  contamination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='  A reference sample was deposited at the same time  than the substrates with the imprinted pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='  Fig-  ure 1 shows the hysteresis loops of the reference and  the patterned samples measured by VSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' The reference  sample was magnetically soft, with an in-plane magnetic  anisotropy of about 40 Oe and a coercive field in the  easy axis (EA) of about 15 Oe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' The hysteresis loops of  the array of magnetic disks are the expected in magnetic  configurations that minimize their stray magnetic fields,  with low remanence, coercivity and comparatively large  saturation fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' Its relative remanence Mr/M is of the  order of 30%, and its coercive field is of about 20 Oe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='  The saturation field of the sample changes from 550 Oe  to more than 700 Oe at two orthogonal directions parallel  to the symmetry axis of the square lattice of the array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='  Scanning electron microspe (SEM) images show that  the lattice is perfectly squared with a lattice parameter,  α, of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='3 µm, confirmed by the x ray diffracted pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='  The magnets have a diamond shape with rounded cor-  ners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' The corners are aligned to the square lattice axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='  The axis of the magnet related to these directions of the  array have not the same length: their proportion ratio is  3  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='  (a) Hysteresis loops of a reference thin film prepared  at the same time that the array of disks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' (b) VSM hysteresis  loops of the 2D array of disks along the (10) and (01) direction  of the square lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='  (a) SEM image of the array of disks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' (b) MFM image  of the array of disks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='  of about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='9 (see figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' The size of the disks was of  805 nm in the long axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' The EA of the sample was par-  allel to this axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' MFM images confirmed the presence  of a single vortex at the top layer of most of the disks in  the remanent state of the samples (see figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='  The thickness of the deposited layer and the quality  of their interfaces was obtained by fitting the reflectivity  curves of the reference sample taken at the Fe and Co res-  onant energies using circularly polarized light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' Figure 3  displays the reflectivity curves and the magnetic dichro-  ism asymmetry with their corresponding fitting curves  obtained at the two resonant energies of Fe (706 eV)  and Co (777 eV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' The fitting of the curves was done  by a home-made code using the methodology presented  in reference [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' The thickness of the cobalt and iron  layers were 176 ˚A and 148 ˚A respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' Their thick-  ness ratio was chosen to be approximately the same that  their magnetization ratio, which is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='84, to have the same  magnetization in the two layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' The thickness of the alu-  minum spacer was of about 22 ˚A, which was large enough  to consider that the magnetic interaction between layers  was entirely dipolar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='  The interfaces of the cobalt and  Fe layers with the nonmagnetic aluminum spacer had a  roughness of about 9 ˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' This value might include some  possible intermixing between layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='  Resonant magnetic reflectivity curves were also ob-  tained from the disks, which are displayed in figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='  Oscillations due to magnetic contrast were visible and  they were used to determine the x ray incident angle un-  der which magnetic contrast was the highest in the range  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='  Fitted reflectivity and magnetic asymmetry curves  of the reference thin film taken at the (a) Co (776 eV) and  (b) Fe (706 eV) edges .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='  Reflectivity curves of the array of disks taken at  the cobalt (776 eV) and iron (706 eV) edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' The position  in qz(θi) chosen for obtained the diffraction patterns with  magnetic contrast are marked in each curve with a vertical  line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='  of large qx values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' However, the curves cannot be fitted  in the same way than the reference sample, likely due  to the discrete nature of the probed surface which might  introduce intensity unrelated to pure reflection from the  disks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='  MAGNETIC SCATTERING  The scattering of x rays is sensitive to the magnetic  moment of the probed material by considering the terms  of the photon scattering that are sensitive to the angular  moment of the electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' These terms are usually too  small but they are enhanced when the photon energy is  close to the absorption edge of the probed element where  its magnetic properties are better manifested, the scat-  tering becoming element specific.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' In this case, the chosen  photon energies were 706 eV and 777 eV, the energies of  the L3 edge for Fe and Co, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' Using circular  polarized light and grazing incidence, the intensity ob-  served is, to a good approximation, proportional to the  scalar product between the scattered magnetic moment  of the electron, ⃗m3d, and the wave vector of the incident  beam, ⃗ki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' In the present experiment, magnetic contrast  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='0  Sil/FelAl/Co  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='5  M/Ms  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='0  M/M  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='5  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='00  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='90°  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='0  1  10  5  0  5  10  1000  500  0  500  1000  H (mT)  H(Oe)(a  (b)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='3 μm0  Fe706ev  log(Reflectivity)  Co 777eV  log(Reflectivity)  2  2  4  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='4  6  c  c  c  6  c  8  10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='30  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='30  q2(R-)  qz(8-)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='0  Fe706eV  Co777eV  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='5  Asymmetry  Asymmetry  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='30  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='30  q2(A-)  q,(R-)Disks  Co 777 eV  c  c  Fe 706 e  8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='30  q2(R~)4  was obtained at the BR directions only, indicating that  this king of scattering corresponds to the term related  to the interference between the charge scattering and the  magnetic scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' This scattered intensity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' IMQi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' has  the following dependence on the scattering vectors and  the magnetic configuration of the disks ([27]):  IMQi = Re  �  F ∗  0 ρ∗ (⃗q) F1  �  ⃗ki · M (⃗q) + ⃗ksct · M (⃗q)  �  ⃗ki · ⃗ksct  ���  P3 ≈ 2Re  �  F ∗  0 ρ∗ (⃗q) F1⃗ki · M (⃗q)  �  P3  (1)  M (⃗q) and ρ∗ (⃗q) are the Fourier transform of the mag-  netic and charge configuration of the array of disks,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' F0  and F1 are the scattering factors for charge and magnetic  scattering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' ⃗ki and ⃗ksct are the wavevectors of the incident  and the scattered beams,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' P3 is the circular  polarization degree and ⃗q = ⃗ksct−⃗ki is the moment trans-  fer vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='  The structure and magnetic characterization of the  samples was done measuring the reflectivity curves over a  relatively wide range of 2θ angles, from 0◦ to 50◦using a  photodiode detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' The light scattered from the two di-  mensional array was detected using a CCD camera placed  at the same location than the photodiode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='  Magnetic  fields were set at constant values for each measurement  using a dedicated electromagnet [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' Magnetic contrast  at the BRs in the CCD images at each magnetic field  intensity was obtained by calculating the circular dichro-  ism: IM(⃗q) = IC+  MQi − IC−  MQi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' Charge scattering was ex-  tracted summing the intensity obtained at the two circu-  lar polarization helicities, IM(⃗q) = IC+  MQi + IC−  MQi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='  The photon energy and incidence angle chosen were  those in which the dichroism contrast occurred at a sim-  ilar angle for the Co and Fe edges to insure that the  probed areas of the sample were nearly the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' The  chosen angle of incidence was θi = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='7◦, which was high  enough to include a large number of BRs in the qx di-  rection, parallel to the direction of the beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' Figure 5  shows the geometry and the axis orientation for the qx  and qy moment transfer vectors used in the CCD images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='  Figure 4 indicates, in the qz scale (qz = 2k0 sin θi, k0 =  2π/λ), the dichroism contrast for that angle of incidence  at the iron and cobalt chosen photon energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='  A to-  tal of 10 snapshots with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='1’ exposure were recorded to  obtain the final Bragg intensity pattern covered by the  CCD camera at each of the applied magnetic fields for  each circular polarization helicity, C+ and C−, of the  incident x-rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' This process was repeated at different  applied fields to complete the two branches of an hys-  teresis loop (HL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' Each of the branches consisted of 30  measurements at constant increments of the applied mag-  netic field, where ∆H =67 Oe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' HL started measuring at  magnetic saturation fields of 1 kOe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' The measurements  were done in two different orientations of the disks arrays  with respect to the magnetic field: 1) at the (11) orienta-  tion of the array, oblique to its magnetic easy axis (EA),  and 2) at the (10) orientation, parallel to the EA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='  Geometry of the scattering experiment: orientation  of the qx and qy axis with respect to the incident, ⃗ki, and  scattered, ⃗ksct, beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='  BRAGG REFLEXIONS  Figure 6 and 7 display the diffraction patterns recorded  on the CCD at the Co and Fe resonant energies, 776 eV  and 706 eV, respectively in the (11) array orientations  with their corresponding profiles along the qy direction  for all the qx values collected in Fe and Co at this ori-  entation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' The figures contain the diffraction pattern due  to the scattering of the charge (IQ (⃗q)) and the magnetic  dichroism (IM (⃗q)) obtained with the sample in magnetic  saturation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' The images have been scaled in the qx and qy  components of the moment transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' Due to the grazing  incidence geometry, the range of qx values collected by  the CCD camera is smaller than in the qy direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' The  dynamic range in the CCD images have been reduced to  enhance the intensity of Bragg reflections at angles far  from the reflected beam, which is the most intense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='  The intensity of the diffraction pattern spreads from  the center of the image, with more intensity at positive  than at negative qx values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' This intensity is also mod-  ulated, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=', its value oscillates from the center of the  pattern, apparently forming parabolic curves with their  vertex located at the positive side of the qx axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' Both  (10) and (11) orientations have a similar intensity distri-  bution pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' There exists marked differences between  the distributed intensity in Co and Fe: the intensity at  qz  k  V  X  (10)  (11)  (01)  7,7  q=ksct- kin  800 nm5  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='  Diffraction pattern of the 2D array of disks taken  at the Co (776 eV) and Fe (706 eV) edges at the (11) orien-  tations due to charge scattering (Q) and magnetic scattering  (M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' The sample was magnetically saturated during image  acquisition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' Charge (Q) and magnetic dichroism (M) scans along  the qy direction for different values of qx in iron and cobalt in  the (11) orientation  the center looks broader in cobalt than in iron when mov-  ing to negative qx values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' In Fe, the BRs with the lowest  intensity near the center of the diffraction pattern forms  an incomplete ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' Cobalt magnetic contrast changes  sign twice counting from the center to the border.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' Fe  changes the sign of its magnetic contrast in few points  just at the center of the image, but there is no change in  contrast at higher qy values as in cobalt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='  The observed modulation in intensity of the BR peaks  cannot be attributed to the form factor of the disks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' Ac-  tually, such a diffraction is missing from the pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' If  it existed, it should form concentric rings from the center  of the pattern since its form factor depends only on the  in-plane coordinates x and y, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=', it is only a function of  qx and qy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' The regular spacing between the BR peaks,  which depends on qx and qy only, avoids any possible de-  formation of these rings due to a sample misalignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='  The size of the observed rings does not correspond to the  expected from the diameter of the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' And the inten-  sity outside the first zero in intensity is far higher than  the expected from the diffraction of a disk, as deduced  from figures 6 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' For instance, the intensity of the  BRs at qx = 4q0 (q0 = 2π  α , and α is the lattice parame-  ter of the array) is comparable to the intensity near the  region close to the reflected beam, BR [0, 0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='  The described intensity distribution in the diffracted  patterns is better explained by assuming that the sur-  face of the disks have a curvature due to a radial de-  crease of their thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' This is the only way to obtain  the observed particular dispersion of the light along the  qx and qy axis and the change in the sign of the magnetic  contrast observed in the magnetic dichroism diffraction  patterns (figures 6 and 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' This change in thickness prob-  ably arises by shadowing effects during the deposition of  the layers into the antidots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' Therefore, the orientation of  the surface at each point of the disk at a certain radial  distance ξ from the center is characterized by an angle γ,  formed between the normal to the surface in this point  and the normal to the sample, and a layer thickness τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='  The steepness of the surface should depend on the rate  at which τ decreases, which is unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' Figure 8 shows  the angles and parameters that describe the proposed  3-dimensional shape of the disks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='  In this model, the oscillations in intensity are due to  the interference between the light scattered at each in-  terface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' Therefore, in the case of a single layer, the scat-  tered light with the lowest intensity have the relation  qzτ = 2πm + π, where m is an integer number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='  The  thickness τ is a function of the position in the disk from  where the scattered light comes, which depends on γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' qz  is also a function of γ due to the way the light is expected  to be scattered from the interfaces, which should be done  mainly in the same direction than the reflected light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' By  rotation of the system of reference to align its z axis to  the normal of the plane at each point of the disk, it is easy  to show that the corresponding moment transfer vector,  ⃗q, of the scattered beam has the following variation with  the angle γ (taking only the linear terms):  qx(8-)  x(A  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='0mA  8x10-404  5  5  CoQ(11)  Co  M (11)  Fe  Q (11)  Fe  M (11)  4:  41  4-  4-  3  3  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='1  3?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='-  21  F心  2  2  1  qy(8")  0  0  0  N  2  2-  2-  2-  3-  3-  3-  3  41  41  4-  4Fe  TT  9x=-4  Fe  Q (11)  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='-3  M(11)  b  9,=-2  Scattering)  q,=-2  Dichroism)  qx=-1  9x=-1  Intensity(Charge  Intensity (Magnetic  0=*b  0=*b  q,=1  q,=1  q,=2  9x=2  9,=3  q,=3  q,=4  4  4 x103  4  9y (A~)  q (A-)  9,=-4  Co  9,=-4  Co  Q (11)  M (11)  9,=-3  E-="b  Scattering)  Intensity (Magnetic Dichroism)  9,=-2  9,=-2  Intensity(Charge  qx=-1  qx=-1  0=b  0=*b  qx=1  a=2l  qx=1  9,=2  =*b  9=3  9x=4  9=4  0  4  0  4  9y (A*)  qy (A-)6  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' Model for the reflection of the x rays from a dome  shaped disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' (a) Definition of γ and ϕ anges;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' (b) Transverse  cut of the model propossed for the structure of the disks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' H  is the the thickness of the layer at the center of the disks,  and τ is the thickness at a distance from the center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' γ is the  angle formed by the normal to the surface at that point with  respect to the z axis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' (c) and (d), dependence of qx, qx and qx  with the angles γ in the longitudinal (x axis) and transverse  (y axis) to the beam directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='  qx ≈ qz0γ cos ϕ  (2)  qy ≈ qz0γ sin ϕ  (3)  qz ≈ qz0 − qx  θi  (4)  qz0 = 2  ���⃗ki  ��� sin θi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' The angle ϕ is formed by the posi-  tion of the point in the plane of the disk with respect to  the x axis (see figure 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' These equalities show that γ is  related to the plane component of the moment transfer  vector, q∥, by the dependence γ =  q∥  qz0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' Note that when  qx = 0, qz = qz0 and, therefore, the variation along qy de-  pends only on the variation of the thickness τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' Also, when  qx > 0 (cos ϕ > 1), qz decreases, and increases when  qx > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' This means that the equality qzτ = 2π (m + 1/2)  is reached at a lower value of γ (qx) than the correspond-  ing in [0, qy] in the former case, but further away in the  qx < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' Moreover, due to the higher reflectivity coeffi-  cients at grazing angles, it is expected that the scattered  intensity decreases as the take off angle of the scattered  beam increases, which occurs at qx < 0 (see figure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' All  of this agrees with what is experimentally observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='  Thanks to this result, it is possible to have access to  the morphology of the disks in a more quantitative way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='  Figure 9 shows the [qx = 0, qy] profiles of the magnetic  dichroism and the charge scattering intensities measured  in cobalt and iron, in the (11) orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' This profile al-  lows a better estimation of the change of the thickness of  the layers across the disks since the change in qz is practi-  cally negligible, all the variations observed are due to the  thickness and the steepness of the disk surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' Also, the  number of points available are much larger than in the  qx direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' In cobalt, the magnetic contrast is perfectly  coupled to the charge scattering: the magnetic contrast  changes sign every time the charge scattering crosses a  point of lowest intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' This is what expected in the  reflection from a single layer, but not from a trilayer sys-  tem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' Actually, the variation in intensity seems to follow  a sin2 (qy) function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' This suggest that the observed scat-  tered intensity is mainly caused by diffuse scattering at  the interfaces of cobalt [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' This is probably the case  since the large incidence angle used (θi = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='7◦), which  gives rise to a low reflectivity coefficient, the relatively  large roughness of the interfaces and the resonant condi-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='  The profiles of iron have a similar oscillation period  than the found in cobalt, but there is one oscillation less.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='  In this case, its magnetic contrast does not change as  in cobalt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' In fact, it is more structured in the region of  highest intensity, in the central region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' There, magnetic  contrast changes, but it remains constant in the rest of  oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' This indicates that the exact understanding  of the light scattered from the iron layer is apparently  more complicated than in cobalt due to its buried condi-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' But this complication affects to the magnetic con-  trast mainly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' The oscillation in intensity of the charge  scattering is compatible with a single layer model, as in  cobalt, which is the most expected behavior at the reso-  nant photon energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='  Then, the highest order of interference occurs at the  point of highest intensity, where the angle γ = 0 and the  thickness is the highest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' This is m = 3 for cobalt and  m = 2 for iron due to the lower thickness of the iron  layer and the higher wavelength at the iron edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' This  agrees with the one less intensity oscillation in iron than  in cobalt and the different distribution of the intensity  oscillation in the diffraction pattern at both absorption  edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' Also, this confirms that the profile along [0, qy]  covers scattered light from all the disk, from the highest  thickness to the null thickness regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' The dependence  of the thickness τ on qy (which is linear on γ) is correlated  to the rate at which the thickness of the layer changes  with the curvature of the surface, which is unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' For  instance, τ will have a quadratic dependence on qy if the  radius of curvature of the surface of the two interfaces is  constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' In that case, the zeros in intensity should occur  at qy values proportional to the square root of the inter-  ference order m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' However, the experimentally observed  relation is close to linear on m (see figure 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' This im-  plies that the curvature of the surface of the disks needs  to be stronger to have a change in the thickness of the  layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' The adjustment of the zeros in the profiles of fig-  ure 9 is done using the equality qzτ = 2π (m + 1/2) and  taking the relation τ = H − σ  2 γ = H − σ qy  2qz0 , where H  is the highest thickness of the disk, and σ a factor that  indicates how fast the layer thickness changes with the  curvature angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' This shows a slightly faster rate in iron  than in cobalt, indicating a larger curvature in cobalt  Z个  (a)  n  (b)  H  y>  Co  Fe  x  (c)  (d)  z1  0  0,  0+2  H  x  x  y  V  qx=-qzo  0×bz+0zb="b  qy=-qzo  Zb="b7  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='  Intensity profile along the qy direction at qx = 0  related to the charge scattering (blue) and the magnetic scat-  tering (red) in cobalt (top) and iron (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='  than in iron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' This relation is the expected since cobalt is  deposited on a curved surface, whereas iron is deposited  in a flat surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' As a consequence of this, there is not a  perfect one-to-one correspondence in the intensity of the  BRs between cobalt and iron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' Iron covers a larger area  of the disk in a smaller range of q∥ than cobalt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' This dif-  ference is not excessively important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' From the previous  adjustment of the [qx = 0, qy] profile, it is estimated a  expansion ratio in cobalt respect to iron of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='  The in-plane component of the moment transfer vec-  tor, ⃗q∥, related to the scattered light from the disks will  be distorted by a non-uniform curvature of the surface,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=', by variations in the γ angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' This will give rise to  a non-uniform distribution of the intensity besides the  caused by the thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' For instance, if the portion of  the area of the disk that is flat is large, most of the inten-  sity will be concentrated in a smaller [qx, qy] area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' Then,  although the BR peaks are correlated to the different re-  gions of the disk, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=', each [qx, qy] coordinate is related to  a [x, y] position in the disk, this correlation is not com-  pletely linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' A precise model of the scattered intensity  is required for that, what will determine, therefore, the  complete shape of the disks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' Note also that the area of  the disk covered by the CCD detector is constrained in  the qx direction, equivalent to the x direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' This re-  gion is delimited by the first interferential zero, meaning  that the total decrease in thickness within it is of the  order of 60 ˚A out of 350 ˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='  HYSTERESIS LOOPS  The one-to-one correlation between the in-plane mo-  ment transfer vector and the in-plane spatial coordinate  of the disks eases the interpretation of the HLs collected  at each BR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' For instance, this explains why XRMS is  specially sensitive to a chiral asymmetry in the disks:  when the vortex is formed, the regions of the disks with  magnetization parallel to the beam have their normal  to their surface mainly pointing transverse to the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='  When there is chiral asymmetry in the magnetic vortex  circulation, each magnetic orientation points in a direc-  tion opposite to the other one giving rise to the resulting  asymmetric magnetic contrast in the qy axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' This expla-  nation is different, but not contradictory, to the origin of  the sensitivity of XRMS to chiral asymmetry proposed in  [31], which still holds and it should be observed in perfect  flat disks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='  In what follows, it will be assumed that each BR posi-  tion [h, k] is related to a region around a position [x, y] in  the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' To describe the different regions of the disk, the  direction of the incident beam is taken as the reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='  This direction is the same than the positive direction of  the applied field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' Therefore, intensity at qx > 0 corre-  sponds to the north (N) side of the disk, qx < 0 to the  south (S) side, qy > 0 to the west (W) side and qy < 0  to the east (E) side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='  The HLs presented here were normalized to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' To im-  prove their visualization, their noise was reduced using  a binomial smoothing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' The smoothing degree was the  same for all the loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' This did not modify in essence  the loops line-shape since the changes in magnetization  should be smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' However, the smoothing was unable  to smearing out all the noise, leaving low frequency oscil-  lations in the magnetization which were obviously more  notorious in those loops with poorer signal to noise ra-  tios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' Although this did not impede to identify the general  trends, it rested accuracy in the value of the onset fields  obtained from them, whose highest accuracy is half the  field step used to measure the HL, which is 33 Oe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='  Note that the HLs are averaged over hundred of disks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='  Therefore, the observed result will depend on the pos-  sible number of magnetic configurations that the vortex  can adopt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' This number obviously decreases when the  symmetry of the system decreases, like the one related  to the sense of circulation of the magnetization in a vor-  tex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='  Figure 10 displays the HLs observed at the E and W  sides of the disk, and located relatively distant from the  center, when the applied field was oriented parallel to the  (11) orientation of the array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' The differences between the  two HLs are due to the broken chiral symmetry of the  magnetic vortex circulation in this orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' The HLs  shows changes in the magnetic susceptibility at some crit-  ical fields which define the onset for the creation, move-  ment and annihilation of the vortex in the disks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' The  field at which the demagnetization of the disk is initiated  is named H0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' The location of the disk where this hap-  Co 11  nsity  Inter  6 mA  4  2  0  2  4  6  Fe 11  6 mA-1  4  2  0  2  4  68  pens is of interest since it sets the circulation sense of the  vortex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' Note that this means that, if the two branches of  the loop are symmetrical, the nucleation occurs always  in the same side making the sense of circulation in the  vortex to invert in each branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='  In the presented ex-  ample, nucleation occurs in the E side, where H0 is the  highest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' Therefore, the circulation is clockwise (CW) in  the downward branch and counter clockwise (CCW) in  the upward branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' The creation of the vortex causes a  fast reduction of the magnetization until a point where  the magnetization reaches a value close to zero and the  magnetic susceptibility changes again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' The field where  this occurs is named Hv0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' Once the vortex is formed,  the region of the disk with opposite magnetization to the  initial one is mainly located in the E side if the circula-  tion is CW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' Therefore, its magnetization will be negative  at Hv0, and positive in the W side, being Hv0 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' The  opposite occurs in the upward branch because vortex cir-  culation inverts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' This makes the branches of the E side  HL to cross each other twice near ±Hv0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' As the field is  increased to magnetized the disk in the opposite direc-  tion, the core of the vortex moves transverse to the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='  This movement is from the E toward the W in the down-  ward branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' This movement starts at a critical field,  named Hv1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' This field has not to be symmetrical to Hv0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='  Also, the core movement to the edge might have a dif-  ferent magnetic susceptibility than the changes produced  during the creation of the vortex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' This makes the HL to  develop lobes near the magnetic saturation regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' The  steepness of the magnetic susceptibility in the region be-  tween Hv0 and Hv1 indicates how much the vortex moves  in that range of fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' Therefore, this field region gives  direct information of the regions of the disk where core  vortex is stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' The HLs of the example shows that the  vortex is relatively stable in the region from where the  HLs are extracted, what is at the region of the disk far  from the center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' The killing of the vortex is usually pro-  duced near saturation fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' The field at which magnetic  saturation is produced is named Hs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' This occurs first in  the E side than in the W side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' Therefore, the magneti-  zation in the W side is always higher than in the E side  in the downward branch, explaining the ”fat” shape of  its HL, whose branches envelopes those of the E side HL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='  By contrast, the magnetization measured at any point in  the center of the disk from the N to the S sides will be  zero if the core of the vortex is in the center of the disk  since their main magnetic component is transverse to the  measured direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='  Figures 11 and 12 shows the HLs of Fe and Co at dif-  ferent BR positions in the (11) sample orientation, giving  a more detailed description of the magnetization at dif-  ferent locations of the disks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='  The BRs are ordered in  the horizontal line from negative qx to positive qx values,  which are related to the magnetization at the regions of  the disk running from S to N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' The first row are the HLs  taken at qy = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=', the HLs located at the center of  the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' The other two rows have increasing qy values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='  They are related to regions of the disk which are increas-  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' On top, HLs collected at BRs at fixed qx but oppo-  site qy in cobalt at the (11) orientation: in red, qy > 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' and  in blue, qy < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' In the middle, a schematics of the related  regions of the disks probed at the corresponding BR orienta-  tions: qy > 0 is the related to the W side of the disk, and  qy < 0 to the E side of the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='  The probed regions are  the overlaps between the disks and the related stripes, in the  corresponding magnetic configuration of the magnetization of  the disks at the fields H0, Hv0 and Hv1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' The differences be-  tween the HLs are due to a fixed magnetic vortex circulation  which is inverted in each branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='  In the drawing, N is on  the right, S is on the left, W is on the top and E is on the  bottom of the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' The beam direction goes from S to the N  direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' On the bottom, the hysteresis loops taken in iron  in the (11) orientation and in similar regions than in cobalt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='  The values of H0, Hv0 and HS signaled in the figure are those  of cobalt in the E side HL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='  ingly further from the center, either moving towards the  E (HLs in blue) or to the W (HLs in red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' The distant  ∆q between BRs along the qx and qy axis is  √  2 π  α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' Some  of the [h, k] values displayed in the N side are not exactly  the same than at the S side because the corresponding  HLs were too distorted to be showed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' They are at BRs  where the magnetic contrast inverts its sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' The HLs  displayed in the third row are taken at qy values that  are further from the center than the allowed qx values  Co (11)  Hs  H  -----  M/M  W  E  1  1  -- Hs\'  1  "H  :Ho  40  0  40  H (mT)  W  E  Ho  Hvo  Hyi  Fe (11)  Hs:  M/Ms  H  vo  40  0  40  H (mT)9  (h ≤ 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='  At the cobalt layer, the HL in BR [0, 0] has a coercive  field, indicating that the core of the vortex avoids the  center of the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' Note that this HL does not resemble  the obtained by VSM (see figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' The magnetic be-  havior in the N and S sides is not symmetric with respect  to the center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' The H0 field is higher in the S side than  in the N side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' Actually, the HL at the extreme S side  does not seem to reach saturation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' There, the coercive  field is null, but it is significant in the N side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' This asym-  metry between the N and S sides occurs also in the HLs  taken at qy ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' The H0 field is higher in the S side of  the E side (HL in blue), decreasing as qy becomes more  negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' The difference in the H0 and Hs fields between  the E (red) and W (blue) side decreases going from the  S side to the center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' The same behavior occurs from the  N side to the center with the important differences that,  in this case, the H0 is significantly higher in the E side  than in the W side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' Moreover, the branches of the HL in  the NE side never cross each other whereas this clearly  happens in the SE side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' Therefore, the N side of the disk  has a lower probability of holding the core of the vortex  than in the S side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' The region where the vortex are more  stable, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=', where little changes in the magnetization oc-  curs, is at the edges of the disk (large |qy| values) since it  is where the susceptibility between Hv0 and Hv1 is flatter  and the distance between both fields is increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' It is  in the NE and NW sides where the distance between the  Hv0 and Hv1 fields is the highest, confirming that most  of the changes in the vortex in the cobalt layer occurs  mainly in the S side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='  The HLs of the iron layer shown in figure 12 are of less  quality than those of cobalt due to the lower scattered in-  tensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' All of them have the downward branch different  than the upward branch, they are not symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' Both  branches cross each other once near H = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' Figure 10 (at  the bottom) shows the shape of these HLs more in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='  The start of the downward branch is similar to that of  cobalt, which is related to the formation of vortices with  the same magnetic circulation sense, CW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' The magneti-  zation of the downward branch falls down to lower values  at H ≥-10 mT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' The reduction of magnetization until sat-  uration from there is done with a slow rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' The value of  Hs is of about 55 mT, similar than the saturation fields  found in cobalt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' The upward branch starts at the same  field H0 than the saturation field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' This is a much higher  H0 field than in cobalt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' The increase in magnetization is  also faster than in the downward branch, stabilizing the  vortex at -20 mT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' Again, the chirality of this vortex is the  same as in cobalt for this orientation of the field, giving  raise to a CCW chirality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' As in cobalt, the onset field H0  is higher in the S side than in the N side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' The HL at [-4,0]  has an slope as if magnetic saturation was not complete,  something that does not happen at the conjugate BR in  [4,0], with lower saturation fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' Therefore, nucleation  starts at the same region than in cobalt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='  The annihilation of the vortex is obviously different for  the two branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' The change in the magnetization at the  field where the vortex in the downward branch was killed  was more important in the SE side of the disk, indicating  that the vortex was preferentially in this region, as in  cobalt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' Saturation occurs first in the NW side for this  branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' For the upward branch, saturation field is higher  than in cobalt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' Saturation occurs at lower fields in the  N side for this loop branch again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' In general, saturation  is produced first in the NW and at a much higher field  in the SE side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' The region of the disk where the change  in the magnetization in the upward branch between Hv0  and Hv1 is small, occurs as well in the edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' But the  range of fields seems to be larger towards the S side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' From  these observations, it seems that the core of vortex in the  cobalt and iron layers stays in similar regions of the disks,  which could be the reason of the sudden anhinilation of  the vortex in the iron layer and the resulting asymmetric  HL branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='  Figures 13 and 14 shows the Fe and Co HLs in the (10)  orientation for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' In this case, there was not a  clear asymmetry in the magnetic vortex circulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' The  distance ∆q between BRs along the qx and qy axis is 2π  α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='  The HL of the cobalt layer at BR [0, 0] is very similar to  the obtained in the (11) orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' The onset field H0  is similar in the N and S sides and smaller than in the  (11) orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' Moreover, the highest H0 seems to be in  the E and W sides, at [0, ±3] and [1, ±3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' There, the HLs  have a different shape than at the other edges, indicating  that it is in this region where the core of the vortex moves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='  There is still an asymmetry between the N and the S  sides, as in the (11) orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' At the S side, the HLs  with qy ̸= 0 shows an imbalance in the chiral symmetry of  the magnetic circulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' The magnetic circulation is, in  this case, CCW in the downward branch and CW in the  upward branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' Surprisingly, such a chiral asymmetry is  not observed in the N side, remarking the asymmetry in  the magnetic behavior between the two sides of the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='  In this side, the downward branch has an onset H0 field  which is clearly different than in the downward branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='  In the iron layer, its HL at BR [0, 0] has a coercive  field which is larger than in cobalt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' The onset field H0  are similar in the N and S sides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' It is also the highest  of the registered for this orientation, but smaller than in  the (11) orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' The asymmetry between the N and  S sides is not so clear as in cobalt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' There is not either a  clear imbalance in the chiral symmetry of the vortex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='  Discussion  Having the same chiral asymmetry in both layers is not  the expected behavior since the nucleation of the vortex  in each layer creates magnetic poles of the same sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='  This makes energetically more favorable for each disk to  do the nucleation at opposite sides of the border of the  disk, inducing an opposite chiral sense in each layer [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='  This is in fact very critical in this case because the re-  gion where the disk begins to demagnetize seems to be  the same in both layers and for the two orientations of the  10  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' Hysteresis loops of the cobalt layer in the (11) orientation, at chosen BRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' The way [h,k] numbers locate the BRs is  described in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' N side is at qx > 0, E side is at qy > 0 (red color HLs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' Hysteresis loops of the iron layer in the (11) orientation, at chosen BRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' The way [h,k] numbers locate the BRs is  described in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' N side is at qx > 0, E side is at qy > 0 (red color HLs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' Hysteresis loops of the cobalt layer in the (10) orientation, at chosen BRs, [qx,qy]=[h,k] and in conjugated sites ([h,±  k]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' N side is at qx > 0, E side is at qy > 0 (red color HLs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='Co -4 0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='Co -2 0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='Co0 0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='Co 2 0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='Co40  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='M/Ms  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='Co -4 4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='Co -2 4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='Co0 4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='Co 2 6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='Co42  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='Co -2-4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='Co 0-4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='M/Ms  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='Co 4-2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='Co -4-4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='Co 2-6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='o-210  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='Co -4 10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content="Co'0 10  " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='Co 2 10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='Co 4 10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='2-8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='Co 0 -24  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='Co  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='Co 4-10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='Co -4-10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='Co 2-10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='M/Ms  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='L  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
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+page_content='FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' Hysteresis loops of the iron layer in the (10) orientation, at chosen BR, [qx,qy]=[h,k] and in conjugated sites ([h,±  k]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='. N side is at qx > 0, E side is at qy > 0 (red color HLs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' Model proposed to explain the chiral asymmetry in  the array of disks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='  saturated magnetization in the disks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' Therefore, the ori-  gin of the asymmetry has to be related to either a higher  energy process that should over compensate the polar  repulsion between the two layers, or to an effective at-  tractive interaction between layers, possibly induced by  the roughness of the interfaces (N´eel peel orange effect  [36]) and/or the observed thickness gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='  The observed type of chiral asymmetry, which changes  of sign depending on the initial magnetization direc-  tion, is induced in submicron magnets by making their  shape non-centrosymetric:  triangles, truncated disks,  asymmetric rings, ”pac-man” shaped disks or asymmet-  rical magnetic moment distribution[2, 15, 37–41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' Chiral  asymmetry occurs in these systems when the applied field  is at an angle with respect to the mirror symmetry axis  of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='  To explain the origin of the chiral asymmetry in the  studied system, we proposed the following model based  on the existence of a non-centrosymmetry in the mag-  nets of the array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' In the (10) field orientation, the cobalt  layer contains an asymmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='Such an asymmetry might  be related to the anisotropy energy which could be asym-  metrically distributed across the area of the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' This  could happen if its shape is not fully symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' In this  case, the S side has possibly a higher anisotropy than  the N side, what is required to preserve mirror symme-  try between the W and E sides, since chiral asymmetry  is not fully manifested in this orientation [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='  In the  (11) orientation, the oblique angle direction of the ap-  plied field breaks such a mirror symmetry causing the  observed chiral asymmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' Then, when the direction of  the field is positive, the moments in the N side have a  weaker anisotropy and align their moments with the ap-  plied field at lower fields than in the S side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' This creates  an in-plane component of the magnetization perpendicu-  lar to the direction of the field at that particular region,  setting the sense of rotation of the magnetization in the  vortex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' When the direction of the field is the opposite,  the orientation of this in-plane perpendicular component  changes its direction as well, changing the sense of the  magnetic circulation of the vortex to the opposite one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='  The model is depicted in figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' This process seems  to be very solid in the cobalt layer since the branches of  its HLs are symmetric at any point in the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' It is also  the layer that register an asymmetry between N a S sides  in the (10) orientation as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='  Figure 16 shows the Fast Fourier Transform (FFT) of  the image obtained by SEM of the sample which included  more than 103 disks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' A visual inspection shows that the  intensity of the peaks deviates from the expected sym-  metric square shape resulting from the FFT of a diamond  shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='  A detailed analysis of the intensity of the BR  peaks in this figure shows that the ideally diamond-shape  disks are actually rhomboids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='  Their shape asymmetry  arises because the distance between opposite sides of the  diamond-like shape of the disks are not exactly the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='  This makes the shape of the disks non-centrosymmetric,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=', there is not perfect mirror symmetry across any of  the symmetry axis oriented along the (10), (01), (11) or  (-1-1) directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' This deviation from centrosymmetry in  the shape of the magnets is much smaller than the non-  centrosymmetry induced in magnets by purpose to fix  their vortex chirality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' This might indicate that a precise  control of the ILL technique can be used to modulate the  chiral properties of magnets arrays by inducing asymme-  tries in their shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='  Since the HL measurement was done in a single cycle,  it is not possible to assert that the different magnetiza-  Fe-2 0  Fe00  Fe10  Fe 3 0  M/Ms  Fe 2 0  e  Fe -2-2  Fe 2-2  Fe 3-2  Fe 02  Fe -2 2  Fe 2 2  M/Ms  Fe32  Fe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='0-2  Fe 1-2  40  0  40  40  0  40  40  0  40  40  0  40  40  0  40  H (mT)  H (mT)  H (mT)  H (mT)  H (mT)12  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='  Fourier transform of a SEM image of the array  including more than 103 disks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' On the bottom, the intensity  of qx scans across the previous image in two opossite values  of qy to evidence the deviation from perfect mirror symmetry  with respect to the (10) and (01) axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' Scans have their qx  zero shifted for better comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' q0 = 2π/α where α is the  lattice constant of the array  tion paths taken in each of the branches of the HL in Fe  in the (11) orientation, and the observed in some HLs in  the (10) orientation, was a systematic and repetitive pro-  cess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' The HL measured by VSM, which covers a much  larger area than the measured by XRMS, did not show  any asymmetry between the branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' Therefore, these  asymmetries are possibly due to a certain stochasticity in  the magnetic inversion process of the disks, which have  to be collective by the nature of the measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='  CONCLUSIONS  In conclusion, we observed a one-to-one correlation be-  tween the in-plane moment vector transfer of the scat-  tered light from an array of submicron magnets and the  spatial location in the submicron magnet from where  the light comes, converting the magnetic contrast of the  diffraction pattern of the array in an image of the magne-  tization of the disk, averaged over the illuminated disks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='  The conditions for this to happen are related to the mor-  phology of the magnets and the configuration of the ex-  periment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' The surface of the magnets must have a certain  curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' In the present experiment, this is obtained us-  ing magnetron sputtering deposition at normal incidence  in patterned holes, which is one of the most used meth-  ods to produce the kind of studied submicron magnet  arrays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' The grazing incidence angles employed were rel-  atively large to allow the collection of scattered light at a  wider range of in-plane moment transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' At such large  angles, the diffuse scattering due to the roughness and  imperfections at the layer interfaces should be important  what, joined to the resonant energies employed, allowed  a better isolation of the targeted layers even if they were  buried under 20 nm thick layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' This kind of magnetic  microscope effect explains why XRMS is specially sen-  sitive to the chiral asymmetry of the magnetization cir-  culation of the vortex in this kind of samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' This was  used to study the chiral asymmetry of the disks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' Thanks  to this effect, it was possible to identify the presence of a  non-centro-symmetry in the magnetization of the sample  that explained the apparition of the chiral asymmetry at  the oblique angle orientation of the field with respect to  the EA axis of the array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' The physical origin of such a  magnetic non-centro-symmetry was attributed to devia-  tions from centro-symmetry in the shape of the magnets,  giving an indication of the sensitivity of the studied sys-  tem to such deviations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' The presented experiment shows  that XRMS can give a collective vision of the stability  and symmetry breaking process in this kind of system  which is complementary to the obtained by other micro-  scopic techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' There is plenty of room to increase  the quality of the data and to increase the information  extracted from the scattered light in the configuration  used in this experiment, specially the related to the mor-  phology of the magnets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='  ACKNOWLEDGMENTS  This  project  has  been  supported  by  Spanish  MINECO  under  grant  PID2019-  104604RB/AEI/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='13039/501100011033,  and  by  Asturias  FICYT  under  grant  AYUD/2021/51185  with the support of FEDER funds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' acknowledge  Basque Country grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' IT1491-22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
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+page_content='  2  2  4 x10  4x10  2  0  2  4  qx(A)  qy=-3qo  qy=3qo  Intensity  3x10°  2  2  9x(A13  [3] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
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+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
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+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
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+page_content=' Erbe, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
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+page_content=' Zimmer, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
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+page_content=' Luber, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' Raberg, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' Schrefl,  and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
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+page_content=' Vavassori, Journal of Physics: Con-  densed Matter 16, R275 (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
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+page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
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+page_content=' Remhof, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' Zabel,  Review of Scientific Instruments 78, 121301 (2007),  https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
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+page_content=' Westphalen, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' Remhof, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' Schumann, and  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content=' Zabel, Journal of Applied Physics 103, 093913 (2008),  https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
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+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
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+page_content=' Srajer, and  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
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+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
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+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFKT4oBgHgl3EQfky6p/content/2301.11851v1.pdf'}
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+Diffusion Properties of a Brownian Ratchet with Coulomb Friction
+Massimiliano Semeraro1, Giuseppe Gonnella1, Eugenio Lippiello2 and Alessandro Sarracino3,
+1Dipartimento Interateneo di Fisica, Universit`a degli Studi di Bari and INFN,
+Sezione di Bari, via Amendola 173, I-70126 Bari, Italy;
+2Dipartimento di Matematica e Fisica, Universit`a della Campania “Luigi Vanvitelli”, 81100 Caserta, Italy;
+3Dipartimento di Ingegneria, Universit`a della Campania “Luigi Vanvitelli”, 81031 Aversa, Italy
+The motion of a Brownian particle in the presence of Coulomb friction and an asymmetric spatial
+potential was evaluated in this study. The system exhibits a ratchet effect, i.e., an average directed
+motion even in the absence of an external force, induced by the coupling of non-equilibrium condi-
+tions with the spatial asymmetry. Both the average motion and the fluctuations of the Brownian
+particle were analysed. The stationary velocity shows a non-monotonic behaviour as a function of
+both the temperature and the viscosity of the bath. The diffusion properties of the particle, which
+show several time regimes, were also investigated. To highlight the role of non-linear friction in the
+dynamics, a comparison is presented with a linear model of a Brownian particle driven by a con-
+stant external force, which allows for analytical treatment. In particular, the study unveils that the
+passage times between different temporal regimes are strongly affected by the presence of Coulomb
+friction.
+I.
+INTRODUCTION
+Ratchet models (or Brownian motors) are systems
+where non-equilibrium conditions can be exploited to ex-
+tract work from random fluctuations [1].
+Due to the
+breaking of temporal and spatial symmetries, even in
+the absence of an external drive, these systems present
+a spontaneous average net drift, which would be forbid-
+den in equilibrium conditions. Several different sources
+of non-equilibrium dynamics can be considered: time-
+dependent forcing, as in flashing ratchet [2]; correlated
+noise [3]; slow relaxation in glasses [4]; dissipative in-
+teractions as in granular systems [5, 6]; the presence of
+velocity-dependent forces [7]; and even the self propulsion
+in active matter systems [8].
+An intriguing example of a force that depends on the
+velocity of the particle is represented by Coulomb (or
+dry) friction, which takes into account the energy dis-
+sipation contribution due to the slipping on a surface.
+This force can be introduced into a Langevin equation
+as a constant-magnitude force whose sign is opposite to
+the particle velocity. The interest in this model was first
+raised by de Gennes in one of his late papers [9] and
+by Hayakawa in [10].
+As mentioned in [9], examples
+of physically relevant situations where the interplay be-
+tween Coulomb friction and Brownian motion can have
+an interesting role are a micron-size solid particle under
+thermal noise and a macroscopic particle on a vibrated
+surface. The stochastic equation for the particle velocity
+under the action of dry friction has been widely studied,
+and some analytical results have been also obtained in
+the absence of spatial potential, in particular, via a path
+integral approach [11], from the Fokker–Planck equation
+that can be solved to obtain the time-dependent propa-
+gator and the particle velocity correlation function [12],
+or even in the presence of an external force [13], or in
+periodically driven systems [14] and in the presence of
+an elastic band [15]. Other studies have focused on the
+issues related to the definition of entropy production in
+these systems [16]. In the specific context of models of
+Brownian motors, the role of Coulomb friction as a source
+of non-equilibrium able to induce a ratchet effect has also
+been investigated in different systems [7, 17, 18], with ex-
+perimental realizations in the context of driven granular
+gases [19, 20].
+As mentioned above, in order to induce a directed mo-
+tion, an asymmetric spatial potential is crucial.
+This
+introduces a coupling between positions and velocities,
+making the problem not analytically tractable. Here, this
+case is studied with extensive numerical simulations with
+a focus on the dynamics of an underdamped Langevin
+equation in the presence of an asymmetric periodic po-
+tential and Coulomb friction. In particular, in Section II,
+the model and its main parameters are introduced. The
+average ratchet velocity is investigated as a function of
+the viscosity and of the temperature of the thermal bath,
+showing that there are optimal values maximising the
+ratchet effect.
+In Section III,
+the diffusion properties
+of the system are considered, investigating the behaviour
+of the position variance and the mean square displace-
+ment (MSD) for a wide range of parameters.
+A simple
+diffusive behaviour at large times is found in the vari-
+ance, while a more complex scenario is observed for the
+MSD due to the presence of different time regimes. In
+Section IV,
+a comparison is presented of some of the
+observed trends, with those relative to an analytically
+solvable model consisting of an underdamped Brownian
+particle driven by a constant external force. In order to
+further deepen the system behaviour,
+the effect of the
+Coulomb friction on the average first exit time from a
+parabolic potential well is also investigated. Finally, in
+Section V, a summary of and comments on our findings
+are presented.
+arXiv:2301.04073v1  [cond-mat.stat-mech]  10 Jan 2023
+
+2
+II.
+LANGEVIN EQUATION WITH COULOMB
+FRICTION AND RATCHET EFFECT
+The system consists of a unitary mass inertial particle
+in one dimension in contact with a thermal bath in the
+presence of both an asymmetric spatial potential and a
+nonlinear velocity-dependent friction force. The model is
+described by the following Langevin equation
+�
+˙x(t) = v(t)
+˙v(t) = −γv(t) − U ′[x(t)] − ασ[v(t)] + √2γT ξ(t),
+(1)
+where x(t) and v(t) are the position and velocity of the
+particle, respectively; γ is the viscous friction coefficient;
+U[x(t)] is an external potential (the prime denoting a
+derivative with respect to x); α is the constant amplitude
+of the Coulomb friction; σ(v) is the sign function (σ(0) =
+0); ξ(t) is white noise with ⟨ξ(t)⟩ = 0 and ⟨ξ(t)ξ(t′)⟩ =
+δ(t − t′); and T is the bath temperature (we take the
+Boltzmann constant kB = 1 throughout the manuscript).
+The spatial potential is chosen as the ratchet potential
+U[x(t)] = sin[x(t)] + µ sin[2x(t)],
+(2)
+where µ is the parameter that introduces the spatial
+asymmetry.
+Note that the term sin(2x) keeps the po-
+tential periodic. For µ = 0 the potential is symmetric,
+and no ratchet effect occurs. This model was first studied
+in [7], where different velocity-dependent friction forces
+were considered and the form of the non-equilibrium gen-
+eralized fluctuation-dissipation relation was investigated.
+As shown in [7], the system (1) does not satisfy the
+detailed balance condition due to the presence of both
+the nonlinear velocity-dependent dry friction and of the
+ratchet potential (2). As discussed in [21], the issue of
+recovering detailed balance in Langevin equations with
+non-linear velocity-dependent forces requires the intro-
+duction of a non-Gaussian thermal bath and a multi-
+plicative noise. The out-of-equilibrium dynamics of the
+system can instead be exploited to induce the ratchet
+effect, namely a finite non-zero particle average velocity
+⟨v(t)⟩ in the stationary state.
+In the following, the analysis obtained from the numer-
+ical integration of the stochastic differential Equation (1)
+via the Euler–Maruyama algorithm [22] with time step
+dt = 10−3 is presented. The code has been written in
+the Python programming language. The stability of re-
+sults for dt ≤ 10−3 has been checked. The simulation’s
+reduced units are provided as follows: the potential pe-
+riodicity ∆x = 2π is the space unit, the potential depth
+∆U is the energy unit, and the inverse of the friction co-
+efficient γ−1 is the time unit; the velocity unit is given by
+γ∆x. In [7], the stationary velocity of the particle was
+studied at fixed values of the friction coefficient γ = 0.05
+and temperature T = 10, finding a maximum for µ = 0.4
+and α = 1.
+Here, we extend the investigation of the
+model and present the complementary analysis by fix-
+ing the asymmetry parameter and the amplitude of dry
+friction and exploring a wide spectrum of T and γ values.
+Results are presented in the contour plot of Figure 1a.
+Quite interestingly, a non-monotonic behaviour of move-
+ment along both the temperature T and the friction co-
+efficient γ axes (see columns and rows) is evident. This
+observation is further clarified by panels (b) and (c), re-
+porting the average velocity as a function of γ and T
+at fixed T and γ, respectively, extracted from the row
+and column of the chart in panel (a) denoted by the blue
+lines. This means that optimal choices of these param-
+eters, such that the system reaches its maximal velocity
+and the ratchet effect is most enhanced, are possible. In
+particular, the figure reveals that at fixed µ and α, any
+choice such that γT ∼ 1 (see darker diagonal) results in
+a maximum ratchet effect. The explanation behind this
+non-trivial phenomenon relies on the two-fold role played
+by the temperature: on the one hand, thermal fluctua-
+tions allow the particle to explore the spatial potential
+so that the ratchet effect can actually take place; on the
+other had, for too-large values of T, the white noise is
+enhanced and the particle dynamics is less affected by
+the presence of the spatial potential, thereby damping
+the ratchet mechanism. The friction coefficient γ plays a
+similar two-fold role: it contributes to the amplitude of
+the random force, as does the temperature, allowing the
+particle to explore the periodic potential, but it also rep-
+resents the viscous friction experienced by the particle,
+which hinders the dynamics.
+(a)
+(b)
+(c)
+(c)
+Figure 1.
+Average ratchet drift vs.
+temperature T and
+friction coefficient γ. (a): Contour plot for the average ratchet
+velocity ⟨v(t)⟩ as a function of the temperature T and the friction
+γ. Parameters: α = 1.0, µ = 0.4. The blue horizontal and vertical
+lines highlight the ⟨v(t)⟩ values at fixed T and γ used to plot panels
+(b) and (c), respectively. (b) and (c): Trend of ⟨v(t)⟩ as a function
+of γ at fixed T = 10.0 and T at fixed γ = 0.05, respectively, as
+extracted from panel (a).
+III.
+DIFFUSION PROPERTIES
+Here, the analysis of the fluctuations around the aver-
+age motion and of the diffusion properties is presented.
+These quantities are relevant to highlighting the role
+of dry friction in the dynamics of the Brownian par-
+
+3
+ticle.
+We focus on the position variance V ar[x(t)] =
+⟨x(t)2⟩−⟨x(t)⟩2 and the MSD ∆(t, t0) = ⟨[x(t)−x(t0)]2⟩
+.
+The position variance V ar[x(t)] is reported in Figure 2,
+for various choices of temperature T (panel (a)) and fric-
+tion coefficient γ (panel (b)) at fixed γ and T, respec-
+tively. The variance is stuck to a constant parameter-
+dependent value at small times as the particle is trapped
+in the potential well.
+This phenomenon resembles the
+dispersionless diffusion regime described in [23], which
+has recently been reconsidered in [24, 25].
+If the sys-
+tem parameters allow the particle to eventually escape
+from the initial potential well, it starts exploring other
+regions, and a diffusive regime V ar[x(t)] ∼ t sets in. The
+diffusion coefficient Dm characterising such a regime was
+measured and is reported in the insets of Figure 2. As in-
+tuitively expected, Dm shows a monotonic trend with T
+and γ, from low values due to the trapping action of the
+potential towards the overdamped prediction D = T/γ
+in both cases (see insets of panels (a) and (b)).
+(a)
+(b)
+~t
+~t
+Figure 2.
+Variance vs.
+temperature T and friction coef-
+ficient γ. Position variance V ar[x(t)] for various choices of tem-
+perature T at fixed γ = 0.05 (panel (a)) and for various choices
+of friction coefficient γ at fixed T = 10 (panel (b)) up to simula-
+tion time t = 106. The insets report the ratio between the mea-
+sured diffusion coefficient Dm and the asymptotic underdamped
+one D = T/γ. Other parameters: α = 1.0, µ = 0.4.
+The MSD instead shows a plateau, followed by diffu-
+sive and ballistic regimes due to the interplay between
+particle fluctuation and ratchet potential. As shown in
+Figure 3a,c one can identify several regimes: (i) a first
+initial ballistic regime due to inertial effects; (ii) a plateau
+regime, whose duration depends on the values of friction
+coefficient γ and temperature T; (iii) a diffusive regime;
+(iv) a final ballistic regime due to the directed motion
+induced by the ratchet effect.
+In order to analyse the
+dependence of the time duration of such regimes on the
+model parameters, we focus on the crossover times for the
+passage from ballistic to diffusive tbd and from diffusive
+to ballistic tdb extracted from the intercept between the
+curves ∼ t or ∼ t2 fitting two consecutive regimes. The
+measured values can be appreciated from Figure 3b,d.
+An interesting non-monotonic behaviour of tdb as func-
+tion of both T and γ appears. As already underlined,
+this occurs because the temperature increase above a
+certain threshold allows thermal fluctuations to play a
+dominant role, making the spatial potential less effective
+and hindering the occurrence of the ratchet mechanism.
+This therefore leads to a larger time for the final ballistic
+regime to set in. Regarding the behaviour of the crossover
+time from the initial ballistic regime due to inertia to
+the intermediate diffusive regime, from the inset of Fig-
+ure 3b, one observes a continuous growth as a function
+of T, while from the inset of Figure 3d, a non-monotonic
+trend as function of γ is apparent. Indeed, as the viscous
+friction γ is increased at fixed temperature, the inertial
+effects become negligible, making the passage to diffusion
+very rapid. On the contrary, an increase in T at fixed γ
+results in a longer inertial regime. These behaviours will
+be reconsidered in the light of the simple constant force
+model discussed in Section IV. Finally, note that for the
+range of parameters investigated, one can clearly iden-
+tify the first crossover time tbd only in a few cases, so one
+cannot comment on its general behaviour.
+(a)
+(c)
+(b)
+(d)
+~t
+~t
+~t2
+~t2
+~t2
+~t2
+Figure 3. Diffusion properties vs. temperature T and fric-
+tion coefficient γ. (a) and (c): Position mean square displace-
+ment ∆(t, 0) for various choices of temperature T at fixed γ = 0.05
+(panel (a)) and friction coefficient γ at fixed T = 10.0 (panel (c))
+up to simulation time t = 106. (b) and (d): Diffusive→ballistic
+crossover times tdb as a function of the temperature T at fixed
+γ = 0.05 (panel (b)) and friction coefficient γ at fixed T = 10.0
+(panel (d)) measured from ∆(t, 0). The insets report the measured
+ballistic→diffusive crossover times tbd. Other parameters: α = 1.0,
+µ = 0.4.
+IV.
+CONSTANT FORCE MODEL
+The ratchet effect is characterised by a spontaneous
+net drift arising from the coupling of non-equilibrium
+conditions with spatial asymmetry. A net average veloc-
+ity can also be trivially induced with only viscous fric-
+tion and no spatial potential, applying a constant ex-
+ternal force. Here this simple case is considered, which
+allows for analytical treatment and comparison of its dif-
+fusional properties with those observed in the ratchet
+system. One finds that some qualitative features, such
+as the several MSD regimes, can be reproduced, while
+
+4
+others cannot, because of the peculiar role of nonlinear
+velocity-dependent forces.
+The constant force model consists of the following un-
+derdamped Langevin equation:
+�
+˙x(t) = v(t)
+˙v(t) = −γv(t) + F + √2γT ξ(t),
+(3)
+where F is a constant external force. This can be easily
+solved, yielding, for the mean velocity and position,
+⟨v(t)⟩ = v0e−γt +
+� t
+0
+dt′e−γ(t−t′)(F +
+�
+2γT ⟨ξ(t′)⟩)
+= v0e−γt + Fe−γt
+� t
+0
+dt′eγt′
+= v0e−γt + F
+γ (1 − e−γt) → F
+γ ,
+(4)
+and
+⟨x(t)⟩ =
+�� t
+0
+dt′ v(t′)
+�
+=
+� t
+0
+dt′ ⟨v(t′)⟩
+=
+� t
+0
+dt′
+��
+v0 − F
+γ
+�
+e−γt′ + F
+γ
+�
+=
+�
+v0 − F
+γ
+� 1 − e−γt
+γ
++ F
+γ t → 1
+γ
+�
+v0 − F
+γ
+�
++ F
+γ t ,
+respectively, where the arrows denote the large time
+limit.
+In order to compare such behaviors with those
+found in the ratchet model, for each choice of temper-
+ature T and friction coefficient γ, the constant force is
+set at F = γ ⟨v(t)⟩, where ⟨v(t)⟩ is the average velocity
+in the corresponding ratchet system. The velocity auto-
+correlation function is given by
+⟨v(t1)v(t2)⟩ = v2
+0e−γ(t1+t2) + v0F
+γ e−γt1(1 − e−γt2)
++ v0F
+γ e−γt2(1 − e−γt1)
++ F 2
+γ2 (1 − e−γt1)(1 − e−γt2)
++ T(e−γ|t1−t2| − e−γ(t1+t2)),
+so that, at equal times t1 = t2 = t, one obtains
+⟨v2(t)⟩ −→ F 2
+γ2 + T.
+Finally, the time integration of the velocity autocorrela-
+tion function yields the MSD
+⟨[x(t) − x(0)]2⟩ =
+�� t
+0
+v(t′)dt′
+� 2
+0
+v(t′′)dt′′
+�
+= v2
+0
+γ2 (e−γt − 1)2 + 2v0F
+γ
+�1 − e−γt
+γ
+t − (e−γt− − 1)2
+γ2
+�
++ F 2
+γ2
+�
+t2 + 2e−γt − 1
+γ
+t + (e−γt − 1)2
+γ2
+�
++ 2T
+γ
+�
+t − 1 − e−γt
+γ
+�
+− T (e−γt − 1)2
+γ2
+.
+(5)
+It is interesting to simplify the above expression in the
+large and small time limits. At large times t ≫ γ−1, the
+MSD can be approximated as
+⟨(x(t) − x(0))2⟩ ≃ v2
+0
+γ2 − 2v0F
+γ3
++ F 2
+γ4 − 3T
+γ2
++ 2
+�v0F
+γ2 − F 2
+γ3 + T
+γ
+�
+t + F 2
+γ2 t2.
+In the opposite limit, t ≪ γ, the exponential expansion
+around t = 0 leads to the expression
+⟨(x(t) − x(0))2⟩ ≃ v2
+0t2+
+�
+−γv2
+0 + v0F + 2
+3γT
+�
+t3+o(t4).
+The previous formulae allow the crossover times to be
+estimated explicitly. Indeed, one easily finds that at large
+times, the passage from a diffusive to the ballistic regime
+occurs at a time
+tdb = γ2
+F 2
+�v0F
+γ2 − F 2
+γ3 + T
+γ
+�
+.
+(6)
+At small times, when v0 ̸= 0, one instead finds that the
+initial ballistic regime changes to diffusion at time
+tbd =
+v2
+0
+−γv2
+0 + v0F + 2
+3T .
+In order to highlight the different contributions from
+the non-linear dry friction and from the potential, we re-
+port in Figure 4a the behaviour of the MSD for the con-
+stant force model compared to the ratchet system. We
+also compute the MSD in the case α = 0 (no Coulomb
+friction) and in the case U = 0 (no potential), in order
+to better understand the role of the different terms con-
+tributing to the dynamics in the Langevin equation. In
+the absence of potential, one observes that the first two
+time regimes are very similar to the case of the ratchet
+model, while, as expected, the final ballistic regime does
+not take place. On the contrary, in the absence of dry
+friction, the particle shows a diffusive behaviour similar
+to the constant force model.
+Figures 4b,c propose in-
+stead a comparison between the trend of the crossover
+times tdb in the constant force model, obtained through
+(6), and the one in the ratchet model, which is numeri-
+cally evaluated. In all cases, one finds a qualitative but
+not quantitative agreement, with some differences wor-
+thy of attention.
+For example, in some cases, as the
+ratchet model leaves the initial ballistic regime and en-
+ters the diffusive one, the constant force model instead
+enters a superdiffusive regime ∼ t3 lasting roughly two
+decades before reaching the final ballistic regime (see in-
+set of panel (a)). An interesting observation regards the
+onset of the ballistic behaviour as a function of both T
+and γ: from Figures 4b,c, one finds that, for small values
+of temperature T and viscous coefficient γ, respectively,
+the crossover times from diffusive to ballistic regimes are
+an order of magnitude smaller in the ratchet model with
+
+5
+respect to the constant force model. This reveals the dra-
+matic role played by non-linear friction in speeding the
+dynamics of the system for a range of parameters.
+(a)
+~t3
+~t2
+(b)
+(c)
+Figure 4. Constant-force model. (a): Mean square displace-
+ment ∆(t, 0) of the constant force model Equation (3) compared
+to those computed in the no-potential (U = 0 in Equation (1)),
+no-Coulomb friction (α = 0 in Equation (1)), and ratchet model
+(Equation (1) with α = 1). Points denote numerical results, and
+the blue solid line is the theoretical expression Equation 5. The
+inset reports the theoretical and numerical ∆(t, 0) for a different
+force, highlighting the ∼ t3 superdiffusive regime.
+(b) and (c):
+Comparison between the computed tdb in the ratchet and in the
+constant force model as a function of the temperature T and of
+the friction coefficient γ, respectively.
+The blue dots are evalu-
+ated through (6), and red dots are numerically estimated. Param-
+eters: α = 1.0, µ = 0.4, γ = 10.0 in panel (b); T = 0.05 in panel
+(c).
+F = 2.28 · 10−4 is the constant force corresponding to the
+T = 10, γ = 0.05, µ = 0.4 ratchet case, while F = 1.0, v0 = 1.0 are
+the parameters chosen for the inset.
+Characteristic Escape Times from a Single Well
+Since the onset of the ratchet effect is related to the
+presence of the spatial potential, also affecting the dif-
+fusion properties, it is interesting to further investigate
+the role of Coulomb friction on the escape time from a
+parabolic well. In particular, we compare the behaviour
+obtained in the ratchet model with that computed for the
+simple constant-force model.
+For the simple Langevin
+equation with no external force, analytical results are re-
+viewed in [26]. In the presence of dry friction and the
+absence of spatial potential, the problem has been ad-
+dressed in [27]. Here, a harmonic approximation kx2/2
+of the ratchet potential around one if its minima is con-
+sidered. The elastic constant k is obtained from a second-
+order expansion of the ratchet potential (2) around one
+of its minima. The average time ⟨te⟩ necessary for the
+particle to reach a distance d from the minimum of the
+potential was computed. Distances are expressed in units
+of d0, which was chosen in such a way that kd2
+0/2 = ∆U,
+with ∆U the ratchet potential depth (see the inset of
+Figure 5 for a graphical depiction). We are interested in
+showing how the mean escape time ⟨te⟩ as function of the
+escape distance d varies in several conditions. As shown
+in Figure 5, the smallest mean exit times are observed in
+the case of the constant force model. More specifically,
+at small values of d/d0, the behaviour for the constant
+force model is very similar to the harmonic case; both
+trends show a saturation, and significant differences only
+arise upon increasing the distance from the bottom of
+the well. On the other hand, the effect of dry friction
+is much stronger, and the marked increase in exit times,
+in the range of explored parameters, seems to exhibit
+an exponential behaviour as a function of the distance
+d/d0. Finally, note that the very long exit times in the
+model with dry friction can be related to the extended
+plateaus observed in the variance, where the particle re-
+mains trapped in the well for long times.
+Figure 5.
+Average escape times.
+Average escape time ⟨te⟩
+for a harmonically confined Brownian particle with and without
+Coulomb friction and constant-force starting at x0 = 0 as a func-
+tion of the right escape point. The inset reports a graphical depic-
+tion of the harmonic approximation introduced in the main text.
+For the sake of clarity, the harmonic potential is horizontally and
+graphically shifted. Parameters: γ = 0.05, T = 10.0, (k = 2.20)
+(α = 1.0). d0 = 1.49 is chosen in such a way that kd2
+0/2 = ∆U,
+with ∆U as the ratchet potential depth for µ = 0.4. F = 2.28·10−4
+is the constant force corresponding to the γ = 0.05, T = 10, µ = 0.4
+ratchet case.
+V.
+CONCLUSIONS
+In this work, a stochastic differential equation featur-
+ing non-linear friction and an asymmetric spatial poten-
+tial has been studied. The system is out of equilibrium
+and shows the occurrence of the ratchet effect, namely
+a net average drift, with a non-monotonic magnitude
+as a function of the bath parameters, temperature, and
+viscous friction. The diffusion properties of the model
+have also been investigated, with a particular focus on
+the position variance and on the MSD, finding the oc-
+
+6
+currence of different regimes and different characteristic
+times separating such time regimes. Finally, the anal-
+ysis proved that the diffusion properties of the ratchet
+model under scrutiny present strong differences with re-
+spect to a simple Brownian particle with inertia under
+the action of an external constant force. Our study con-
+tributes by shedding light on some important dynami-
+cal features of systems characterized by the presence of
+both non-linear friction and fluctuations, which play a
+central role in many natural phenomena, from biological
+molecular motors at the cellular scale to earthquakes and
+avalanches at geophysical scales, and in experimental ap-
+plications, in particular for nano- and micro-friction such
+as for nanometer contacts in the context of micro- and
+nanomachines [28].
+We plan to extend the study of this model to the frame-
+work of stochastic thermodynamics, addressing the inter-
+esting issues related to the definition of entropy produc-
+tion and fluctuating efficiency, in future works.
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+
diff --git a/VNE2T4oBgHgl3EQftwhp/content/tmp_files/load_file.txt b/VNE2T4oBgHgl3EQftwhp/content/tmp_files/load_file.txt
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf,len=478
+page_content='Diffusion Properties of a Brownian Ratchet with Coulomb Friction  Massimiliano Semeraro1, Giuseppe Gonnella1, Eugenio Lippiello2 and Alessandro Sarracino3,  1Dipartimento Interateneo di Fisica, Universit`a degli Studi di Bari and INFN,  Sezione di Bari, via Amendola 173, I-70126 Bari, Italy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='  2Dipartimento di Matematica e Fisica, Universit`a della Campania “Luigi Vanvitelli”, 81100 Caserta, Italy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='  3Dipartimento di Ingegneria, Universit`a della Campania “Luigi Vanvitelli”, 81031 Aversa, Italy  The motion of a Brownian particle in the presence of Coulomb friction and an asymmetric spatial  potential was evaluated in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' The system exhibits a ratchet effect, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=', an average directed  motion even in the absence of an external force, induced by the coupling of non-equilibrium condi-  tions with the spatial asymmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' Both the average motion and the fluctuations of the Brownian  particle were analysed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' The stationary velocity shows a non-monotonic behaviour as a function of  both the temperature and the viscosity of the bath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' The diffusion properties of the particle, which  show several time regimes, were also investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' To highlight the role of non-linear friction in the  dynamics, a comparison is presented with a linear model of a Brownian particle driven by a con-  stant external force, which allows for analytical treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' In particular, the study unveils that the  passage times between different temporal regimes are strongly affected by the presence of Coulomb  friction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='  INTRODUCTION  Ratchet models (or Brownian motors) are systems  where non-equilibrium conditions can be exploited to ex-  tract work from random fluctuations [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='  Due to the  breaking of temporal and spatial symmetries, even in  the absence of an external drive, these systems present  a spontaneous average net drift, which would be forbid-  den in equilibrium conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' Several different sources  of non-equilibrium dynamics can be considered: time-  dependent forcing, as in flashing ratchet [2];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' correlated  noise [3];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' slow relaxation in glasses [4];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' dissipative in-  teractions as in granular systems [5, 6];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' the presence of  velocity-dependent forces [7];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' and even the self propulsion  in active matter systems [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='  An intriguing example of a force that depends on the  velocity of the particle is represented by Coulomb (or  dry) friction, which takes into account the energy dis-  sipation contribution due to the slipping on a surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='  This force can be introduced into a Langevin equation  as a constant-magnitude force whose sign is opposite to  the particle velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' The interest in this model was first  raised by de Gennes in one of his late papers [9] and  by Hayakawa in [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='  As mentioned in [9], examples  of physically relevant situations where the interplay be-  tween Coulomb friction and Brownian motion can have  an interesting role are a micron-size solid particle under  thermal noise and a macroscopic particle on a vibrated  surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' The stochastic equation for the particle velocity  under the action of dry friction has been widely studied,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='  and some analytical results have been also obtained in  the absence of spatial potential,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' in particular,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' via a path  integral approach [11],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' from the Fokker–Planck equation  that can be solved to obtain the time-dependent propa-  gator and the particle velocity correlation function [12],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='  or even in the presence of an external force [13],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' or in  periodically driven systems [14] and in the presence of  an elastic band [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' Other studies have focused on the  issues related to the definition of entropy production in  these systems [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' In the specific context of models of  Brownian motors, the role of Coulomb friction as a source  of non-equilibrium able to induce a ratchet effect has also  been investigated in different systems [7, 17, 18], with ex-  perimental realizations in the context of driven granular  gases [19, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='  As mentioned above, in order to induce a directed mo-  tion, an asymmetric spatial potential is crucial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='  This  introduces a coupling between positions and velocities,  making the problem not analytically tractable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' Here, this  case is studied with extensive numerical simulations with  a focus on the dynamics of an underdamped Langevin  equation in the presence of an asymmetric periodic po-  tential and Coulomb friction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' In particular, in Section II,  the model and its main parameters are introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' The  average ratchet velocity is investigated as a function of  the viscosity and of the temperature of the thermal bath,  showing that there are optimal values maximising the  ratchet effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='  In Section III,  the diffusion properties  of the system are considered, investigating the behaviour  of the position variance and the mean square displace-  ment (MSD) for a wide range of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='  A simple  diffusive behaviour at large times is found in the vari-  ance, while a more complex scenario is observed for the  MSD due to the presence of different time regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' In  Section IV,  a comparison is presented of some of the  observed trends, with those relative to an analytically  solvable model consisting of an underdamped Brownian  particle driven by a constant external force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' In order to  further deepen the system behaviour,  the effect of the  Coulomb friction on the average first exit time from a  parabolic potential well is also investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' Finally, in  Section V, a summary of and comments on our findings  are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='04073v1  [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='stat-mech]  10 Jan 2023  2  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='  LANGEVIN EQUATION WITH COULOMB  FRICTION AND RATCHET EFFECT  The system consists of a unitary mass inertial particle  in one dimension in contact with a thermal bath in the  presence of both an asymmetric spatial potential and a  nonlinear velocity-dependent friction force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' The model is  described by the following Langevin equation  �  ˙x(t) = v(t)  ˙v(t) = −γv(t) − U ′[x(t)] − ασ[v(t)] + √2γT ξ(t),  (1)  where x(t) and v(t) are the position and velocity of the  particle, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' γ is the viscous friction coefficient;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='  U[x(t)] is an external potential (the prime denoting a  derivative with respect to x);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' α is the constant amplitude  of the Coulomb friction;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' σ(v) is the sign function (σ(0) =  0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' ξ(t) is white noise with ⟨ξ(t)⟩ = 0 and ⟨ξ(t)ξ(t′)⟩ =  δ(t − t′);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' and T is the bath temperature (we take the  Boltzmann constant kB = 1 throughout the manuscript).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='  The spatial potential is chosen as the ratchet potential  U[x(t)] = sin[x(t)] + µ sin[2x(t)],  (2)  where µ is the parameter that introduces the spatial  asymmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='  Note that the term sin(2x) keeps the po-  tential periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' For µ = 0 the potential is symmetric,  and no ratchet effect occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' This model was first studied  in [7], where different velocity-dependent friction forces  were considered and the form of the non-equilibrium gen-  eralized fluctuation-dissipation relation was investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='  As shown in [7], the system (1) does not satisfy the  detailed balance condition due to the presence of both  the nonlinear velocity-dependent dry friction and of the  ratchet potential (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' As discussed in [21], the issue of  recovering detailed balance in Langevin equations with  non-linear velocity-dependent forces requires the intro-  duction of a non-Gaussian thermal bath and a multi-  plicative noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' The out-of-equilibrium dynamics of the  system can instead be exploited to induce the ratchet  effect, namely a finite non-zero particle average velocity  ⟨v(t)⟩ in the stationary state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='  In the following, the analysis obtained from the numer-  ical integration of the stochastic differential Equation (1)  via the Euler–Maruyama algorithm [22] with time step  dt = 10−3 is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' The code has been written in  the Python programming language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' The stability of re-  sults for dt ≤ 10−3 has been checked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' The simulation’s  reduced units are provided as follows: the potential pe-  riodicity ∆x = 2π is the space unit, the potential depth  ∆U is the energy unit, and the inverse of the friction co-  efficient γ−1 is the time unit;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' the velocity unit is given by  γ∆x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' In [7], the stationary velocity of the particle was  studied at fixed values of the friction coefficient γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='05  and temperature T = 10, finding a maximum for µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='4  and α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='  Here, we extend the investigation of the  model and present the complementary analysis by fix-  ing the asymmetry parameter and the amplitude of dry  friction and exploring a wide spectrum of T and γ values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='  Results are presented in the contour plot of Figure 1a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='  Quite interestingly, a non-monotonic behaviour of move-  ment along both the temperature T and the friction co-  efficient γ axes (see columns and rows) is evident.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' This  observation is further clarified by panels (b) and (c), re-  porting the average velocity as a function of γ and T  at fixed T and γ, respectively, extracted from the row  and column of the chart in panel (a) denoted by the blue  lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' This means that optimal choices of these param-  eters, such that the system reaches its maximal velocity  and the ratchet effect is most enhanced, are possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' In  particular, the figure reveals that at fixed µ and α, any  choice such that γT ∼ 1 (see darker diagonal) results in  a maximum ratchet effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' The explanation behind this  non-trivial phenomenon relies on the two-fold role played  by the temperature: on the one hand, thermal fluctua-  tions allow the particle to explore the spatial potential  so that the ratchet effect can actually take place;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' on the  other had, for too-large values of T, the white noise is  enhanced and the particle dynamics is less affected by  the presence of the spatial potential, thereby damping  the ratchet mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' The friction coefficient γ plays a  similar two-fold role: it contributes to the amplitude of  the random force, as does the temperature, allowing the  particle to explore the periodic potential, but it also rep-  resents the viscous friction experienced by the particle,  which hinders the dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='  (a)  (b)  (c)  (c)  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='  Average ratchet drift vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='  temperature T and  friction coefficient γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' (a): Contour plot for the average ratchet  velocity ⟨v(t)⟩ as a function of the temperature T and the friction  γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' Parameters: α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='0, µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' The blue horizontal and vertical  lines highlight the ⟨v(t)⟩ values at fixed T and γ used to plot panels  (b) and (c), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' (b) and (c): Trend of ⟨v(t)⟩ as a function  of γ at fixed T = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='0 and T at fixed γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='05, respectively, as  extracted from panel (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='  DIFFUSION PROPERTIES  Here, the analysis of the fluctuations around the aver-  age motion and of the diffusion properties is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='  These quantities are relevant to highlighting the role  of dry friction in the dynamics of the Brownian par-  3  ticle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='  We focus on the position variance V ar[x(t)] =  ⟨x(t)2⟩−⟨x(t)⟩2 and the MSD ∆(t, t0) = ⟨[x(t)−x(t0)]2⟩  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='  The position variance V ar[x(t)] is reported in Figure 2,  for various choices of temperature T (panel (a)) and fric-  tion coefficient γ (panel (b)) at fixed γ and T, respec-  tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' The variance is stuck to a constant parameter-  dependent value at small times as the particle is trapped  in the potential well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='  This phenomenon resembles the  dispersionless diffusion regime described in [23], which  has recently been reconsidered in [24, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='  If the sys-  tem parameters allow the particle to eventually escape  from the initial potential well, it starts exploring other  regions, and a diffusive regime V ar[x(t)] ∼ t sets in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' The  diffusion coefficient Dm characterising such a regime was  measured and is reported in the insets of Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' As in-  tuitively expected, Dm shows a monotonic trend with T  and γ, from low values due to the trapping action of the  potential towards the overdamped prediction D = T/γ  in both cases (see insets of panels (a) and (b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='  (a)  (b)  ~t  ~t  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='  Variance vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='  temperature T and friction coef-  ficient γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' Position variance V ar[x(t)] for various choices of tem-  perature T at fixed γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='05 (panel (a)) and for various choices  of friction coefficient γ at fixed T = 10 (panel (b)) up to simula-  tion time t = 106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' The insets report the ratio between the mea-  sured diffusion coefficient Dm and the asymptotic underdamped  one D = T/γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' Other parameters: α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='0, µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='  The MSD instead shows a plateau, followed by diffu-  sive and ballistic regimes due to the interplay between  particle fluctuation and ratchet potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' As shown in  Figure 3a,c one can identify several regimes: (i) a first  initial ballistic regime due to inertial effects;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' (ii) a plateau  regime, whose duration depends on the values of friction  coefficient γ and temperature T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' (iii) a diffusive regime;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='  (iv) a final ballistic regime due to the directed motion  induced by the ratchet effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='  In order to analyse the  dependence of the time duration of such regimes on the  model parameters, we focus on the crossover times for the  passage from ballistic to diffusive tbd and from diffusive  to ballistic tdb extracted from the intercept between the  curves ∼ t or ∼ t2 fitting two consecutive regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' The  measured values can be appreciated from Figure 3b,d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='  An interesting non-monotonic behaviour of tdb as func-  tion of both T and γ appears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' As already underlined,  this occurs because the temperature increase above a  certain threshold allows thermal fluctuations to play a  dominant role, making the spatial potential less effective  and hindering the occurrence of the ratchet mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='  This therefore leads to a larger time for the final ballistic  regime to set in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' Regarding the behaviour of the crossover  time from the initial ballistic regime due to inertia to  the intermediate diffusive regime, from the inset of Fig-  ure 3b, one observes a continuous growth as a function  of T, while from the inset of Figure 3d, a non-monotonic  trend as function of γ is apparent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' Indeed, as the viscous  friction γ is increased at fixed temperature, the inertial  effects become negligible, making the passage to diffusion  very rapid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' On the contrary, an increase in T at fixed γ  results in a longer inertial regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' These behaviours will  be reconsidered in the light of the simple constant force  model discussed in Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' Finally, note that for the  range of parameters investigated, one can clearly iden-  tify the first crossover time tbd only in a few cases, so one  cannot comment on its general behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='  (a)  (c)  (b)  (d)  ~t  ~t  ~t2  ~t2  ~t2  ~t2  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' Diffusion properties vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' temperature T and fric-  tion coefficient γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' (a) and (c): Position mean square displace-  ment ∆(t, 0) for various choices of temperature T at fixed γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='05  (panel (a)) and friction coefficient γ at fixed T = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='0 (panel (c))  up to simulation time t = 106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' (b) and (d): Diffusive→ballistic  crossover times tdb as a function of the temperature T at fixed  γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='05 (panel (b)) and friction coefficient γ at fixed T = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='0  (panel (d)) measured from ∆(t, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' The insets report the measured  ballistic→diffusive crossover times tbd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' Other parameters: α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='0,  µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='  CONSTANT FORCE MODEL  The ratchet effect is characterised by a spontaneous  net drift arising from the coupling of non-equilibrium  conditions with spatial asymmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' A net average veloc-  ity can also be trivially induced with only viscous fric-  tion and no spatial potential, applying a constant ex-  ternal force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' Here this simple case is considered, which  allows for analytical treatment and comparison of its dif-  fusional properties with those observed in the ratchet  system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' One finds that some qualitative features, such  as the several MSD regimes, can be reproduced, while  4  others cannot, because of the peculiar role of nonlinear  velocity-dependent forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='  The constant force model consists of the following un-  derdamped Langevin equation:  �  ˙x(t) = v(t)  ˙v(t) = −γv(t) + F + √2γT ξ(t),  (3)  where F is a constant external force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' This can be easily  solved, yielding, for the mean velocity and position,  ⟨v(t)⟩ = v0e−γt +  � t  0  dt′e−γ(t−t′)(F +  �  2γT ⟨ξ(t′)⟩)  = v0e−γt + Fe−γt  � t  0  dt′eγt′  = v0e−γt + F  γ (1 − e−γt) → F  γ ,  (4)  and  ⟨x(t)⟩ =  �� t  0  dt′ v(t′)  �  =  � t  0  dt′ ⟨v(t′)⟩  =  � t  0  dt′  ��  v0 − F  γ  �  e−γt′ + F  γ  �  =  �  v0 − F  γ  � 1 − e−γt  γ  + F  γ t → 1  γ  �  v0 − F  γ  �  + F  γ t ,  respectively, where the arrows denote the large time  limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='  In order to compare such behaviors with those  found in the ratchet model, for each choice of temper-  ature T and friction coefficient γ, the constant force is  set at F = γ ⟨v(t)⟩, where ⟨v(t)⟩ is the average velocity  in the corresponding ratchet system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' The velocity auto-  correlation function is given by  ⟨v(t1)v(t2)⟩ = v2  0e−γ(t1+t2) + v0F  γ e−γt1(1 − e−γt2)  + v0F  γ e−γt2(1 − e−γt1)  + F 2  γ2 (1 − e−γt1)(1 − e−γt2)  + T(e−γ|t1−t2| − e−γ(t1+t2)),  so that, at equal times t1 = t2 = t, one obtains  ⟨v2(t)⟩ −→ F 2  γ2 + T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='  Finally, the time integration of the velocity autocorrela-  tion function yields the MSD  ⟨[x(t) − x(0)]2⟩ =  �� t  0  v(t′)dt′  � 2  0  v(t′′)dt′′  �  = v2  0  γ2 (e−γt − 1)2 + 2v0F  γ  �1 − e−γt  γ  t − (e−γt− − 1)2  γ2  �  + F 2  γ2  �  t2 + 2e−γt − 1  γ  t + (e−γt − 1)2  γ2  �  + 2T  γ  �  t − 1 − e−γt  γ  �  − T (e−γt − 1)2  γ2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='  (5)  It is interesting to simplify the above expression in the  large and small time limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' At large times t ≫ γ−1, the  MSD can be approximated as  ⟨(x(t) − x(0))2⟩ ≃ v2  0  γ2 − 2v0F  γ3  + F 2  γ4 − 3T  γ2  + 2  �v0F  γ2 − F 2  γ3 + T  γ  �  t + F 2  γ2 t2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='  In the opposite limit, t ≪ γ, the exponential expansion  around t = 0 leads to the expression  ⟨(x(t) − x(0))2⟩ ≃ v2  0t2+  �  −γv2  0 + v0F + 2  3γT  �  t3+o(t4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='  The previous formulae allow the crossover times to be  estimated explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' Indeed, one easily finds that at large  times, the passage from a diffusive to the ballistic regime  occurs at a time  tdb = γ2  F 2  �v0F  γ2 − F 2  γ3 + T  γ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='  (6)  At small times, when v0 ̸= 0, one instead finds that the  initial ballistic regime changes to diffusion at time  tbd =  v2  0  −γv2  0 + v0F + 2  3T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='  In order to highlight the different contributions from  the non-linear dry friction and from the potential, we re-  port in Figure 4a the behaviour of the MSD for the con-  stant force model compared to the ratchet system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' We  also compute the MSD in the case α = 0 (no Coulomb  friction) and in the case U = 0 (no potential), in order  to better understand the role of the different terms con-  tributing to the dynamics in the Langevin equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' In  the absence of potential, one observes that the first two  time regimes are very similar to the case of the ratchet  model, while, as expected, the final ballistic regime does  not take place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' On the contrary, in the absence of dry  friction, the particle shows a diffusive behaviour similar  to the constant force model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='  Figures 4b,c propose in-  stead a comparison between the trend of the crossover  times tdb in the constant force model, obtained through  (6), and the one in the ratchet model, which is numeri-  cally evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' In all cases, one finds a qualitative but  not quantitative agreement, with some differences wor-  thy of attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='  For example, in some cases, as the  ratchet model leaves the initial ballistic regime and en-  ters the diffusive one, the constant force model instead  enters a superdiffusive regime ∼ t3 lasting roughly two  decades before reaching the final ballistic regime (see in-  set of panel (a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' An interesting observation regards the  onset of the ballistic behaviour as a function of both T  and γ: from Figures 4b,c, one finds that, for small values  of temperature T and viscous coefficient γ, respectively,  the crossover times from diffusive to ballistic regimes are  an order of magnitude smaller in the ratchet model with  5  respect to the constant force model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' This reveals the dra-  matic role played by non-linear friction in speeding the  dynamics of the system for a range of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='  (a)  ~t3  ~t2  (b)  (c)  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' Constant-force model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' (a): Mean square displace-  ment ∆(t, 0) of the constant force model Equation (3) compared  to those computed in the no-potential (U = 0 in Equation (1)),  no-Coulomb friction (α = 0 in Equation (1)), and ratchet model  (Equation (1) with α = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' Points denote numerical results, and  the blue solid line is the theoretical expression Equation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' The  inset reports the theoretical and numerical ∆(t, 0) for a different  force, highlighting the ∼ t3 superdiffusive regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='  (b) and (c):  Comparison between the computed tdb in the ratchet and in the  constant force model as a function of the temperature T and of  the friction coefficient γ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='  The blue dots are evalu-  ated through (6), and red dots are numerically estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' Param-  eters: α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='0, µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='4, γ = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='0 in panel (b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='05 in panel  (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='  F = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='28 · 10−4 is the constant force corresponding to the  T = 10, γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='05, µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='4 ratchet case, while F = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='0, v0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='0 are  the parameters chosen for the inset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='  Characteristic Escape Times from a Single Well  Since the onset of the ratchet effect is related to the  presence of the spatial potential, also affecting the dif-  fusion properties, it is interesting to further investigate  the role of Coulomb friction on the escape time from a  parabolic well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' In particular, we compare the behaviour  obtained in the ratchet model with that computed for the  simple constant-force model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='  For the simple Langevin  equation with no external force, analytical results are re-  viewed in [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' In the presence of dry friction and the  absence of spatial potential, the problem has been ad-  dressed in [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' Here, a harmonic approximation kx2/2  of the ratchet potential around one if its minima is con-  sidered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' The elastic constant k is obtained from a second-  order expansion of the ratchet potential (2) around one  of its minima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' The average time ⟨te⟩ necessary for the  particle to reach a distance d from the minimum of the  potential was computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' Distances are expressed in units  of d0, which was chosen in such a way that kd2  0/2 = ∆U,  with ∆U the ratchet potential depth (see the inset of  Figure 5 for a graphical depiction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' We are interested in  showing how the mean escape time ⟨te⟩ as function of the  escape distance d varies in several conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' As shown  in Figure 5, the smallest mean exit times are observed in  the case of the constant force model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' More specifically,  at small values of d/d0, the behaviour for the constant  force model is very similar to the harmonic case;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' both  trends show a saturation, and significant differences only  arise upon increasing the distance from the bottom of  the well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' On the other hand, the effect of dry friction  is much stronger, and the marked increase in exit times,  in the range of explored parameters, seems to exhibit  an exponential behaviour as a function of the distance  d/d0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' Finally, note that the very long exit times in the  model with dry friction can be related to the extended  plateaus observed in the variance, where the particle re-  mains trapped in the well for long times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='  Average escape times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='  Average escape time ⟨te⟩  for a harmonically confined Brownian particle with and without  Coulomb friction and constant-force starting at x0 = 0 as a func-  tion of the right escape point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' The inset reports a graphical depic-  tion of the harmonic approximation introduced in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='  For the sake of clarity, the harmonic potential is horizontally and  graphically shifted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' Parameters: γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='05, T = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='0, (k = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='20)  (α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' d0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='49 is chosen in such a way that kd2  0/2 = ∆U,  with ∆U as the ratchet potential depth for µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' F = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='28·10−4  is the constant force corresponding to the γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='05, T = 10, µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='4  ratchet case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='  CONCLUSIONS  In this work, a stochastic differential equation featur-  ing non-linear friction and an asymmetric spatial poten-  tial has been studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' The system is out of equilibrium  and shows the occurrence of the ratchet effect, namely  a net average drift, with a non-monotonic magnitude  as a function of the bath parameters, temperature, and  viscous friction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' The diffusion properties of the model  have also been investigated, with a particular focus on  the position variance and on the MSD, finding the oc-  6  currence of different regimes and different characteristic  times separating such time regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' Finally, the anal-  ysis proved that the diffusion properties of the ratchet  model under scrutiny present strong differences with re-  spect to a simple Brownian particle with inertia under  the action of an external constant force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' Our study con-  tributes by shedding light on some important dynami-  cal features of systems characterized by the presence of  both non-linear friction and fluctuations, which play a  central role in many natural phenomena, from biological  molecular motors at the cellular scale to earthquakes and  avalanches at geophysical scales, and in experimental ap-  plications, in particular for nano- and micro-friction such  as for nanometer contacts in the context of micro- and  nanomachines [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='  We plan to extend the study of this model to the frame-  work of stochastic thermodynamics, addressing the inter-  esting issues related to the definition of entropy produc-  tion and fluctuating efficiency, in future works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
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+page_content=' Marchesoni, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' Artificial Brownian motors:  Controlling transport on the nanoscale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
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+page_content='  [2] Reimann, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
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+page_content=' H¨anggi, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' Introduction to the physics of  Brownian motors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
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+page_content=' Ratchet mo-  tion induced by deterministic and correlated stochastic  forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
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+page_content=' Sarracino, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
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+page_content=' Langevin equation with Coulomb friction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' D Nonlinear Phenom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' 2005, 205, 48–56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
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+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' Cohen, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
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+page_content=' Touchette, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='  A path integral  approach to random motion with nonlinear friction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' 2009, 43, 025003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
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+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' Van der Straeten, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' Just, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' Brownian  motion with dry friction: Fokker–Planck approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' 2010, 43, 445002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='  [13] Baule, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' Touchette, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' Cohen, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' Stick–slip motion of  solids with dry friction subject to random vibrations and  an external field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' Nonlinearity 2010, 24, 351.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='  [14] Pototsky,  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='  Marchesoni,  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='  Periodically driven  Brownian motion with dry friction and ultrarelativistic  Langevin equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' E 2013, 87, 032132.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='  [15] Feghhi,  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='  Tichy,  W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='  Lau,  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='  Pulling a har-  monically bound particle subjected to Coulombic fric-  tion: A nonequilibrium analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
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+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' Puglisi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' Entropy production for velocity-  dependent macroscopic forces: The problem of dissipa-  tion without fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' EPL (Europhys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
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+page_content='  [17] Sarracino, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
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+page_content=' Gnoli, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' Puglisi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' Ratchet effect driven  by Coulomb friction: the asymmetric Rayleigh piston.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' E 2013, 87, 040101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='  [18] Manacorda, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
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+page_content=' Puglisi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' Sarracino, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' Coulomb fric-  tion driving Brownian motors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='  2014, 62, 505.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='  [19] Gnoli, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' Petri, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' Dalton, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' Pontuale, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' Gradenigo,  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
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+page_content=' Sarracino, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' Puglisi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' Brownian ratchet in a ther-  mal bath driven by coulomb friction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
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+page_content=' Puglisi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
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+page_content=' Granular Brownian  motion with dry friction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' EPL (Europhys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
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+page_content=' Goychuk, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' Non-linear Brow-  nian motion:  the problem of obtaining the thermal  Langevin equation for a non-Gaussian bath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
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+page_content=' Theory Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
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+page_content=' Bruti-Liberati, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='  Numerical solution of  stochastic differential equations with jumps in finance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='  Springer Science & Business Media: Berlin/Heidelberg,  Germany, 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' Volume 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
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+page_content=' A Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='  2014, 48, 013001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content='  [27] Chen, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
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+page_content=' Just, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' First-passage time of Brownian mo-  tion with dry friction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
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+page_content='  Zapperi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
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+page_content='  Colloquium:  Modeling friction:  From  nanoscale to mesoscale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQftwhp/content/2301.04073v1.pdf'}
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diff --git a/VtAyT4oBgHgl3EQf8_oB/content/tmp_files/2301.00864v1.pdf.txt b/VtAyT4oBgHgl3EQf8_oB/content/tmp_files/2301.00864v1.pdf.txt
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@@ -0,0 +1,2355 @@
+Bond-based nonlocal models by nonlocal operator method in
+symmetric support domain
+Huilong Rena, Timon Rabczukb, Xiaoying Zhuanga,c,∗, Zhiyuan Lid
+aInstitute of Photonics, Department of Mathematics and Physics, Leibniz University Hannover,Germany
+bInstitute of Structural Mechanics, Bauhaus-Universit at Weimar, Germany
+cState Key Laboratory of Disaster Reduction in Civil Engineering, College of Civil Engineering,Tongji
+University, Shanghai 200092, China
+dCollege of Mechanics and Materials, Hohai University, Nanjing, 211100, China
+Abstract
+This paper is concerned with the energy decomposition of various nonlocal models, including
+elasticity, thin plates, and gradient elasticity, to arrive at bond-based nonlocal models in
+which the bond force depends only on the deformation of a single bond. By assuming an
+appropriate form of bond force and using energy equivalence between local and nonlocal
+models, several very concise bond-based models are derived. We also revisit the nonlocal
+operator methods and study the simplified form of second-order NOM in the symmetric
+support domain. A bent-bond consisting of three points is proposed to describe the curvature
+and moment. To model the damage, a rule based on Griffith theory for the critical normal
+strain of the bond is proposed in analogy to the phase field model, which can be applied
+individually to each bond and provides strain localization. With this rule, the crack direction
+can be automatically predicted by simply cutting the bond, giving comparable results to
+the phase field method. At the same time, a damage rule for critical shear strains in shear
+fractures is proposed. Furthermore, an incremental form of the plasticity model for bond
+reaction force is derived. Several numerical examples are presented to further validate the
+nonlocal bond-based models.
+Keywords: energy decomposition, bond-based, bent bond, tensile damage, shear dam-
+age, fracture, phase field
+Preprint submitted to Elsevier
+January 4, 2023
+arXiv:2301.00864v1  [physics.class-ph]  2 Jan 2023
+
+1. Introduction
+Material damage and structural failure are an enduring problem in many engineering
+applications. Inadequate description can lead to serious hazards and financial losses. The
+problem arises from the fact that the complicated mechanisms in the damage process are
+difficult to predict both in theoretical models and by numerical methods. Nevertheless, great
+efforts have been made in the last decades to develop many robust numerical methods, e.g.
+Damage mechanics [1, 2], phase field method [3, 4], extended finite element method [5, 6],
+meshless methods [7, 8], cracking particle method [9, 10], virtual crack closure technique[11],
+Peridynamics (PD) [12] and many others. These methods generally fall into two categories:
+Using an additional field to represent the crack or modifying the topology to form a crack
+surface. The additional field in the phase field method forms the topology of the crack
+surface without modifying the mesh, and it is numerically stable and smooth, but at the
+expense of solving another field. The method based on topology modification can directly
+form a sharp crack surface, but the geometric operation can lead to instability problems.
+These two categories seem to be quite different, but both can be based on Griffith theory,
+according to which the creation of new free surfaces is due to the conversion of reduced
+potential energy into surface energy [13].
+Two representative examples of fracture modeling are phase field methods and nonlocal
+methods.
+The phase field method can handle many difficult engineering problems in a
+relatively simple theoretical framework, see for example [14, 15, 16, 17, 18, 19]. Peridynamics,
+as a nonlocal theory, has some advantages in topology modification because it separately
+accounts for interactions in a finite-size domain. In other words, the strain energy density is
+distributed in the domain rather than in a point of no size. The severing of individual bonds
+provides great physical intuition for cracking. Examples include bond-based PD [20], bond-
+based PD with shear deformation [21], extended bond-based PD [22, 23, 24], the conjugate
+∗Institute of Photonics, Department of Mathematics and Physics, Leibniz University Hannover,Germany.
+zhuang@iop.uni-hannover.de;
+Email addresses: huilong.ren@iop.uni-hannover.de (Huilong Ren),
+timon.rabczuk@uni-weimar.de (Timon Rabczuk), zhuang@iop.uni-hannover.de
+(Xiaoying Zhuang),
+lizhiyuan1007@163.com (Zhiyuan Li)
+2
+
+bond pair-based PD [25], the bond-based micropolar PD [26] and so on. In the bond-based
+PD, the energy carried by a bond is actually separated from each other, so they have good
+numerical stability in fracture modeling. The state-based PD [12] can deal with continuous
+problems, but the bond cutting process is not very stable since all bonds in the horizon are
+fully coupled. As a generalization of the dual-horizon peridynamics [27], in Ref. [28, 29, 30]
+the nonlocal operator method (NOM) is proposed. It provides a rule for converting many
+local models into their corresponding nonlocal forms, as well as introducing a variational
+framework for solving many difficult problems [31, 32, 33].
+Bond-based models provide great flexibility in fracture modeling. NOM expresses the
+derivatives of a function over the collective information in the support. It does not require
+that the support domain be constant or have regular shapes. The general form of NOM
+seems complicated, and in particular, it cannot be used directly to model fracture by cutting
+bonds. Considering the advantages of bond-based peridynamics, such as automatic crack
+development and crack direction determination, we want to develop some general bond-
+based models for various mechanical problems so that this property can be used for fracture
+modeling. The cutting of bonds to form fractures is based on the critical strain in BB-PD.
+The exact critical strain based on Griffith theory in nonlocal models is still not consistent
+with respect to load-displacement curves compared to other methods such as the phase field
+method. Therefore, two main objectives of the research are a) to derive bond-based nonlocal
+models based on NOM in symmetric support and b) to determine an appropriate critical
+stretch based on Griffith theory.
+The remainder of this paper is organized as follows. Section 2 explains the motivation
+from the perspective of orthogonal energy decomposition. Considering the relatively com-
+plicated form of NOM, a simplified second-order NOM with symmetric support is presented
+in Section 3. In the same section, the weighted bond-based nonlocal bar and nonlocal beam
+are derived. In Section 4, the bond-based nonlocal elasticity is derived in 2D and 3D using
+energy equivalence in detail. In analogy to the phase field method, a simple but effective
+bond-cutting criterion based on normal or shear strains is proposed. The plastic model for
+the bond- element in nonlocal elasticity is derived. In Section 5, using the second order
+3
+
+NOM in ideal support, the nonlocal bond-based isotropic thin plate model and the nonlocal
+bond-based gradient elasticity are derived. Three numerical experiments are presented in
+Section 6, including the nonlocal simple support beam and crack propagation in single-edge-
+notched plate under tension/shear boundary conditions and the Kalthoff-Winkler test with
+tension and shear fractures. Some conclusions and outlook are given in Section 7.
+2. Motivation: energy orthogonal decomposition
+Miehe [3] proposed a thermodynamic consistent phase field model for brittle fracture.
+The success of this model lies in the orthogonal decomposition of strain energy density of
+isotropic linear elasticity
+ψ = 1
+2σ : ε = 1
+2(
+3
+�
+i=1
+σini ⊗ ni) : (
+3
+�
+i=1
+εini ⊗ ni) = 1
+2
+�
+iσiεi,
+(1)
+where the stress tensor σ = �3
+i=1 σini ⊗ ni and strain tensor ε = �3
+i=1 εini ⊗ ni are
+formulated based on the eigenvalue decomposition of a matrix. For linear isotropic elas-
+ticity, the orthogonal decomposition of stress tensor and strain tensor are co-axial and the
+positive/negative parts of stress tensor can be written as
+σ± =∂ψ±
+e
+∂ε = λ⟨ε1 + ε2 + ε3⟩±I
++ 2µ(⟨ε1⟩±n1 ⊗ n1 + ⟨ε2⟩±n2 ⊗ n2 + ⟨ε3⟩±n3 ⊗ n3),
+(2)
+where I is the identity matrix, n1, n2, n3 are the eigenvectors of the associated principal
+strains ε1, ε2, ε3 of ε and ⟨x⟩± = (x±|x|)/2. One disadvantage of the phase field model is the
+complicated calculation of partial derivatives of eigenvalues and eigenvectors with respect
+to strain tensors.
+One of the most successful Peridynamics is the bond-based version. In bond-based PD,
+the strain energy carried by a point is
+ψ = 1
+2
+�
+S
+cs2dV,
+(3)
+where s is the extensional strain for the bond and c is the material parameter. The energy for
+each bond is independent of each other. In state-based PD, the bond-force depends on the
+4
+
+state (e.g. stress tensor). Breaking bond leads to singularity of shape tensor. The singularity
+does not occur to bond-based PD owing to the independence of all bonds. Although the
+original bond-based PD has limitations such as Poisson ratio restriction, it is quite robust
+for the modeling of tensile fractures.
+The local elasticity and nonlocal elasticity model are related by energy equivalence. As
+shown in [34], based on the local models, many nonlocal models can be derived by variational
+derivations. Let us focus on the linear elasticity and the strain energy density is written as
+ε : C : ε, where ε := 1
+2(∇u + ∇uT) is the strain tensor described by displacement gradient
+∇u, and C is the 4-th order material tensor. Then we use nonlocal gradient to replace strain
+tensor, e.g. ε →
+�
+Siω(rij)uij ⊗gijdVj, where uij = uj −ui, ω(rij) is the weight function and
+gij is a function of rij, the equivalent nonlocal energy density can be conceptually written
+as
+(
+�
+Siω(rij)uij ⊗ gijdV ) : C : (
+�
+Si ω(rij)uij ⊗ gijdV )
+(4a)
+?=
+�
+Si(ω(rij)uij ⊗ gij) : ¯C(rij) : (uij ⊗ gij)dV,
+(4b)
+where ¯C(rij) is the material tensor for a single bond rij.
+Each integral form contains
+infinite terms and the multiplication of two integral forms leads to more infinite terms.
+It would be great when Eq.4b is equal to Eq.4a. This condition is similar to find when
+(�n
+i=1 ai) · (�n
+i=1 bi)
+?= �n
+i=1 ai · bi. Mathematically, the derivation requires the orthogonal
+condition, that is ai · bj = δij.
+Local model is formulated based on differential equations and nonlocal theories such as
+peridynamics are expressed by integral equation
+∇ · σ + b = ρ¨u,
+�
+SifijdVj + b = ρ¨ui.
+(5)
+The most distinct difference between local model and nonlocal model is the way to view
+internal force, as shown in Fig.4. The former is defined on point structure without shape
+size while the latter introduces a neighborhood of finite size to account for the nearby
+interaction explicitly.
+In dual-horizon PD, the shape of the horizon has great flexibility,
+although the circular horizon is prefered. By distributing internal force on the finite size
+5
+
+-f
+f
+-f
+f
+-f
+f
+f
+f
+n
+s
+(a)                                               (b)                                                  (c)
+Figure 1: Internal forces (a) between virtual segments in local theory, (b) between two micro-volumes in
+nonlocal theory. (c) bond-force decomposition f = fs + fn.
+Point
+Figure 2: Shapes of support.
+horizon domain while other physical quantities such as density is the same as conventional
+local theory, the finite size horizon offers advantage to manipulating the internal force when
+discontinuity or damage happens. Traditional local theories are formulated on point sets.
+Mathematically, a point is infinitesimally small and has no specific shape. It is inconvenient
+to break one point into two points. The closest shape to a point is circular domain in 2D or
+spherical domain in 3D when ideal symmetry of a geometric object is considered. Indeed, for
+complete symmetric horizon domain or support domain, the nonlocal models can be greatly
+simplified at least for the nonlocal operator methods. By doing this, it is possible to derive
+some bond-based versions of nonlocal models.
+In the current research, the aim is the simplification of NOM by considering a symmetric
+support or horizon and therefore the dual-horizon is the same as the conventional horizon.
+Furthermore, the derivation of bond-based solid, thin plate and gradient solid is pursued,
+and for each bond, there is no need to consider the reaction force and direct force. The
+calculation of internal force of each material point is independent of other points, which
+offers some merit for implementation.
+6
+
+3. Nonlocal operator method
+3.1. Revisit of NOM in symmetric support
+In a support, the nonlocal derivatives are calculated by
+˜∂ui = (ui,x, ui,y, ui,xx, ui,xy, ui,yy)T :=
+�
+Si
+ω (rij) Ki · pijuijdVj
+(6a)
+with
+uij = uj − ui = u⟨r⟩, rij = r = (rx, ry) = (xij, yij) = rn
+(6b)
+pij =
+�
+xij, yij, x2
+ij/2, xijyij, y2
+ij/2
+�T
+(6c)
+Ki =
+��
+Si
+ω (rij) pij ⊗ pT
+ijdVj
+�−1
+,
+(6d)
+where n is the unit direction of bond r and r = ||r|| and ω(r) is the weight function, for
+example, ω(r) = 1, ω(r) = 1/r2. Herein, both □ij and □⟨r⟩ are used to denote the physical
+quantities related to a bond and □⟨r⟩ is used when the bond pair is not explicitly specified.
+When the support domain is a circular area, the expression of Ki can be greatly simplified.
+With some mathematical manipulation, the nonlocal gradient and nonlocal Hessian become
+˜∂u =
+�
+S
+ω(r)u⟨r⟩
+�
+(rx, ry)
+π
+� δ
+0 ω(r)r3 dr
+,
+1
+π
+� δ
+0 ω(r)r5 dr
+(3r2
+x − r2
+y, 4rxry, 3r2
+y − r2
+x)
+�
+dV.
+For single bond, its contribution to the nonlocal derivative can be simply written as
+˜∂u⟨r⟩ = ω(r)u⟨r⟩
+�
+(rx, ry)
+π
+� δ
+0 ω(r)r3 dr
+�
+��
+�
+g
+,
+1
+π
+� δ
+0 ω(r)r5 dr
+(3r2
+x − r2
+y, 4rxry, 3r2
+y − r2
+x)
+�
+��
+�
+⃗h
+�
+.
+Based on above formula, we extract bond gradient vector g and bond curvature tensor h2d
+from vector ⃗h in 2D, which are defined as
+g = rn
+αg
+with αg =
+�
+�
+�
+�
+�
+π
+� δ
+0 ω(r)r3 dr in 2D
+4π
+3
+� δ
+0 ω(r)r4 dr in 3D
+,
+(7)
+h2d =
+r2
+π
+� δ
+0 ω(r)r5 dr
+�
+�4n2
+x − 1
+4nxny
+4nxny
+4n2
+y − 1
+�
+� =
+r2
+π
+� δ
+0 ω(r)r5 dr
+�
+4n ⊗ n − I
+�
+,
+(8)
+7
+
+where n = (nx, ny) = r/r. Similarly, the matrix form of the curvature in 3D is
+h3d =
+3r2
+4π
+� δ
+0 ω(r)r6 dr
+�
+5n ⊗ n − I
+�
+.
+(9)
+For circular or spherical support, the shape tensor can be written by the identity matrix
+�
+S
+ω(r)r ⊗ rdV = αsI
+(10)
+with coefficients defined as α2d =
+� δ
+0 ω(r)πr3dr, α3d =
+� δ
+0 ω(r) 4
+3πr4dV .
+The nonlocal gradient, nonlocal divergence and nonlocal curl operator using explicit bond
+notations can be rewritten as
+˜∇ ∗ ui :=
+�
+Si
+ω(rij)gij ∗ uijdVj,
+(11)
+where ∗ ∈ {⊗, ·, ×} and gij is defined in Eq.7 for bond rij.
+For the case of gradient operator, the nonloal gradient of a vector field can also be written
+as
+∇u =
+�
+S
+3ω(r)
+4π
+� δ
+0 ω(r)r4 dr
+(rx, ry, rz) ⊗ u⟨r⟩dV =
+�
+S
+ω(r)g ⊗ u⟨r⟩dV.
+(12)
+The Hessian of a scalar field in 2D has the form
+∇∇u =
+�
+S
+ω(r)u⟨r⟩
+π
+� δ
+0 ω(r)r5 dr
+�
+�3r2
+x − r2
+y
+4rxry
+4rxry
+3r2
+y − r2
+x
+�
+� dV
+=
+�
+S
+ω(r)h ⊗ u⟨r⟩dV
+(13)
+=
+�
+S+ ω(r)h ⊗ (u⟨r⟩ + u⟨−r⟩)dV.
+(14)
+In the last step, h is invariant for both r and −r and the symmetricity of S is considered.
+The half support S+ is defined based on the symmetric support domain as shown in Fig.3.
+In the definition of curvature, only a half support is required.
+Traditional NOM deals with the gradient or Hessian tensor at a point as a whole. In
+this sense, all bonds in support domain are coupled. In the spirit of bond-based PD, it is
+8
+
+i
+j
+j'
+r
+-r
+δ
+  ij--bond
+ijj'--bent bond
+S
+S+
+S-
+Figure 3: Normal bond and bending bond. S = S+ ∪ S−. j′ ∈ S− is the symmetric point of j ∈ S+ with
+respect to center point i, ijj′ for a bent bond.
+natural to define the derivatives for each individual bond. Based on Eq.7, the bond gradient
+on single bond is defined as
+∇u⟨r⟩ = r
+r2u⟨r⟩ = u⟨r⟩n
+r .
+(15)
+Based on Eq.8, the curvature of a pair-wised bond in 2D is defined as
+∇∇u⟨r⟩ = (u⟨r⟩ + u⟨−r⟩)
+r2
+(4n ⊗ n − I).
+(16)
+Above definition is reasonable because u⟨r⟩
+r2 (4n⊗n−I) ≈ (∇∇u : (r⊗r))/r2(4n⊗n−I) =
+(∇∇u : (n ⊗ n))(4n ⊗ n − I), which depends on the bond direction and second-order
+derivatives.
+Conventionally, NOM can derive the nonlocal model from the local energy functional
+by using the nonlocal gradient or nonlocal Hessian. Consider a general field with energy
+density φ being a function of ∇u and ∇2u, the total potential energy in domain is
+Ψ =
+�
+V
+φ(∇u, ∇2u)dV.
+(17)
+The variation of the energy functional
+δΨ =
+�
+V
+∂φ
+∂∇u : ∇δu +
+∂φ
+∂∇2u˙:∇2δudV
+=
+�
+V
+σi :
+�
+Si
+ω(r)δuij ⊗ gij + Σi˙:
+�
+Si
+ω(r)δuij ⊗ hijdVj
+=
+�
+V
+�
+Si
+ω(r)σi · gij · δuij +
+�
+Si
+ω(r)Σi : hij · δuijdVj.
+(18)
+9
+
+where σ := ∂φ
+∂ε, ε = 1
+2(∇u+∇uT) and Σ :=
+∂φ
+∂∇ε. For the cases of the general linear elasticity
+and the general linear gradient elasticity, the material constitutions are
+σ = C : ε, or σij = Cijklεkl
+(19a)
+Σ = D : ∇ε, or Σijk = Dijklmn∂lεmn,
+(19b)
+where C, D are material tensors.
+The nonlocal governing equations for elasticity and gradient elasticity are
+�
+S
+(ω(rij)σi · gij − ω(rji)σj · gji) + bi = ρ¨ui,
+(20a)
+�
+S
+(ω(rij)(σi · gij + Σi : hij) − ω(rji)(σj · gji + Σj : hji)) + bi = ρ¨ui.
+(20b)
+Based on the material constitutive of thin plate
+M =
+�
+� Mxx
+Mxy
+Mxy
+Myy
+�
+� = D0 (νtr(κ)I2×2 + (1 − ν)κ) ,
+(21)
+the nonlocal governing equation of nonlocal thin plate is
+�
+Si
+(fij + fji) + q = ρ ¨wi, with fij = ω(rij)Mij : hij,
+(22)
+where D0 =
+Et3
+12(1−ν2) and t is the thickness of the plate.
+The above equation depends on the state quantity defined in the support domain, where
+the internal force of each bond is fully coupled. Is it possible to derive the force or mo-
+ment depending on the bond only by using the bond derivatives given in Eq.15 and Eq.16?
+The answer is the decoupled NOM. In the following sections, the conditions of decoupled
+NOM for nonlocal elasticity, nonlocal thin plate and nonlocal gradient elasticity in different
+dimensional spaces will be explored.
+3.2. Decoupled NOM in 1D
+In order to illustrate the decoupled NOM, let us consider the NOM in 1D and derive the
+nonlocal bar/beam models. For the case in one-dimensional space, the second-order NOM
+10
+
+can be derived based on the Taylor series in 1D as
+uij = u′
+ixij + 1
+2u′′
+i x2
+ij
+(23a)
+� δ
+−δ
+ω(rij)uijx2 = 1
+2u′′
+� δ
+−δ
+(ω(rij)x4)dx
+(23b)
+u′′ =
+� δ
+−δ ω(rij)uijx2dx
+� δ
+0 ω(rij)x4dx
+=
+� δ
+0 ω(rij)(uij + uij′)x2dx
+� δ
+0 ω(rij)x4dx
+.
+(23c)
+For the case of one dimension, the gradient and curvature of a particle can be derived
+similarly as
+du
+dx =
+1
+� δ
+−δ ω(x)x2dx
+� δ
+−δ
+ω(x)uijxdx
+d2u
+dx2 =
+1
+� δ
+0 ω(x)x4dx
+� δ
+0
+ω(x)(uij + uij′)x2dx.
+(24)
+For each direction, the bond individual gradient and curvature are
+du
+dx⟨x⟩ = uijx
+x2
+= uij
+x , d2u
+dx2⟨x⟩ = (uij + uij′)x2
+x4
+= (uij + uij′)
+x2
+.
+(25)
+3.2.1. One-dimensional nonlocal bar
+Consider a one-dimensional nonlocal bar model with elastic modulus of E and section
+area of A, we assume the bond energy density as
+φij = 1
+2eijfij|rij| = 1
+2ω(rij)cu2
+ij/|rij|,
+(26)
+where eij = uij/|rij| is the relative strain and fij = ω(rij)ceij is the bond force. The energy
+equivalence between local model and nonlocal model requires
+� δ
+−δ
+φijdx = 1
+2EAε2
+i ,
+(27)
+where εi is the local strain at point xi.
+In order to derive the specific form of bond force, a displacement field u(x) = ax with
+constant gradient u′(x) = a is assumed. Let ui = 0, uj = ax, then uij = ax, eij = uij/|x| =
+a sign(x), fij = ω(x)ceij. In 1D, only the elongation is involved. The strain energy carried
+by a bond due to bond force and displacement becomes
+φij = 1
+2eijfij|rij| = 1
+2a2|x|cω(x).
+(28)
+11
+
+Here the process of doing work is considered, e.g. bond force fij acting on distance eij|rij|.
+The energy equivalent in Eq.27 is calculated as
+� δ
+−δ
+1
+2eijfij|x|dx =
+� δ
+−δ
+1
+2a2|x|cω(x)dx = 1
+2EAa2 → c =
+EA
+2
+� δ
+0 ω(x)xdx
+.
+(29)
+The bond force in 1D is the variation of bond energy
+fij = δφij
+δuij
+=
+EA
+2
+� δ
+0 ω(x)xdx
+ω(rij)uij
+|rij|
+.
+(30)
+Another scheme to consider the bond energy is
+φij = 1
+2eijfij = 1
+2a2cω(x).
+(31)
+The energy equivalence leads to
+� δ
+−δ
+1
+2eijfijdx =
+� δ
+−δ
+1
+2a2cω(x)dx = 1
+2EAa2 → c =
+EA
+2
+� δ
+0 ω(x)dx
+.
+(32)
+The bond force of bond ij is
+fij = δφij
+δuij
+=
+EA
+2
+� δ
+0 ω(x)dx
+ω(rij)uij
+r2
+ij
+.
+(33)
+It should be noted that Eq.30 and Eq.33 are equivalent when the weight function in Eq.33
+is set as |rij|ω(rij). The direct bond force is added to material point i and the reaction
+bond force is added to j. However, the bond ij of i and bond ji of j are the same, and the
+calculation of bond ji gives i a reaction bond force −fji, which satisfies −fji = fij. Hence,
+the governing equation of nonlocal bar is
+� δ
+−δ
+2fijdx + b = ρ¨ui.
+(34)
+3.2.2. One-dimensional nonlocal beam
+Consider a one-dimensional nonlocal beam model with elastic modulus denoted by E
+and the second moment of area of the beam’s cross section denoted by I, we can assume the
+bond bending energy density as
+φij = 1
+2κijmij = 1
+2ω(rij)cκ2
+ij,
+(35)
+12
+
+where κij is the relative curvature and mij = ω(rij)cκij is the moment. The energy equiva-
+lence between local model and nonlocal model requires
+� δ
+−δ
+1
+2κijmijdx = 1
+2EIκ2
+i ,
+(36)
+where κi is the local curvature at point xi.
+Let us assume a deflection field u(x) = x2/2 with constant curvature κ = u′′(x) = 1.
+Let xi = 0, xj = x, then uij = x2/2, κij = (uij + uij′)/x2 = 1, mij = ω(x)cκij = cω(x),
+1
+4κijmij = 1
+4cω(x). The bending energy carried by a bond due to curvature is
+1
+2κijmij = 1
+2cω(x).
+(37)
+The equivalent of bending energy in support to the local model can be simplified as
+� δ
+0
+1
+2κijmijdx =
+� δ
+0
+1
+2cω(x)dx = 1
+2EIκ2 = 1
+2EI → c =
+EI
+� δ
+0 ω(x)dx
+.
+(38)
+For a homogeneous beam with thickness h, the coefficient c of different weight functions
+can be written as
+ω(x)EI
+� δ
+0 ω(x)dx
+=
+�
+�
+�
+�
+�
+Eh3
+12δ
+if ω(r) = 1
+Eh3r
+24δ2
+if ω(r) = r.
+(39a)
+Formerly, the nonlocal curvature and moment are defined as
+κij = (uij + uij′)
+r2
+ij
+, mij = ω(rij)(uij + uij′)
+r2
+ij
+EI
+� δ
+0 ω(x)dx
+.
+(40)
+The bent energy of bent bond is the multiplication of double volume ∆x2 and the bent
+energy density φij as
+φij∆x2 = 1
+2κijmij∆x2 = 1
+2
+EIω(rij)
+� δ
+0 ω(x)dx
+(uij + uij′)2
+r4
+ij
+∆x2,
+(41)
+where ∆x is the volume of the material point.
+The variation of φij∆x2 reads
+13
+
+δφij∆x2 = EIω(rij)
+� δ
+0 ω(x)dx
+(uij + uij′)
+r4
+ij
+∆x2
+�
+��
+�
+force due to bent: fijj′
+·(δuj + δuj′ − 2δui)
+= fijj′ · δuj + fijj′ · δuj′ − 2fijj′ · δui.
+(42)
+Therefore, the bond forces adding to i, j, j′ due to bond curvature energy are −2fijj′, fijj′, fijj′,
+respectively.
+When an implicit algorithm is used, the tangent stiffness matrix is also required, which
+can be written by a second variation of φij.
+δ2φij∆x2 = EIω(rij)(∆x)2
+r4
+ij
+� δ
+0 ω(x)dx
+(δuj + δuj′ − 2δui)2
+=
+�
+�
+�
+�
+�
+δui
+δuj
+δuj′
+�
+�
+�
+�
+�
+T
+(EIω(rij)(∆x)2
+r4
+ij
+� δ
+0 ω(x)dx
+)
+�
+�
+�
+�
+�
+4
+−2
+−2
+−2
+1
+1
+−2
+1
+1
+�
+�
+�
+�
+�
+�
+��
+�
+Kijj′
+�
+�
+�
+�
+�
+δui
+δuj
+δuj′
+�
+�
+�
+�
+� .
+(43)
+4. Nonlocal isotropic elasticity
+4.1. Bond force in 3D
+γ
+l
+f
+fs(γ )  
+fn(l )  
+(a)                                                           (b)
+xj(t1)
+rij
+xi
+uij
+uij
++
+θ
+c1
+c2
+rij
+xj(t2)
+uij
+ij
+ij
+ij
+ij
+r
+Figure 4: Bond deformation with rotations, shear stiffness and extension stiffness.
+Consider the strain tensor project on bond direction nij = (cos θ sin φ, sin θ sin φ, cos φ)
+in spherical polar coordinate based φ ∈ [0, π), θ ∈ [0, 2π), the extension strain and shear
+strain along the bond direction is
+lij = (εi · nij) · (nij ⊗ nij)
+(44a)
+γij = (εi · nij) · (I − nij ⊗ nij).
+(44b)
+14
+
+The relative strain vector εn = εi · nij = lij + γij and the relative displacement is uij =
+εnrij = (lij + γij)rij.
+We assume the bond force be the form
+fij = ω(rij)(c1lij + c2γij).
+(45)
+The energy density carried by a bond’s deformation is
+wij = 1
+2fij · uij = ω(rij)(c1lij + c2γij) · (lij + γij)rij = 1
+2ω(rij)rij(c1lij · lij + c2γij · γij).
+Then the nonlocal strain energy density at a point in support domain equalizes to the local
+strain energy density
+W =
+�
+Si
+wijdVj =
+�
+Si
+1
+2rijω(rij)(c1lij · lij + c2γij · γij)dVj
+= Wlocal = 1
+2σ : ε = (λ Tr(ε)I + 2µε) : ε,
+(46)
+where λ, µ are Lame constants.
+For any ε, using undetermined coefficient method yields
+c1 =
+E
+α(1 − 2ν), c2 =
+E(1 − 4ν)
+α(ν + 1)(1 − 2ν),
+(47)
+where α =
+� δ
+0
+4
+3πr3ω(r)dr, and elastic modulus E and ν are used to replace the Lame
+constants by λ =
+E
+(1−2ν)(1+ν), µ =
+E
+2(1+ν).
+When the weight function ω(rij) = 1, the coefficients become
+c1 =
+3E
+πδ4(1 − 2ν), c2 =
+3E(1 − 4ν)
+πδ4(ν + 1)(1 − 2ν),
+(48)
+which are the same as the extended bond-based PD in [22]. Here, the values of c1 or c2 is
+one half of those in [22] because of the consideration of direct bond force and reaction bond
+forces. In sum, the bond deformation and bond force considering the weight function are
+lij = uij · nij
+rij
+nij
+(49a)
+γij = uij
+rij
+− lij = uij
+rij
+− uij · nij
+rij
+nij
+(49b)
+fij = ω(rij)(c1lij + c2γij).
+(49c)
+15
+
+And the corresponding governing equations are
+�
+Si
+2fijdVj + b = ρ¨ui.
+(50)
+4.2. Bond force in 2D
+For the case of plane stress condition, the material constitutive in local form is
+σ =
+E
+1 − ν2(νtrϵI2x2 + (1 − ν)ε).
+(51)
+The equivalence of strain energy density for arbitrary strain tensor leads to
+c1 =
+E
+α(1 − ν), c2 = E(1 − 3ν)
+α(1 − ν2) ,
+(52)
+where α =
+� δ
+0 πω(r)r2dr
+Similarly, for plane strain condition, the material constitutive in local form is
+σ =
+E
+(1 + ν)(1 − 2ν)(νtrϵI2x2 + (1 − 2ν)ε).
+(53)
+The energy equivalent gives the coefficients as
+c1 =
+E
+α(1 − ν − 2ν2), c2 =
+E(1 − 4ν)
+α(1 − ν − 2ν2),
+(54)
+where α =
+� δ
+0 πω(r)r2dr.
+By assuming appropriate bond deformation and bond force and considering the weighted
+energy equivalent to the local energy, the weighted bond-based nonlocal elasticity in 1D,2D
+and 3D are derived. By doing so, the energy for each bond depends on the bond deformation
+only, the energy for each bond is separable while the collective energy of all bonds recovers
+the isotropic elasticity.
+Remarks on implementation: In the definition of bond force previously, each bond
+is independent of each other, in which the interference between each bond is minimized,
+which can greatly improve the numerical stability when cutting bonds.
+Meanwhile, the
+definition depends on the spherical support or horizon. For the convenience of numerical
+implementation, we assume that each particle has the same volume and support radius. The
+interested domain is discretized into uniform particles or lattices. For different particles, the
+bonds in the same direction and radius have the same coefficients.
+16
+
+4.3. Two damage rules based on critical energy release rate
+In this subsection, by relating the critical shear strain or critical normal strain to the
+energy release rate, two damage rules are proposed.
+S-
+S+ u
+-u
+i
+j
+k
+l
+S-
+S+
+i
+j
+k
+l
+u
+-u
+(a) Model I: Opening
+(b) Model II: In-plane shear
+Figure 5: Deformation of bond with shear deformation or tensile deformation.
+4.3.1. Critical normal strain damage rule
+Bond-based peridynamic models have advantages such as independent bond energy and
+simple damage criterion based on critical stretch.
+Although the direct neighbor cutting
+operation perturbates the system greatly, the numerical stability is well preserved. There
+are several obstacles in removing bonds. Even for the simplest bond-based model enriched
+with rotation for open-mode fracture, at the crack front tip, the deformation for each bond
+is complicated. As shown in Fig.5(a), bond ik is tensile and bond ij has shear deformation
+or mixed deformation. It is doubtful to apply a criterion of stretch rule or rotation rule to
+cut the bond ik since ik falls in the open-mode fracture. It is obvious that for the simplest
+fracture mode, the shear bonds and tensile bonds coexist. For the case denoted in Fig.5(b),
+some bonds have compressive shear deformation and the situation is even worse because
+of the perturbation of sudden removed internal bond force. The bond removing technique
+is more based on a geometrical and intuitive operation but lacks sound theoretical basis.
+Some authors in their work [23, 22, 35, 24] by examining the shear deformation state and
+tensile deformation state and sorting these states, cut the most damage-prone bonds and
+then iterate globally. These methods are relatively complicated since they depend heavily on
+the bond sequences and cannot be analyzed theoretically. In addition, the assembly of the
+global tangent stiffness matrix and the related solving techniques are much more expensive
+than the explicit time integration scheme.
+17
+
+In order to model fracture automatically, the breaking of bonds should be as simple as
+possible. We borrow some ideas from the phase field scheme [3]. The phase field model
+considers the principal strains, which is independent of shear strain in that direction. The
+degradation of strain energy by principal strain has good numerical stability. In the bond-
+based nonlocal elasticity, each bond usually has both axial deformation and shear deforma-
+tion. On the other hand, the magnitude of shear deformation (i.e. the rotation) is not well
+predicted since the rigid rotation may be significant. However, in the sense of discretization,
+the bonds loop in “every” direction in support. It is not required to calculate the eigenvalue
+decomposition of strain tensors. But when the bond direction points to the direction of
+principal strain, the situation is quite similar to the phase field model. Unlike the bond-
+based model based on the stretch contributed partially by shear deformation, or the critical
+rotation model [35, 24], we remove the bond by considering only a bond-directional strain
+while ignoring the shear deformation. The change of strain energy in this sense is quite
+similar to the phase field model by considering only the energy on principle strain direction.
+For each bond direction, the interaction status is determined through a bond status
+parameter given by
+µ(xj − xi) =
+�
+�
+�
+�
+�
+1, sn(t) < sc
+n
+0, max
+0≤t≤Tsn(t) ≥ sc
+n,
+(55)
+where sn(t) is the strain at time t along the initial bond direction and sc
+n is the critical bond
+stretch determined by the Griffith energy release rate. The local damage is evaluated as
+φ(xi) = 1 −
+�
+Si µ(xj − xi)dVj
+�
+Si dVj
+.
+(56)
+When considering the deformation along the principal strain direction, the deformation
+can be simplified into 1D with cross-section area A. Consider the deformation in 1D, in order
+to form a crack surface, half support should be cut. The equivalence of fracture energy and
+strain energy in half support is
+1
+2GcA = 1
+2Kε2Aδ → ε =
+�
+Gc
+Kδ,
+(57)
+18
+
+where K =
+E
+3(1−2ν) is the bulk modulus of the material. Therefore, we select the critical
+normal strain as
+sc
+n =
+�
+Gc
+Kδ =
+�
+3(1 − 2ν)Gc
+Eδ
+.
+(58)
+This rule derived from the 1 dimensional case appears to be simple, but it can achieve almost
+the same accurate result as the phase field method by finite element methods in some cases,
+which will be shown in the numerical examples.
+4.3.2. Critical shear strain damage rule
+Similar to the principal strain direction, the other direction is the maximal shear strain
+direction. For many materials, the shear strain gives rise to the shear fractures. Consider
+the deformation in 1D, the equivalence of fracture energy and strain energy in half support
+is
+1
+2GIIA = 1
+2µε2
+sAδ → εs =
+�
+GII
+µδ ,
+(59)
+where GII is the energy release rate for mode II fracture and µ the shear modulus. Therefore,
+the critical shear strain is selected as
+sc
+t =
+�
+GII
+µδ =
+�
+2(1 + ν)GII
+Eδ
+.
+(60)
+4.4. Plasticity for bond element
+Let (n, m, t) be a set of orthogonal local axes, with en, em, et being the normal vector, the
+shear-direction and out-of-plane direction of the bond element, respectively. The kinematics
+of a bond element is
+εn = len + γem
+(61)
+where l = uij
+r · en, γ = uij
+r · em.
+In a local coordinate system, the strain and force are
+ε = (l, γ)T, σ = (σ, τ)T =
+�
+�c1
+0
+0
+c2
+�
+�
+�
+��
+�
+E0
+�
+�l
+γ
+�
+�
+(62)
+19
+
+where the vector σ, ε represent the force vector and strain vector; E0 is the second-order
+material tensor in bond local coordinates.
+For elastoplastic models, the constitutive relation of a bond element in local coordinate
+can be expressed in rate form as
+˙ε = ˙εe + ˙εp, ˙σ = E0 · ( ˙ε − ˙εp),
+(63)
+where εe and εp being the elastic and plastic parts of the strain tensor.
+Without loss of generality, the plastic strain rate is given by the following flow rule based
+on the plastic potential function f p(σ, q)
+˙εp = ˙λ ∂f p
+∂σ
+����
+Λp
+, ˙κ = − ˙λ∂f p
+∂q ,
+(64)
+for the plastic multiplier ˙λ satisfying the classical Karush-Kuhn-Tucker conditions
+˙λ ≥ 0, f(σ, q) ≤ 0, ˙λf(σ, q) = 0,
+(65)
+where a force-based yield function f(σ, q) ≤ 0, with q being the force-like internal variable
+(yield force) conjugate to the strain-like one κ which measures the plastic state; Λp := ∂fp
+∂σ
+is the plastic flow direction.
+For associated plasticity, the potential function f p(σ, q) is
+identical or proportional to the yield function f(σ, q).
+Then the force state rate can be written as
+˙σ = E0 · ( ˙ε − ˙λΛp).
+(66)
+Plastic yielding occurs when the yield condition f(σ, q) = 0 is activated, i.e.
+˙λ > 0.
+Follow from the consistency condition ˙f = 0 gives
+˙λ =
+Λ · E0 · ˙ε
+Λ · E0 · Λp + hHhp,
+(67)
+for the derivative Λ := ∂f
+∂σ and h = − ∂f
+∂q of yield function f(σ, q) and hardening/softening
+modulus H := ∂q
+∂κ.
+20
+
+The corresponding constitutive relation in rate form then reads
+˙σ = E · ( ˙ε − ˙εp) = Eep · ˙ε,
+(68)
+where the second-order elastoplasticity tangent Eep is expressed as
+Eep = E0 − (E0 · Λp) ⊗ (Λ · E0)
+Λ · E0 · Λp + hHhp .
+(69)
+The yield function and plastic potential function are assumed to have the same form
+f p = f(σ, τ, q) = aσ + b
+√
+τ 2 − q(κ).
+(70)
+Then
+Λ = (a, b sign(τ)).
+(71)
+The elastoplasticity tangent becomes
+Eep = ω(r)
+�
+�
+�c1
+0
+0
+c2
+�
+� −
+1
+a2c1 + b2c2
+�
+�
+a2c2
+1
+abc1c2 sign(τ)
+abc1c2 sign(τ)
+b2c2
+2
+�
+�
+�
+.
+(72)
+5. Higher-order nonlocal bond-based models
+The bond-based nonlocal model is not restricted in first-order. By making use of the bent-
+bond, the bond-based plate model and bond-based gradient elastic model will be derived in
+the following.
+5.1. Nonlocal isotropic thin plate
+Assuming deflection field w(x, y) = 1
+2(ax2 + by2 + 2cxy). κ = ∇∇w, the second-gradient
+of deflection field, can be written as
+κ =
+�
+�κ11
+κ12
+κ12
+κ22
+�
+� .
+(73)
+Along with bond direction n = (cos θ, sin θ), the orthogonal decomposition of bond
+curvature tensor is
+κn = (wij + wij′)/r2
+ij(4n ⊗ n − I)
+= 3(wij + wij′)/r2
+ij(n ⊗ n)
+�
+��
+�
+κnn
++ (−(wij + wij′)/r2
+ij)(I − n ⊗ n)
+�
+��
+�
+κns
+.
+(74)
+21
+
+The moment force for single bent bond is assumed as
+Mn = ω(rij)(c1κnn + c2κns),
+(75)
+where c1, c2 are the material parameters to be determined. The total energy carried by a
+point
+Wnonlocal =
+�
+S+
+1
+2(Mn) : (κn)dV
+=
+�
+S+
+1
+2ω(r)(wij + wij′)2/r4
+ij(9c1 + c2)dV.
+(76)
+Wnonlocal = Wlocal := 1
+2M : κ for any field yields (9c1 + c2) = 16D0
+3α , ν = 1/3, where α =
+� δ
+0 ω(r)πr dr. Therefore, the equivalent curvature and moment for a bond are
+κij = (wij + wij′)/r2
+ij
+(77a)
+mij = ω(r)16D0
+3α (wij + wij′)/r2 = ω(r)Et3
+2α (wij + wij′)/r2
+ij,
+(77b)
+where D0 =
+Et3
+12(1−ν2) =
+3
+32Et3 and t is the thickness of the plate. This is the bond-based
+version of nonlocal thin plate. Only the Poisson ratio of 1/3 can be modeled.
+The corresponding bond force can be derived by considering the first variation of the
+bond energy
+fijj′ = ω(rij)Et3
+2α
+(wij + wij′)
+r4
+ij
+.
+(78)
+5.1.1. Cohesive damage model for bent bond
+Cohesive damage model of bending bond. In order to introduce the localization, the
+moment is calculated as
+mij = c0sign(κij) min(|κij|, κ2
+crit
+|κij|),
+(79)
+where κcrit is the critical curvature when softening of force occurs.
+5.2. Nonlocal isotropic gradient elasticity
+Similar to the nonlocal thin plate, we consider one field in gradient elasticity.
+For
+any bond, the local coordinate system can be expressed with orthogonal unit basis vec-
+22
+
+κ/κcrit
+m/mmax
+1
+-1
+-5
+5
+-1.0
+-0.5
+0.5
+1.0
+Figure 6: Moment and curvature relation in a cohesive model with κcrit and mcrit being critical curvature
+and critical moment.
+xi
+xj(t1)
+rij
+uij
+uij
++
+γ
+c1
+c2
+κs
+κn
+κ
+fκ
+fκs(κs)  
+fκn(κn)  
+(a)                                                           (b)
+rij
+xj(t2)
+Figure 7: Bond deformation with rotations, shear curvature stiffness and extension curvature stiffness.
+tors n1, n2, n3 as
+n1 = (cos θ sin φ, sin θ sin φ, cos φ),
+n2 = (cos θ cos φ, cos φ sin θ, − sin φ), n3 = (− sin θ, cos θ, 0).
+(80)
+For any field u(x, y, z) = 1
+2(ax2 + by2 + cz2 + 2dxy + 2fxz + 2gyz), the curvature tensor
+κ = ((a, d, f), (d, b, g), (f, g, c)).
+The orthogonal decomposition of nonlocal Hessian on a bond is
+κn = (uij + uij′)/r2
+ij(5n1 ⊗ n1 − I)
+= 4(uij + uij′)/r2
+ijn1 ⊗ n1
+�
+��
+�
+κ1
++ (−(uij + uij′)/r2
+ij)n2 ⊗ n2
+�
+��
+�
+κ2
++ (−(uij + uij′)/r2
+ij)n3 ⊗ n3
+�
+��
+�
+κ3
+.
+23
+
+The moment of the bond is assumed as
+Mn = ω(rij)
+�
+c14(uij + uij′)/r2
+ijn1 ⊗ n1
+�
+��
+�
+M1
++ c2(−(uij + uij′)/r2
+ij)n2 ⊗ n2
+�
+��
+�
+M2
++ c2(−(uij + uij′)/r2
+ij)n3 ⊗ n3
+�
+��
+�
+M3
+�
+,
+(81)
+where c1, c2 are the unknown curvature stiffness and here we assume the stiffnesses in M2
+and M3 are the same.
+The bent energy carried by a bond is
+wn = 1
+2Mn : κn = ω(rij)(8c1 + c2)(uij + uij′)2/r4
+ij.
+(82)
+The bent energy carried by a point is the summation of all bent bonds:
+W =
+�
+S+ wndV = 1
+15α(8c1 + c2)(3a2 + 2a(b + c) + 3
+b2 + 2bc + 3c2 + 4(d2 + f 2 + g2)),
+(83)
+where α =
+� δ
+0 πr2ω(r)dr.
+Assume the material constitution for gradient strain energy be
+M = ℓ2(λTr(κ)I + 2µκ).
+(84)
+The gradient strain energy in local theory for any curvature deformation can be simplied as
+Wlocal = 1
+2M : κ = 1
+2ℓ2�
+a2(λ + 2µ) + 2aλ(b + c)
++b2(λ + 2µ) + 2bcλ + c2λ + 2c2µ + 4d2µ + 4f 2µ + 4g2µ
+�
+.
+(85)
+The energy equivalence W = Wlocal for any κ leads to
+(8c1 + c2) = 15ℓ2µ
+2α , λ = µ → ν = 1
+4.
+(86)
+Therefore, the curvature and moment force for u field is
+κu
+ij = (uij + uij′)/r2
+ij, mu
+ij = ω(rij)15ℓ2µ
+α
+(uij + uij′)/r2
+ij.
+(87)
+24
+
+The curvature force follows the direction of u field.
+Therefore, the bent-bond energy becomes
+φij = 1
+2ω(rij)15ℓ2µ
+α
+(uij + uij′)2/r4
+ij.
+(88)
+And the corresponding bond force
+f u
+ijj′ = ω(rij)15ℓ2µ
+α
+(uij + uij′)/r4
+ij.
+(89)
+For field v, w, the same conclusions can be obtained. In sum, the curvature and momen-
+tum of bond-based gradient elasticity is
+fijj′ = ω(rij)15ℓ2µ
+α
+(uij + uij′)/r4
+ij.
+(90)
+The governing equations become
+�
+Si
+2fijdVj
+�
+��
+�
+first order contribution
++
+�
+S+
+i
+(2fijj′ + fjii′ + fj′ii′)dVj
+�
+��
+�
+second order contribution
++b = ρ¨ui.
+(91)
+6. Numerical examples
+The numerical examples are carried out based on a Verlet-velocity explicit time inte-
+gration algorithm. The quasi-static condition for some cases is achieved by applying the
+velocity boundary conditions gradually. The reaction forces are retrieved by summing the
+internal forces of the selected particle set before applying the boundary conditions.
+6.1. Simple support beam
+The bent bond defined by three points is immune to the first order derivative in complete
+support.
+However, for material points near the boundaries, the support domain is not
+complete. Additional particles outside the boundaries are added to make the support domain
+complete. The simple support boundary is fulfilled by
+w(0) = w(L) = 0.
+25
+
+The function of additional particles is to make sure the half support S+ is well defined. The
+full implementation code of the simple support beam can be found by the Github link (???).
+The material parameters of this example are E = 30 × 109 Pa, beam length L = 1 and
+thickness h = 0.05. A damping (fdamp = −200 ˙w) is used to converge the dynamic solution
+to the static result. With damping effect, the evolution of deflection of the middle point is
+plotted in Fig.8. The final deflection of the beam is plotted in Fig.9. It can be observed
+that the numerical solution for N = 25 material points and δ = 2 is quite close to the exact
+solution. As shown in Fig.9, the deflection of beam with a discretization N = 100 agrees
+very well with the exact solution, in which the L2 norm of error is 3.01 × 10−4.
+The influence of support size is investigated in Fig.10. With the increase of support
+size, the beam becomes slightly stiffer. The influence of the weight function ω(r) = rn, n =
+(0, 1, 2, 3, 4) is plotted in Fig.11. Here the support size is selected as δ = 3∆x. one can see
+that the weight function in this case has significant influence on the deflection.
+0.02
+0.04
+0.06
+0.08
+Time (s)
+- 1.4
+- 1.2
+- 1.0
+- 0.8
+- 0.6
+- 0.4
+- 0.2
+w(L/2)/wmax
+Figure 8: The evolution of deflection of midpoint .
+In order to model fracture in a thin beam, we applied the cohesive damage rule to model
+the fracture. We select the critical curvature tensor as κcrit = 4× 10−4 and use the damping
+coefficient p = 300 for reducing oscillation. The damage distribution and displacement field
+at the t = 0.03 seconds are shown in Fig.12 and Fig.13, respectively. It can be observed that
+the damage happens at the center of the beam.
+6.2. Single-edge-notched tension test
+In this subsection, we model the single-edge-notched tension test, which is a squared
+plate with initial notched crack as shown in Fig.21. The material parameters are set as
+26
+
+0.2
+0.4
+0.6
+0.8
+1.0
+L (m)
+- 1.0
+- 0.8
+- 0.6
+- 0.4
+- 0.2
+w/wmax
+Analytical solution
+Bond-based beam,N=100,n=2
+Bond-based beam,N=25,n=2
+Figure 9: The comparison of exact solution and bond based beam N = 100 and N = 25 .
+0.2
+0.4
+0.6
+0.8
+1.0
+L (m)
+- 1.0
+- 0.8
+- 0.6
+- 0.4
+- 0.2
+0.2
+w/wmax
+Analytical solution
+Bond-based beam,n=2
+Bond-based beam,n=3
+Bond-based beam,n=4
+Figure 10: The influence of support size in a bond based beam .
+0.2
+0.4
+0.6
+0.8
+1.0
+L (m)
+- 1.0
+- 0.8
+- 0.6
+- 0.4
+- 0.2
+0.2
+w/wmax
+Analytical solution
+Bond-based beam,ω(r)=1
+Bond-based beam,ω(r)=r
+Bond-based beam,ω(r)=r2
+Bond-based beam,ω(r)=r3
+Bond-based beam,ω(r)=r4
+Figure 11: The influence of weight function in a bond based beam.
+27
+
+0.2
+0.4
+0.6
+0.8
+1.0
+L (m)
+0.2
+0.4
+0.6
+0.8
+1.0
+Damage
+T=0.005 s
+T=0.006 s
+Figure 12: Damage in bond based beam.
+0.2
+0.4
+0.6
+0.8
+1.0
+L (m)
+-1.5
+-1.0
+-0.5
+w/wmax
+Analytical solution
+T=0.005 s
+T=0.006 s
+Figure 13: Deflection of bond based beam.
+λ = 121.1538 kN/mm2 and µ = 80.7692 kN/mm2 for elastic constants, Gc = 2.7 × 10−3
+kN/mm for critical energy release rate. These parameters are identical to that used in the
+small strain brittle fracture phase field in Ref [3]. Two displacement conditions are tested:
+Case a) for tensile boundary condition and Case b) for shear boundary conditions. The plate
+is discretized into 100 × 100 material points. The displacement load is monotonic applied
+with velocity boundary condition defined by
+v(t) =
+�
+�
+�
+�
+�
+t
+t0vmax
+if t < t0
+vmax
+otherwise
+with t0 = 1.0 × 10−5s and vmax = 2 m/s.
+In the case of tensile load, two discretizations are employed. The load curves for the
+tensile boundary are plotted in Fig.17.
+The maximal normal strain criterion is derived
+in a simple way but it is very effective in application. From the test of tensile boundary
+condition, the load-curve matched the result by FEM phase field model very well. The study
+28
+
+0.5                 0.5
+0.5                 0.5
+a)
+b)
+initial crack
+Figure 14: Single-edge-notched test. Geometry and Case a for tensile boundary condition and Case b for
+shear boundary condition.
+Figure 15: Single-edge-notched tension test, damage patterns for 60x60 particles and 120x120 particles.
+on different discretization shows that the damage model is insensitive to the discretization.
+It is interesting that the fracture model by explicit time integration without using damping
+agrees so well with the phase field method in the static case.
+The evolutions of displacement field and velocity field are plotted in Fig.16. The crack
+initiates when the boundary displacement reaches uy = 3.96 × 10−3 mm as indicated by
+the irregular velocity field around the crack tip in Fig.16(e). In the stage of stable crack
+propagation, the significant velocity wave due to cutting bond can be observed in Fig.16(f).
+The velocity field is greatly interfered by the fracture, but the displacement field is stable.
+The final result for shear tests with a discretization of 120×120 is depicted in Fig.18. In
+Fig.19, the damage patterns for different discretization subjected to shear loading condition
+are plotted. With finer discretization, the resolution of fracture becomes sharper.
+The displacement curve for shear test is plotted in Fig.20. Before the crack is initialized,
+the result by NOM agrees with the FEM result. The crack began to initialize when the
+29
+
+4.40×10-7
+1.32×10-6
+2.20×10-6
+3.08×10-6
+3.96×10-6
+4.60×10-7
+1.38×10-6
+2.30×10-6
+3.22×10-6
+4.14×10-6
+5.70×10-7
+1.71×10-6
+2.85×10-6
+3.99×10-6
+5.13×10-6
+0
+1.3×10-6
+2.6×10-6
+3.9×10-6
+5.2×10-6
+-0.168
+-0.120
+-0.072
+-0.024
+0.024
+-0.25
+-0.15
+-0.05
+0.05
+0.15
+-2.4
+-1.2
+0
+1.2
+2.4
+-3.6
+-1.2
+1.2
+3.6
+6.0
+(a)   
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+(b)  
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+   (c)  
+ 
+ 
+ 
+ 
+ 
+ 
+             (d)
+(e)  
+ 
+ 
+ 
+ 
+ 
+ 
+             (f)  
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ (g)  
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+(h)
+Figure 16: Single-edge-notched tension test: first row (a,b,c,d) denotes displacement field in y-direction and
+the second row (e,f,g,h) velocity in x-direction. Two figures of the same column correspond to the same
+time.
+0.001
+0.002
+0.003
+0.004
+0.005
+0.006
+Uy (mm)
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+FN (kN)
+NOM 120x120
+NOM 60x60
+FEM-PF
+Figure 17: Single-edge-notched tension test, load curves for NOM and phase field by FEM.
+5.20×10-7
+1.04×10-6
+1.56×10-6
+2.08×10-6
+2.60×10-6
+3.12×10-6
+3.64×10-6
+4.16×10-6
+4.68×10-6
+5.20×10-6
+-3.×10-7
+-2.×10-7
+-1.×10-7
+0
+1.×10-7
+2.×10-7
+3.×10-7
+4.×10-7
+5.×10-7
+6.×10-7
+0.062
+0.124
+0.186
+0.248
+0.310
+0.372
+0.434
+0.496
+0.558
+0.620
+Figure 18: Single-edge-notched plate subjected to shear boundary condition: Displacement field in x-
+direction and y-direction and the damage distribution.
+30
+
+Figure 19: Damage subjected to shear load.
+0.002
+0.004
+0.006
+0.008
+0.010
+0.012
+Ux (mm)
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+Fx (kN)
+NOM 120x120
+NOM 60x60
+FEM-PF
+Figure 20: Single-edge-notched shear test: load curves for NOM and phase field by FEM.
+displacement ux = 0.009 mm for both FEM and NOM. With the increase of load, the bond
+cutting process becomes irregular and the reaction force oscillates. This is partially because
+the particle distribution in support no longer aligns with the crack surface. It also reveals the
+complicated stress state due to perturbation of broken bonds. The overall fracture pattern
+agrees with that by the FEM phase field method. Due to the discrete feature of the current
+method, the crack surface is not as smooth as that by continuum methods such as the phase
+field.
+6.3. Critical shear damage criterion
+The material parameters are the same as previous sections except that the energy release
+rate for mode II is selected as GII = 3×10−3 kN/mm. The critical shear stretch is calculated
+based on Eq.60. Fig.22 and Fig.23 are the resultant reaction forces in x and y directions of
+material points at the top of the plate. For the pure shear boundary conditions, the force
+in y direction is neglectable compared to that in x direction. The peak reaction force is
+proportional to the square root of GII. It is also observed that the load curve is insensitive
+31
+
+0.5                 0.5
+0.5                 0.5
+u
+initial crack
+uy
+        
+     ux
+u
+uy
+        
+     ux
+Figure 21: Single-edge-notched test based on shear damage criterion: displacement boundary.
+0.005
+0.010
+0.015
+0.020
+0.025
+Ux (mm)
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+Fx (kN)
+120x120,GII
+60x60,GII
+120x120,4GII
+60x60,4GII
+Figure 22: Single-edge-notched shear test: load curves for the case of ux : uy = 2 : 0.
+0.005
+0.010
+0.015
+0.020
+0.025
+Ux (mm)
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+Fy (kN)
+120x120,GII
+60x60,GII
+120x120,4GII
+60x60,4GII
+Figure 23: Single-edge-notched shear test: load curves for the case of ux : uy = 2 : 0.
+32
+
+Figure 24: Single-edge-notched shear strain: fracture patterns on different displacement boundaries.
+to the discretization. For the case of discretized by 120x120 and 4GII, the peak reaction force
+is F max
+x
+= 1.29 kN when displacement reaches ux = 2.253×10−2 mm, which corresponds the
+external work approximately as Wext = 1
+2F max
+x
+ux = 1.455 × 10−2 J. The energy consumed
+by forming fracture surface is 2× (4GII) × lcrack = 2× (4× 3× 10−3) × 0.5 = 1.2× 10−2 J, in
+which lcrack = 0.5 mm is the crack length and multiplying 2 denotes two crack surfaces. The
+fracture energy is slightly less than the total external work Wext. The result is reasonable
+because the kinetic energy and strain energy comprise of certain portion of the total energy.
+We also test the influence of the loading angle.
+By adjusting the ratio of ux : uy,
+different paths of shear crack can be observed, as shown in Fig.24. Interestingly, the crack
+path direction is quite close to the displacement direction. The crack paths of the case with
+ux : uy = 2 : 1 and the case with ux : uy = 2 : −1 are symmetric with respect to the horizontal
+line. It can be concluded that the critical shear strain damage rule can automatically find
+the direction of maximal shear strain and form a shear crack path consistently.
+6.4. Kalthoff-Winkler experiments
+The Kalthoff-Winkler experiment [36] is a classical benchmark problem for dynamic
+fracture modeling [37, 38, 39]. For different impact velocities, the fracture can be brittle or
+ductile. For low impact velocities, the dynamic brittle fracture propagates from the crack
+tip at an angle of around 70◦ vs. the direction of the initially horizontal crack. When the
+impact velocity is increased further, a ductile failure (or shear fracture) phenomenon occurs
+and a shear band is observed. The dimension of the plate is 0.2×0.1 m2 as shown in Fig.25.
+The material parameters are E = 190 GPa, ν = 0.3, Gc = 2.4 × 104 J/m2. Consider the
+symmetry, only half of the plate is modeled. The plate is discretized with 200x200 particles.
+33
+
+ux : uy =2: -1Ux : uy = 1 : 2ux : uy = 1 : 1ux : uy = 2: 1ux : uy =2: 00.05m
+0.075m
+0.075m
+0.2m
+0.1m
+v0
+v0
+0.05m
+Figure 25: Setup for the Kalthoff-Winkler experiment.
+The support radius is selected as l = 3∆x m. The initial crack is represented by modifying
+the neighbors in support. The number of neighbors for each particle is restricted to 28. The
+brittle failure at low impact velocity is studied with critial normal strain criterion. In this
+case, the velocity applied on the top of the plate starts from 0 to vy = 20m/s in a period of
+10−7 s and remains constant thereafter [40]. For the case of shear fracture at higher impact
+velocity vy = 39 m/s, the critical shear damage criterion is employed and energy release rate
+of mode-II fracture is selected as GII = 4Gc.
+In low impact velocity, the displacement field and velocity field in x-direction at different
+times are depicted in Fig.26. It can be observed that the crack initiated at 24µs and finished
+at 82µs. Around the crack tip, the breaking of bonds causes obvious ossilation of velocity.
+The final crack path of tensile crack is shown in Fig.27. For the higher impact velocity, the
+displacement field and velocity are plotted in Fig.28. The shear crack starts to propagate at
+14.4µs, following the direction of initial crack. At the final stages, crack branching of shear
+fractures is observed as shown in Fig.29.
+7. Conclusions
+In this work, we have proposed several bond-based models for solids, thin plates, and
+gradient solids in different dimensional spaces. The main motivation is to derive the force
+34
+
+Figure 26:
+Kalthoff-Winkler test vy
+=
+20 m/s:
+ux (first row) and vx (second row) at times
+(24µs, 33.7µs, 62.6µs, 82µs).
+0.087
+0.174
+0.261
+0.348
+0.435
+0.522
+0.609
+0.696
+0.783
+0.870
+Figure 27: Kalthoff-Winkler test vy = 20 m/s: tensile fractures.
+Figure 28:
+Kalthoff-Winkler test vy
+=
+39 m/s:
+ux (first row) and vx (second row) at times
+(14.4µs, 19.3µs, 33.7µs, 43.4µs).
+35
+
+0.000340
+0.000610
+0.00110
+0.00144
+0.000306
+0.000549
+0.00099
+0.00132
+0.000272
+0.000488
+0.00088
+0.00120
+0.000238
+0.000427
+0.00077
+0.00108
+0.000204
+0.000366
+0.00066
+0.00096
+0.000170
+0.000305
+0.00055
+0.00084
+0.000136
+0.000244
+0.00044
+0.00072
+0.000102
+0.000183
+0.00033
+0.00060
+0.000068
+0.000122
+0.00022
+0.00048
+0.000034
+0.000061
+0.00011
+0.00036
+21.0
+28.8
+ 55
+75
+17.5
+25.2
+ 44 
+- 60 
+14.0
+21.6
+ 33
+- 45 
+10.5
+18.0
+22
+30 
+7.0 
+14.4
+11
+15
+3.5 
+10.8
+0
+0
+0
+7.2 
+11
+-15
+-3.5
+3.6
+-22 
+-30
+-7.0 
+0
+-33
+-45
+-10.5
+-3.6
+-44
+-600.00050
+0.000670
+0.00112
+0.00125
+0.00045
+0.000603
+0.00096
+0.00100
+0.00040
+0.000536
+0.00080
+0.00075
+0.00035
+0.000469
+0.00064
+0.00050
+0.00030
+0.000402
+0.00048
+0.00025
+0.00025
+0.000335
+0.00032
+0
+0.00020
+0.000268
+0.00016
+-0.00025
+0.00015
+0.000201
+0
+-0.00050
+0.00010
+0.000134
+-0.00016
+-0.00075
+0.00005
+0.000067
+-0.00032
+-0.00100
+47.0
+ 45
+75
+114
+42.3
+ 40
+ 60 
+95
+37.6
+35
+45
+76
+32.9
+30
+ 30 
+ 57
+28.2
+25
+15
+38
+23.5
+20
+0
+19
+18.8
+15
+-15
+0
+14.1
+10
+30
+-19
+ 9.4 
+5
+-45
+86-
+4.7
+0
+-60
+-570.087
+0.174
+0.261
+0.348
+0.435
+0.522
+0.609
+0.696
+0.783
+0.870
+Figure 29: Kalthoff-Winkler test vy = 39 m/s: shear fractures.
+constitution that depends only on the bond deformation, while the collective deformations
+recover the traditional local theory by using an energy equivalence principle and assuming
+a constant, fully symmetric support region. For the bond-based NOM, a weight function is
+introduced to define the bond forces. To account for the bending effect due to curvature,
+the nonlocal curvature and the nonlocal moment are defined by introducing a bent bond.
+The bent bond is defined symmetrically by including three points. It is shown that the thin
+plate model based on bonds has a Poisson’s ratio restriction. A simple bond-based gradient
+elasticity is also developed based on the equivalence of the gradient deformation energy.
+The bond-based elasticity accounts for the strain extension and shear deformation of a
+bond and has no Poisson’s ratio constraint. By introducing a weight function, the distribu-
+tion of nonlocal bond strain energy can be regulated. At the same time, a cohesive damage
+model for bent bonds is proposed, which weakens the bond force when the threshold value
+of bond strain or bond curvature is reached. This setting provides a simple rule to localize
+the strain without cutting the bond. In addition, a plasticity model is derived based on the
+increase in bond force for a single bond.
+Several numerical examples are presented, including a simple support beam and a 2D
+solid plate with shear or tensile damage patterns. Although the numerical examples use
+explicit time integration, the implicit implementation is straightforward for static problems.
+For each bond element, one can calculate the second variation and convert the tangent
+stiffness matrix in the local coordinate system to the global coordinate system. Last but not
+least, a simple rule for the critical normal strain and critical shear strain for bond cutting
+36
+
+in tensile and shear fracture modeling is proposed, which is numerically stable and easy to
+implement and can achieve comparable results to the phase field method.
+Acknowledgments
+The first author gratefully acknowledges financial support from the EU project enti-
+tled ”Computational Modeling, Topological Optimization and Design of Flexoelectric Nano
+Energy Harvesters” (ERC COTOFLEXI 802205).
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+
diff --git a/VtAyT4oBgHgl3EQf8_oB/content/tmp_files/load_file.txt b/VtAyT4oBgHgl3EQf8_oB/content/tmp_files/load_file.txt
new file mode 100644
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+page_content='Bond-based nonlocal models by nonlocal operator method in  symmetric support domain  Huilong Rena,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' Timon Rabczukb,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' Xiaoying Zhuanga,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' Zhiyuan Lid  aInstitute of Photonics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' Department of Mathematics and Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' Leibniz University Hannover,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='Germany  bInstitute of Structural Mechanics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' Bauhaus-Universit at Weimar,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' Germany  cState Key Laboratory of Disaster Reduction in Civil Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' College of Civil Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='Tongji  University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' Shanghai 200092,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' China  dCollege of Mechanics and Materials,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' Hohai University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' Nanjing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' 211100,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' China  Abstract  This paper is concerned with the energy decomposition of various nonlocal models,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' including  elasticity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' thin plates,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' and gradient elasticity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' to arrive at bond-based nonlocal models in  which the bond force depends only on the deformation of a single bond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' By assuming an  appropriate form of bond force and using energy equivalence between local and nonlocal  models, several very concise bond-based models are derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' We also revisit the nonlocal  operator methods and study the simplified form of second-order NOM in the symmetric  support domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' A bent-bond consisting of three points is proposed to describe the curvature  and moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' To model the damage, a rule based on Griffith theory for the critical normal  strain of the bond is proposed in analogy to the phase field model, which can be applied  individually to each bond and provides strain localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' With this rule, the crack direction  can be automatically predicted by simply cutting the bond, giving comparable results to  the phase field method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' At the same time, a damage rule for critical shear strains in shear  fractures is proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' Furthermore, an incremental form of the plasticity model for bond  reaction force is derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' Several numerical examples are presented to further validate the  nonlocal bond-based models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  Keywords: energy decomposition, bond-based, bent bond, tensile damage, shear dam-  age, fracture, phase field  Preprint submitted to Elsevier  January 4, 2023  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='00864v1  [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='class-ph]  2 Jan 2023  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' Introduction  Material damage and structural failure are an enduring problem in many engineering  applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' Inadequate description can lead to serious hazards and financial losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' The  problem arises from the fact that the complicated mechanisms in the damage process are  difficult to predict both in theoretical models and by numerical methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' Nevertheless, great  efforts have been made in the last decades to develop many robust numerical methods, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  Damage mechanics [1, 2], phase field method [3, 4], extended finite element method [5, 6],  meshless methods [7, 8], cracking particle method [9, 10], virtual crack closure technique[11],  Peridynamics (PD) [12] and many others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' These methods generally fall into two categories:  Using an additional field to represent the crack or modifying the topology to form a crack  surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' The additional field in the phase field method forms the topology of the crack  surface without modifying the mesh, and it is numerically stable and smooth, but at the  expense of solving another field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' The method based on topology modification can directly  form a sharp crack surface, but the geometric operation can lead to instability problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  These two categories seem to be quite different, but both can be based on Griffith theory,  according to which the creation of new free surfaces is due to the conversion of reduced  potential energy into surface energy [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  Two representative examples of fracture modeling are phase field methods and nonlocal  methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  The phase field method can handle many difficult engineering problems in a  relatively simple theoretical framework, see for example [14, 15, 16, 17, 18, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' Peridynamics,  as a nonlocal theory, has some advantages in topology modification because it separately  accounts for interactions in a finite-size domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' In other words, the strain energy density is  distributed in the domain rather than in a point of no size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' The severing of individual bonds  provides great physical intuition for cracking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' Examples include bond-based PD [20], bond-  based PD with shear deformation [21], extended bond-based PD [22, 23, 24], the conjugate  ∗Institute of Photonics, Department of Mathematics and Physics, Leibniz University Hannover,Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  zhuang@iop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='uni-hannover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='de;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  Email addresses: huilong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='ren@iop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='uni-hannover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='de (Huilong Ren),  timon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='rabczuk@uni-weimar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='de (Timon Rabczuk), zhuang@iop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='uni-hannover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='de  (Xiaoying Zhuang),  lizhiyuan1007@163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='com (Zhiyuan Li)  2  bond pair-based PD [25], the bond-based micropolar PD [26] and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' In the bond-based  PD, the energy carried by a bond is actually separated from each other, so they have good  numerical stability in fracture modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' The state-based PD [12] can deal with continuous  problems, but the bond cutting process is not very stable since all bonds in the horizon are  fully coupled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' As a generalization of the dual-horizon peridynamics [27], in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' [28, 29, 30]  the nonlocal operator method (NOM) is proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' It provides a rule for converting many  local models into their corresponding nonlocal forms, as well as introducing a variational  framework for solving many difficult problems [31, 32, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  Bond-based models provide great flexibility in fracture modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' NOM expresses the  derivatives of a function over the collective information in the support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' It does not require  that the support domain be constant or have regular shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' The general form of NOM  seems complicated, and in particular, it cannot be used directly to model fracture by cutting  bonds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' Considering the advantages of bond-based peridynamics, such as automatic crack  development and crack direction determination, we want to develop some general bond-  based models for various mechanical problems so that this property can be used for fracture  modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' The cutting of bonds to form fractures is based on the critical strain in BB-PD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  The exact critical strain based on Griffith theory in nonlocal models is still not consistent  with respect to load-displacement curves compared to other methods such as the phase field  method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' Therefore, two main objectives of the research are a) to derive bond-based nonlocal  models based on NOM in symmetric support and b) to determine an appropriate critical  stretch based on Griffith theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  The remainder of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' Section 2 explains the motivation  from the perspective of orthogonal energy decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' Considering the relatively com-  plicated form of NOM, a simplified second-order NOM with symmetric support is presented  in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' In the same section, the weighted bond-based nonlocal bar and nonlocal beam  are derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' In Section 4, the bond-based nonlocal elasticity is derived in 2D and 3D using  energy equivalence in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' In analogy to the phase field method, a simple but effective  bond-cutting criterion based on normal or shear strains is proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' The plastic model for  the bond- element in nonlocal elasticity is derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' In Section 5, using the second order  3  NOM in ideal support, the nonlocal bond-based isotropic thin plate model and the nonlocal  bond-based gradient elasticity are derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' Three numerical experiments are presented in  Section 6, including the nonlocal simple support beam and crack propagation in single-edge-  notched plate under tension/shear boundary conditions and the Kalthoff-Winkler test with  tension and shear fractures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' Some conclusions and outlook are given in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' Motivation: energy orthogonal decomposition  Miehe [3] proposed a thermodynamic consistent phase field model for brittle fracture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  The success of this model lies in the orthogonal decomposition of strain energy density of  isotropic linear elasticity  ψ = 1  2σ : ε = 1  2(  3  �  i=1  σini ⊗ ni) : (  3  �  i=1  εini ⊗ ni) = 1  2  �  iσiεi,  (1)  where the stress tensor σ = �3  i=1 σini ⊗ ni and strain tensor ε = �3  i=1 εini ⊗ ni are  formulated based on the eigenvalue decomposition of a matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' For linear isotropic elas-  ticity, the orthogonal decomposition of stress tensor and strain tensor are co-axial and the  positive/negative parts of stress tensor can be written as  σ± =∂ψ±  e  ∂ε = λ⟨ε1 + ε2 + ε3⟩±I  + 2µ(⟨ε1⟩±n1 ⊗ n1 + ⟨ε2⟩±n2 ⊗ n2 + ⟨ε3⟩±n3 ⊗ n3),  (2)  where I is the identity matrix, n1, n2, n3 are the eigenvectors of the associated principal  strains ε1, ε2, ε3 of ε and ⟨x⟩± = (x±|x|)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' One disadvantage of the phase field model is the  complicated calculation of partial derivatives of eigenvalues and eigenvectors with respect  to strain tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  One of the most successful Peridynamics is the bond-based version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' In bond-based PD,  the strain energy carried by a point is  ψ = 1  2  �  S  cs2dV,  (3)  where s is the extensional strain for the bond and c is the material parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' The energy for  each bond is independent of each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' In state-based PD, the bond-force depends on the  4  state (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' stress tensor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' Breaking bond leads to singularity of shape tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' The singularity  does not occur to bond-based PD owing to the independence of all bonds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' Although the  original bond-based PD has limitations such as Poisson ratio restriction, it is quite robust  for the modeling of tensile fractures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  The local elasticity and nonlocal elasticity model are related by energy equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' As  shown in [34], based on the local models, many nonlocal models can be derived by variational  derivations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' Let us focus on the linear elasticity and the strain energy density is written as  ε : C : ε, where ε := 1  2(∇u + ∇uT) is the strain tensor described by displacement gradient  ∇u, and C is the 4-th order material tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' Then we use nonlocal gradient to replace strain  tensor, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' ε →  �  Siω(rij)uij ⊗gijdVj, where uij = uj −ui, ω(rij) is the weight function and  gij is a function of rij, the equivalent nonlocal energy density can be conceptually written  as  (  �  Siω(rij)uij ⊗ gijdV ) : C : (  �  Si ω(rij)uij ⊗ gijdV )  (4a)  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='=  �  Si(ω(rij)uij ⊗ gij) : ¯C(rij) : (uij ⊗ gij)dV,  (4b)  where ¯C(rij) is the material tensor for a single bond rij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  Each integral form contains  infinite terms and the multiplication of two integral forms leads to more infinite terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  It would be great when Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='4b is equal to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='4a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' This condition is similar to find when  (�n  i=1 ai) · (�n  i=1 bi)  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='= �n  i=1 ai · bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' Mathematically, the derivation requires the orthogonal  condition, that is ai · bj = δij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  Local model is formulated based on differential equations and nonlocal theories such as  peridynamics are expressed by integral equation  ∇ · σ + b = ρ¨u,  �  SifijdVj + b = ρ¨ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  (5)  The most distinct difference between local model and nonlocal model is the way to view  internal force, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' The former is defined on point structure without shape  size while the latter introduces a neighborhood of finite size to account for the nearby  interaction explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  In dual-horizon PD, the shape of the horizon has great flexibility,  although the circular horizon is prefered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' By distributing internal force on the finite size  5  f  f  f  f  f  f  f  f  n  s  (a)                                               (b)                                                  (c)  Figure 1: Internal forces (a) between virtual segments in local theory, (b) between two micro-volumes in  nonlocal theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' (c) bond-force decomposition f = fs + fn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  Point  Figure 2: Shapes of support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  horizon domain while other physical quantities such as density is the same as conventional  local theory, the finite size horizon offers advantage to manipulating the internal force when  discontinuity or damage happens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' Traditional local theories are formulated on point sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  Mathematically, a point is infinitesimally small and has no specific shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' It is inconvenient  to break one point into two points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' The closest shape to a point is circular domain in 2D or  spherical domain in 3D when ideal symmetry of a geometric object is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' Indeed, for  complete symmetric horizon domain or support domain, the nonlocal models can be greatly  simplified at least for the nonlocal operator methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' By doing this, it is possible to derive  some bond-based versions of nonlocal models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  In the current research, the aim is the simplification of NOM by considering a symmetric  support or horizon and therefore the dual-horizon is the same as the conventional horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  Furthermore, the derivation of bond-based solid, thin plate and gradient solid is pursued,  and for each bond, there is no need to consider the reaction force and direct force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' The  calculation of internal force of each material point is independent of other points, which  offers some merit for implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  6  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' Nonlocal operator method  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' Revisit of NOM in symmetric support  In a support, the nonlocal derivatives are calculated by  ˜∂ui = (ui,x, ui,y, ui,xx, ui,xy, ui,yy)T :=  �  Si  ω (rij) Ki · pijuijdVj  (6a)  with  uij = uj − ui = u⟨r⟩, rij = r = (rx, ry) = (xij, yij) = rn  (6b)  pij =  �  xij, yij, x2  ij/2, xijyij, y2  ij/2  �T  (6c)  Ki =  ��  Si  ω (rij) pij ⊗ pT  ijdVj  �−1  ,  (6d)  where n is the unit direction of bond r and r = ||r|| and ω(r) is the weight function, for  example, ω(r) = 1, ω(r) = 1/r2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' Herein, both □ij and □⟨r⟩ are used to denote the physical  quantities related to a bond and □⟨r⟩ is used when the bond pair is not explicitly specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  When the support domain is a circular area, the expression of Ki can be greatly simplified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  With some mathematical manipulation, the nonlocal gradient and nonlocal Hessian become  ˜∂u =  �  S  ω(r)u⟨r⟩  �  (rx, ry)  π  � δ  0 ω(r)r3 dr  ,  1  π  � δ  0 ω(r)r5 dr  (3r2  x − r2  y, 4rxry, 3r2  y − r2  x)  �  dV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  For single bond, its contribution to the nonlocal derivative can be simply written as  ˜∂u⟨r⟩ = ω(r)u⟨r⟩  �  (rx, ry)  π  � δ  0 ω(r)r3 dr  �  ��  �  g  ,  1  π  � δ  0 ω(r)r5 dr  (3r2  x − r2  y, 4rxry, 3r2  y − r2  x)  �  ��  �  ⃗h  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  Based on above formula, we extract bond gradient vector g and bond curvature tensor h2d  from vector ⃗h in 2D, which are defined as  g = rn  αg  with αg =  �  �  �  �  �  π  � δ  0 ω(r)r3 dr in 2D  4π  3  � δ  0 ω(r)r4 dr in 3D  ,  (7)  h2d =  r2  π  � δ  0 ω(r)r5 dr  �  �4n2  x − 1  4nxny  4nxny  4n2  y − 1  �  � =  r2  π  � δ  0 ω(r)r5 dr  �  4n ⊗ n − I  �  ,  (8)  7  where n = (nx, ny) = r/r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' Similarly, the matrix form of the curvature in 3D is  h3d =  3r2  4π  � δ  0 ω(r)r6 dr  �  5n ⊗ n − I  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  (9)  For circular or spherical support, the shape tensor can be written by the identity matrix  �  S  ω(r)r ⊗ rdV = αsI  (10)  with coefficients defined as α2d =  � δ  0 ω(r)πr3dr, α3d =  � δ  0 ω(r) 4  3πr4dV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  The nonlocal gradient, nonlocal divergence and nonlocal curl operator using explicit bond  notations can be rewritten as  ˜∇ ∗ ui :=  �  Si  ω(rij)gij ∗ uijdVj,  (11)  where ∗ ∈ {⊗, ·, ×} and gij is defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='7 for bond rij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  For the case of gradient operator, the nonloal gradient of a vector field can also be written  as  ∇u =  �  S  3ω(r)  4π  � δ  0 ω(r)r4 dr  (rx, ry, rz) ⊗ u⟨r⟩dV =  �  S  ω(r)g ⊗ u⟨r⟩dV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  (12)  The Hessian of a scalar field in 2D has the form  ∇∇u =  �  S  ω(r)u⟨r⟩  π  � δ  0 ω(r)r5 dr  �  �3r2  x − r2  y  4rxry  4rxry  3r2  y − r2  x  �  � dV  =  �  S  ω(r)h ⊗ u⟨r⟩dV  (13)  =  �  S+ ω(r)h ⊗ (u⟨r⟩ + u⟨−r⟩)dV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  (14)  In the last step, h is invariant for both r and −r and the symmetricity of S is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  The half support S+ is defined based on the symmetric support domain as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  In the definition of curvature, only a half support is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  Traditional NOM deals with the gradient or Hessian tensor at a point as a whole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' In  this sense, all bonds in support domain are coupled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=" In the spirit of bond-based PD, it is  8  i  j  j'  r  r  δ  ij--bond  ijj'--bent bond  S  S+  S-  Figure 3: Normal bond and bending bond." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' S = S+ ∪ S−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' j′ ∈ S− is the symmetric point of j ∈ S+ with  respect to center point i, ijj′ for a bent bond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  natural to define the derivatives for each individual bond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' Based on Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='7, the bond gradient  on single bond is defined as  ∇u⟨r⟩ = r  r2u⟨r⟩ = u⟨r⟩n  r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  (15)  Based on Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='8, the curvature of a pair-wised bond in 2D is defined as  ∇∇u⟨r⟩ = (u⟨r⟩ + u⟨−r⟩)  r2  (4n ⊗ n − I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  (16)  Above definition is reasonable because u⟨r⟩  r2 (4n⊗n−I) ≈ (∇∇u : (r⊗r))/r2(4n⊗n−I) =  (∇∇u : (n ⊗ n))(4n ⊗ n − I), which depends on the bond direction and second-order  derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  Conventionally, NOM can derive the nonlocal model from the local energy functional  by using the nonlocal gradient or nonlocal Hessian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' Consider a general field with energy  density φ being a function of ∇u and ∇2u, the total potential energy in domain is  Ψ =  �  V  φ(∇u, ∇2u)dV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  (17)  The variation of the energy functional  δΨ =  �  V  ∂φ  ∂∇u : ∇δu +  ∂φ  ∂∇2u˙:∇2δudV  =  �  V  σi :  �  Si  ω(r)δuij ⊗ gij + Σi˙:  �  Si  ω(r)δuij ⊗ hijdVj  =  �  V  �  Si  ω(r)σi · gij · δuij +  �  Si  ω(r)Σi : hij · δuijdVj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  (18)  9  where σ := ∂φ  ∂ε, ε = 1  2(∇u+∇uT) and Σ :=  ∂φ  ∂∇ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' For the cases of the general linear elasticity  and the general linear gradient elasticity, the material constitutions are  σ = C : ε, or σij = Cijklεkl  (19a)  Σ = D : ∇ε, or Σijk = Dijklmn∂lεmn,  (19b)  where C, D are material tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  The nonlocal governing equations for elasticity and gradient elasticity are  �  S  (ω(rij)σi · gij − ω(rji)σj · gji) + bi = ρ¨ui,  (20a)  �  S  (ω(rij)(σi · gij + Σi : hij) − ω(rji)(σj · gji + Σj : hji)) + bi = ρ¨ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  (20b)  Based on the material constitutive of thin plate  M =  �  � Mxx  Mxy  Mxy  Myy  �  � = D0 (νtr(κ)I2×2 + (1 − ν)κ) ,  (21)  the nonlocal governing equation of nonlocal thin plate is  �  Si  (fij + fji) + q = ρ ¨wi, with fij = ω(rij)Mij : hij,  (22)  where D0 =  Et3  12(1−ν2) and t is the thickness of the plate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  The above equation depends on the state quantity defined in the support domain, where  the internal force of each bond is fully coupled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' Is it possible to derive the force or mo-  ment depending on the bond only by using the bond derivatives given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='15 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='16?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  The answer is the decoupled NOM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' In the following sections, the conditions of decoupled  NOM for nonlocal elasticity, nonlocal thin plate and nonlocal gradient elasticity in different  dimensional spaces will be explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' Decoupled NOM in 1D  In order to illustrate the decoupled NOM, let us consider the NOM in 1D and derive the  nonlocal bar/beam models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' For the case in one-dimensional space, the second-order NOM  10  can be derived based on the Taylor series in 1D as  uij = u′  ixij + 1  2u′′  i x2  ij  (23a)  � δ  −δ  ω(rij)uijx2 = 1  2u′′  � δ  −δ  (ω(rij)x4)dx  (23b)  u′′ =  � δ  −δ ω(rij)uijx2dx  � δ  0 ω(rij)x4dx  =  � δ  0 ω(rij)(uij + uij′)x2dx  � δ  0 ω(rij)x4dx  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  (23c)  For the case of one dimension, the gradient and curvature of a particle can be derived  similarly as  du  dx =  1  � δ  −δ ω(x)x2dx  � δ  −δ  ω(x)uijxdx  d2u  dx2 =  1  � δ  0 ω(x)x4dx  � δ  0  ω(x)(uij + uij′)x2dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  (24)  For each direction, the bond individual gradient and curvature are  du  dx⟨x⟩ = uijx  x2  = uij  x , d2u  dx2⟨x⟩ = (uij + uij′)x2  x4  = (uij + uij′)  x2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  (25)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' One-dimensional nonlocal bar  Consider a one-dimensional nonlocal bar model with elastic modulus of E and section  area of A, we assume the bond energy density as  φij = 1  2eijfij|rij| = 1  2ω(rij)cu2  ij/|rij|,  (26)  where eij = uij/|rij| is the relative strain and fij = ω(rij)ceij is the bond force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' The energy  equivalence between local model and nonlocal model requires  � δ  −δ  φijdx = 1  2EAε2  i ,  (27)  where εi is the local strain at point xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  In order to derive the specific form of bond force, a displacement field u(x) = ax with  constant gradient u′(x) = a is assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' Let ui = 0, uj = ax, then uij = ax, eij = uij/|x| =  a sign(x), fij = ω(x)ceij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' In 1D, only the elongation is involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' The strain energy carried  by a bond due to bond force and displacement becomes  φij = 1  2eijfij|rij| = 1  2a2|x|cω(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  (28)  11  Here the process of doing work is considered, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' bond force fij acting on distance eij|rij|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  The energy equivalent in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='27 is calculated as  � δ  −δ  1  2eijfij|x|dx =  � δ  −δ  1  2a2|x|cω(x)dx = 1  2EAa2 → c =  EA  2  � δ  0 ω(x)xdx  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  (29)  The bond force in 1D is the variation of bond energy  fij = δφij  δuij  =  EA  2  � δ  0 ω(x)xdx  ω(rij)uij  |rij|  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  (30)  Another scheme to consider the bond energy is  φij = 1  2eijfij = 1  2a2cω(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  (31)  The energy equivalence leads to  � δ  −δ  1  2eijfijdx =  � δ  −δ  1  2a2cω(x)dx = 1  2EAa2 → c =  EA  2  � δ  0 ω(x)dx  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  (32)  The bond force of bond ij is  fij = δφij  δuij  =  EA  2  � δ  0 ω(x)dx  ω(rij)uij  r2  ij  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  (33)  It should be noted that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='30 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='33 are equivalent when the weight function in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='33  is set as |rij|ω(rij).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' The direct bond force is added to material point i and the reaction  bond force is added to j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' However, the bond ij of i and bond ji of j are the same, and the  calculation of bond ji gives i a reaction bond force −fji, which satisfies −fji = fij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' Hence,  the governing equation of nonlocal bar is  � δ  −δ  2fijdx + b = ρ¨ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  (34)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' One-dimensional nonlocal beam  Consider a one-dimensional nonlocal beam model with elastic modulus denoted by E  and the second moment of area of the beam’s cross section denoted by I, we can assume the  bond bending energy density as  φij = 1  2κijmij = 1  2ω(rij)cκ2  ij,  (35)  12  where κij is the relative curvature and mij = ω(rij)cκij is the moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' The energy equiva-  lence between local model and nonlocal model requires  � δ  −δ  1  2κijmijdx = 1  2EIκ2  i ,  (36)  where κi is the local curvature at point xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  Let us assume a deflection field u(x) = x2/2 with constant curvature κ = u′′(x) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  Let xi = 0, xj = x, then uij = x2/2, κij = (uij + uij′)/x2 = 1, mij = ω(x)cκij = cω(x),  1  4κijmij = 1  4cω(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' The bending energy carried by a bond due to curvature is  1  2κijmij = 1  2cω(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  (37)  The equivalent of bending energy in support to the local model can be simplified as  � δ  0  1  2κijmijdx =  � δ  0  1  2cω(x)dx = 1  2EIκ2 = 1  2EI → c =  EI  � δ  0 ω(x)dx  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  (38)  For a homogeneous beam with thickness h, the coefficient c of different weight functions  can be written as  ω(x)EI  � δ  0 ω(x)dx  =  �  �  �  �  �  Eh3  12δ  if ω(r) = 1  Eh3r  24δ2  if ω(r) = r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  (39a)  Formerly, the nonlocal curvature and moment are defined as  κij = (uij + uij′)  r2  ij  , mij = ω(rij)(uij + uij′)  r2  ij  EI  � δ  0 ω(x)dx  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  (40)  The bent energy of bent bond is the multiplication of double volume ∆x2 and the bent  energy density φij as  φij∆x2 = 1  2κijmij∆x2 = 1  2  EIω(rij)  � δ  0 ω(x)dx  (uij + uij′)2  r4  ij  ∆x2,  (41)  where ∆x is the volume of the material point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  The variation of φij∆x2 reads  13  δφij∆x2 = EIω(rij)  � δ  0 ω(x)dx  (uij + uij′)  r4  ij  ∆x2  �  ��  �  force due to bent: fijj′  (δuj + δuj′ − 2δui)  = fijj′ · δuj + fijj′ · δuj′ − 2fijj′ · δui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  (42)  Therefore, the bond forces adding to i, j, j′ due to bond curvature energy are −2fijj′, fijj′, fijj′,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  When an implicit algorithm is used, the tangent stiffness matrix is also required, which  can be written by a second variation of φij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  δ2φij∆x2 = EIω(rij)(∆x)2  r4  ij  � δ  0 ω(x)dx  (δuj + δuj′ − 2δui)2  =  �  �  �  �  �  δui  δuj  δuj′  �  �  �  �  �  T  (EIω(rij)(∆x)2  r4  ij  � δ  0 ω(x)dx  )  �  �  �  �  �  4  −2  −2  −2  1  1  −2  1  1  �  �  �  �  �  �  ��  �  Kijj′  �  �  �  �  �  δui  δuj  δuj′  �  �  �  �  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  (43)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' Nonlocal isotropic elasticity  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' Bond force in 3D  γ  l  f  fs(γ )  fn(l )  (a)                                                           (b)  xj(t1)  rij  xi  uij  uij  +  θ  c1  c2  rij  xj(t2)  uij  ij  ij  ij  ij  r  Figure 4: Bond deformation with rotations, shear stiffness and extension stiffness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  Consider the strain tensor project on bond direction nij = (cos θ sin φ, sin θ sin φ, cos φ)  in spherical polar coordinate based φ ∈ [0, π), θ ∈ [0, 2π), the extension strain and shear  strain along the bond direction is  lij = (εi · nij) · (nij ⊗ nij)  (44a)  γij = (εi · nij) · (I − nij ⊗ nij).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  (44b)  14  The relative strain vector εn = εi · nij = lij + γij and the relative displacement is uij =  εnrij = (lij + γij)rij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  We assume the bond force be the form  fij = ω(rij)(c1lij + c2γij).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  (45)  The energy density carried by a bond’s deformation is  wij = 1  2fij · uij = ω(rij)(c1lij + c2γij) · (lij + γij)rij = 1  2ω(rij)rij(c1lij · lij + c2γij · γij).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  Then the nonlocal strain energy density at a point in support domain equalizes to the local  strain energy density  W =  �  Si  wijdVj =  �  Si  1  2rijω(rij)(c1lij · lij + c2γij · γij)dVj  = Wlocal = 1  2σ : ε = (λ Tr(ε)I + 2µε) : ε,  (46)  where λ, µ are Lame constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  For any ε, using undetermined coefficient method yields  c1 =  E  α(1 − 2ν), c2 =  E(1 − 4ν)  α(ν + 1)(1 − 2ν),  (47)  where α =  � δ  0  4  3πr3ω(r)dr, and elastic modulus E and ν are used to replace the Lame  constants by λ =  E  (1−2ν)(1+ν), µ =  E  2(1+ν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  When the weight function ω(rij) = 1, the coefficients become  c1 =  3E  πδ4(1 − 2ν), c2 =  3E(1 − 4ν)  πδ4(ν + 1)(1 − 2ν),  (48)  which are the same as the extended bond-based PD in [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' Here, the values of c1 or c2 is  one half of those in [22] because of the consideration of direct bond force and reaction bond  forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' In sum, the bond deformation and bond force considering the weight function are  lij = uij · nij  rij  nij  (49a)  γij = uij  rij  − lij = uij  rij  − uij · nij  rij  nij  (49b)  fij = ω(rij)(c1lij + c2γij).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  (49c)  15  And the corresponding governing equations are  �  Si  2fijdVj + b = ρ¨ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  (50)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' Bond force in 2D  For the case of plane stress condition, the material constitutive in local form is  σ =  E  1 − ν2(νtrϵI2x2 + (1 − ν)ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  (51)  The equivalence of strain energy density for arbitrary strain tensor leads to  c1 =  E  α(1 − ν), c2 = E(1 − 3ν)  α(1 − ν2) ,  (52)  where α =  � δ  0 πω(r)r2dr  Similarly, for plane strain condition, the material constitutive in local form is  σ =  E  (1 + ν)(1 − 2ν)(νtrϵI2x2 + (1 − 2ν)ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  (53)  The energy equivalent gives the coefficients as  c1 =  E  α(1 − ν − 2ν2), c2 =  E(1 − 4ν)  α(1 − ν − 2ν2),  (54)  where α =  � δ  0 πω(r)r2dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  By assuming appropriate bond deformation and bond force and considering the weighted  energy equivalent to the local energy, the weighted bond-based nonlocal elasticity in 1D,2D  and 3D are derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' By doing so, the energy for each bond depends on the bond deformation  only, the energy for each bond is separable while the collective energy of all bonds recovers  the isotropic elasticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  Remarks on implementation: In the definition of bond force previously, each bond  is independent of each other, in which the interference between each bond is minimized,  which can greatly improve the numerical stability when cutting bonds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  Meanwhile, the  definition depends on the spherical support or horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' For the convenience of numerical  implementation, we assume that each particle has the same volume and support radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' The  interested domain is discretized into uniform particles or lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' For different particles, the  bonds in the same direction and radius have the same coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  16  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' Two damage rules based on critical energy release rate  In this subsection, by relating the critical shear strain or critical normal strain to the  energy release rate, two damage rules are proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  S-  S+ u  u  i  j  k  l  S-  S+  i  j  k  l  u  u  (a) Model I: Opening  (b) Model II: In-plane shear  Figure 5: Deformation of bond with shear deformation or tensile deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' Critical normal strain damage rule  Bond-based peridynamic models have advantages such as independent bond energy and  simple damage criterion based on critical stretch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  Although the direct neighbor cutting  operation perturbates the system greatly, the numerical stability is well preserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' There  are several obstacles in removing bonds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' Even for the simplest bond-based model enriched  with rotation for open-mode fracture, at the crack front tip, the deformation for each bond  is complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='5(a), bond ik is tensile and bond ij has shear deformation  or mixed deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' It is doubtful to apply a criterion of stretch rule or rotation rule to  cut the bond ik since ik falls in the open-mode fracture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' It is obvious that for the simplest  fracture mode, the shear bonds and tensile bonds coexist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' For the case denoted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='5(b),  some bonds have compressive shear deformation and the situation is even worse because  of the perturbation of sudden removed internal bond force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' The bond removing technique  is more based on a geometrical and intuitive operation but lacks sound theoretical basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  Some authors in their work [23, 22, 35, 24] by examining the shear deformation state and  tensile deformation state and sorting these states, cut the most damage-prone bonds and  then iterate globally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' These methods are relatively complicated since they depend heavily on  the bond sequences and cannot be analyzed theoretically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' In addition, the assembly of the  global tangent stiffness matrix and the related solving techniques are much more expensive  than the explicit time integration scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  17  In order to model fracture automatically, the breaking of bonds should be as simple as  possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' We borrow some ideas from the phase field scheme [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' The phase field model  considers the principal strains, which is independent of shear strain in that direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' The  degradation of strain energy by principal strain has good numerical stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' In the bond-  based nonlocal elasticity, each bond usually has both axial deformation and shear deforma-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' On the other hand, the magnitude of shear deformation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' the rotation) is not well  predicted since the rigid rotation may be significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' However, in the sense of discretization,  the bonds loop in “every” direction in support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' It is not required to calculate the eigenvalue  decomposition of strain tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' But when the bond direction points to the direction of  principal strain, the situation is quite similar to the phase field model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' Unlike the bond-  based model based on the stretch contributed partially by shear deformation, or the critical  rotation model [35, 24], we remove the bond by considering only a bond-directional strain  while ignoring the shear deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' The change of strain energy in this sense is quite  similar to the phase field model by considering only the energy on principle strain direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  For each bond direction, the interaction status is determined through a bond status  parameter given by  µ(xj − xi) =  �  �  �  �  �  1, sn(t) < sc  n  0, max  0≤t≤Tsn(t) ≥ sc  n,  (55)  where sn(t) is the strain at time t along the initial bond direction and sc  n is the critical bond  stretch determined by the Griffith energy release rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' The local damage is evaluated as  φ(xi) = 1 −  �  Si µ(xj − xi)dVj  �  Si dVj  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  (56)  When considering the deformation along the principal strain direction, the deformation  can be simplified into 1D with cross-section area A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' Consider the deformation in 1D, in order  to form a crack surface, half support should be cut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' The equivalence of fracture energy and  strain energy in half support is  1  2GcA = 1  2Kε2Aδ → ε =  �  Gc  Kδ,  (57)  18  where K =  E  3(1−2ν) is the bulk modulus of the material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' Therefore, we select the critical  normal strain as  sc  n =  �  Gc  Kδ =  �  3(1 − 2ν)Gc  Eδ  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  (58)  This rule derived from the 1 dimensional case appears to be simple, but it can achieve almost  the same accurate result as the phase field method by finite element methods in some cases,  which will be shown in the numerical examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' Critical shear strain damage rule  Similar to the principal strain direction, the other direction is the maximal shear strain  direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' For many materials, the shear strain gives rise to the shear fractures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' Consider  the deformation in 1D, the equivalence of fracture energy and strain energy in half support  is  1  2GIIA = 1  2µε2  sAδ → εs =  �  GII  µδ ,  (59)  where GII is the energy release rate for mode II fracture and µ the shear modulus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' Therefore,  the critical shear strain is selected as  sc  t =  �  GII  µδ =  �  2(1 + ν)GII  Eδ  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  (60)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' Plasticity for bond element  Let (n, m, t) be a set of orthogonal local axes, with en, em, et being the normal vector, the  shear-direction and out-of-plane direction of the bond element, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' The kinematics  of a bond element is  εn = len + γem  (61)  where l = uij  r · en, γ = uij  r · em.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  In a local coordinate system, the strain and force are  ε = (l, γ)T, σ = (σ, τ)T =  �  �c1  0  0  c2  �  �  �  ��  �  E0  �  �l  γ  �  �  (62)  19  where the vector σ, ε represent the force vector and strain vector;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' E0 is the second-order  material tensor in bond local coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  For elastoplastic models, the constitutive relation of a bond element in local coordinate  can be expressed in rate form as  ˙ε = ˙εe + ˙εp, ˙σ = E0 · ( ˙ε − ˙εp),  (63)  where εe and εp being the elastic and plastic parts of the strain tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  Without loss of generality,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' the plastic strain rate is given by the following flow rule based  on the plastic potential function f p(σ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' q)  ˙εp = ˙λ ∂f p  ∂σ  ����  Λp  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' ˙κ = − ˙λ∂f p  ∂q ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  (64)  for the plastic multiplier ˙λ satisfying the classical Karush-Kuhn-Tucker conditions  ˙λ ≥ 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' f(σ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' q) ≤ 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' ˙λf(σ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' q) = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  (65)  where a force-based yield function f(σ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' q) ≤ 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' with q being the force-like internal variable  (yield force) conjugate to the strain-like one κ which measures the plastic state;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' Λp := ∂fp  ∂σ  is the plastic flow direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  For associated plasticity, the potential function f p(σ, q) is  identical or proportional to the yield function f(σ, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  Then the force state rate can be written as  ˙σ = E0 · ( ˙ε − ˙λΛp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  (66)  Plastic yielding occurs when the yield condition f(σ, q) = 0 is activated, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  ˙λ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  Follow from the consistency condition ˙f = 0 gives  ˙λ =  Λ · E0 · ˙ε  Λ · E0 · Λp + hHhp,  (67)  for the derivative Λ := ∂f  ∂σ and h = − ∂f  ∂q of yield function f(σ, q) and hardening/softening  modulus H := ∂q  ∂κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  20  The corresponding constitutive relation in rate form then reads  ˙σ = E · ( ˙ε − ˙εp) = Eep · ˙ε,  (68)  where the second-order elastoplasticity tangent Eep is expressed as  Eep = E0 − (E0 · Λp) ⊗ (Λ · E0)  Λ · E0 · Λp + hHhp .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  (69)  The yield function and plastic potential function are assumed to have the same form  f p = f(σ, τ, q) = aσ + b  √  τ 2 − q(κ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  (70)  Then  Λ = (a, b sign(τ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  (71)  The elastoplasticity tangent becomes  Eep = ω(r)  �  �  �c1  0  0  c2  �  � −  1  a2c1 + b2c2  �  �  a2c2  1  abc1c2 sign(τ)  abc1c2 sign(τ)  b2c2  2  �  �  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  (72)  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' Higher-order nonlocal bond-based models  The bond-based nonlocal model is not restricted in first-order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' By making use of the bent-  bond, the bond-based plate model and bond-based gradient elastic model will be derived in  the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' Nonlocal isotropic thin plate  Assuming deflection field w(x, y) = 1  2(ax2 + by2 + 2cxy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' κ = ∇∇w, the second-gradient  of deflection field, can be written as  κ =  �  �κ11  κ12  κ12  κ22  �  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  (73)  Along with bond direction n = (cos θ, sin θ), the orthogonal decomposition of bond  curvature tensor is  κn = (wij + wij′)/r2  ij(4n ⊗ n − I)  = 3(wij + wij′)/r2  ij(n ⊗ n)  �  ��  �  κnn  + (−(wij + wij′)/r2  ij)(I − n ⊗ n)  �  ��  �  κns  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  (74)  21  The moment force for single bent bond is assumed as  Mn = ω(rij)(c1κnn + c2κns),  (75)  where c1, c2 are the material parameters to be determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' The total energy carried by a  point  Wnonlocal =  �  S+  1  2(Mn) : (κn)dV  =  �  S+  1  2ω(r)(wij + wij′)2/r4  ij(9c1 + c2)dV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  (76)  Wnonlocal = Wlocal := 1  2M : κ for any field yields (9c1 + c2) = 16D0  3α , ν = 1/3, where α =  � δ  0 ω(r)πr dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' Therefore, the equivalent curvature and moment for a bond are  κij = (wij + wij′)/r2  ij  (77a)  mij = ω(r)16D0  3α (wij + wij′)/r2 = ω(r)Et3  2α (wij + wij′)/r2  ij,  (77b)  where D0 =  Et3  12(1−ν2) =  3  32Et3 and t is the thickness of the plate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' This is the bond-based  version of nonlocal thin plate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' Only the Poisson ratio of 1/3 can be modeled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  The corresponding bond force can be derived by considering the first variation of the  bond energy  fijj′ = ω(rij)Et3  2α  (wij + wij′)  r4  ij  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  (78)  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' Cohesive damage model for bent bond  Cohesive damage model of bending bond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' In order to introduce the localization, the  moment is calculated as  mij = c0sign(κij) min(|κij|, κ2  crit  |κij|),  (79)  where κcrit is the critical curvature when softening of force occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' Nonlocal isotropic gradient elasticity  Similar to the nonlocal thin plate, we consider one field in gradient elasticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  For  any bond, the local coordinate system can be expressed with orthogonal unit basis vec-  22  κ/κcrit  m/mmax  1  1  5  5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='0  Figure 6: Moment and curvature relation in a cohesive model with κcrit and mcrit being critical curvature  and critical moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  xi  xj(t1)  rij  uij  uij  +  γ  c1  c2  κs  κn  κ  fκ  fκs(κs)  fκn(κn)  (a)                                                           (b)  rij  xj(t2)  Figure 7: Bond deformation with rotations, shear curvature stiffness and extension curvature stiffness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  tors n1, n2, n3 as  n1 = (cos θ sin φ, sin θ sin φ, cos φ),  n2 = (cos θ cos φ, cos φ sin θ, − sin φ), n3 = (− sin θ, cos θ, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  (80)  For any field u(x, y, z) = 1  2(ax2 + by2 + cz2 + 2dxy + 2fxz + 2gyz), the curvature tensor  κ = ((a, d, f), (d, b, g), (f, g, c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  The orthogonal decomposition of nonlocal Hessian on a bond is  κn = (uij + uij′)/r2  ij(5n1 ⊗ n1 − I)  = 4(uij + uij′)/r2  ijn1 ⊗ n1  �  ��  �  κ1  + (−(uij + uij′)/r2  ij)n2 ⊗ n2  �  ��  �  κ2  + (−(uij + uij′)/r2  ij)n3 ⊗ n3  �  ��  �  κ3  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  23  The moment of the bond is assumed as  Mn = ω(rij)  �  c14(uij + uij′)/r2  ijn1 ⊗ n1  �  ��  �  M1  + c2(−(uij + uij′)/r2  ij)n2 ⊗ n2  �  ��  �  M2  + c2(−(uij + uij′)/r2  ij)n3 ⊗ n3  �  ��  �  M3  �  ,  (81)  where c1, c2 are the unknown curvature stiffness and here we assume the stiffnesses in M2  and M3 are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  The bent energy carried by a bond is  wn = 1  2Mn : κn = ω(rij)(8c1 + c2)(uij + uij′)2/r4  ij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  (82)  The bent energy carried by a point is the summation of all bent bonds:  W =  �  S+ wndV = 1  15α(8c1 + c2)(3a2 + 2a(b + c) + 3  b2 + 2bc + 3c2 + 4(d2 + f 2 + g2)),  (83)  where α =  � δ  0 πr2ω(r)dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  Assume the material constitution for gradient strain energy be  M = ℓ2(λTr(κ)I + 2µκ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  (84)  The gradient strain energy in local theory for any curvature deformation can be simplied as  Wlocal = 1  2M : κ = 1  2ℓ2�  a2(λ + 2µ) + 2aλ(b + c)  +b2(λ + 2µ) + 2bcλ + c2λ + 2c2µ + 4d2µ + 4f 2µ + 4g2µ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  (85)  The energy equivalence W = Wlocal for any κ leads to  (8c1 + c2) = 15ℓ2µ  2α , λ = µ → ν = 1  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  (86)  Therefore, the curvature and moment force for u field is  κu  ij = (uij + uij′)/r2  ij, mu  ij = ω(rij)15ℓ2µ  α  (uij + uij′)/r2  ij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  (87)  24  The curvature force follows the direction of u field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  Therefore, the bent-bond energy becomes  φij = 1  2ω(rij)15ℓ2µ  α  (uij + uij′)2/r4  ij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  (88)  And the corresponding bond force  f u  ijj′ = ω(rij)15ℓ2µ  α  (uij + uij′)/r4  ij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  (89)  For field v, w, the same conclusions can be obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' In sum, the curvature and momen-  tum of bond-based gradient elasticity is  fijj′ = ω(rij)15ℓ2µ  α  (uij + uij′)/r4  ij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  (90)  The governing equations become  �  Si  2fijdVj  �  ��  �  first order contribution  +  �  S+  i  (2fijj′ + fjii′ + fj′ii′)dVj  �  ��  �  second order contribution  +b = ρ¨ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  (91)  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' Numerical examples  The numerical examples are carried out based on a Verlet-velocity explicit time inte-  gration algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' The quasi-static condition for some cases is achieved by applying the  velocity boundary conditions gradually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' The reaction forces are retrieved by summing the  internal forces of the selected particle set before applying the boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' Simple support beam  The bent bond defined by three points is immune to the first order derivative in complete  support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  However, for material points near the boundaries, the support domain is not  complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' Additional particles outside the boundaries are added to make the support domain  complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' The simple support boundary is fulfilled by  w(0) = w(L) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  25  The function of additional particles is to make sure the half support S+ is well defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' The  full implementation code of the simple support beam can be found by the Github link (?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='??' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  The material parameters of this example are E = 30 × 109 Pa, beam length L = 1 and  thickness h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' A damping (fdamp = −200 ˙w) is used to converge the dynamic solution  to the static result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' With damping effect, the evolution of deflection of the middle point is  plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' The final deflection of the beam is plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' It can be observed  that the numerical solution for N = 25 material points and δ = 2 is quite close to the exact  solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='9, the deflection of beam with a discretization N = 100 agrees  very well with the exact solution, in which the L2 norm of error is 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='01 × 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  The influence of support size is investigated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' With the increase of support  size, the beam becomes slightly stiffer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' The influence of the weight function ω(r) = rn, n =  (0, 1, 2, 3, 4) is plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' Here the support size is selected as δ = 3∆x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' one can see  that the weight function in this case has significant influence on the deflection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='08  Time (s)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='2  w(L/2)/wmax  Figure 8: The evolution of deflection of midpoint .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  In order to model fracture in a thin beam, we applied the cohesive damage rule to model  the fracture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' We select the critical curvature tensor as κcrit = 4× 10−4 and use the damping  coefficient p = 300 for reducing oscillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' The damage distribution and displacement field  at the t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='03 seconds are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='12 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='13, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' It can be observed that  the damage happens at the center of the beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' Single-edge-notched tension test  In this subsection, we model the single-edge-notched tension test, which is a squared  plate with initial notched crack as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' The material parameters are set as  26  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
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+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='0  L (m)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
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+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='2  w/wmax  Analytical solution  Bond-based beam,N=100,n=2  Bond-based beam,N=25,n=2  Figure 9: The comparison of exact solution and bond based beam N = 100 and N = 25 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
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+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='0  L (m)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
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+page_content='2  w/wmax  Analytical solution  Bond-based beam,n=2  Bond-based beam,n=3  Bond-based beam,n=4  Figure 10: The influence of support size in a bond based beam .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
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+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='0  L (m)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
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+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='2  w/wmax  Analytical solution  Bond-based beam,ω(r)=1  Bond-based beam,ω(r)=r  Bond-based beam,ω(r)=r2  Bond-based beam,ω(r)=r3  Bond-based beam,ω(r)=r4  Figure 11: The influence of weight function in a bond based beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  27  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
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+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='0  L (m)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='0  Damage  T=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='005 s  T=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='006 s  Figure 12: Damage in bond based beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='0  L (m)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='5  w/wmax  Analytical solution  T=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='005 s  T=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='006 s  Figure 13: Deflection of bond based beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  λ = 121.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='1538 kN/mm2 and µ = 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='7692 kN/mm2 for elastic constants, Gc = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='7 × 10−3  kN/mm for critical energy release rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' These parameters are identical to that used in the  small strain brittle fracture phase field in Ref [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' Two displacement conditions are tested:  Case a) for tensile boundary condition and Case b) for shear boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' The plate  is discretized into 100 × 100 material points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' The displacement load is monotonic applied  with velocity boundary condition defined by  v(t) =  �  �  �  �  �  t  t0vmax  if t < t0  vmax  otherwise  with t0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='0 × 10−5s and vmax = 2 m/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  In the case of tensile load, two discretizations are employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' The load curves for the  tensile boundary are plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  The maximal normal strain criterion is derived  in a simple way but it is very effective in application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' From the test of tensile boundary  condition, the load-curve matched the result by FEM phase field model very well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' The study  28  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='5                 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='5                 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='5  a)  b)  initial crack  Figure 14: Single-edge-notched test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' Geometry and Case a for tensile boundary condition and Case b for  shear boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  Figure 15: Single-edge-notched tension test, damage patterns for 60x60 particles and 120x120 particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  on different discretization shows that the damage model is insensitive to the discretization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  It is interesting that the fracture model by explicit time integration without using damping  agrees so well with the phase field method in the static case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  The evolutions of displacement field and velocity field are plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' The crack  initiates when the boundary displacement reaches uy = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='96 × 10−3 mm as indicated by  the irregular velocity field around the crack tip in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='16(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' In the stage of stable crack  propagation, the significant velocity wave due to cutting bond can be observed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='16(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  The velocity field is greatly interfered by the fracture, but the displacement field is stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  The final result for shear tests with a discretization of 120×120 is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' In  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='19, the damage patterns for different discretization subjected to shear loading condition  are plotted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' With finer discretization, the resolution of fracture becomes sharper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  The displacement curve for shear test is plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' Before the crack is initialized,  the result by NOM agrees with the FEM result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' The crack began to initialize when the  29  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='40×10-7  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='32×10-6  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='20×10-6  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
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+page_content='96×10-6  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='60×10-7  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='38×10-6  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='30×10-6  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='22×10-6  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
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+page_content='0  (a)  (b)  (c)  (d)  (e)  (f)  (g)  (h)  Figure 16: Single-edge-notched tension test: first row (a,b,c,d) denotes displacement field in y-direction and  the second row (e,f,g,h) velocity in x-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' Two figures of the same column correspond to the same  time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
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+page_content='006  Uy (mm)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
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+page_content='7  FN (kN)  NOM 120x120  NOM 60x60  FEM-PF  Figure 17: Single-edge-notched tension test, load curves for NOM and phase field by FEM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
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+page_content='620  Figure 18: Single-edge-notched plate subjected to shear boundary condition: Displacement field in x-  direction and y-direction and the damage distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  30  Figure 19: Damage subjected to shear load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='002  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='004  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
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+page_content='008  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
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+page_content='012  Ux (mm)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='6  Fx (kN)  NOM 120x120  NOM 60x60  FEM-PF  Figure 20: Single-edge-notched shear test: load curves for NOM and phase field by FEM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  displacement ux = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='009 mm for both FEM and NOM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' With the increase of load, the bond  cutting process becomes irregular and the reaction force oscillates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' This is partially because  the particle distribution in support no longer aligns with the crack surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' It also reveals the  complicated stress state due to perturbation of broken bonds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' The overall fracture pattern  agrees with that by the FEM phase field method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' Due to the discrete feature of the current  method, the crack surface is not as smooth as that by continuum methods such as the phase  field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' Critical shear damage criterion  The material parameters are the same as previous sections except that the energy release  rate for mode II is selected as GII = 3×10−3 kN/mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' The critical shear stretch is calculated  based on Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='22 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='23 are the resultant reaction forces in x and y directions of  material points at the top of the plate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' For the pure shear boundary conditions, the force  in y direction is neglectable compared to that in x direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' The peak reaction force is  proportional to the square root of GII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' It is also observed that the load curve is insensitive  31  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='5                 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='5                 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='5  u  initial crack  uy  ux  u  uy  ux  Figure 21: Single-edge-notched test based on shear damage criterion: displacement boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
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+page_content='020  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='025  Ux (mm)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='2  Fx (kN)  120x120,GII  60x60,GII  120x120,4GII  60x60,4GII  Figure 22: Single-edge-notched shear test: load curves for the case of ux : uy = 2 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='005  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='010  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
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+page_content='020  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='025  Ux (mm)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='2  Fy (kN)  120x120,GII  60x60,GII  120x120,4GII  60x60,4GII  Figure 23: Single-edge-notched shear test: load curves for the case of ux : uy = 2 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  32  Figure 24: Single-edge-notched shear strain: fracture patterns on different displacement boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  to the discretization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' For the case of discretized by 120x120 and 4GII, the peak reaction force  is F max  x  = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='29 kN when displacement reaches ux = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='253×10−2 mm, which corresponds the  external work approximately as Wext = 1  2F max  x  ux = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='455 × 10−2 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' The energy consumed  by forming fracture surface is 2× (4GII) × lcrack = 2× (4× 3× 10−3) × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='5 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='2× 10−2 J, in  which lcrack = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='5 mm is the crack length and multiplying 2 denotes two crack surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' The  fracture energy is slightly less than the total external work Wext.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' The result is reasonable  because the kinetic energy and strain energy comprise of certain portion of the total energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  We also test the influence of the loading angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  By adjusting the ratio of ux : uy,  different paths of shear crack can be observed, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' Interestingly, the crack  path direction is quite close to the displacement direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' The crack paths of the case with  ux : uy = 2 : 1 and the case with ux : uy = 2 : −1 are symmetric with respect to the horizontal  line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' It can be concluded that the critical shear strain damage rule can automatically find  the direction of maximal shear strain and form a shear crack path consistently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' Kalthoff-Winkler experiments  The Kalthoff-Winkler experiment [36] is a classical benchmark problem for dynamic  fracture modeling [37, 38, 39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' For different impact velocities, the fracture can be brittle or  ductile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' For low impact velocities, the dynamic brittle fracture propagates from the crack  tip at an angle of around 70◦ vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' the direction of the initially horizontal crack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' When the  impact velocity is increased further, a ductile failure (or shear fracture) phenomenon occurs  and a shear band is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' The dimension of the plate is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='2×0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='1 m2 as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  The material parameters are E = 190 GPa, ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='3, Gc = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='4 × 104 J/m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' Consider the  symmetry, only half of the plate is modeled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' The plate is discretized with 200x200 particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  33  ux : uy =2: -1Ux : uy = 1 : 2ux : uy = 1 : 1ux : uy = 2: 1ux : uy =2: 00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='05m  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='075m  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='075m  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='2m  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='1m  v0  v0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='05m  Figure 25: Setup for the Kalthoff-Winkler experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  The support radius is selected as l = 3∆x m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' The initial crack is represented by modifying  the neighbors in support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' The number of neighbors for each particle is restricted to 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' The  brittle failure at low impact velocity is studied with critial normal strain criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' In this  case, the velocity applied on the top of the plate starts from 0 to vy = 20m/s in a period of  10−7 s and remains constant thereafter [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' For the case of shear fracture at higher impact  velocity vy = 39 m/s, the critical shear damage criterion is employed and energy release rate  of mode-II fracture is selected as GII = 4Gc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  In low impact velocity, the displacement field and velocity field in x-direction at different  times are depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' It can be observed that the crack initiated at 24µs and finished  at 82µs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' Around the crack tip, the breaking of bonds causes obvious ossilation of velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  The final crack path of tensile crack is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' For the higher impact velocity, the  displacement field and velocity are plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' The shear crack starts to propagate at  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='4µs, following the direction of initial crack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' At the final stages, crack branching of shear  fractures is observed as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' Conclusions  In this work, we have proposed several bond-based models for solids, thin plates, and  gradient solids in different dimensional spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' The main motivation is to derive the force  34  Figure 26:  Kalthoff-Winkler test vy  =  20 m/s:  ux (first row) and vx (second row) at times  (24µs, 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='7µs, 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='6µs, 82µs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
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+page_content='870  Figure 27: Kalthoff-Winkler test vy = 20 m/s: tensile fractures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  Figure 28:  Kalthoff-Winkler test vy  =  39 m/s:  ux (first row) and vx (second row) at times  (14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
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+page_content='4µs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
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+page_content='870  Figure 29: Kalthoff-Winkler test vy = 39 m/s: shear fractures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  constitution that depends only on the bond deformation, while the collective deformations  recover the traditional local theory by using an energy equivalence principle and assuming  a constant, fully symmetric support region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' For the bond-based NOM, a weight function is  introduced to define the bond forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' To account for the bending effect due to curvature,  the nonlocal curvature and the nonlocal moment are defined by introducing a bent bond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  The bent bond is defined symmetrically by including three points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' It is shown that the thin  plate model based on bonds has a Poisson’s ratio restriction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' A simple bond-based gradient  elasticity is also developed based on the equivalence of the gradient deformation energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  The bond-based elasticity accounts for the strain extension and shear deformation of a  bond and has no Poisson’s ratio constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' By introducing a weight function, the distribu-  tion of nonlocal bond strain energy can be regulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' At the same time, a cohesive damage  model for bent bonds is proposed, which weakens the bond force when the threshold value  of bond strain or bond curvature is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' This setting provides a simple rule to localize  the strain without cutting the bond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' In addition, a plasticity model is derived based on the  increase in bond force for a single bond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  Several numerical examples are presented, including a simple support beam and a 2D  solid plate with shear or tensile damage patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' Although the numerical examples use  explicit time integration, the implicit implementation is straightforward for static problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  For each bond element, one can calculate the second variation and convert the tangent  stiffness matrix in the local coordinate system to the global coordinate system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' Last but not  least, a simple rule for the critical normal strain and critical shear strain for bond cutting  36  in tensile and shear fracture modeling is proposed, which is numerically stable and easy to  implement and can achieve comparable results to the phase field method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  Acknowledgments  The first author gratefully acknowledges financial support from the EU project enti-  tled ”Computational Modeling, Topological Optimization and Design of Flexoelectric Nano  Energy Harvesters” (ERC COTOFLEXI 802205).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
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+page_content=' Gradient damage vs phase-field approaches for fracture:  Similarities and differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
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+page_content=' Dynamic crack propagation based on loss of hyperbolicity  and a new discontinuous enrichment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' International Journal for Numerical Methods in Engineering,  58(12):1873–1905, 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  [39] JH Song, PMA Areias, and T Belytschko.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' A method for dynamic crack and shear band propagation  with phantom nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  International Journal for Numerical Methods in Engineering, 67(6):868–893,  2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  [40] Christian Miehe, M Hofacker, L-M Sch¨anzel, and Fadi Aldakheel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' Phase field modeling of fracture  in multi-physics problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' part ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' coupled brittle-to-ductile failure criteria and crack propagation in  thermo-elastic–plastic solids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content=' Computer Methods in Applied Mechanics and Engineering, 294:486–522,  2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
+page_content='  39' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQf8_oB/content/2301.00864v1.pdf'}
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+ALMA: Automata Learner using Modulo 2
+Multiplicity Automata
+Nevin George
+Yale University, New Haven CT 06511, USA
+nevin.george@yale.edu
+Abstract. We present ALMA (Automata Learner using modulo 2 Mul-
+tiplicity Automata), a Java-based tool that can learn any automaton
+accepting regular languages of finite or infinite words with an imple-
+mentable membership query function. Users can either pass as input their
+own membership query function, or use the predefined membership query
+functions for modulo 2 multiplicity automata and non-deterministic Büchi
+automata. While learning, ALMA can output the state of the observation
+table after every equivalence query, and upon termination, it can output
+the dimension, transition matrices, and final vector of the learned mod-
+ulo 2 multiplicity automaton. Users can test whether a word is accepted
+by performing a membership query on the learned automaton.
+ALMA follows the polynomial-time learning algorithm of Beimel et.
+al. (Learning functions represented as multiplicity automata. J. ACM
+47(3), 2000), which uses membership and equivalence queries and repre-
+sents hypotheses using modulo 2 multiplicity automata. ALMA also im-
+plements a polynomial-time learning algorithm for strongly unambigu-
+ous Büchi automata by Angluin et. al. (Strongly unambiguous Büchi
+automata are polynomially predictable with membership queries. CSL
+2020), and a minimization algorithm for modulo 2 multiplicity automata
+by Sakarovitch (Elements of Automata Theory. 2009).
+Keywords: automata theory · finite automata · Büchi automata · mul-
+tiplicity automata · learning
+1
+Introduction
+Angluin’s exact learning model [1] has been studied extensively in the context
+of learning theory, and it can be used to learn automata representing regular
+languages of finite and infinite words. In the model, a learner interacts with an
+oracle to learn a regular language using membership and equivalence queries.
+In a membership query, the learner learns from the oracle whether a word is in
+the language. In an equivalence query, the learner forms a hypothesis on what
+the language is, and the oracle either confirms that the hypothesis is correct
+or returns a counterexample, i.e, a word for which the hypothesis and language
+differ.
+arXiv:2301.04077v1  [cs.FL]  10 Jan 2023
+
+2
+N. George
+As one application of the exact learning model, Beimel et. al. [4] detail a
+polynomial-time algorithm to learn multiplicity automata using membership and
+equivalence queries. We provide a high-level overview of the algorithm: the algo-
+rithm begins with a trivial hypothesis of the language, represented as a multiplic-
+ity automaton of dimension 1 defined over some field K. On each iteration of a
+loop, the algorithm performs an equivalence query. If the hypothesis is equivalent
+to the language, the algorithm terminates and outputs the hypothesis. Other-
+wise, the algorithm receives a counterexample from the oracle, which is used to
+improve the hypothesis. The algorithm terminates after d iterations, where d is
+the dimension of the smallest possible multiplicity automaton that can represent
+the language.
+Strongly unambiguous Büchi automata (SUBAs) (defined in Section 2.1) are
+a type of non-deterministic Büchi automaton (NBA) first introduced by Bosquet
+and Löding [6]. SUBAs are useful for modeling reactive systems, as they are fully
+expressive, i.e., they can represent any regular ω-language, and can often rep-
+resent regular ω-languages more succinctly than other NBA representations [2].
+Angluin et. al. [2] also showed that SUBAs are learnable in polynomial time,
+further increasing their importance.
+In this paper, we present ALMA, a Java-based tool that implements the al-
+gorithm of Beimel et. al. and extends it to learn any arbitrary automaton repre-
+senting regular languages of finite or infinite words with an implementable mem-
+bership query function. Membership query functions have been implemented for
+M2MAs and NBAs, and users can enter as input any arbitrary membership
+query function that 1) takes as input any possible word that can be formed from
+the given alphabet, and 2) outputs 0 or 1 to indicate whether the word is in the
+language. To improve the run time of learning SUBAs, ALMA does not use the
+general learning algorithm but rather the SUBA learning algorithm of Angluin
+et. al. [2]. Also when learning M2MAs, ALMA first implements the algorithm of
+Sakarovitch [10] to minimize the M2MA before running the learning algorithm.
+This is because in the SUBA learning algorithm, converting the input SUBA
+into an equivalent M2MA incurs a quadratic increase size. Minimizing M2MAs
+before learning enables the algorithm to learn significantly larger SUBAs, es-
+pecially because empirically the M2MAs have been seen to often minimize to
+M2MAs of much smaller dimension. The effect of the minimization algorithm on
+a sample of input SUBAs can be seen in Table 2 in Section 5.
+Usefulness/Novelty In the verification community, M2MAs can be useful for
+representing regular ω-languages, as they are learnable in polynomial time and
+often relatively succinct. They are also useful for tasks such as model checking,
+since performing operations such as intersection, union, complementation, empti-
+ness, and equivalence on M2MAs is cheap [3]. Through the minimization and
+learning algorithms, ALMA enables researchers in the verification community to
+easily test and verify these properties of M2MAs, promoting further exploration
+into these useful automata. As an example, ALMA was used in the paper by An-
+gluin et. al. [3] to explore the suitableness of representing regular ω-languages
+
+ALMA: Automata Learner using Modulo 2 Multiplicity Automata
+3
+using M2MAs. The authors used ALMA to convert SUBAs, NBAs, and DBAs
+into equivalent M2MAs, and they compared the succinctness of these M2MAs
+with that of DFAs accepting the same language. ALMA can similarly be used
+by other researchers to gain insights into M2MAs and their benefits/drawbacks
+as compared to other representations.
+ALMA is the first publicly available implementation of the novel SUBA learn-
+ing algorithm by Angluin et. al. [2]. Using ALMA, users can run the algorithm to
+learn any regular ω-language. Since learned automata are represented as M2MAs,
+users can use the many desirable properties of M2MAs to gain insights into
+the initial SUBA and regular ω-language. In addition, since membership query
+functions are often relatively easy to implement and ALMA already provides a
+membership query function for the general NBA case, the scope of what ALMA
+can learn is large, promoting its usefulness in a wide variety of settings.
+Many tools such as ROLL (ω-Regular Language Learning Library) [9] and
+libalf [5] already exist that can learn automata representing regular languages
+of finite and infinite words. However, ALMA is the first tool that uses M2MAs
+to represent hypotheses and the learned automaton in the learning algorithm.
+ALMA’s usefulness lies not with necessarily being the fastest tool available to
+learn regular languages, but with exploring the practical benefits of M2MAs and
+the features of the algorithms by Beimel et. al., Angluin et. al., and Sakarovitch.
+2
+Useful Definitions
+2.1
+Finite and Büchi Automata
+Let Σ be a finite alphabet. Then Σ∗ and Σω are the sets of all finite and infinite
+words, respectively, that can be formed using elements from Σ. A finite language
+is a subset of Σ∗, and an ω-language is a subset of Σω. If w is a word in Σ∗ or
+Σω, let |w| be the length of w and w[i] be the i’th character of w.
+A finite-state automaton A is represented as a tuple (Σ, Q, I, ∆, F), where
+Σ is the alphabet, Q is the finite set of states, I ⊆ Q is the set of initial states,
+∆ ⊆ Q × Σ × Q is the set of transitions, and F ⊆ Q is the set of final states.
+The automaton A is deterministic if every pair (q, σ) ∈ Q × Σ appears as the
+first two elements in at most one triple in ∆.
+A run on A for a word w is a series of states q0, q1, . . . ∈ Q such that ∀i
+satisfying 1 ≤ i ≤ |w|, (qi−1, w[i], qi) ∈ ∆. A run for a finite word is final if it
+ends in a final state. For infinite words, a run is final if it passes infinitely often
+through a final state. A run is accepting if it is final and begins at an initial state.
+The automaton A accepts the finite/infinite word w if there exists an accepting
+run for w.
+Non-deterministic finite automata (NFAs) and non-deterministic Büchi au-
+tomata (NBAs) are automata accepting finite and infinite words, respectively.
+Deterministic finite automata (DFAs) are deterministic NFAs, and deterministic
+Büchi automata (DBAs) are deterministic NBAs. Unambiguous finite automata
+
+4
+N. George
+(UFAs) and unambiguous Büchi automata (UBAs) are NFAs and NBAs, respec-
+tively, for which every word has at most one accepting run. A UBA is a strongly
+unambiguous Büchi automaton (SUBA) if every word has at most one final run.
+2.2
+Modulo 2 Multiplicity Automata
+Assume a field K and some dimension d. A multiplicity automaton A is repre-
+sented as a tuple (Σ, vI, {µσ}σ∈Σ, vF ), where Σ is the alphabet, vI is the initial
+vector, each µσ is a transition matrix, and vF is the final vector. vI and vF have
+dimension d × 1, and each µσ has dimension d × d. The set of states is all row
+vectors v ∈ {0, 1}d, and the initial state is v⊤
+I . For a given word w = σ1σ2 . . . σn,
+let µ(w) = µσ1µσ2 . . . µσn. The set of reachable states is all vectors of the form
+v⊤
+I µ(w). Associated with the automaton A is a function fA : Σ∗ → K, where
+∀w ∈ Σ∗,
+fA(w) = v⊤
+I µ(w)vF .
+A modulo 2 multiplicity automaton (M2MA) is a multiplicity automaton where
+K = GF(2) and all calculations are done modulo 2. An M2MA A accepts a word
+w ∈ Σ∗ if and only if fA(w) = 1.
+As an example, consider the following M2MA M adapted from Angluin et.
+al. [2].
+M = ({a, b},
+�1 0 0�⊤ , {µa, µb},
+�1 1 0�⊤}
+where
+µa =
+�
+�
+0 0 1
+1 0 0
+1 1 1
+�
+� and µb =
+�
+�
+0 1 0
+1 0 1
+1 1 0
+�
+� .
+M is equivalent to the DFA in Fig. 1. We explain Fig. 1: the computation begins
+at the initial state
+�
+1 0 0
+�
+. On reading an a/b from the initial state, the DFA
+visits the states
+�1 0 0�
+µa =
+�1 0 0�
+�
+�
+0 0 1
+1 0 0
+1 1 1
+�
+� =
+�0 0 1�
+�1 0 0�
+µb =
+�1 0 0�
+�
+�
+0 1 0
+1 0 1
+1 1 0
+�
+� =
+�0 1 0�
+.
+Other states are visited similarly by multiplying the current state vector by µa
+or µb.
+�1 0 0�
+,
+�0 1 0�
+, and
+�1 0 1�
+are accepting states since
+�1 0 0� �1 1 0�⊤ =
+�0 1 0� �1 1 0�⊤ =
+�1 0 1� �1 1 0�⊤ = 1,
+where vF =
+�1 1 0�⊤.
+M2MAs have many important properties described in Section 1 that make
+them useful in verification communities, e.g., learnable in polynomial time and
+cheap intersection, union, complementation, emptiness, and equivalence. For
+
+ALMA: Automata Learner using Modulo 2 Multiplicity Automata
+5
+100
+start
+010
+101
+001
+111
+110
+b
+b
+b
+a
+a
+a
+b
+b
+a
+a
+a
+b
+Fig. 1. DFA for the M2MA M
+more information on M2MAs, we recommend reading the paper by Angluin
+et. al. [3], which studies these properties of M2MAs extensively and performs
+a detailed analysis of M2MAs’ ability to succinctly represent regular finite and
+ω-languages.
+2.3
+L$ Language
+Since Büchi automata accept infinite words and M2MAs accept finite words,
+Büchi automata and M2MAs cannot accept words from the same language L.
+However, Büchi [7] showed that two regular ω-languages are equivalent if and
+only if they agree on a set of ultimately periodic words, i.e., words of the form
+u(v)ω where u and v are finite words and v is non-empty. We then consider
+the language of finite words L$ = {u$v | u(v)ω ∈ L}, which Calbrix et. al. [8]
+showed is regular. If a Büchi automaton accepts an infinite language L, a finite
+automaton is said to also represent L if it accepts L$. The L$ language is used
+by Angluin et. al. [2] in their SUBA learning algorithm in order to obtain a
+finite automaton equivalent to the initial SUBA, and the learned M2MA accepts
+words from L$.
+3
+Usage
+3.1
+Access and How to Run
+ALMA is an open-source library freely available at the following GitHub reposi-
+tory: https://github.com/nevingeorge/Learning-Automata. The repository
+contains the executables, source files, and sample input files for ALMA, and the
+README document contains detailed information on how to use the executables
+and the required format for the input files.
+ALMA is a Java-based tool designed to be used on the command line. To
+run for example the executable M2MA.jar, a user should enter the command java
+-jar M2MA.jar within a Terminal/Command Prompt window. This will start the
+program, and the program will then print instructions on how to input the
+desired input file and flags.
+
+6
+N. George
+3.2
+Executables
+ALMA consists of five main executables: 1) M2MA.jar, 2) SUBA.jar, 3) mini-
+mize.jar, 4) NBA.jar, and 5) arbitrary.jar. We provide a basic overview of each
+of the jar files and their use cases. For each executable, the output is always
+the dimension, final vector, and transition matrices of the learned/minimized
+M2MA. Also after the algorithms terminate, each executable allows users to
+check whether a word is accepted by the outputted M2MA. Infinite words are
+represented using the L$ language explained in Section 2.3.
+M2MA.jar is used to minimize and learn M2MAs. The alphabet, dimension,
+final vector, and transition matrices of the input M2MA must be specified (the
+initial vector is always the vector with all zeros except for a 1 in the first row).
+SUBA.jar is used to learn SUBAs using the algorithm of Angluin et. al. [2].
+The alphabet, number of states, final states, and transitions of the input SUBA
+must be specified (the only initial state is the first state).
+minimize.jar is used to minimize M2MAs using the algorithm of Sakarovitch [10].
+The input is the same as that for M2MA.jar. minimize.jar can also be used to
+output the minimized M2MA equivalent to the input SUBA in the SUBA learn-
+ing algorithm (i.e., it runs every step of the SUBA learning algorithm except
+for learning the final M2MA). In this case, the input is the same as that for
+SUBA.jar.
+NBA.jar is used to learn NBAs. The alphabet, number of states, final states,
+and transitions of the input NBA must be specified (the only initial state is
+the first state). NBA.jar uses approximate equivalence queries (described in Sec-
+tion 4.5), which requires as input the number of tests to run, maximum length
+of a test, and maximum limit on the number of equivalence queries.
+arbitrary.jar is used to learn arbitrary automata representing regular languages
+of finite and infinite words with an implementable membership query function.
+The membership query function must be defined in MQ.java. The only con-
+straints on the function is that it must accept as input any possible word formed
+from letters in the alphabet, and it must output 0 or 1 depending on whether the
+word is contained in the language. arbitrary.jar also uses approximate equivalence
+queries, which requires the same input parameters as described for NBA.jar.
+3.3
+Flags
+Users can enter the following optional flags to add or remove information from
+the output of the algorithm.
+
+ALMA: Automata Learner using Modulo 2 Multiplicity Automata
+7
+-v: The algorithm outputs the state of the observation table (a matrix consist-
+ing of the words whose membership in the language is known) after every
+equivalence query in the learning algorithm.
+-m: The algorithm outputs detailed information on the progress of the mini-
+mization algorithm. It gives status updates at different points in the algo-
+rithm, such as when it finishes creating the state/co-state spaces and the
+observation table. It also outputs the initial M2MA to be minimized and the
+final minimized observation table.
+-d: When running minimize.jar on a SUBA, the algorithm outputs only the di-
+mension of the minimized M2MA instead of the dimension, final vector, and
+transition matrices of the M2MA.
+-a: After minimizing the M2MA, the algorithm outputs the number of states of
+the minimal deterministic finite automaton (DFA) that represents the same
+language as the M2MA.
+4
+Implementation Details
+4.1
+Architecture
+Fig. 2. M2MA.jar Architecture
+Fig. 3. SUBA.jar Architecture
+Fig. 4. minimize.jar Architecture
+
+M2MA
+M2MA
+Learning
+Final
+Ready for
+Minimization
+input
+Algorithm
+Check
+Operations
+AlgorithmSUBA
+SUBA
+UFA to
+M2MA.jar
+input
+to UFA
+M2MAM2MA
+M2MA
+Final
+Ready for
+Minimization
+input
+Check
+Operations
+Algorithm8
+N. George
+Fig. 5. NBA.jar and arbitrary.jar Architecture
+Figures 2-5 provide a high-level overview of the architecture for each of the
+five executables. The learning algorithm of Beimel et. al. [4], described in more
+detail in Section 4.3, is implemented in every executable except for minimize.jar.
+The M2MA minimization algorithm of Sakarovitch [10] (described in Section 4.2)
+is implemented in M2MA.jar, SUBA.jar, and minimize.jar. All five executables first
+take in as input from the command line the name of the input file and optional
+flags. The input is passed to a parser, which then reads the input and sends the
+parameters of the automaton to be learned/minimized to the algorithms. The
+executables also all run final checks (described in Section 4.6) on the outputted
+M2MA. Once the checks complete, users can test whether a word is accepted by
+the automaton.
+4.2
+M2MA Minimization Algorithm
+The M2MA minimization algorithm, implemented in the minimize function of
+M2MA.java, is described in abstract terms in the work by Sakarovitch [10] and
+in more concrete detail in the appendix of [3]. The implemented minimization
+algorithm follows the pseudo-code in the appendix of [3] closely.
+The algorithm requires a heavy use of linear algebra, much of which is done
+using the Apache Commons Math package. While testing the executables, how-
+ever, many of the unminimized M2MAs were seen to be relatively sparse with
+few 1’s in the transition matrices. To take advantage of the sparseness, ALMA
+implements custom linear algebra functions that use a sparse representation of
+matrices. Matrix rows are represented using arrays containing the locations of
+the 1’s in the row. For example, the row
+�0 1 0 0 0 1�
+is represented as
+�2 6�
+,
+indicating that the row has a 1 in columns 2 and 6. Matrix multiplication, dot
+products, and tests for linear independence are implemented using the sparse
+matrix representation. Since the minimization function deals with M2MAs, the
+custom linear algebra functions perform all calculations modulo 2, which further
+improves the run time.
+4.3
+General Learning Algorithm
+The learning algorithm of Beimel et. al. [4] is the core algorithm underpinning
+ALMA, and it is implemented in the learn function of M2MA.java. We provide
+
+NBA
+input
+Learning
+Final
+Ready for
+Algorithm
+Check
+Operations
+Arbitrary
+inputALMA: Automata Learner using Modulo 2 Multiplicity Automata
+9
+a high-level overview of the algorithm in Section 1, and more details on the
+algorithm are found in [4]. Like the minimization algorithm, the linear algebra
+is implemented using a combination of the Apache Commons Math package and
+custom functions using the sparse representation for matrices.
+The default membership query function the learning algorithm uses is that
+for M2MAs. As described in Section 2.2, an M2MA A = (Σ, vI, {µσ}σ∈Σ, vF ) ac-
+cepts a word w if v⊤
+I µ(w)vF = 1. Equivalence queries for M2MAs are easy, since
+as a by-product of running the minimization algorithm we get a complete obser-
+vation table for the automaton. The equivalence query function checks whether
+the current hypothesis agrees with the M2MA being learned on every word in
+the observation table, as well as all possible one-letter extensions of the words.
+If they agree on every word and one-letter extension, the algorithm terminates;
+otherwise, the algorithm returns a word for which they disagree on as a coun-
+terexample.
+4.4
+SUBA Learning Algorithm
+The SUBA learning algorithm of Angluin et. al. [2] works as follows: the input
+SUBA is first converted into an equivalent UFA using a simple construction by
+Bosquet and Löding [6]. If the SUBA has n states, the constructed UFA has
+size 2n2 + n. Next, the UFA is converted to an equivalent M2MA of the same
+size. Lastly, the M2MA is learned using the algorithm of Beimel et. al. [4]. The
+algorithms for the SUBA to UFA and UFA to M2MA conversions are found in
+SUBA.java, and the outputted M2MA is sent to M2MA.java to be minimized and
+learned.
+4.5
+Learning NBA and Arbitrary Automata
+In M2MA.jar, minimize.jar, and SUBA.jar, the learning algorithm runs member-
+ship and equivalence queries on M2MAs. The learning algorithm for NBA.jar,
+however, uses a membership query function designed specifically for NBAs which
+we describe in this section. arbitrary.jar passes to the learning algorithm a mem-
+bership query function specified by the user from MQ.java. Also since the equiva-
+lence query function for M2MAs doesn’t work in the general setting, NBA.jar and
+arbitrary.jar use approximate equivalence queries that rely on testing a random
+sample of words from the language.
+NBA Membership Query Function The NBA membership query function
+is described in Algorithm 1. As explained in Section 2.1, an NBA accepts a word
+if there exists a path for the word that begins at an initial state and passes
+infinitely often through a final state. The input parameters u, v represent a word
+w = u$v in the L$ language.
+In Algorithm 1, the reachable function finds all the states reachable from a
+set of initial states S on reading some positive number of v’s. On every iteration
+of the loop defined in line 16, the function finds the states reachable on reading
+
+10
+N. George
+Algorithm 1 NBA Membership Query Function
+1: function main(u, v)
+2:
+Su ← states reachable from the initial state on reading u
+3:
+Suv ← Su∪ reachable(Su, v)
+4:
+for all s ∈ Suv do
+5:
+S′
+uv ← reachable({s}, v)
+6:
+if s ∈ S′
+uv & passed a final state to reach s then
+7:
+return 1
+8:
+end if
+9:
+end for
+10:
+return 0
+11: end function
+12:
+13: function reachable(S, v)
+14:
+Sv ← states reachable from a state in S on reading v
+15:
+S′
+v ← ∅
+16:
+repeat
+17:
+Sv ← Sv ∪ S′
+v
+18:
+S′
+v ← states reachable from a state in Sv on reading v
+19:
+until S′
+v ⊆ Sv
+20:
+return Sv
+21: end function
+another v, and the function terminates after an iteration of the loop where no
+new states are found.
+In the main function, in lines 2-3 Algorithm 1 stores in Suv all the states
+reachable from the initial state on reading a u and a non-negative number of v’s.
+Then in lines 4-10, the algorithm determines whether there is an accepting loop
+on any of the states in Suv (i.e., a path from a state in Suv to itself that passes
+through a final state).
+Run Time Analysis Let n be the number of states, m be the number of
+transitions, and l be the length of u$v. Line 2 runs in O(nml) - we read each of
+the O(l) characters in u one at a time, and with each character we calculate the
+new states that can be reached using the O(m) transitions from the O(n) states
+reached so far. reachable runs in O(n2ml) - the loop runs at most n times
+since there are at most n states to add to S, and line 18 runs in O(nml). The
+loop in line 4 terminates after O(n) iterations, and since reachable is called
+in line 5, the loop runs in O(n3ml). Therefore, Algorithm 1 runs in O(n3ml).
+Approximate Equivalence Queries The approximate equivalence query func-
+tion is defined in arbitrary.java. Along with the hypothesis, it requires two param-
+eters n and l as input. The function generates n random words of length at most
+l, and it tests whether the hypothesis and automaton being learned agrees on
+these words. Increasing n improves the accuracy of the function at the expense
+
+ALMA: Automata Learner using Modulo 2 Multiplicity Automata
+11
+of the run time. In the input file, users also give a limit m on the number of
+equivalence queries that can be run. Since there are no guarantees on the size of
+the learned M2MA, the parameter m prevents the algorithm from running for
+an arbitrarily long length of time.
+4.6
+Checks
+The code performs checks at various points of the algorithm. If a check fails, the
+code throws an exception and terminates the program. Example checks include
+checking the validity of the input (e.g., correct number of transition matrices,
+matrices only contain 0’s and 1’s, etc.), whether a matrix is invertible before
+running a linear equation solver, and if the dimension of the minimized M2MA
+equals that of the final learned M2MA.
+After learning an M2MA, the code checks whether the outputted M2MA
+agrees with the input automaton on the membership of a random sample of
+words. By default, the code generates 1000 random words of length at most 25,
+but these constants can be modified. To perform this check, the code uses the
+previously described membership query functions for M2MAs, NBAs, and arbi-
+trary automata, as well as a membership query function for SUBAs described in
+a paper by Bosquet and Löding [6]. Users can also manually confirm whether the
+outputted M2MA accepts words from the language in the “Ready for Operations”
+phase of the executables described in Section 4.1.
+5
+Experimental Evaluation
+The GitHub repository contains many input files that can be used to test each
+of the executables. There exist input files for many different sizes of M2MAs,
+SUBAs, and NBAs, as well as input files for different edge cases (e.g., M2MAs
+of dimension 1).
+Table 1. M2MA.jar Run Time
+M2MA Dimension
+Average Run Time
+10
+0.33s
+20
+2.44s
+30
+9.76s
+40
+28.57s
+50
+82.57s
+60
+177.00s
+70
+355.09s
+80
+657.82s
+90
+1084.85s
+100
+1819.80s
+
+12
+N. George
+Table 2. SUBA.jar Run Time
+SUBA Language
+SUBA Size
+Unminimized
+M2MA Dim
+Learned
+M2MA Dim
+Run Time
+aΣ∗(Σ∗bΣ∗)ω
+2
+10
+5
+0.20s
+Σ∗aΣ5abω
+8
+136
+10
+0.30s
+((a + b)∗(a(a + b)a(a + b)c
++b(a + b)b(a + b)d))ω
+9
+171
+92
+50.86s
+(a∗a4b)ω
+5
+55
+32
+0.57s
+(a∗a5b)ω
+6
+78
+44
+2.04s
+(a∗a6b)ω
+7
+105
+58
+6.72s
+aω
+1
+3
+3
+0.05s
+(ab5)ω
+6
+78
+43
+0.45s
+(ab10)ω
+11
+253
+133
+25.11s
+(ab15)ω
+16
+528
+273
+411.48s
+(ab20)ω
+21
+903
+463
+3064.32s
+Table 3. NBA.jar Run Time
+NBA Size
+Number of
+tests/EQ
+Learned M2MA
+Dimension
+Run Time
+2
+10000
+3
+1.02s
+4
+10000
+7
+0.25s
+6
+10000
+24
+1.80s
+8
+100000
+27
+14.00s
+10
+100000
+83
+983.39s
+Tables 1-3 give a sense of the run times for M2MA.jar, SUBA.jar, and NBA.jar.
+The experiments were run on a standard laptop with 8 GB RAM, 8 cores, and
+a CPU frequency of 3200 MHz.
+Table 1 details the run time of M2MA.jar. Fifty random M2MAs with the
+alphabet {a, b, c} were generated for each of the dimensions 10, 20, . . . , 50, and
+the average run times were calculated. For the dimensions 60, 70, . . . , 100, ten
+random M2MAs were generated. These results can be reproduced using the
+executable M2MA_experiments.jar in the Experimental Evaluation folder of the
+GitHub repository. Instructions on how to use the jar file are printed to standard
+output once the executable is run.
+Table 2 details the run time of SUBA.jar. The SUBA input files used to gen-
+erate the results are found in the Experimental Evaluation folder of the GitHub
+repository.
+Table 3 details the run time of NBA.jar. The NBAs in the table were randomly
+generated, with one NBA generated per size. Since NBA.jar uses approximate
+equivalence queries, the table contains a column for the number of tests to run
+for every equivalence query. The maximum length of a test for every NBA in the
+
+ALMA: Automata Learner using Modulo 2 Multiplicity Automata
+13
+table is 25. The NBA input files used to generate the results are found in the
+Experimental Evaluation folder of the GitHub repository.
+Practical Capabilities The run times for M2MA.jar increase at a roughly
+cubic rate. For smaller M2MAs, the program runs quickly, but for M2MAs of
+dimension larger than 100, the program can take hours to terminate. For the
+purpose of using ALMA to explore the properties of M2MAs, it is unlikely that
+one will need to work with M2MAs of dimension much larger than 100, so the
+program’s run time should not pose a significant constraint.
+One may think that experimenting with the SUBA learning algorithm, which
+incurs a 2n2 + n blow up in size going from the initial SUBA to the converted
+M2MA, will be impractical when dealing with SUBAs of size even greater than
+10. However, as can be seen in Table 2, the unminimized M2MA in the SUBA
+learning algorithm often minimizes to a much smaller M2MA, which is why the
+tool implements the minimization algorithm of Sakarovitch [10]. For example,
+the SUBA of size 21 representing the language (ab20)ω converts into a large
+unminimized M2MA of dimension 903. However, it minimizes to an M2MA of
+dimension 463, and SUBA.jar runs in less than an hour for this SUBA.
+The run times for NBA.jar depend heavily on the number of tests performed
+every equivalence query. To get decent approximations on the learned M2MA
+dimension for NBAs of size larger than 10, the number of tests per equivalence
+query should be at least on the order of 10 to the 5th or 6th power. One can try
+to change the maximum length of the tests to get better results. For randomly
+generated NBAs of size much larger than 10, the trade-off between the run
+time and accuracy becomes much more apparent, and the number of tests per
+equivalence query may have to be decreased for the run time to be practical.
+6
+Conclusion
+ALMA has limitations - for example, the run time of M2MA.jar becomes im-
+practical for random M2MAs of size much larger than 100, and ALMA doesn’t
+implement an exact equivalence query function for learning NBAs and arbitrary
+automata. However, ALMA is highly efficient for most standard use cases, and
+it can be used to promote further research into M2MAs and their properties. For
+example, in the paper by Angluin et. al. [3], ALMA is used to find the dimension
+of the minimum M2MA that can represent a regular ω-language. Angluin et. al.
+compare this dimension with the size of other finite automata that represent the
+same language to analyze the succinctness of the M2MA representation. Along
+with serving as a useful tool for investigating M2MAs, ALMA confirms the theo-
+retical results of Beimel et. al. [4] and Angluin et. al [2,3], and is the first publicly
+available tool that can be used to explore these learning algorithms.
+Acknowledgements I would like to thank Dana Angluin, Timos Antonopoulos,
+and Dana Fisman for their help with this paper and all the feedback and advice
+
+14
+N. George
+they gave. Dana Angluin especially helped me significantly during the entire
+process of creating ALMA, and I would like to thank her greatly for her support
+and mentorship.
+References
+1. Angluin, D.: Queries and concept learning. Machine Learning 2(4), 319–342 (1987)
+2. Angluin, D., Antonopoulos, T., Fisman, D.: Strongly unambiguous Büchi automata
+are polynomially predictable with membership queries. In: 28th EACSL Annual
+Conference on Computer Science Logic, CSL. pp. 8:1–8:17 (2020)
+3. Angluin, D., Antonopoulos, T., Fisman, D., George, N.: Representing regular lan-
+guages of infinite words using mod 2 multiplicity automata. In: Foundations of
+Software Science and Computation Structures - 25th International Conference,
+FOSSACS. p. 1–20 (2022)
+4. Beimel, A., Bergadano, F., Bshouty, N.H., Kushilevitz, E., Varricchio, S.: Learning
+functions represented as multiplicity automata. J. ACM 47(3), 506–530 (May 2000)
+5. Bollig, B., Katoen, J.P., Kern, C., Leucker, M., Neider, D., Piegdon, D.R.: libalf:
+The automata learning framework. In: Computer Aided Verification - 22nd Inter-
+national Conference, CAV. pp. 360–364 (2010)
+6. Bousquet, N., Löding, C.: Equivalence and inclusion problem for strongly un-
+ambiguous Büchi automata. In: Language and Automata Theory and Appli-
+cations, 4th International Conference, LATA. Proceedings. pp. 118–129 (2010).
+https://doi.org/10.1007/978-3-642-13089-2_10
+7. Büchi, J.: On a decision method in restricted second order arithmetic. In: Interna-
+tional Congress on Logic, Methodology and Philosophy. pp. 1–11. Stanford Univ.
+Press (1962)
+8. Calbrix, H., Nivat, M., Podelski, A.: Ultimately periodic words of rational w-
+languages. In: Proceedings of the 9th International Conference on Mathematical
+Foundations of Programming Semantics. pp. 554–566. Springer-Verlag (1994)
+9. Li, Y., Sun, X., Turrini, A., Chen, Y.F., Xu, J.: Roll 1.0: ω-regular language learning
+library. In: Tools and Algorithms for the Construction and Analysis of Systems -
+25th International Conference, TACAS, Part I. pp. 365–371 (2019)
+10. Sakarovitch, J.: Elements of Automata Theory. Cambridge University Press, USA
+(2009)
+
diff --git a/WNE2T4oBgHgl3EQfuAju/content/tmp_files/load_file.txt b/WNE2T4oBgHgl3EQfuAju/content/tmp_files/load_file.txt
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@@ -0,0 +1,500 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf,len=499
+page_content='ALMA: Automata Learner using Modulo 2  Multiplicity Automata  Nevin George  Yale University, New Haven CT 06511, USA  nevin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='george@yale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='edu  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' We present ALMA (Automata Learner using modulo 2 Mul-  tiplicity Automata), a Java-based tool that can learn any automaton  accepting regular languages of finite or infinite words with an imple-  mentable membership query function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' Users can either pass as input their  own membership query function, or use the predefined membership query  functions for modulo 2 multiplicity automata and non-deterministic Büchi  automata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' While learning, ALMA can output the state of the observation  table after every equivalence query, and upon termination, it can output  the dimension, transition matrices, and final vector of the learned mod-  ulo 2 multiplicity automaton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' Users can test whether a word is accepted  by performing a membership query on the learned automaton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='  ALMA follows the polynomial-time learning algorithm of Beimel et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' (Learning functions represented as multiplicity automata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' ACM  47(3), 2000), which uses membership and equivalence queries and repre-  sents hypotheses using modulo 2 multiplicity automata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' ALMA also im-  plements a polynomial-time learning algorithm for strongly unambigu-  ous Büchi automata by Angluin et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' (Strongly unambiguous Büchi  automata are polynomially predictable with membership queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' CSL  2020), and a minimization algorithm for modulo 2 multiplicity automata  by Sakarovitch (Elements of Automata Theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='  Keywords: automata theory · finite automata · Büchi automata · mul-  tiplicity automata · learning  1  Introduction  Angluin’s exact learning model [1] has been studied extensively in the context  of learning theory, and it can be used to learn automata representing regular  languages of finite and infinite words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' In the model, a learner interacts with an  oracle to learn a regular language using membership and equivalence queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='  In a membership query, the learner learns from the oracle whether a word is in  the language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' In an equivalence query, the learner forms a hypothesis on what  the language is, and the oracle either confirms that the hypothesis is correct  or returns a counterexample, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='e, a word for which the hypothesis and language  differ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='04077v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='FL]  10 Jan 2023  2  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' George  As one application of the exact learning model, Beimel et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' [4] detail a  polynomial-time algorithm to learn multiplicity automata using membership and  equivalence queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' We provide a high-level overview of the algorithm: the algo-  rithm begins with a trivial hypothesis of the language, represented as a multiplic-  ity automaton of dimension 1 defined over some field K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' On each iteration of a  loop, the algorithm performs an equivalence query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' If the hypothesis is equivalent  to the language, the algorithm terminates and outputs the hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' Other-  wise, the algorithm receives a counterexample from the oracle, which is used to  improve the hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' The algorithm terminates after d iterations, where d is  the dimension of the smallest possible multiplicity automaton that can represent  the language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='  Strongly unambiguous Büchi automata (SUBAs) (defined in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='1) are  a type of non-deterministic Büchi automaton (NBA) first introduced by Bosquet  and Löding [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' SUBAs are useful for modeling reactive systems, as they are fully  expressive, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=', they can represent any regular ω-language, and can often rep-  resent regular ω-languages more succinctly than other NBA representations [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='  Angluin et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' [2] also showed that SUBAs are learnable in polynomial time,  further increasing their importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='  In this paper, we present ALMA, a Java-based tool that implements the al-  gorithm of Beimel et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' and extends it to learn any arbitrary automaton repre-  senting regular languages of finite or infinite words with an implementable mem-  bership query function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' Membership query functions have been implemented for  M2MAs and NBAs, and users can enter as input any arbitrary membership  query function that 1) takes as input any possible word that can be formed from  the given alphabet, and 2) outputs 0 or 1 to indicate whether the word is in the  language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' To improve the run time of learning SUBAs, ALMA does not use the  general learning algorithm but rather the SUBA learning algorithm of Angluin  et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' Also when learning M2MAs, ALMA first implements the algorithm of  Sakarovitch [10] to minimize the M2MA before running the learning algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='  This is because in the SUBA learning algorithm, converting the input SUBA  into an equivalent M2MA incurs a quadratic increase size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' Minimizing M2MAs  before learning enables the algorithm to learn significantly larger SUBAs, es-  pecially because empirically the M2MAs have been seen to often minimize to  M2MAs of much smaller dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' The effect of the minimization algorithm on  a sample of input SUBAs can be seen in Table 2 in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='  Usefulness/Novelty In the verification community, M2MAs can be useful for  representing regular ω-languages, as they are learnable in polynomial time and  often relatively succinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' They are also useful for tasks such as model checking,  since performing operations such as intersection, union, complementation, empti-  ness, and equivalence on M2MAs is cheap [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' Through the minimization and  learning algorithms, ALMA enables researchers in the verification community to  easily test and verify these properties of M2MAs, promoting further exploration  into these useful automata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' As an example, ALMA was used in the paper by An-  gluin et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' [3] to explore the suitableness of representing regular ω-languages  ALMA: Automata Learner using Modulo 2 Multiplicity Automata  3  using M2MAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' The authors used ALMA to convert SUBAs, NBAs, and DBAs  into equivalent M2MAs, and they compared the succinctness of these M2MAs  with that of DFAs accepting the same language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' ALMA can similarly be used  by other researchers to gain insights into M2MAs and their benefits/drawbacks  as compared to other representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='  ALMA is the first publicly available implementation of the novel SUBA learn-  ing algorithm by Angluin et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' Using ALMA, users can run the algorithm to  learn any regular ω-language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' Since learned automata are represented as M2MAs,  users can use the many desirable properties of M2MAs to gain insights into  the initial SUBA and regular ω-language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' In addition, since membership query  functions are often relatively easy to implement and ALMA already provides a  membership query function for the general NBA case, the scope of what ALMA  can learn is large, promoting its usefulness in a wide variety of settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='  Many tools such as ROLL (ω-Regular Language Learning Library) [9] and  libalf [5] already exist that can learn automata representing regular languages  of finite and infinite words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' However, ALMA is the first tool that uses M2MAs  to represent hypotheses and the learned automaton in the learning algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='  ALMA’s usefulness lies not with necessarily being the fastest tool available to  learn regular languages, but with exploring the practical benefits of M2MAs and  the features of the algorithms by Beimel et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=', Angluin et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=', and Sakarovitch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='  2  Useful Definitions  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='1  Finite and Büchi Automata  Let Σ be a finite alphabet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' Then Σ∗ and Σω are the sets of all finite and infinite  words, respectively, that can be formed using elements from Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' A finite language  is a subset of Σ∗, and an ω-language is a subset of Σω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' If w is a word in Σ∗ or  Σω, let |w| be the length of w and w[i] be the i’th character of w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='  A finite-state automaton A is represented as a tuple (Σ, Q, I, ∆, F), where  Σ is the alphabet, Q is the finite set of states, I ⊆ Q is the set of initial states,  ∆ ⊆ Q × Σ × Q is the set of transitions, and F ⊆ Q is the set of final states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='  The automaton A is deterministic if every pair (q, σ) ∈ Q × Σ appears as the  first two elements in at most one triple in ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='  A run on A for a word w is a series of states q0, q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' ∈ Q such that ∀i  satisfying 1 ≤ i ≤ |w|, (qi−1, w[i], qi) ∈ ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' A run for a finite word is final if it  ends in a final state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' For infinite words, a run is final if it passes infinitely often  through a final state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' A run is accepting if it is final and begins at an initial state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='  The automaton A accepts the finite/infinite word w if there exists an accepting  run for w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='  Non-deterministic finite automata (NFAs) and non-deterministic Büchi au-  tomata (NBAs) are automata accepting finite and infinite words, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='  Deterministic finite automata (DFAs) are deterministic NFAs, and deterministic  Büchi automata (DBAs) are deterministic NBAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' Unambiguous finite automata  4  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' George  (UFAs) and unambiguous Büchi automata (UBAs) are NFAs and NBAs, respec-  tively, for which every word has at most one accepting run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' A UBA is a strongly  unambiguous Büchi automaton (SUBA) if every word has at most one final run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='2  Modulo 2 Multiplicity Automata  Assume a field K and some dimension d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' A multiplicity automaton A is repre-  sented as a tuple (Σ, vI, {µσ}σ∈Σ, vF ), where Σ is the alphabet, vI is the initial  vector, each µσ is a transition matrix, and vF is the final vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' vI and vF have  dimension d × 1, and each µσ has dimension d × d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' The set of states is all row  vectors v ∈ {0, 1}d, and the initial state is v⊤  I .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' For a given word w = σ1σ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' σn,  let µ(w) = µσ1µσ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' µσn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' The set of reachable states is all vectors of the form  v⊤  I µ(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' Associated with the automaton A is a function fA : Σ∗ → K, where  ∀w ∈ Σ∗,  fA(w) = v⊤  I µ(w)vF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='  A modulo 2 multiplicity automaton (M2MA) is a multiplicity automaton where  K = GF(2) and all calculations are done modulo 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' An M2MA A accepts a word  w ∈ Σ∗ if and only if fA(w) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='  As an example, consider the following M2MA M adapted from Angluin et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='  M = ({a, b},  �1 0 0�⊤ , {µa, µb},  �1 1 0�⊤}  where  µa =  �  �  0 0 1  1 0 0  1 1 1  �  � and µb =  �  �  0 1 0  1 0 1  1 1 0  �  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='  M is equivalent to the DFA in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' We explain Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' 1: the computation begins  at the initial state  �  1 0 0  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' On reading an a/b from the initial state, the DFA  visits the states  �1 0 0�  µa =  �1 0 0�  �  �  0 0 1  1 0 0  1 1 1  �  � =  �0 0 1�  �1 0 0�  µb =  �1 0 0�  �  �  0 1 0  1 0 1  1 1 0  �  � =  �0 1 0�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='  Other states are visited similarly by multiplying the current state vector by µa  or µb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='  �1 0 0�  ,  �0 1 0�  , and  �1 0 1�  are accepting states since  �1 0 0� �1 1 0�⊤ =  �0 1 0� �1 1 0�⊤ =  �1 0 1� �1 1 0�⊤ = 1,  where vF =  �1 1 0�⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='  M2MAs have many important properties described in Section 1 that make  them useful in verification communities, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=', learnable in polynomial time and  cheap intersection, union, complementation, emptiness, and equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' For  ALMA: Automata Learner using Modulo 2 Multiplicity Automata  5  100  start  010  101  001  111  110  b  b  b  a  a  a  b  b  a  a  a  b  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' DFA for the M2MA M  more information on M2MAs, we recommend reading the paper by Angluin  et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' [3], which studies these properties of M2MAs extensively and performs  a detailed analysis of M2MAs’ ability to succinctly represent regular finite and  ω-languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='3  L$ Language  Since Büchi automata accept infinite words and M2MAs accept finite words,  Büchi automata and M2MAs cannot accept words from the same language L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='  However, Büchi [7] showed that two regular ω-languages are equivalent if and  only if they agree on a set of ultimately periodic words, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=', words of the form  u(v)ω where u and v are finite words and v is non-empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' We then consider  the language of finite words L$ = {u$v | u(v)ω ∈ L}, which Calbrix et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' [8]  showed is regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' If a Büchi automaton accepts an infinite language L, a finite  automaton is said to also represent L if it accepts L$.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' The L$ language is used  by Angluin et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' [2] in their SUBA learning algorithm in order to obtain a  finite automaton equivalent to the initial SUBA, and the learned M2MA accepts  words from L$.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='  3  Usage  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='1  Access and How to Run  ALMA is an open-source library freely available at the following GitHub reposi-  tory: https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='com/nevingeorge/Learning-Automata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' The repository  contains the executables, source files, and sample input files for ALMA, and the  README document contains detailed information on how to use the executables  and the required format for the input files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='  ALMA is a Java-based tool designed to be used on the command line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' To  run for example the executable M2MA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='jar, a user should enter the command java  jar M2MA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='jar within a Terminal/Command Prompt window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' This will start the  program, and the program will then print instructions on how to input the  desired input file and flags.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='  6  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' George  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='2  Executables  ALMA consists of five main executables: 1) M2MA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='jar, 2) SUBA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='jar, 3) mini-  mize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='jar, 4) NBA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='jar, and 5) arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='jar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' We provide a basic overview of each  of the jar files and their use cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' For each executable, the output is always  the dimension, final vector, and transition matrices of the learned/minimized  M2MA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' Also after the algorithms terminate, each executable allows users to  check whether a word is accepted by the outputted M2MA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' Infinite words are  represented using the L$ language explained in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='  M2MA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='jar is used to minimize and learn M2MAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' The alphabet, dimension,  final vector, and transition matrices of the input M2MA must be specified (the  initial vector is always the vector with all zeros except for a 1 in the first row).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='  SUBA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='jar is used to learn SUBAs using the algorithm of Angluin et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='  The alphabet, number of states, final states, and transitions of the input SUBA  must be specified (the only initial state is the first state).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='  minimize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='jar is used to minimize M2MAs using the algorithm of Sakarovitch [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='  The input is the same as that for M2MA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='jar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' minimize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='jar can also be used to  output the minimized M2MA equivalent to the input SUBA in the SUBA learn-  ing algorithm (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=', it runs every step of the SUBA learning algorithm except  for learning the final M2MA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' In this case, the input is the same as that for  SUBA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='jar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='  NBA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='jar is used to learn NBAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' The alphabet, number of states, final states,  and transitions of the input NBA must be specified (the only initial state is  the first state).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' NBA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='jar uses approximate equivalence queries (described in Sec-  tion 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='5), which requires as input the number of tests to run, maximum length  of a test, and maximum limit on the number of equivalence queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='  arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='jar is used to learn arbitrary automata representing regular languages  of finite and infinite words with an implementable membership query function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='  The membership query function must be defined in MQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='java.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' The only con-  straints on the function is that it must accept as input any possible word formed  from letters in the alphabet, and it must output 0 or 1 depending on whether the  word is contained in the language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='jar also uses approximate equivalence  queries, which requires the same input parameters as described for NBA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='jar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='3  Flags  Users can enter the following optional flags to add or remove information from  the output of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='  ALMA: Automata Learner using Modulo 2 Multiplicity Automata  7  v: The algorithm outputs the state of the observation table (a matrix consist-  ing of the words whose membership in the language is known) after every  equivalence query in the learning algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='  m: The algorithm outputs detailed information on the progress of the mini-  mization algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' It gives status updates at different points in the algo-  rithm, such as when it finishes creating the state/co-state spaces and the  observation table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' It also outputs the initial M2MA to be minimized and the  final minimized observation table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='  d: When running minimize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='jar on a SUBA, the algorithm outputs only the di-  mension of the minimized M2MA instead of the dimension, final vector, and  transition matrices of the M2MA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='  a: After minimizing the M2MA, the algorithm outputs the number of states of  the minimal deterministic finite automaton (DFA) that represents the same  language as the M2MA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='  4  Implementation Details  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='1  Architecture  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' M2MA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='jar Architecture  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' SUBA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='jar Architecture  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' minimize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='jar Architecture  M2MA  M2MA  Learning  Final  Ready for  Minimization  input  Algorithm  Check  Operations  AlgorithmSUBA  SUBA  UFA to  M2MA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='jar  input  to UFA  M2MAM2MA  M2MA  Final  Ready for  Minimization  input  Check  Operations  Algorithm8  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' George  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' NBA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='jar and arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='jar Architecture  Figures 2-5 provide a high-level overview of the architecture for each of the  five executables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' The learning algorithm of Beimel et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' [4], described in more  detail in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='3, is implemented in every executable except for minimize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='jar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='  The M2MA minimization algorithm of Sakarovitch [10] (described in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='2)  is implemented in M2MA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='jar, SUBA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='jar, and minimize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='jar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' All five executables first  take in as input from the command line the name of the input file and optional  flags.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' The input is passed to a parser, which then reads the input and sends the  parameters of the automaton to be learned/minimized to the algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' The  executables also all run final checks (described in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='6) on the outputted  M2MA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' Once the checks complete, users can test whether a word is accepted by  the automaton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='2  M2MA Minimization Algorithm  The M2MA minimization algorithm, implemented in the minimize function of  M2MA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='java, is described in abstract terms in the work by Sakarovitch [10] and  in more concrete detail in the appendix of [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' The implemented minimization  algorithm follows the pseudo-code in the appendix of [3] closely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='  The algorithm requires a heavy use of linear algebra, much of which is done  using the Apache Commons Math package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' While testing the executables, how-  ever, many of the unminimized M2MAs were seen to be relatively sparse with  few 1’s in the transition matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' To take advantage of the sparseness, ALMA  implements custom linear algebra functions that use a sparse representation of  matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' Matrix rows are represented using arrays containing the locations of  the 1’s in the row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' For example, the row  �0 1 0 0 0 1�  is represented as  �2 6�  ,  indicating that the row has a 1 in columns 2 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' Matrix multiplication, dot  products, and tests for linear independence are implemented using the sparse  matrix representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' Since the minimization function deals with M2MAs, the  custom linear algebra functions perform all calculations modulo 2, which further  improves the run time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='3  General Learning Algorithm  The learning algorithm of Beimel et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' [4] is the core algorithm underpinning  ALMA, and it is implemented in the learn function of M2MA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='java.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' We provide  NBA  input  Learning  Final  Ready for  Algorithm  Check  Operations  Arbitrary  inputALMA: Automata Learner using Modulo 2 Multiplicity Automata  9  a high-level overview of the algorithm in Section 1, and more details on the  algorithm are found in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' Like the minimization algorithm, the linear algebra  is implemented using a combination of the Apache Commons Math package and  custom functions using the sparse representation for matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='  The default membership query function the learning algorithm uses is that  for M2MAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' As described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='2, an M2MA A = (Σ, vI, {µσ}σ∈Σ, vF ) ac-  cepts a word w if v⊤  I µ(w)vF = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' Equivalence queries for M2MAs are easy, since  as a by-product of running the minimization algorithm we get a complete obser-  vation table for the automaton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' The equivalence query function checks whether  the current hypothesis agrees with the M2MA being learned on every word in  the observation table, as well as all possible one-letter extensions of the words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='  If they agree on every word and one-letter extension, the algorithm terminates;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='  otherwise, the algorithm returns a word for which they disagree on as a coun-  terexample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='4  SUBA Learning Algorithm  The SUBA learning algorithm of Angluin et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' [2] works as follows: the input  SUBA is first converted into an equivalent UFA using a simple construction by  Bosquet and Löding [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' If the SUBA has n states, the constructed UFA has  size 2n2 + n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' Next, the UFA is converted to an equivalent M2MA of the same  size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' Lastly, the M2MA is learned using the algorithm of Beimel et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' The  algorithms for the SUBA to UFA and UFA to M2MA conversions are found in  SUBA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='java, and the outputted M2MA is sent to M2MA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='java to be minimized and  learned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='5  Learning NBA and Arbitrary Automata  In M2MA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='jar, minimize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='jar, and SUBA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='jar, the learning algorithm runs member-  ship and equivalence queries on M2MAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' The learning algorithm for NBA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='jar,  however, uses a membership query function designed specifically for NBAs which  we describe in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='jar passes to the learning algorithm a mem-  bership query function specified by the user from MQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='java.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' Also since the equiva-  lence query function for M2MAs doesn’t work in the general setting, NBA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='jar and  arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='jar use approximate equivalence queries that rely on testing a random  sample of words from the language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='  NBA Membership Query Function The NBA membership query function  is described in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' As explained in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='1, an NBA accepts a word  if there exists a path for the word that begins at an initial state and passes  infinitely often through a final state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' The input parameters u, v represent a word  w = u$v in the L$ language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='  In Algorithm 1, the reachable function finds all the states reachable from a  set of initial states S on reading some positive number of v’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' On every iteration  of the loop defined in line 16, the function finds the states reachable on reading  10  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' George  Algorithm 1 NBA Membership Query Function  1: function main(u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' v)  2:  Su ← states reachable from the initial state on reading u  3:  Suv ← Su∪ reachable(Su,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' v)  4:  for all s ∈ Suv do  5:  S′  uv ← reachable({s},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' v)  6:  if s ∈ S′  uv & passed a final state to reach s then  7:  return 1  8:  end if  9:  end for  10:  return 0  11: end function  12:  13: function reachable(S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' v)  14:  Sv ← states reachable from a state in S on reading v  15:  S′  v ← ∅  16:  repeat  17:  Sv ← Sv ∪ S′  v  18:  S′  v ← states reachable from a state in Sv on reading v  19:  until S′  v ⊆ Sv  20:  return Sv  21: end function  another v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' and the function terminates after an iteration of the loop where no  new states are found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='  In the main function, in lines 2-3 Algorithm 1 stores in Suv all the states  reachable from the initial state on reading a u and a non-negative number of v’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='  Then in lines 4-10, the algorithm determines whether there is an accepting loop  on any of the states in Suv (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=', a path from a state in Suv to itself that passes  through a final state).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='  Run Time Analysis Let n be the number of states, m be the number of  transitions, and l be the length of u$v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' Line 2 runs in O(nml) - we read each of  the O(l) characters in u one at a time, and with each character we calculate the  new states that can be reached using the O(m) transitions from the O(n) states  reached so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' reachable runs in O(n2ml) - the loop runs at most n times  since there are at most n states to add to S, and line 18 runs in O(nml).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' The  loop in line 4 terminates after O(n) iterations, and since reachable is called  in line 5, the loop runs in O(n3ml).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' Therefore, Algorithm 1 runs in O(n3ml).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='  Approximate Equivalence Queries The approximate equivalence query func-  tion is defined in arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='java.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' Along with the hypothesis, it requires two param-  eters n and l as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' The function generates n random words of length at most  l, and it tests whether the hypothesis and automaton being learned agrees on  these words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' Increasing n improves the accuracy of the function at the expense  ALMA: Automata Learner using Modulo 2 Multiplicity Automata  11  of the run time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' In the input file, users also give a limit m on the number of  equivalence queries that can be run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' Since there are no guarantees on the size of  the learned M2MA, the parameter m prevents the algorithm from running for  an arbitrarily long length of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='6  Checks  The code performs checks at various points of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' If a check fails, the  code throws an exception and terminates the program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' Example checks include  checking the validity of the input (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=', correct number of transition matrices,  matrices only contain 0’s and 1’s, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' ), whether a matrix is invertible before  running a linear equation solver, and if the dimension of the minimized M2MA  equals that of the final learned M2MA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='  After learning an M2MA, the code checks whether the outputted M2MA  agrees with the input automaton on the membership of a random sample of  words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' By default, the code generates 1000 random words of length at most 25,  but these constants can be modified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' To perform this check, the code uses the  previously described membership query functions for M2MAs, NBAs, and arbi-  trary automata, as well as a membership query function for SUBAs described in  a paper by Bosquet and Löding [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' Users can also manually confirm whether the  outputted M2MA accepts words from the language in the “Ready for Operations”  phase of the executables described in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='  5  Experimental Evaluation  The GitHub repository contains many input files that can be used to test each  of the executables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' There exist input files for many different sizes of M2MAs,  SUBAs, and NBAs, as well as input files for different edge cases (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=', M2MAs  of dimension 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' M2MA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='jar Run Time  M2MA Dimension  Average Run Time  10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='33s  20  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='44s  30  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='76s  40  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='57s  50  82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='57s  60  177.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='00s  70  355.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='09s  80  657.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='82s  90  1084.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='85s  100  1819.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='80s  12  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' George  Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' SUBA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='jar Run Time  SUBA Language  SUBA Size  Unminimized  M2MA Dim  Learned  M2MA Dim  Run Time  aΣ∗(Σ∗bΣ∗)ω  2  10  5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='20s  Σ∗aΣ5abω  8  136  10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='30s  ((a + b)∗(a(a + b)a(a + b)c  +b(a + b)b(a + b)d))ω  9  171  92  50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='86s  (a∗a4b)ω  5  55  32  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='57s  (a∗a5b)ω  6  78  44  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='04s  (a∗a6b)ω  7  105  58  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='72s  aω  1  3  3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='05s  (ab5)ω  6  78  43  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='45s  (ab10)ω  11  253  133  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='11s  (ab15)ω  16  528  273  411.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='48s  (ab20)ω  21  903  463  3064.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='32s  Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' NBA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='jar Run Time  NBA Size  Number of  tests/EQ  Learned M2MA  Dimension  Run Time  2  10000  3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='02s  4  10000  7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='25s  6  10000  24  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='80s  8  100000  27  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='00s  10  100000  83  983.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='39s  Tables 1-3 give a sense of the run times for M2MA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='jar, SUBA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='jar, and NBA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='jar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='  The experiments were run on a standard laptop with 8 GB RAM, 8 cores, and  a CPU frequency of 3200 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='  Table 1 details the run time of M2MA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='jar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' Fifty random M2MAs with the  alphabet {a, b, c} were generated for each of the dimensions 10, 20, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' , 50, and  the average run times were calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' For the dimensions 60, 70, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' , 100, ten  random M2MAs were generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' These results can be reproduced using the  executable M2MA_experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='jar in the Experimental Evaluation folder of the  GitHub repository.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' Instructions on how to use the jar file are printed to standard  output once the executable is run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='  Table 2 details the run time of SUBA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='jar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' The SUBA input files used to gen-  erate the results are found in the Experimental Evaluation folder of the GitHub  repository.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='  Table 3 details the run time of NBA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='jar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' The NBAs in the table were randomly  generated, with one NBA generated per size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' Since NBA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='jar uses approximate  equivalence queries, the table contains a column for the number of tests to run  for every equivalence query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' The maximum length of a test for every NBA in the  ALMA: Automata Learner using Modulo 2 Multiplicity Automata  13  table is 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' The NBA input files used to generate the results are found in the  Experimental Evaluation folder of the GitHub repository.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='  Practical Capabilities The run times for M2MA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='jar increase at a roughly  cubic rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' For smaller M2MAs, the program runs quickly, but for M2MAs of  dimension larger than 100, the program can take hours to terminate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' For the  purpose of using ALMA to explore the properties of M2MAs, it is unlikely that  one will need to work with M2MAs of dimension much larger than 100, so the  program’s run time should not pose a significant constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='  One may think that experimenting with the SUBA learning algorithm, which  incurs a 2n2 + n blow up in size going from the initial SUBA to the converted  M2MA, will be impractical when dealing with SUBAs of size even greater than  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' However, as can be seen in Table 2, the unminimized M2MA in the SUBA  learning algorithm often minimizes to a much smaller M2MA, which is why the  tool implements the minimization algorithm of Sakarovitch [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' For example,  the SUBA of size 21 representing the language (ab20)ω converts into a large  unminimized M2MA of dimension 903.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' However, it minimizes to an M2MA of  dimension 463, and SUBA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='jar runs in less than an hour for this SUBA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='  The run times for NBA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='jar depend heavily on the number of tests performed  every equivalence query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' To get decent approximations on the learned M2MA  dimension for NBAs of size larger than 10, the number of tests per equivalence  query should be at least on the order of 10 to the 5th or 6th power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' One can try  to change the maximum length of the tests to get better results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' For randomly  generated NBAs of size much larger than 10, the trade-off between the run  time and accuracy becomes much more apparent, and the number of tests per  equivalence query may have to be decreased for the run time to be practical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='  6  Conclusion  ALMA has limitations - for example, the run time of M2MA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='jar becomes im-  practical for random M2MAs of size much larger than 100, and ALMA doesn’t  implement an exact equivalence query function for learning NBAs and arbitrary  automata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' However, ALMA is highly efficient for most standard use cases, and  it can be used to promote further research into M2MAs and their properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' For  example, in the paper by Angluin et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' [3], ALMA is used to find the dimension  of the minimum M2MA that can represent a regular ω-language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' Angluin et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='  compare this dimension with the size of other finite automata that represent the  same language to analyze the succinctness of the M2MA representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' Along  with serving as a useful tool for investigating M2MAs, ALMA confirms the theo-  retical results of Beimel et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' [4] and Angluin et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' al [2,3], and is the first publicly  available tool that can be used to explore these learning algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content='  Acknowledgements I would like to thank Dana Angluin, Timos Antonopoulos,  and Dana Fisman for their help with this paper and all the feedback and advice  14  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' George  they gave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
+page_content=' Dana Angluin especially helped me significantly during the entire  process of creating ALMA, and I would like to thank her greatly for her support  and mentorship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
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+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
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+page_content=' Springer-Verlag (1994)  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfuAju/content/2301.04077v1.pdf'}
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+arXiv:2301.03646v1  [math.GR]  9 Jan 2023
+ABELIAN CONGRUENCES AND SOLVABILITY
+IN MOUFANG LOOPS
+ALEˇS DR´APAL AND PETR VOJTˇECHOVSK´Y
+Abstract. In groups, an abelian normal subgroup induces an abelian congruence. We
+construct a class of centrally nilpotent Moufang loops containing an abelian normal subloop
+that does not induce an abelian congruence. On the other hand, we prove that in 6-divisible
+Moufang loops, every abelian normal subloop induces an abelian congruence.
+In loops, congruence solvability adopted from the universal-algebraic commutator theory
+of congruence modular varieties is strictly stronger than classical solvability adopted from
+group theory.
+It is an open problem whether the two notions of solvability coincide in
+Moufang loops. We prove that they coincide in 6-divisible Moufang loops and in Moufang
+loops of odd order. In fact, we show that every Moufang loop of odd order is congruence
+solvable, thus strengthening Glauberman’s Odd Order Theorem for Moufang loops.
+We investigate abelian normal subloops and the theory of solvability in Moufang loops.
+There are two notions of solvability in loop theory, one adopted from solvability in group
+theory, called classical solvability here, and another adopted from commutator theory in
+congruence modular varieties, called congruence solvability here.
+When translated into the language of normal series Q = Q0 ≥ Q1 ≥ · · · ≥ Qn = 1, the
+difference between the two solvability notions is that classical solvability requires all factors
+Qi/Qi+1 to be abelian (that is, commutative groups), while congruence solvability requires
+a potentially stronger condition, namely that every factor Qi/Qi+1 induces an abelian con-
+gruence of Q/Qi+1. Whether a normal subloop of a loop Q is merely abelian or whether it
+induces an abelian congruence of Q can be seen on the level of multiplication tables, which
+must have a more rigid structure in the latter case, cf. Subsection 2.2.
+Every congruence solvable loop is classically solvable but the converse is not true. The
+two notions of solvability coincide in groups. More generally, they coincide in any loop Q
+in which every abelian normal subloop induces an abelian congruence of Q. The situation
+is delicate, however, since it is certainly possible for a loop Q to be congruence solvable, yet
+posses an abelian normal subloop that does not induce an abelian congruence of Q.
+It is an open problem whether the two notions of solvability coincide in Moufang loops. In
+this paper we offer a general construction of centrally nilpotent Moufang loops that contain
+an abelian normal subloop that does not induce an abelian congruence, cf. Proposition 3.1.
+On the other hand, we show that if Q is a 3-divisible Moufang loop and X is a 2-divisible
+abelian normal subloop of Q, then X induces an abelian congruence of Q, cf. Theorem 4.3.
+In particular, in a 6-divisible Moufang loop, every abelian normal subloop induces an abelian
+2020 Mathematics Subject Classification. 20N05.
+Key words and phrases. Moufang loop, extra loop, solvability, congruence solvability, abelian congruence,
+pseudoautomorphism, semiautomorphism.
+A. Dr´apal supported by the INTER-EXCELLENCE project LTAUSA19070 of MˇSMT Czech Republic.
+P. Vojtˇechovsk´y supported by the Simons Foundation Mathematics and Physical Sciences Collaboration
+Grant for Mathematicians no. 855097 and by the PROF grant of the University of Denver.
+1
+
+congruence, cf. Corollary 4.4, and hence the two notions of solvability coincide in 6-divisible
+Moufang loops, cf. Corollary 4.5. Up to that point, the exposition is self-contained (modulo
+basic results from loop theory) and the arguments are elementary in nature.
+This is in
+contrast with other results in Moufang loops, such as the Lagrange Theorem, whose proofs
+rely on the classification of finite simple groups.
+Continuing, we build upon recent deep results of Cs¨org˝o [6] on the nucleus in Moufang
+loops and prove that every Moufang loop of odd order is congruence solvable, cf. Theorem
+5.3. This strengthens a well-known result of Glauberman [14] that states that every Moufang
+loop of odd order is classically solvable. Theorem 5.3 implies the finitary version of Corollary
+4.5.
+Background material on loops, Moufang loops and divisibility in power associative loops
+is collected in Section 1. Abelianess and solvability for loops are discussed in Section 2. An
+elementary proof of the Cauchy property for p = 3 in Moufang loops is given in Section 6 as
+an appendix.
+1. Background on loops and Moufang loops
+1.1. Loops. See [4] or [21] for an introduction to the theory of loops. A loop Q is a magma
+(Q, ·, 1) with identity element 1 such that all left translations Lx : Q → Q, Lx(y) = x · y and
+all right translations Rx : Q → Q, Rx(y) = y · x are permutations of Q.
+We mostly denote the multiplication operation · by juxtaposition and we take advantage
+of · to indicate priority of multiplications in products, e.g., x · yz stands for x · (y · z). The
+implicit division operations will be denoted by x\y = L−1
+x (y) and y/x = R−1
+x (y).
+A mapping f : Q1 → Q2 of loops is a homomorphism if f(xy) = f(x)f(y) for all x, y ∈ Q1.
+Then the identities f(x\y) = f(x)\f(y) and f(x/y) = f(x)/f(y) are automatically satisfied.
+Let Sym(Q) denote the symmetric group on Q.
+The multiplication group of Q is the
+subgroup
+Mlt(Q) = ⟨Lx, Rx : x ∈ Q⟩
+of Sym(Q). The inner mapping group Inn(Q) of Q is the stabilizer of 1 in Mlt(Q). It is well
+known that
+Inn(Q) = ⟨Tx, Rx,y, Lx,y : x, y ∈ Q⟩,
+where
+Tx = R−1
+x Lx,
+Rx,y = R−1
+xy RyRx,
+and
+Lx,y = L−1
+xy LxLy.
+In groups, the inner mapping group is the familiar inner automorphism group. However,
+inner mappings of loops need not be automorphisms.
+A subloop X of a loop Q is normal, denoted by X ⊴ Q, if it is a kernel of a loop homo-
+morphism. It turns out that a subloop X ≤ Q is normal if and only if f(X) = X for every
+f ∈ Inn(Q).
+The left nucleus, middle nucleus and the right nucleus of Q are the respective subloops
+Nucℓ(Q) = {x ∈ Q : x(yz) = (xy)z for all y, z ∈ Q},
+Nucm(Q) = {x ∈ Q : y(xz) = (yx)z for all y, z ∈ Q},
+Nucr(Q) = {x ∈ Q : y(zx) = (yz)x for all y, z ∈ Q}.
+The nucleus Nuc(Q) of Q is the intersection of the above three nuclei. A subloop X ≤ Q is
+nuclear if X ≤ Nuc(Q).
+2
+
+Lemma 1.1. Let X be a normal subloop of a loop Q such that X ≤ Nucm(Q) ∩ Nucr(Q).
+Then for every u ∈ Q the inner mapping Tu restricts to an automorphism of X
+Proof. Since X ⊴ Q, every inner mapping restricts to a permutation of X, and we only need
+to show that Tu(xy) = Tu(x)Tu(y) holds for all x, y ∈ X. Hence we need to verify (u·xy)/u =
+((ux)/u)((uy)/u) for every x, y ∈ X. This is equivalent to u · xy = ((ux)/u)((uy)/u) · u.
+Since y and (uy)/u are elements of X ≤ Nucm(Q) ∩ Nucr(Q), we calculate
+((ux)/u)((uy)/u) · u = ((ux)/u) · ((uy)/u)u = ((ux)/u) · uy
+= ((ux)/u)u · y = (ux)y = u · xy.
+□
+Note that Lemma 1.1 also holds under the dual assumption X ⊴ Q and X ≤ Nucℓ(Q) ∩
+Nucm(Q). See [7, Lemma 1.7] for a slightly stronger statement.
+Corollary 1.2. Let X be a nuclear normal subloop of a loop Q. Then every inner mapping
+of Q restricts to an automorphism of X.
+Proof. Since X ≤ Nucℓ(Q) ∩ Nucr(Q), for every u, v ∈ Q the inner mappings Lu,v, Ru,v
+restrict to the identity mapping on X. We are done by Lemma 1.1.
+□
+The center of Q is the subloop
+Z(Q) = {x ∈ Nuc(Q) : xy = yx for all y ∈ Q}.
+A central subloop of Q is a subloop of Z(Q). Every central subloop of Q is normal in Q.
+A loop Q is power associative if every element of Q generates an associative subloop of
+Q, that is, a subgroup. In particular, the powers xi of elements are well-defined in power
+associative loops, x−1 = x\1 = 1/x, etc. A loop Q is diassociative if any two elements of Q
+generate a subgroup.
+A permutation f of Q is a (left) pseudoautomorphism of Q if there exists c ∈ Q such that
+cf(x) · f(y) = cf(xy)
+for every x, y ∈ Q. The element c is then called a (left) companion of f. The set of all pairs
+(c, f) ∈ Q × Sym(Q), where f is a pseudoautomorphism of Q and c is a companion of f,
+forms a group Psaℓ(Q) under the operations
+(1.1)
+(c, f)(d, g) = (cf(d), fg)
+and
+(c, f)−1 = (f −1(c\1), f −1).
+A permutation f ∈ Sym(Q) is a semiautomorphism of Q if f(1) = 1 and
+f(x · yx) = f(x) · f(y)f(x)
+holds for all x, y ∈ Q. If f is a semiautomorphism of a power associative loop Q then an
+inductive argument shows that f(xi) = f(x)i for every i ∈ Z.
+A triple (f, g, h) of permutations of Q is an autotopism of Q if f(x)g(y) = h(xy) holds for
+all x, y ∈ Q. The autotopisms of Q form a group Atp(Q) under componentwise composition,
+the autotopism group of Q. The following well-known result describes all autotopisms with
+a trivial component.
+Lemma 1.3. Let Q be a loop. Then:
+(i) (idQ, g, h) ∈ Atp(Q) iff g = h = Rx for some x ∈ Nucr(Q).
+(ii) (f, idQ, h) ∈ Atp(Q) iff f = h = Lx for some x ∈ Nucℓ(Q).
+(iii) (f, g, idQ) ∈ Atp(Q) iff f = R−1
+x
+and g = Lx for some x ∈ Nucm(Q).
+3
+
+Corollary 1.4. Let Q be a loop. Then:
+(i) If Nucr(Q) = 1 and (f, g, h) ∈ Atp(Q) then g and h are determined by f.
+(ii) If Nucℓ(Q) = 1 and (f, g, h) ∈ Atp(Q) then f and h are determined by g.
+(iii) If Nucm(Q) = 1 and (f, g, h) ∈ Atp(Q) then f and g are determined by h.
+Proof. Let us prove (i), the other parts being similar. Suppose that Nucr(Q) = 1 and (f, g, h),
+(f, g, h) ∈ Atp(Q). Then (idQ, g−1g, h−1h) = (f, g, h)−1(f, g, h) ∈ Atp(Q). By Lemma 1.3,
+g−1g = h−1h = Rx for some x ∈ Nucr(Q). Since Nucr(Q) = 1, we have Rx = R1 = idQ, and
+g = g, h = h follow.
+□
+1.2. Divisibility in power associative loops. For an integer d > 1 and a power associa-
+tive loop Q, consider the mapping
+(1.2)
+hd : Q → Q,
+x �→ xd.
+In general, injectivity and surjectivity of hd are unrelated properties already in groups. (In
+the group of nonzero complex numbers under multiplication, hd is surjective but not injective.
+In the additive group of integers, hd is injective but not surjective.)
+A power associative loop Q is d-divisible (resp. uniquely d-divisible) if the mapping hd of
+(1.2) is surjective (resp. bijective).
+Lemma 1.5. Let Q be a finite power associative loop, d > 1 an integer and hd as in (1.2).
+The following conditions are equivalent:
+(i) hd is surjective on Q,
+(ii) hd is injective on Q,
+(iii) Q contains no nonidentity element of order dividing d,
+(iv) Q contains no element of prime order dividing d.
+Proof. Thanks to finiteness, (i) and (ii) are equivalent. If Q contains an element x ̸= 1 of
+order dividing d, then hd(x) = xd = 1 = hd(1) and hd is not injective. Hence (ii) implies
+(iii). Clearly, (iii) implies (iv). In fact, (iii) and (iv) are equivalent since if 1 ̸= x ∈ Q is
+such that |x| divides d and p is a prime dividing |x|, then the cyclic group ⟨x⟩ contains an
+element of order p (dividing d). Finally, suppose that (iii) holds, let x ∈ Q and consider the
+cyclic group C = ⟨x⟩. Let k = gcd(|C|, d). If k > 1 then C contains a nonidentity element of
+order k dividing d, a contradiction. Thus gcd(|C|, d) = 1 and hd restricts to a permutation
+of C. In particular, there is y ∈ C such that hd(y) = x, so hd is surjective on Q.
+□
+Given a prime p, we say that a finite power associative loop Q has the Cauchy property for
+p if whenever p divides |Q| then there is x ∈ Q such that |x| = p. A finite power associative
+loop Q is said to have the elementwise Lagrange property if |x| divides |Q| for every x ∈ Q.
+Lemma 1.6. Let Q be a finite power associative loop and let d > 1.
+(i) Suppose that Q has the Cauchy property for every prime p dividing d. If Q is uniquely
+d-divisible then |Q| is coprime to d.
+(ii) Supose that Q has the elementwise Lagrange property. If |Q| is coprime to d then Q
+is uniquely d-divisible.
+Proof. (i) Suppose that Q has the Cauchy property for every prime dividing d, and also
+assume that |Q| is not coprime to d. Let p be any common prime divisor of d and |Q|. By
+assumption, there is x ∈ Q such that |x| = p. By Lemma 1.5, Q is not uniquely d-divisible.
+4
+
+(ii) Suppose that Q has the elementwise Lagrange property, and also assume that Q is
+not uniquely d-divisible. By Lemma 1.5, there is a prime p dividing d and some x ∈ Q such
+that |x| = p. By assumption, |x| divides |Q|, which implies that |Q| is not coprime to d.
+□
+1.3. Moufang loops. A loop Q is Moufang if it satisfies any one of the equivalent Moufang
+identities
+xy · zx = (x · yz)x,
+xy · zx = x(yz · x),
+x(y · zy) = (xy · z)y,
+x(y · xz) = (xy · x)z.
+(1.3)
+We start by summarizing several well-known results for Moufang loops.
+By Moufang Theorem [18, 8], if three elements x, y and z of a Moufang loop associate, that
+is, x(yz) = (xy)z, then the subloop ⟨x, y, z⟩ is a group. Consequently, Moufang loops are
+diassociative, power associative, satisfy the flexible law x(yx) = (xy)x, the inverse properties
+x−1(xy) = y = (yx)x−1, and so on.
+The four nuclei of a Moufang loop Q coincide and form a normal subloop of Q.
+All inner mappings of a Moufang loop can be seen as pseudoautomorphisms, with suitable
+companions. In particular,
+(1.4)
+(x−3, Tx)
+is an element of Psaℓ(Q) in a Moufang loop Q. Moreover, every pseudoautomorphism of a
+Moufang loop is a semiautomorphism.
+We proceed to less familiar results on Moufang loops.
+Lemma 1.7. Let Q be a Moufang loop. Then
+(1.5)
+x−1(xy · z) = yx−1 · xz
+and
+(z · yx)x−1 = zx · x−1y
+for every x, y, z ∈ Q.
+Proof. Note that xy · z = x(yx−1)x · z = x(yx−1 · xz) by diassociativity and the Moufang
+identities (1.3). Multiplying by x−1 on the left then yields the first identity. The second
+identity follows dually.
+□
+Lemma 1.8. Let Q be a Moufang loop, c ∈ Q and f ∈ Sym(Q). Then (c, f) ∈ Psaℓ(Q) if
+and only if
+xc−1 · cy = f(f −1(x)f −1(y))
+for all x, y ∈ Q.
+Proof. Indeed, cf(x)·f(y) = cf(xy) if and only if f(xy) = c−1(cf(x)·f(y)) = f(x)c−1·cf(y),
+by (1.5). We are done upon substituting f −1(x) for x and f −1(y) for y.
+□
+Proposition 1.9. Let Q be a Moufang loop. Then
+xa−3 · a3y = T −1
+a (Ta(x)Ta(y))
+for all a, x, y ∈ Q.
+Proof. We have (a−3, Ta) ∈ Psaℓ(Q) by (1.4).
+By (1.1), (a−3, Ta)−1 = (T −1
+a (a3), T −1
+a ) =
+(a3, T −1
+a ). We are done by Lemma 1.8.
+□
+5
+
+1.4. Lagrange and Cauchy properties for Moufang loops. Finally, we present a few
+results on d-divisible Moufang loops, taking advantage of Lemma 1.6.
+It is not difficult to show that finite Moufang loops have the elementwise Lagrange prop-
+erty:
+Lemma 1.10. Let Q be a finite power associative loop satisfying the right power alternative
+identity (abi)bj = abi+j for every i, j ∈ Z. Then |x| divides |Q| for every x ∈ Q.
+Proof. Let x ∈ Q and X = ⟨x⟩. It suffices to show that the right cosets of X partition Q.
+Suppose that aX ∩bX ̸= ∅. Then axi = bxj for some i, j ∈ Z, therefore a = (bxj)x−i = bxj−i
+by the right power alternative law, and thus aX = (bxj−i)X = {(bxj−i)xk : k ∈ Z} =
+{bxj−i+k : k ∈ Z} = bX.
+□
+In general, Moufang loops do not satisfy the Cauchy property for every prime p. For
+instance, the smallest nonassociative simple Moufang loop of order 120 contains no element
+of order 5 [22] and therefore it violates the Cauchy property for p = 5. But the Cauchy
+property holds in Moufang loops for the primes p = 2 and p = 3:
+Theorem 1.11. Ever finite Moufang loop satisfies the Cauchy property for p = 2 and p = 3.
+Proof. For p = 2, the standard group-theoretic argument works. Let Q be a power associative
+loop of even order. The mapping J : Q → Q, x �→ x−1 is an involution and therefore has
+only orbits of sizes 1 and 2. Since |Q| is even and J(1) = 1, there must be 1 ̸= x ∈ Q such
+that J(x) = x, that is, |x| = 2.
+For p = 3, we will give an elementary argument, but we postpone it until Section 6 so as
+not to distract from the exposition here.
+□
+Remark 1.12. Both Lemma 1.10 and Theorem 1.11 follow from results in the theory of
+Moufang loops whose only known proofs depend on the classification of finite simple Moufang
+loops [19] and hence also on the classification of finite simple groups. Lemma 1.10 is an
+immediate consequence of the Lagrange Theorem for Moufang loops, cf. [12, 15]. For the
+Cauchy property, Grishkov and Zavarnitsine proved in [16] that every Moufang loop of order
+2a3bm with m coprime to 6 contains (Sylow) subloops of orders 2a and 3b. Therefore, if Q
+is a Moufang loop whose order is divisible by p ∈ {2, 3}, it contains a subloop X of order pc
+for some c > 0, then any element 1 ̸= x ∈ X generates a cyclic group ⟨x⟩ of p-power order
+by Lemma 1.10, and the group ⟨x⟩ then certainly contains an element of order p.
+Combining Lemma 1.6, Lemma 1.10 and Theorem 1.11, we get:
+Proposition 1.13. Let Q be a finite Moufang loop and let d = 2a3b > 1. Then Q is uniquely
+d-divisible if and only if |Q| is coprime to d.
+2. Centrality, nilpotency, abelianess and solvability for loops
+In [11], Freese and McKenzie developed commutator theory for congruence modular vari-
+eties. Their commutator of two congruences α, β in an algebra Q will be denoted by [α, β]Q.
+The smallest congruence on Q will be denoted by ⊥Q = {(x, x) : x ∈ Q} and the largest
+congruence on Q will be denoted by ⊤Q = {(x, y) : x, y ∈ Q}.
+The commutator theory of [11] was specialized to the variety of loops in [24]. Although
+we will not need to work with the exact form of the commutator of loop congruences (and
+instead take advantage of Theorem 2.8), we give it here for the sake of completeness. In
+6
+
+[24, Theorem 2.1], the commutator of loop congruences was expressed as the congruence
+generated by certain pairs of evaluated total inner mappings. The technical complication
+with total inner mappings has been recently removed by Barnes who obtained the following
+description of the commutator of loop congruences in her PhD thesis [2]:
+Theorem 2.1 (Barnes). Let α, β be congruences of a loop Q. Then the commutator [α, β]Q
+is the congruence of Q generated by all pairs
+(Tu1(a), Tv1(a)),
+(Lu1,u2(a), Lv1,v2(a)),
+(Ru1,u2(a), Rv1,v2(a)),
+where (1, a) ∈ α and (u1, v1), (u2, v2) ∈ β.
+In loops, there is a one-to-one correspondence between congruences and normal subloops.
+Given a normal subloop X of a loop Q, the congruence αX induced by X is the equivalence
+relation on Q with equivalence classes {aX : x ∈ Q}. Conversely, given a congruence α of a
+loop Q, the normal subloop of Q induced by α is the equivalence class of α containing 1.
+We will therefore write [X, Y ]Q for the commutator of normal subloops X, Y of Q, by
+which we mean the normal subloop of Q induced by the congruence [αX, αY ]Q. Theorem 2.1
+can then be routinely translated to the context of normal subloops:
+Theorem 2.2. Let X, Y be normal subloops of a loop Q. Then the commutator [X, Y ]Q is
+the normal subloop of Q generated by all quotients
+Tu1(a)/Tv1(a),
+Lu1,u2(a)/Lv1,v2(a),
+Ru1,u2(a)/Rv1,v2(a),
+where a ∈ X and u1/v1, u2/v2 ∈ Y .
+The commutator theory of [11] gives rise naturally to theories of central nilpotency and
+solvability. In loops, the central nilpotency theory of [11] coincides with the classical nilpo-
+tency theory adopted from groups. But the solvability theory of [11] is strictly stronger in
+loops than the classical solvability theory adopted from groups. Here are more details:
+2.1. Centrality and central nilpotency. A congruence α of an algebra Q is central if
+[α, ⊤Q]Q = ⊥Q. Passing to normal subloops, a normal subloop X of a loop Q is then said to
+be central if [X, Q]Q = 1. Fortunately, this agrees with the traditional definition of centrality,
+because a normal subloop X of Q satisfies [X, Q]Q = 1 if and only if X ≤ Z(Q).
+Definition 2.3. Given a commutative group (X, +, 0), a loop (F, ·, 1) and a mapping θ :
+F × F → X satisfying θ1,r = 0 = θr,1 for every r ∈ F, the loop defined on F × X by
+(r, x)(s, y) = (rs, x + y + θr,s)
+is a central extension of X by F.
+Theorem 2.4 ([25, Theorem 4.2]). Let X be a normal subloop of a loop Q. Then X is
+central in Q (that is, [X, Q]Q = 1) if and only if Q is isomorphic to a central extension of
+X by Q/X.
+A loop Q is an iterated central extension if it is either an abelian group or there exists a
+central subloop X of Q such that Q/X is an iterated central extension.
+Definition 2.5. A central series for a loop Q is a series
+Q = Q0 ≥ Q1 ≥ · · · ≥ Qn = 1,
+7
+
+such that for every 0 ≤ i < n, Qi+1 is a normal subloop of Q and the factor Qi/Qi+1 is central
+in Q/Qi+1 (that is, [Qi/Qi+1, Q/Qi+1]Q/Qi+1 = 1, or, equivalently, Qi/Qi+1 ≤ Z(Q/Qi+1)).
+A loop Q is (centrally) nilpotent if it contains a central series.
+Theorem 2.6 ([25, Corollary 5.2]). A loop is centrally nilpotent if and only if it is an iterated
+central extension.
+2.2. Abelianess. A congruence α of an algebra Q is abelian if [α, α]Q = ⊥Q. An algebra Q
+is abelian if the congruence ⊤Q is abelian, that is, [⊤Q, ⊤Q]Q = ⊥Q.
+A loop Q is therefore abelian if [Q, Q]Q = 1. It is well-known that a group is abelian if
+and only if it is a commutative group. More generally, a loop is abelian if and only if it is a
+commutative group, cf. [24].
+A conflict arises when the “abelian” terminology is used for a normal subloop X of a loop
+Q, since X can be seen either as a congruence of Q or as a loop in its own right. We will
+therefore be more careful in that context and say that a normal subloop X of a loop Q is
+abelian in Q or that it induces an abelian congruence of Q if [X, X]Q = 1, while we say that
+a normal subloop X of a loop Q is abelian if [X, X]X = 1. Thus, for instance, the phrase
+“X is an abelian normal subloop of Q” means that X is a commutative group and X is a
+normal subloop of Q.
+Every normal subloop X of Q that induces an abelian congruence of Q is an abelian
+normal subloop of Q. The converse is also true in the variety of groups, cf. Lemma 2.9, but
+there are numerous examples of loops Q with an abelian normal subloop X that does not
+induce an abelian congruence of Q. See [24] for examples of order 8.
+Definition 2.7. Given a commutative group (X, +, 0), a loop (F, ·, 1) and mappings ϕ, ψ :
+F × F → Aut(X) and θ : F × F → X, the loop defined on F × X by
+(r, x)(s, y) = (rs, ϕr,s(x) + ψr,s(y) + θr,s)
+is an abelian extension of X by F if ϕr,1 = idX = ψ1,r and θ1,r = 0 = θr,1 for every r ∈ F.
+Central extensions are therefore those abelian extensions in which the automorphisms ϕr,s
+and ψr,s are trivial.
+Theorem 2.8 ([25, Theorem 4.1]). Let X be a normal subloop of a loop Q. Then X is
+abelian in Q (that is, [X, X]Q = 1) if and only if Q is isomorphic to an abelian extension of
+X by Q/X.
+The above external description of abelian extensions can be rewritten internally as follows.
+Let (X, ·, 1) be an abelian normal subloop of a loop (Q, ·, 1). Let U be a (left) transversal
+to X in Q such that 1 ∈ U. Then Q is an abelian extension of X by Q/X if there exist ϕ,
+ψ : U × U → Aut(X) and θ : U × U → X satisfying ϕr,1 = idX = ψ1,r and θ1,r = 1 = θr,1 for
+every r ∈ U, and
+(2.1)
+rx · sy = ur,s · ϕr,s(x)ψr,s(y)θr,s
+holds for every r, s ∈ U and x, y ∈ X, where ur,s is the unique element of U ∩ (rs)X.
+There is a substantial difference between abelian normal subloops of Q and normal subloops
+of Q that are abelian in Q. This can be illustrated by considering the multiplication table
+of Q.
+Let X be a commutative group.
+For X to be an abelian normal subloop of Q,
+nothing else is required but that Q is a disjoint union of {uX : u ∈ U} for some subset
+8
+
+U ⊆ Q, the multiplication table of X is reproduced in the subsquare X × X, and for every
+r, s ∈ U the subsquare rX × sX is a latin square with entries running through ur,sX, where
+ur,s ∈ U ∩(rs)X. However, for X to induce an abelian congruence of Q, the structure of the
+subsquare rX × sX must be much more rigid, conforming to (2.1).
+Using the notion of abelian extension, it is easy to show that every abelian normal subgroup
+of a group Q is abelian in Q. Here is a more general result:
+Lemma 2.9. Let Q be a loop and let X be an abelian normal subloop of Q such that X ≤
+Nucm(Q) ∩ Nucr(Q). Then X induces an abelian congruence of Q.
+Proof. By Theorem 2.8, it suffices to show that Q is an abelian extension of X by Q/X.
+Let U be a transversal to X in Q. For r, s ∈ U, let ϕr,s be the restriction of T −1
+s
+to X, let
+ψr,s = idX ∈ Aut(X), let ur,s ∈ U ∩ (rs)X and let θr,s = ur,s\(rs) ∈ X. By Lemma 1.1,
+ϕr,s ∈ Aut(X).
+Let x, y ∈ X. Note that sT −1
+s (x) = s(s\(xs)) = xs. Using the fact that x, y, T −1
+s (x) ∈
+X ≤ Nucm(Q) ∩ Nucr(Q), we have
+rx · sy = r(x · sy) = r(xs · y) = r(sT −1
+s (x) · y) = r(s · T −1
+s (x)y) = rs · T −1
+s (x)y
+= ur,sθr,s · T −1
+s (x)y = ur,s · θr,s(T −1
+s (x)y) = utr,s · T −1
+s (x)yθr,s,
+where the last step follows from the fact that θr,s, T −1
+s (x) and y lie in the abelian group
+X. We have obtained an instance of (2.1), proving that Q an abelian extension of X by
+Q/X.
+□
+2.3. Classical solvability and congruence solvability. The history of the notion of
+solvability in loop theory is convoluted.
+Albert defined solvable loops in [1, p. 412] as loops whose composition factors have no
+notrivial subloops, mimicking a result from groups that states that a finite group is solvable
+if and only if each of its composition factors is isomorphic to a group of prime order. Albert’s
+definition of solvability has been abandoned.
+Bruck introduced the notion of a derived subloop in [3, p. 268]. The derived subloop Q′
+of a loop Q is the smallest normal subloop H of Q such that Q/H is a commutative group.
+The derived series of Q is then the (possibly infinite) series
+Q = Q0 ≥ Q1 ≥ · · · ≥ Qn ≥ · · ·
+such that for every i ≥ 0, Qi+1 is the derived subloop of Qi. Bruck then defines solvable
+loops as loops whose derived series reaches the trivial subloop 1 in finitely many steps.
+Another definition of solvability for loops was given by Glauberman. A loop Q is said to
+be solvable in [14, p. 397] if there exists a series
+Q = Q0 ≥ Q1 ≥ · · · ≥ Qn = 1
+such that for every 0 ≤ i < n, Qi+1 is a normal subloop of Qi and the factor Qi/Qi+1 is a
+commutative group. By adopting the standard proof from group theory, it is not difficult to
+show that Bruck’s and Glauberman’s definitions of solvability for loops are equivalent. In
+particular, Glauberman’s definition of solvability will not be affected if we demand that for
+every 0 ≤ i < n, Qi+1 is a normal subloop of Q, not just a normal subloop of Qi.
+The commutator theory of Freese and McKenzie [11] naturally leads to a definition of
+solvability in congruence modular varieties.
+In loops, their concept of solvability, called
+9
+
+congruence solvability in Definition 2.10, is strictly stronger than the equivalent solvabil-
+ity concepts of Bruck and Glauberman, called classical solvability in Definition 2.10. The
+terminology comes from [25].
+Definition 2.10. A classically solvable series for a loop Q is a series
+Q = Q0 ≥ Q1 ≥ · · · ≥ Qn = 1,
+such that for every 0 ≤ i < n, Qi+1 is a normal subloop of Q and the factor Qi/Qi+1 is
+abelian (that is, a commutative group). A loop Q is classically solvable if it contains a
+classically solvable series.
+A congruence solvable series for a loop Q is a series
+Q = Q0 ≥ Q1 ≥ · · · ≥ Qn = 1
+such that for every 0 ≤ i < n, Qi+1 is a normal subloop of Q and the factor Qi/Qi+1 is
+abelian in Q/Qi+1 (that is, Qi/Qi+1 induces an abelian congruence of Q/Qi+1). A loop Q is
+congruence solvable if it contains a congruence solvable series.
+Obviously, every congruence solvable loop is classically solvable. The converse is true for
+groups but not for loops, with small counterexamples easy to construct, cf. [11, 24]. The
+following problem is open:
+Problem 2.11. Is every classically solvable Moufang loop congruence solvable?
+Towards a solution of Problem 2.11, we prove in Sections 4 and 5 that if Q is a 6-divisible
+classically solvable Moufang loop or a Moufang loop of odd order, then Q is congruence
+solvable.
+Since every central series is a congruence solvable series, we have:
+Theorem 2.12. Centrally nilpotent loops are congruence solvable and classically solvable.
+Call a loop Q an iterated abelian extension if either Q is a commutative group, or Q is
+an abelian extension of a commutative group X by some loop that is an iterated abelian
+extension. It was shown in [25, Corollary 5.1] that iterated abelian extensions of loops are
+precisely congruence solvable loops. For the sake of completeness, let us give a short proof
+here:
+Proposition 2.13. A loop is congruence solvable if and only if it is an iterated abelian
+extension.
+Proof. Let Q = Q0 > Q1 > · · · > Qn = 1 be a congruence solvable series for Q. We prove
+by induction on n that Q is an iterated abelian extension. If n = 1 then the series becomes
+Q > 1 and Q is a commutative group, hence a congruence solvable loop. If n > 1, let
+X = Qn−1 ⊴ Q and note that Q > X > 1. Since X and 1 are adjacent terms in the original
+series, X/1 = X is abelian in Q/1 = Q. By Theorem 2.8, Q is an abelian extension of X by
+Q/X. It remains to show that Q/X is an iterated abelian extension. Consider the series
+(2.2)
+Q/X = Q0/X > Q1/X > · · · > Qn−1/X = X/X.
+Since Qi ⊴Q, we have Qi/X ⊴Q/X by the Correspondence Theorem. As Qi/Qi+1 is abelian
+in Q/Qi+1, the factor loop (Qi/X)/(Qi+1/X) ∼= Qi/Qi+1 is abelian in (Q/X)/(Qi+1/X) ∼=
+Q/Qi+1. Hence (2.2) is a congruence solvable series, Q/X is congruence solvable and, by the
+induction assumption, Q/X is an iterated abelian extension.
+10
+
+Conversely, suppose that Q is an iterated abelian extension constructed in n steps, each
+an abelian extension. We prove by induction on n that Q is congruence solvable. If n = 0
+then Q is a commutative group and we are done. Else let Q be an abelian extension of a
+commutative group X by Q/X, where Q/X is constructed by n − 1 abelian extensions. By
+Theorem 2.8, X is abelian in Q. By the induction assumption, Q/X is congruence solvable.
+Let Q/X = Q0/X > · · · > Qm/X = X/X be a congruence solvable series. Consider the
+series
+(2.3)
+Q = Q0 > Q1 > · · · > Qm−1 > Qm = X > 1.
+Since Qi/X ⊴ Q/X, we have Qi ⊴ Q. Moreover, Qi/Qi+1 ∼= (Qi/X)/(Qi+1/X) is abelian
+in (Q/X)/(Qi+1/X) ∼= Q/Qi+1 for every i < m. The last inclusion in (2.3) is X > 1, and
+certainly X/1 = X is abelian in Q = Q/1 as we have already shown. Therefore (2.3) is a
+congruence solvable series and Q is congruence solvable.
+□
+3. Moufang loops with an abelian normal subgroup that does not induce
+an abelian congruence
+By inspecting the library of small Moufang loops available in the GAP [13] package LOOPS
+[20], it was observed in [24] that there exists a Moufang loop Q of order 16 with an abelian
+normal subloop X (isomorphic to C2×C4) such that X does not induce an abelian congruence
+of Q.
+In this section we offer a general construction of Moufang loops Q containing an abelian
+normal subloop X that does not induce an abelian congruence of Q.
+Recall that for a vector space V over a field F, a mapping q : V → F is a quadratic
+form if q(λu) = λ2u for every λ ∈ F, u ∈ V , and if h : V × V → F defined by h(u, v) =
+q(u + v) − q(u) − q(v) is a bilinear form. The form h is referred to as the associated bilinear
+form.
+Also recall that a loop Q is an extra loop if it satisfies the identity x(y · zx) = (xy · z)x.
+Fenyves proved in [10] that extra loops are Moufang.
+Proposition 3.1. Let W = (W, +) be a commutative group with subgroups F ≤ B ≤ W.
+Suppose that F = {0, 1} and W = W/B is an elementary abelian 2-group. Let q : W → F
+be a quadratic form with associated bilinear form h : W × W → F. Let q : W → F and h :
+W × W → F be defined by q(u) = q(u) and h(u, v) = h(u, v). Denote by Q = Q(F, B, W, q)
+the magma defined on F × W by
+(3.1)
+(i, u) · (j, v) = (i + j, u + v + jq(u) + ih(u, v)).
+Then:
+(i) Q is a centrally nilpotent loop, a central extension of the commutative group B by the
+elementary abelian 2-group F × W,
+(ii) Q is congruence solvable and hence classically solvable,
+(iii) Q is an extra loop,
+(iv) Q is a group if and only if the quadratic form q is linear,
+(v) X = 0 × W is an abelian normal subloop of Q,
+(vi) if Q is not a group, then the congruence of Q induced by X is not abelian.
+11
+
+Proof. (i) The commutative group W is (isomorphic to) a central extension of B by W, with
+multiplication on W × B given by
+(u, a)(v, b) = (u + v, a + b + θu,v),
+where θ : W × W → B is some group cocycle. The multiplication formula (3.1) on F × W
+can then be seen as a multiplication on F × W × B, namely
+(i, u, a)(j, v, b) = (i + j, u + v, a + b + θu,v + jq(u) + ih(u, v)).
+Hence Q is isomorphic to the magma defined on (F × W) × B by
+((i, u), a)((j, v), b) = ((i, u) + (j, v), a + b + θ(i,u),(j,v)),
+where θ(i,u),(j,v) = θu,v + jq(u) + ih(u, v). Since θ(0,0),(j,v) = θ0,v + jq(0) = 0 and θ(i,u),(0,0) =
+θu,0 + ih(u, 0) = 0, we see at once that θ is a loop cocycle and Q is a central extension of
+the commutative group B by the elementary abelian 2-group F ×W. Hence Q is a centrally
+nilpotent loop, finishing the proof of part (i). Part (ii) then follows from Theorem 2.12.
+(iii) Chein an Robinson characterized extra loops as Moufang loops in which squares
+are in the nucleus [5].
+In the isomorphic copy of Q from part (i) we have ((i, u), a)2 =
+((0, 0), 2a+θ(i,u),(i,u)) ∈ 0×B ≤ Z(Q) ≤ Nuc(Q). To prove that Q is extra, it therefore suffices
+to check one of the Moufang identities, say (xy·x)z = x(y·xz). By definition, q(b+v) = q(v)
+and h(b+v, w) = h(v, w) for every b ∈ B and v, w ∈ W. Note that 2W ⊆ B since W = W/B
+is an elementary abelian 2-group. Finally observe that h(u, u) = h(u, u) = q(0) + 2q(u) = 0
+and h(u, u + v) = h(u, u) + h(u, v) = h(u, v) for every u, v ∈ Q. Let x = (i, u), y = (j, v) and
+z = (k, w) ∈ F × W. We then calculate
+((i, u)(j, v) · (i, u))(k, w)
+= (i + j, u + v + jq(u) + ih(u, v))(i, u) · (k, w)
+= (j, 2u + v + jq(u) + ih(u, v) + iq(u + v) + (i + j)h(v, u)) · (k, w)
+= (j, 2u + v + jq(u) + iq(u + v) + jh(u, v)) · (k, w)
+= (j + k, 2u + v + w, jq(u) + iq(u + v) + jh(u, v) + kq(v) + jh(v, w)).
+On the other hand
+(i, u)((j, v) · (i, u)(k, w))
+= (i, u) · (j, v)(i + k, u + w + kq(u) + ih(u, w))
+= (i, u) · (i + j + k, u + v + w + kq(u) + ih(u, w) + (i + k)q(v) + jh(v, u + w))
+= (j + k, 2u+v+w+kq(u)+ih(u, w)+(i+k)q(v)+jh(v, u+w)+(i+j+k)q(u)+ih(u, v+w))
+= (j + k, 2u + v + w + (i + j)q(u) + ih(u, v) + (i + k)q(v) + jh(v, u + w)).
+These two products agree in the first coordinate. Upon canceling like terms in the second
+coordinate (while taking advantage of bilinearity of h), only iq(u + v) remains in the first
+product, while iq(u) + ih(u, v) + iq(v) remains in the second product.
+Since h(u, v) =
+q(u + v) + q(u) + q(v), we are done.
+(iv) Suppose that q is linear. Then h = 0 and the cocycle of (i) reduces to θ(i,u),(j,v) =
+θu,v + jq(u). Since θ is a group cocycle, it satisfies the group cocycle identity
+θu,v + θu+v,w = θv,w + θu,v+w.
+12
+
+We then have
+θ(i,u),(j,v) + θ(i,u)+(j,v),(k,w) = θu,v + θu+v,w + jq(u) + kq(u + v)
+= θv,w + θu,v+w + kq(v) + (j + k)q(u)
+= θ(j,v),(k,w) + θ(i,u),(j,v)+(k,w),
+which is a group cocycle identity for θ, so Q is a group.
+Conversely, if Q is a group then for all u, v ∈ W the product
+(1, 0)(0, u) · (1, v) = (1, u)(1, v) = (0, u + v + q(u) + h(u, v))
+is equal to
+(1, 0) · (0, u)(1, v) = (1, 0)(1, u + v + q(u)) = (0, u + v + q(u)),
+so h = 0 and q is linear.
+(v) On X = 0 × W, the multiplication formula (3.1) reduces to
+(0, u)(0, v) = (0, u + v),
+and X is therefore isomorphic to the commutative group W. The mapping f : Q → F,
+(i, u) �→ i is clearly a homomorphism with kernel equal to X, which shows that X is a
+normal subloop of Q.
+(vi) Suppose that X induces an abelian congruence of Q. By Theorem 2.8, Q is an abelian
+extension of X = W by F, and there exist ϕ, ψ : F × F → Aut(X) and θ : F × F → X such
+that ϕi,0 = ψ0,i = idX, θi,0 = θ0,i = 0 and
+(i, u)(j, v) = (i + j, ϕi,j(u) + ψi,j(v) + θi,j).
+Since (0, u)(1, v) = (1, u+v+q(u)) by (3.1) and ϕ0,1(u)+ψ0,1(v)+θ0,1 = ϕ0,1(u)+v, we must
+have ϕ0,1(u) = u + q(u). As ϕ0,1 is an automorphism of X, we deduce (u + v) + q(u + v) =
+(u + q(u)) + (v + q(v)) for all u, v ∈ W. This shows that q is linear and Q is a group by
+(iii).
+□
+The nonassociative loops Q(F, B, W, q) afforded by Proposition 3.1 demonstrate that it
+is possible for a Moufang loop to be congruence solvable (even centrally nilpotent) and yet
+posses an abelian normal subloop that does not induce an abelian congruence.
+Example 3.2. Smallest examples of interest are obtained from Proposition 3.1 when q is
+a nonlinear quadratic form on a vector space W/B of dimension two, which forces (up to
+isomorphism) either F = B = 0 × 0 × C2 ≤ W = C2 × C2 × C2 or F = B = 0 × C2 ≤ W =
+C2 × C4. We then obtain a Moufang loop Q of order 16 with an abelian normal subloop X
+(isomorphic to C2 × C2 × C2 or to C2 × C4) that does not induce an abelian congruence of
+Q. This covers the example that was found in [24] by an exhaustive search.
+4. Moufang loops in which classical solvability and congruence solvability
+coincide
+We start with the following easy fact:
+Lemma 4.1. Let X be a 2-divisible commutative group. Then every semiautomorphism of
+X is an automorphism of X.
+13
+
+Proof. Let f be a semiautomorphism of X. Recall that f(xn) = f(x)n for every x ∈ X
+and n ∈ Z.
+Let x, y ∈ X.
+By 2-divisibility, there is u ∈ X such that x = u2.
+Then
+f(xy) = f(u2y) = f(uyu) = f(u)f(y)f(u) = f(u)2f(y) = f(u2)f(y) = f(x)f(y).
+□
+Lemma 4.2. Let Q be a Moufang loop and X a 2-divisible abelian normal subgroup of Q.
+Then every inner mapping of Q restricts to an automorphism of X.
+Proof. Let f ∈ Inn(Q). By the introductory remarks in Subsection 1.3, f is a pseudoauto-
+morphism of Q and hence a semiautomorphism of Q. By Lemma 4.1, the restriction f|X of
+f to X is an automorphism of X.
+□
+Theorem 4.3. Let Q be a 3-divisible Moufang loop and let X be a 2-divisible normal subgroup
+of Q. Then the congruence {aX : a ∈ Q} on Q induced by X is an abelian congruence of Q.
+Proof. Let U be a transversal to X containing 1, let r, s ∈ U and x, y ∈ X. There are
+uniquely determined u = ur,s ∈ U and z ∈ X such that rs = uz. We wish to apply Theorem
+2.8 and hence to find ϕr,s, ψr,s ∈ Aut(X) and θr,s ∈ X such that rx·sy = u·ϕr,s(x)ψr,s(y)θr,s
+as in (2.1). In addition, we must verify ϕr,1 = idX = ψ1,r and θ1,r = θr,1 = 1. We will build
+the automorphisms in several steps. Consider
+f1 = (Ts)|X.
+By Lemma 4.2, f1 ∈ Aut(X). We have f1(y)s = Ts(y)s = sys−1s = sy, so
+rx · sy = rx · f1(y)s.
+Let
+f2 = (L−1
+s−1rL−1
+s Lr)|X.
+Since f2(1) = (s−1r)−1(s−1r) = 1, f2 is a restriction of an inner mapping of Q to X. By
+Lemma 4.2, f2 ∈ Aut(X). Moreover, we have s · (s−1r)f2(x) = rx. Therefore
+rx · sy = rx · f1(y)s = (s · (s−1r)f2(x))(f1(y)s) = s((s−1r)f2(x) · f1(y))s,
+where we have used a Moufang identity (1.3) in the last step. By the identity (1.5), we have
+uv · w = u(u−1 · (uv · w)) = u(vu−1 · uw) for any u, v, w ∈ Q. In particular, with u = s−1r,
+v = f2(x) ∈ X and w = f1(y) ∈ X, we obtain
+rx · sy = s(uv · w)s = s(u(vu−1 · uw))s = su · (vu−1 · uw)s,
+where we have again used a Moufang identity in the last step. Since Q is 3-divisible, there
+is a ∈ Q such that u = a3. Proposition 1.9 then yields
+vu−1 · uw = va−3 · a3w = T −1
+a (Ta(v)Ta(w)) = vw,
+where in the last step we used Ta|X ∈ Aut(X), by Lemma 4.2. So far we showed
+rx · sy = su · (vw)s = s(s−1r) · (vw)s = r · (vw)s = r · (f2(x)f1(y))s.
+Now, sf −1
+1 (x0) = ss−1x0s = x0s for any x0 ∈ Q and thus
+rx · sy = r · (f2(x)f1(y))s = r · sf −1
+1 (f2(x)f1(y)) = r · s(f −1
+1 f2(x) · y),
+taking advantage of f1 ∈ Aut(X). Consider
+f3 = (L−1
+rs LrLs)|X
+14
+
+and note that f3(1) = (rs)−1(rs) = 1, which implies f3 ∈ Aut(X) as usual. Moreover,
+rs · f3(x0) = r · sx0 for any x0 ∈ Q, and hence
+rx · sy = r · s(f −1
+1 f2(x) · y) = rs · f3(f −1
+1 f2(x) · y) = rs · (f3f −1
+1 f2(x) · f3(y)),
+using f3 ∈ Aut(X). Recall that rs = uz with u ∈ U, z ∈ X, and consider
+f4 = (L−1
+z L−1
+u Lrs)|X.
+We have f4(1) = z−1(u−1(rs)) = 1 (since rs = uz) and f4 ∈ Aut(X). Moreover, u·zf4(x0) =
+rs · x0 for any x0 ∈ Q, and thus
+rx · sy=rs · (f3f −1
+1 f2(x) · f3(y))=u · zf4(f3f −1
+1 f2(x) · f3(y))=u · f4f3f −1
+1 f2(x)f4f3(y)z,
+where we have used f4 ∈ Aut(X) and commutativity of the group X in the last step.
+Rewriting, we have
+rx · sy = u · ϕr,s(x)ψr,s(y)θr,s,
+where ϕr,s = f4f3f −1
+1 f2 ∈ Aut(X), ψr,s = f4f3 ∈ Aut(X) and θr,s = z = u−1(rs) ∈ X.
+If s = 1, we observe f1 = (T1)|X = idX, f2 = (L−1
+r L−1
+1 Lr)|X = idX, f3 = (L−1
+r LrL1)|X =
+idX, u ∈ (r1)X ∩ U = {r}, θr,1 = z = 1, f4 = (L−1
+1 L−1
+r Lr)|X = idX and therefore ϕr,1 = id.
+If r = 1, we observe f3 = (L−1
+s L1Ls)|X = idX, u ∈ (1s)X ∩ U = {s}, θ1,s = z = 1,
+f4 = (L−1
+1 L−1
+s Ls)|X = idX and therefore ψ1,s = idX.
+□
+Recall the power mapping hd of (1.2) and note that h6 = h3h2 = h2h3 in a power associative
+loop. Hence a power associative loop is 6-divisible if and only if it is 2-divisible and 3-divisible.
+We therefore deduce from Theorem 4.3:
+Corollary 4.4. Let Q be a 6-divisible Moufang loop and let X be an abelian normal subloop
+of Q. Then X induces an abelian congruence of Q.
+Since in the two definitions of solvability (Definition 2.10), the only difference is whether
+the quotients Qi/Qi+1 are merely commutative groups or whether they induce an abelian
+congruence, we have:
+Corollary 4.5. Let Q be a 6-divisible Moufang loop. Then Q is solvable if and only if it is
+congruence solvable.
+5. Moufang loops of odd order are congruence solvable
+We proceed to show that Moufang loops of odd order are congruence solvable.
+This
+strengthens Glauberman’s Odd Order Theorem for Moufang loops [14, Theorem 16]:
+Theorem 5.1 (Glauberman). A Moufang loop of odd order is classically solvable.
+Our proof is based on the recent, deep result of Cs¨org˝o [6, Theorem 5.1]:
+Theorem 5.2 (Cs¨org˝o). A nontrivial Moufang loop of odd order has a nontrivial nucleus.
+A subloop X of a loop Q is characteristic if f(X) = X for every f ∈ Aut(Q). Unlike in
+groups, a characteristic subloop of Q is not necessarily a normal subloop of Q. It is clear
+from the definition of the nucleus that Nuc(Q) is a characteristic subloop of Q.
+Theorem 5.3. A Moufang loop of odd order is congruence solvable.
+15
+
+Proof. We proceed by induction on the order of the Moufang loop Q. There is nothing to
+prove when Q = 1, so suppose that Q is nontrivial. In every Moufang loop, the nucleus is a
+normal subloop. By Theorem 5.2, 1 < N = Nuc(Q) ⊴ Q. Since N is a group of odd order,
+it is solvable by the Odd Order Theorem for groups [9].
+Let X be a minimal characteristic subgroup of N. Let f ∈ Inn(Q). By Corollary 1.2, f
+restricts to an automorphism of N. Since X is characteristic in N, f(X) = X. Hence X ⊴Q.
+A variation on a standard group-theoretic argument, cf. [23, Theorem 5.24], now shows that
+X is an abelian group. (Consider the derived subgroup X′ of the solvable group X.)
+Altogether, we have shown that X is an abelian normal subgroup of Q and 1 < X ≤
+Nuc(Q). By Lemma 2.9, X induces an abelian congruence of Q. By Theorem 2.8, Q is an
+abelian extension of X by Q/X. By the induction assumption, Q/X is congruence solvable.
+By Proposition 2.13, Q/X is an iterated abelian extension, hence Q is an iterated abelian
+extension, and Q is congruence solvable by Proposition 2.13 again.
+□
+Note that in the finite case, Theorem 5.3 implies Corollary 4.5. Indeed, if Q is a finite
+6-divisible Moufang loop then it is of odd order by Proposition 1.13, congruence solvable by
+Theorem 5.3 and thus also classically solvable.
+6. The Cauchy property for p = 3 in Moufang loops
+Here we give an elementary proof of the Cauchy property for p = 3 in Moufang loops. We
+start with a lemma motivated by triality for Moufang loops, cf. [17]. Let
+Com(Q) = {x ∈ Q : xy = yx for all y ∈ Q}
+be the commutant of Q.
+In addition to the already introduced bijections Lx, Rx and Tx = R−1
+x Lx, consider also
+Mx = RxLx. In diassociative loops, we have Ln
+x = Lxn, Mx = LxRx, etc.
+Lemma 6.1. Let Q be a diassociative loop. Then a mapping
+σ : {Lx, Rx, L−1
+x , R−1
+x
+: x ∈ Q} → Mlt(Q)
+is well-defined by σ(Lx) = Rx, σ(L−1
+x ) = R−1
+x , σ(Rx) = M−1
+x
+and σ(R−1
+x ) = Mx if and only
+if x3 = 1 for every x ∈ Com(Q).
+Proof. Note that if Lx ∈ {Ly, Ry, L−1
+y , R−1
+y } then x ∈ {y, y−1}, and similarly for Rx. To check
+that σ is well-defined, we therefore need to establish the following implications: (a) Lx = L−1
+x
+implies Rx = R−1
+x , (b) Rx = R−1
+x
+implies M−1
+x
+= Mx, (c) Lx = Rx implies Rx = M−1
+x , (d)
+Lx = R−1
+x
+implies Rx = Mx, (e) L−1
+x
+= Rx implies R−1
+x
+= M−1
+x , and (f) L−1
+x
+= R−1
+x
+implies
+R−1
+x
+= Mx.
+The implication (a) is immediate, for if Lx = L−1
+x
+then x = x−1 and Rx = R−1
+x . The
+argument for (b) is similar. The implications (c) and (f) are equivalent, and so are the
+implications (d) and (e).
+Concerning (c), suppose that Lx = Rx, that is, x ∈ Com(Q). We will show that then
+Rx = M−1
+x
+iff x3 = 1. Certainly if Rx = M−1
+x
+then evaluating at 1 yields x = x−2, that is,
+x3 = 1. Conversely, if x3 = 1 then R3
+x = idQ and Rx = R−2
+x
+= R−1
+x L−1
+x
+= M−1
+x .
+Finally, for (d), suppose that Lx = R−1
+x , that is, x2 = 1 and x ∈ Com(Q). We will again
+show that Rx = Mx iff x3 = 1. Certainly if Rx = Mx then Rx = RxLx, Lx = idQ, x = 1 and
+x3 = 1. Conversely, if x3 = 1 then from x2 = 1 we deduce x = 1 and Rx = Mx.
+□
+16
+
+Corollary 6.2. If Q is a Moufang loop with trivial nucleus then the mapping σ of Lemma
+6.1 is well-defined.
+Proof. Let x ∈ Com(Q). Then Tx = idQ, so in particular Tx is an automorphism of Q. Since
+(x−3, Tx) ∈ Psaℓ(Q) by (1.4), we have x−3Tx(y) · Tx(z) = x−3Tx(yz) = x−3(Tx(y)Tx(z)) for
+all y, z ∈ Q. Therefore x−3 ∈ Nuc(Q) = 1.
+□
+Lemma 6.3. Let Q be a Moufang loop with trivial nucleus. Then there exists a unique σ ∈
+Aut(Mlt(Q)) such that σ(Lx) = Rx and σ(Rx) = M−1
+x , for every x ∈ Q. This automorphism
+satisfies σ3 = idMlt(Q), and if ϕ is an inner mapping of Q with companion c (when seen as a
+pseudoautomorphism) then σ(ϕ) = R−1
+c ϕ.
+Proof. The maping σ of Lemma 6.1 is well-defned by Corollary 6.2. Our first goal is to show
+that it is possible to extend it into an automorphism of Mlt(Q). For that it suffices to verify
+that
+(6.1)
+ψ1 · · · ψk = idQ
+⇔
+σ(ψ1) · · ·σ(ψk) = idQ
+whenever each ψi, 1 ≤ i ≤ k, belongs to {L±1
+x , R±1
+x
+: x ∈ Q}.
+The first Moufang identity of (1.3) can be rewritten as (Lx, Rx, Mx) ∈ Atp(Q). Substi-
+tuting yx−1 for y in the second identity of (1.5) yields (zy)x−1 = zx · x−1(yx−1), which says
+(Rx, M−1
+x , R−1
+x ) ∈ Atp(Q). Taking inverses into consideration, we see that all four triples
+(6.2)
+(Lx, Rx, Mx), (L−1
+x , R−1
+x , M−1
+x ), (Rx, M−1
+x , R−1
+x ) and (R−1
+x , Mx, Rx),
+are autotopisms of Q, for every x ∈ Q.
+Now, autotopisms may be chosen from this list in such a way that the first coordinate is
+equal to ψi and the second coordinate is equal to σ(ψi), 1 ≤ i ≤ k. By composing these
+autotopisms we obtain an autotopism in which the first coordinate is equal to ψ1 · · · ψk and
+the second coordinate is equal to σ(ψ1) · · ·σ(ψk). Since all the (equal) nuclei of Q are trivial,
+Corollary 1.4 implies that the first coordinate is trivial if and only if the second coordinate
+is trivial. We proved σ ∈ Aut(Mlt(Q)).
+Let us establish σ3 = idMlt(Q).
+We have σ(Lx) = Rx, σ(Rx) = M−1
+x
+= R−1
+x L−1
+x
+and
+σ(M−1
+x ) = σ(R−1
+x )σ(L−1
+x ) = MxR−1
+x
+= LxRxR−1
+x
+= Lx. This shows that the automorphism
+σ3 is identical on a generating set of Mlt(Q), hence also on Mlt(Q).
+Finally, let ϕ ∈ Inn(Q). Then ϕ is a pseudoautomorphism with some companion c ∈ Q.
+This can also be expressed as (Lcϕ, ϕ, Lcϕ) ∈ Atp(Q). Let us compose the autotopisms
+of (6.2) so that the first coordinate of the resulting autotopism (f, g, h) is equal to Lcϕ.
+Certainly g = σ(f). By Corollary 1.4, (f, g, h) = (Lcϕ, ϕ, Lcϕ). Thus Rcσ(ϕ) = σ(Lc)σ(ϕ) =
+σ(Lcϕ) = σ(f) = g = ϕ and σ(ϕ) = R−1
+c ϕ follows.
+□
+Proposition 6.4. Let Q be a finite Moufang loop of order divisible by three. If the nucleus
+of Q is trivial, then Q possesses a proper subloop of order divisible by three.
+Proof. Let σ be as in Lemma 6.3, H = ⟨σ⟩, and let G = Mlt(Q) ⋉ H be the semidirect
+product defined by the natural action of H on Mlt(Q). Hence λσi · ρσj = λσi(ρ) · σi+j for
+all λ, ρ ∈ Mlt(Q) and i, j ∈ Z.
+By Lemma 6.3, σ3 = idMlt(Q). Should we have σ = idMlt(Q), then Rx = σ(Rx) = Mx =
+RxLx would imply Lx = idQ, x = 1 and Q = 1, a contradiction. Hence |H| = 3. Consider
+a 3-Sylow subgroup S of G that contains H. Let P = Mlt(Q) ∩ S. As S contains H, the
+underlying set of S is equal to P ×H. Since | Mlt(Q)| = |Q|·| Inn(Q)| in any loop, and since
+17
+
+3 divides |Q| here, it follows that 3 divides | Mlt(Q)|. Then |S| > 3 and P > 1. As Mlt(Q)
+is normal in G, the group P = Mlt(Q) ∩ S is normal in S. The center Z(P) ̸= 1 of P is also
+normal in S, being a characteristic subgroup of P. The automorphism σ acts by conjugation
+on Z(P). Since σ3 = idMlt(Q) and Z(P) ̸= 1 is a 3-group, the conjugation by σ also fixes
+some 1 ̸= ψ ∈ Z(P). Passing to a suitable power of ψ, we can assume that |ψ| = 3. Note
+that ψσ = σψ = σ(ψ)σ implies σ(ψ) = ψ.
+Write ψ = Lxϕ for some x ∈ Q and ϕ ∈ Inn(Q). Let c be the companion of ϕ. By
+Lemma 6.3, Lxϕ = ψ = σ(ψ) = σ(Lxϕ) = σ(Lx)σ(ϕ) = RxR−1
+c ϕ, hence Lx = RxR−1
+c ,
+c = 1 (so ϕ is an automorphism), Lx = Rx and x ∈ Com(Q). By Corollary 6.2 and Lemma
+6.1, x3 = 1. If x ̸= 1 then ⟨x⟩ is the sought-after subloop. Else x = 1 and ψ = ϕ is an
+automorphism of order 3. Since 3 divides |Q|, it then also divides the order of the proper
+subloop Fix(ϕ) = {u ∈ Q : ϕ(u) = u}.
+□
+We are ready to prove the Cauchy property for p = 3 in Moufang loops. Let Q be a
+finite Moufang loop whose order is divisible by 3. We proceed by induction on |Q|. Let
+N = Nuc(Q). If N = 1 then Proposition 6.4 yields a proper subloop of Q whose order is
+divisible by 3, and we are done by the induction assumption. Suppose from now on that
+N ̸= 1. If 3 divides |N| then N contains an element of order 3 since N is a group. Else 3
+divides |Q/N|, and by the induction assumption there is x ∈ Q such that xN is of order 3
+in Q/N. Since xN is the homomorphic image of x under the natural projection modulo N,
+the order of xN divides the order of x. A suitable power of x is then of order 3.
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+Programming,
+Version
+4.4.12;
+2008,
+http://www.gap-system.org.
+[14] G. Glauberman, On loops of odd order. II., J. Algebra 8 (1968), 393–414.
+[15] A.N. Grishkov and A.V. Zavarnitsine, Lagrange’s theorem for Moufang loops, Math. Proc. Cambridge
+Philos. Soc. 139 (2005), no. 1, 41–57.
+[16] A.N. Grishkov and A.V. Zavarnitsine, Sylow’s theorem for Moufang loops, J. Algebra 321 (2009), no.
+7, 1813–1825.
+[17] J.I. Hall, Moufang Loops and Groups with Triality are Essentially the Same Thing, Mem. Amer. Math.
+Soc. 260 (2019), no. 1252.
+18
+
+[18] R. Moufang, Zur Struktur von Alternativk¨orpern, Math. Ann. 110 (1935), no. 1, 416–430.
+[19] M.W. Liebeck, The classification of finite simple Moufang loops, Math. Proc. Cambridge Philos. Soc.
+102 (July 1987), issue 1 , 33–47.
+[20] G.P.
+Nagy
+and
+P.
+Vojtˇechovsk´y,
+LOOPS,
+version
+3.4.1,
+package
+for
+GAP,
+https://github.com/gap-packages/loops
+[21] H.O. Pflugfelder, Quasigroups and loops: introduction, Sigma Series in Pure Mathematics 7, Heldermann
+Verlag, Berlin, 1990.
+[22] L.J. Paige, A class of simple Moufang loops, Proc. Amer. Math. Soc. 7 (1956), 471–482.
+[23] J.J. Rotman, An introduction to the theory of groups, fourth edition, Graduate Texts in Mathematics,
+Springer-Verlag, 1995.
+[24] D. Stanovsk´y and P. Vojtˇechovsk´y, Commutator theory for loops, J. Algebra 399 (2014), 290–322.
+[25] D. Stanovsk´y and P. Vojtˇechovsk´y, Abelian extensions and solvable loops, Results Math. 66 (2014),
+367–384.
+(Dr´apal) Dept. of Mathematics, Charles University, Sokolovsk´a 83, 186 75 Praha 8, Czech
+Republic
+Email address, Dr´apal: drapal@karlin.mff.cuni.cz
+(Vojtˇechovsk´y) Dept. of Mathematics, University of Denver, 2390 S. York St., Denver, CO
+80208, USA
+Email address, Vojtˇechovsk´y: petr@math.du.edu
+19
+
diff --git a/btE2T4oBgHgl3EQfFgYY/content/tmp_files/load_file.txt b/btE2T4oBgHgl3EQfFgYY/content/tmp_files/load_file.txt
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@@ -0,0 +1,892 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf,len=891
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='03646v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='GR]  9 Jan 2023  ABELIAN CONGRUENCES AND SOLVABILITY  IN MOUFANG LOOPS  ALEˇS DR´APAL AND PETR VOJTˇECHOVSK´Y  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' In groups, an abelian normal subgroup induces an abelian congruence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' We  construct a class of centrally nilpotent Moufang loops containing an abelian normal subloop  that does not induce an abelian congruence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' On the other hand, we prove that in 6-divisible  Moufang loops, every abelian normal subloop induces an abelian congruence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  In loops, congruence solvability adopted from the universal-algebraic commutator theory  of congruence modular varieties is strictly stronger than classical solvability adopted from  group theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  It is an open problem whether the two notions of solvability coincide in  Moufang loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' We prove that they coincide in 6-divisible Moufang loops and in Moufang  loops of odd order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' In fact, we show that every Moufang loop of odd order is congruence  solvable, thus strengthening Glauberman’s Odd Order Theorem for Moufang loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  We investigate abelian normal subloops and the theory of solvability in Moufang loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  There are two notions of solvability in loop theory, one adopted from solvability in group  theory, called classical solvability here, and another adopted from commutator theory in  congruence modular varieties, called congruence solvability here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  When translated into the language of normal series Q = Q0 ≥ Q1 ≥ · · · ≥ Qn = 1, the  difference between the two solvability notions is that classical solvability requires all factors  Qi/Qi+1 to be abelian (that is, commutative groups), while congruence solvability requires  a potentially stronger condition, namely that every factor Qi/Qi+1 induces an abelian con-  gruence of Q/Qi+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Whether a normal subloop of a loop Q is merely abelian or whether it  induces an abelian congruence of Q can be seen on the level of multiplication tables, which  must have a more rigid structure in the latter case, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  Every congruence solvable loop is classically solvable but the converse is not true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' The  two notions of solvability coincide in groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' More generally, they coincide in any loop Q  in which every abelian normal subloop induces an abelian congruence of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' The situation  is delicate, however, since it is certainly possible for a loop Q to be congruence solvable, yet  posses an abelian normal subloop that does not induce an abelian congruence of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  It is an open problem whether the two notions of solvability coincide in Moufang loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' In  this paper we offer a general construction of centrally nilpotent Moufang loops that contain  an abelian normal subloop that does not induce an abelian congruence, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  On the other hand, we show that if Q is a 3-divisible Moufang loop and X is a 2-divisible  abelian normal subloop of Q, then X induces an abelian congruence of Q, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  In particular, in a 6-divisible Moufang loop, every abelian normal subloop induces an abelian  2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' 20N05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Moufang loop, extra loop, solvability, congruence solvability, abelian congruence,  pseudoautomorphism, semiautomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Dr´apal supported by the INTER-EXCELLENCE project LTAUSA19070 of MˇSMT Czech Republic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Vojtˇechovsk´y supported by the Simons Foundation Mathematics and Physical Sciences Collaboration  Grant for Mathematicians no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' 855097 and by the PROF grant of the University of Denver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  1  congruence, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='4, and hence the two notions of solvability coincide in 6-divisible  Moufang loops, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Up to that point, the exposition is self-contained (modulo  basic results from loop theory) and the arguments are elementary in nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  This is in  contrast with other results in Moufang loops, such as the Lagrange Theorem, whose proofs  rely on the classification of finite simple groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  Continuing, we build upon recent deep results of Cs¨org˝o [6] on the nucleus in Moufang  loops and prove that every Moufang loop of odd order is congruence solvable, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Theorem  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' This strengthens a well-known result of Glauberman [14] that states that every Moufang  loop of odd order is classically solvable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='3 implies the finitary version of Corollary  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  Background material on loops, Moufang loops and divisibility in power associative loops  is collected in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Abelianess and solvability for loops are discussed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' An  elementary proof of the Cauchy property for p = 3 in Moufang loops is given in Section 6 as  an appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Background on loops and Moufang loops  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' See [4] or [21] for an introduction to the theory of loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' A loop Q is a magma  (Q, ·, 1) with identity element 1 such that all left translations Lx : Q → Q, Lx(y) = x · y and  all right translations Rx : Q → Q, Rx(y) = y · x are permutations of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  We mostly denote the multiplication operation · by juxtaposition and we take advantage  of · to indicate priority of multiplications in products, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=', x · yz stands for x · (y · z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' The  implicit division operations will be denoted by x\\y = L−1  x (y) and y/x = R−1  x (y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  A mapping f : Q1 → Q2 of loops is a homomorphism if f(xy) = f(x)f(y) for all x, y ∈ Q1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  Then the identities f(x\\y) = f(x)\\f(y) and f(x/y) = f(x)/f(y) are automatically satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  Let Sym(Q) denote the symmetric group on Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  The multiplication group of Q is the  subgroup  Mlt(Q) = ⟨Lx, Rx : x ∈ Q⟩  of Sym(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' The inner mapping group Inn(Q) of Q is the stabilizer of 1 in Mlt(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' It is well  known that  Inn(Q) = ⟨Tx, Rx,y, Lx,y : x, y ∈ Q⟩,  where  Tx = R−1  x Lx,  Rx,y = R−1  xy RyRx,  and  Lx,y = L−1  xy LxLy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  In groups, the inner mapping group is the familiar inner automorphism group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' However,  inner mappings of loops need not be automorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  A subloop X of a loop Q is normal, denoted by X ⊴ Q, if it is a kernel of a loop homo-  morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' It turns out that a subloop X ≤ Q is normal if and only if f(X) = X for every  f ∈ Inn(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  The left nucleus, middle nucleus and the right nucleus of Q are the respective subloops  Nucℓ(Q) = {x ∈ Q : x(yz) = (xy)z for all y, z ∈ Q},  Nucm(Q) = {x ∈ Q : y(xz) = (yx)z for all y, z ∈ Q},  Nucr(Q) = {x ∈ Q : y(zx) = (yz)x for all y, z ∈ Q}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  The nucleus Nuc(Q) of Q is the intersection of the above three nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' A subloop X ≤ Q is  nuclear if X ≤ Nuc(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  2  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Let X be a normal subloop of a loop Q such that X ≤ Nucm(Q) ∩ Nucr(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  Then for every u ∈ Q the inner mapping Tu restricts to an automorphism of X  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Since X ⊴ Q, every inner mapping restricts to a permutation of X, and we only need  to show that Tu(xy) = Tu(x)Tu(y) holds for all x, y ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Hence we need to verify (u·xy)/u =  ((ux)/u)((uy)/u) for every x, y ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' This is equivalent to u · xy = ((ux)/u)((uy)/u) · u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  Since y and (uy)/u are elements of X ≤ Nucm(Q) ∩ Nucr(Q), we calculate  ((ux)/u)((uy)/u) · u = ((ux)/u) · ((uy)/u)u = ((ux)/u) · uy  = ((ux)/u)u · y = (ux)y = u · xy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  □  Note that Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='1 also holds under the dual assumption X ⊴ Q and X ≤ Nucℓ(Q) ∩  Nucm(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' See [7, Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='7] for a slightly stronger statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Let X be a nuclear normal subloop of a loop Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Then every inner mapping  of Q restricts to an automorphism of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Since X ≤ Nucℓ(Q) ∩ Nucr(Q), for every u, v ∈ Q the inner mappings Lu,v, Ru,v  restrict to the identity mapping on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' We are done by Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  □  The center of Q is the subloop  Z(Q) = {x ∈ Nuc(Q) : xy = yx for all y ∈ Q}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  A central subloop of Q is a subloop of Z(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Every central subloop of Q is normal in Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  A loop Q is power associative if every element of Q generates an associative subloop of  Q, that is, a subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' In particular, the powers xi of elements are well-defined in power  associative loops, x−1 = x\\1 = 1/x, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' A loop Q is diassociative if any two elements of Q  generate a subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  A permutation f of Q is a (left) pseudoautomorphism of Q if there exists c ∈ Q such that  cf(x) · f(y) = cf(xy)  for every x, y ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' The element c is then called a (left) companion of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' The set of all pairs  (c, f) ∈ Q × Sym(Q), where f is a pseudoautomorphism of Q and c is a companion of f,  forms a group Psaℓ(Q) under the operations  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='1)  (c, f)(d, g) = (cf(d), fg)  and  (c, f)−1 = (f −1(c\\1), f −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  A permutation f ∈ Sym(Q) is a semiautomorphism of Q if f(1) = 1 and  f(x · yx) = f(x) · f(y)f(x)  holds for all x, y ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' If f is a semiautomorphism of a power associative loop Q then an  inductive argument shows that f(xi) = f(x)i for every i ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  A triple (f, g, h) of permutations of Q is an autotopism of Q if f(x)g(y) = h(xy) holds for  all x, y ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' The autotopisms of Q form a group Atp(Q) under componentwise composition,  the autotopism group of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' The following well-known result describes all autotopisms with  a trivial component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Let Q be a loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Then:  (i) (idQ, g, h) ∈ Atp(Q) iff g = h = Rx for some x ∈ Nucr(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  (ii) (f, idQ, h) ∈ Atp(Q) iff f = h = Lx for some x ∈ Nucℓ(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  (iii) (f, g, idQ) ∈ Atp(Q) iff f = R−1  x  and g = Lx for some x ∈ Nucm(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  3  Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Let Q be a loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Then:  (i) If Nucr(Q) = 1 and (f, g, h) ∈ Atp(Q) then g and h are determined by f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  (ii) If Nucℓ(Q) = 1 and (f, g, h) ∈ Atp(Q) then f and h are determined by g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  (iii) If Nucm(Q) = 1 and (f, g, h) ∈ Atp(Q) then f and g are determined by h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Let us prove (i), the other parts being similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Suppose that Nucr(Q) = 1 and (f, g, h),  (f, g, h) ∈ Atp(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Then (idQ, g−1g, h−1h) = (f, g, h)−1(f, g, h) ∈ Atp(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' By Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='3,  g−1g = h−1h = Rx for some x ∈ Nucr(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Since Nucr(Q) = 1, we have Rx = R1 = idQ, and  g = g, h = h follow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  □  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Divisibility in power associative loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' For an integer d > 1 and a power associa-  tive loop Q, consider the mapping  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='2)  hd : Q → Q,  x �→ xd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  In general, injectivity and surjectivity of hd are unrelated properties already in groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' (In  the group of nonzero complex numbers under multiplication, hd is surjective but not injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  In the additive group of integers, hd is injective but not surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=')  A power associative loop Q is d-divisible (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' uniquely d-divisible) if the mapping hd of  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='2) is surjective (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' bijective).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Let Q be a finite power associative loop, d > 1 an integer and hd as in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  The following conditions are equivalent:  (i) hd is surjective on Q,  (ii) hd is injective on Q,  (iii) Q contains no nonidentity element of order dividing d,  (iv) Q contains no element of prime order dividing d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Thanks to finiteness, (i) and (ii) are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' If Q contains an element x ̸= 1 of  order dividing d, then hd(x) = xd = 1 = hd(1) and hd is not injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Hence (ii) implies  (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Clearly, (iii) implies (iv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' In fact, (iii) and (iv) are equivalent since if 1 ̸= x ∈ Q is  such that |x| divides d and p is a prime dividing |x|, then the cyclic group ⟨x⟩ contains an  element of order p (dividing d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Finally, suppose that (iii) holds, let x ∈ Q and consider the  cyclic group C = ⟨x⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Let k = gcd(|C|, d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' If k > 1 then C contains a nonidentity element of  order k dividing d, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Thus gcd(|C|, d) = 1 and hd restricts to a permutation  of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' In particular, there is y ∈ C such that hd(y) = x, so hd is surjective on Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  □  Given a prime p, we say that a finite power associative loop Q has the Cauchy property for  p if whenever p divides |Q| then there is x ∈ Q such that |x| = p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' A finite power associative  loop Q is said to have the elementwise Lagrange property if |x| divides |Q| for every x ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Let Q be a finite power associative loop and let d > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  (i) Suppose that Q has the Cauchy property for every prime p dividing d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' If Q is uniquely  d-divisible then |Q| is coprime to d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  (ii) Supose that Q has the elementwise Lagrange property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' If |Q| is coprime to d then Q  is uniquely d-divisible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' (i) Suppose that Q has the Cauchy property for every prime dividing d, and also  assume that |Q| is not coprime to d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Let p be any common prime divisor of d and |Q|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' By  assumption, there is x ∈ Q such that |x| = p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' By Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='5, Q is not uniquely d-divisible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  4  (ii) Suppose that Q has the elementwise Lagrange property, and also assume that Q is  not uniquely d-divisible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' By Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='5, there is a prime p dividing d and some x ∈ Q such  that |x| = p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' By assumption, |x| divides |Q|, which implies that |Q| is not coprime to d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  □  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Moufang loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' A loop Q is Moufang if it satisfies any one of the equivalent Moufang  identities  xy · zx = (x · yz)x,  xy · zx = x(yz · x),  x(y · zy) = (xy · z)y,  x(y · xz) = (xy · x)z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='3)  We start by summarizing several well-known results for Moufang loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  By Moufang Theorem [18, 8], if three elements x, y and z of a Moufang loop associate, that  is, x(yz) = (xy)z, then the subloop ⟨x, y, z⟩ is a group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Consequently, Moufang loops are  diassociative, power associative, satisfy the flexible law x(yx) = (xy)x, the inverse properties  x−1(xy) = y = (yx)x−1, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  The four nuclei of a Moufang loop Q coincide and form a normal subloop of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  All inner mappings of a Moufang loop can be seen as pseudoautomorphisms, with suitable  companions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' In particular,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='4)  (x−3, Tx)  is an element of Psaℓ(Q) in a Moufang loop Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Moreover, every pseudoautomorphism of a  Moufang loop is a semiautomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  We proceed to less familiar results on Moufang loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Let Q be a Moufang loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Then  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='5)  x−1(xy · z) = yx−1 · xz  and  (z · yx)x−1 = zx · x−1y  for every x, y, z ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Note that xy · z = x(yx−1)x · z = x(yx−1 · xz) by diassociativity and the Moufang  identities (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Multiplying by x−1 on the left then yields the first identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' The second  identity follows dually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  □  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Let Q be a Moufang loop, c ∈ Q and f ∈ Sym(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Then (c, f) ∈ Psaℓ(Q) if  and only if  xc−1 · cy = f(f −1(x)f −1(y))  for all x, y ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Indeed, cf(x)·f(y) = cf(xy) if and only if f(xy) = c−1(cf(x)·f(y)) = f(x)c−1·cf(y),  by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' We are done upon substituting f −1(x) for x and f −1(y) for y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  □  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Let Q be a Moufang loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Then  xa−3 · a3y = T −1  a (Ta(x)Ta(y))  for all a, x, y ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' We have (a−3, Ta) ∈ Psaℓ(Q) by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  By (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='1), (a−3, Ta)−1 = (T −1  a (a3), T −1  a ) =  (a3, T −1  a ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' We are done by Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  □  5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Lagrange and Cauchy properties for Moufang loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Finally, we present a few  results on d-divisible Moufang loops, taking advantage of Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  It is not difficult to show that finite Moufang loops have the elementwise Lagrange prop-  erty:  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Let Q be a finite power associative loop satisfying the right power alternative  identity (abi)bj = abi+j for every i, j ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Then |x| divides |Q| for every x ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Let x ∈ Q and X = ⟨x⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' It suffices to show that the right cosets of X partition Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  Suppose that aX ∩bX ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Then axi = bxj for some i, j ∈ Z, therefore a = (bxj)x−i = bxj−i  by the right power alternative law, and thus aX = (bxj−i)X = {(bxj−i)xk : k ∈ Z} =  {bxj−i+k : k ∈ Z} = bX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  □  In general, Moufang loops do not satisfy the Cauchy property for every prime p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' For  instance, the smallest nonassociative simple Moufang loop of order 120 contains no element  of order 5 [22] and therefore it violates the Cauchy property for p = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' But the Cauchy  property holds in Moufang loops for the primes p = 2 and p = 3:  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Ever finite Moufang loop satisfies the Cauchy property for p = 2 and p = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' For p = 2, the standard group-theoretic argument works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Let Q be a power associative  loop of even order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' The mapping J : Q → Q, x �→ x−1 is an involution and therefore has  only orbits of sizes 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Since |Q| is even and J(1) = 1, there must be 1 ̸= x ∈ Q such  that J(x) = x, that is, |x| = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  For p = 3, we will give an elementary argument, but we postpone it until Section 6 so as  not to distract from the exposition here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  □  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Both Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='10 and Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='11 follow from results in the theory of  Moufang loops whose only known proofs depend on the classification of finite simple Moufang  loops [19] and hence also on the classification of finite simple groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='10 is an  immediate consequence of the Lagrange Theorem for Moufang loops, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' [12, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' For the  Cauchy property, Grishkov and Zavarnitsine proved in [16] that every Moufang loop of order  2a3bm with m coprime to 6 contains (Sylow) subloops of orders 2a and 3b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Therefore, if Q  is a Moufang loop whose order is divisible by p ∈ {2, 3}, it contains a subloop X of order pc  for some c > 0, then any element 1 ̸= x ∈ X generates a cyclic group ⟨x⟩ of p-power order  by Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='10, and the group ⟨x⟩ then certainly contains an element of order p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  Combining Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='6, Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='10 and Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='11, we get:  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Let Q be a finite Moufang loop and let d = 2a3b > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Then Q is uniquely  d-divisible if and only if |Q| is coprime to d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Centrality, nilpotency, abelianess and solvability for loops  In [11], Freese and McKenzie developed commutator theory for congruence modular vari-  eties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Their commutator of two congruences α, β in an algebra Q will be denoted by [α, β]Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  The smallest congruence on Q will be denoted by ⊥Q = {(x, x) : x ∈ Q} and the largest  congruence on Q will be denoted by ⊤Q = {(x, y) : x, y ∈ Q}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  The commutator theory of [11] was specialized to the variety of loops in [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Although  we will not need to work with the exact form of the commutator of loop congruences (and  instead take advantage of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='8), we give it here for the sake of completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' In  6  [24, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='1], the commutator of loop congruences was expressed as the congruence  generated by certain pairs of evaluated total inner mappings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' The technical complication  with total inner mappings has been recently removed by Barnes who obtained the following  description of the commutator of loop congruences in her PhD thesis [2]:  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='1 (Barnes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Let α, β be congruences of a loop Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Then the commutator [α, β]Q  is the congruence of Q generated by all pairs  (Tu1(a), Tv1(a)),  (Lu1,u2(a), Lv1,v2(a)),  (Ru1,u2(a), Rv1,v2(a)),  where (1, a) ∈ α and (u1, v1), (u2, v2) ∈ β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  In loops, there is a one-to-one correspondence between congruences and normal subloops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  Given a normal subloop X of a loop Q, the congruence αX induced by X is the equivalence  relation on Q with equivalence classes {aX : x ∈ Q}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Conversely, given a congruence α of a  loop Q, the normal subloop of Q induced by α is the equivalence class of α containing 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  We will therefore write [X, Y ]Q for the commutator of normal subloops X, Y of Q, by  which we mean the normal subloop of Q induced by the congruence [αX, αY ]Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='1  can then be routinely translated to the context of normal subloops:  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Let X, Y be normal subloops of a loop Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Then the commutator [X, Y ]Q is  the normal subloop of Q generated by all quotients  Tu1(a)/Tv1(a),  Lu1,u2(a)/Lv1,v2(a),  Ru1,u2(a)/Rv1,v2(a),  where a ∈ X and u1/v1, u2/v2 ∈ Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  The commutator theory of [11] gives rise naturally to theories of central nilpotency and  solvability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' In loops, the central nilpotency theory of [11] coincides with the classical nilpo-  tency theory adopted from groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' But the solvability theory of [11] is strictly stronger in  loops than the classical solvability theory adopted from groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Here are more details:  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Centrality and central nilpotency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' A congruence α of an algebra Q is central if  [α, ⊤Q]Q = ⊥Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Passing to normal subloops, a normal subloop X of a loop Q is then said to  be central if [X, Q]Q = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Fortunately, this agrees with the traditional definition of centrality,  because a normal subloop X of Q satisfies [X, Q]Q = 1 if and only if X ≤ Z(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Given a commutative group (X, +, 0), a loop (F, ·, 1) and a mapping θ :  F × F → X satisfying θ1,r = 0 = θr,1 for every r ∈ F, the loop defined on F × X by  (r, x)(s, y) = (rs, x + y + θr,s)  is a central extension of X by F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='4 ([25, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Let X be a normal subloop of a loop Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Then X is  central in Q (that is, [X, Q]Q = 1) if and only if Q is isomorphic to a central extension of  X by Q/X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  A loop Q is an iterated central extension if it is either an abelian group or there exists a  central subloop X of Q such that Q/X is an iterated central extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' A central series for a loop Q is a series  Q = Q0 ≥ Q1 ≥ · · · ≥ Qn = 1,  7  such that for every 0 ≤ i < n, Qi+1 is a normal subloop of Q and the factor Qi/Qi+1 is central  in Q/Qi+1 (that is, [Qi/Qi+1, Q/Qi+1]Q/Qi+1 = 1, or, equivalently, Qi/Qi+1 ≤ Z(Q/Qi+1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  A loop Q is (centrally) nilpotent if it contains a central series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='6 ([25, Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' A loop is centrally nilpotent if and only if it is an iterated  central extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Abelianess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' A congruence α of an algebra Q is abelian if [α, α]Q = ⊥Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' An algebra Q  is abelian if the congruence ⊤Q is abelian, that is, [⊤Q, ⊤Q]Q = ⊥Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  A loop Q is therefore abelian if [Q, Q]Q = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' It is well-known that a group is abelian if  and only if it is a commutative group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' More generally, a loop is abelian if and only if it is a  commutative group, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  A conflict arises when the “abelian” terminology is used for a normal subloop X of a loop  Q, since X can be seen either as a congruence of Q or as a loop in its own right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' We will  therefore be more careful in that context and say that a normal subloop X of a loop Q is  abelian in Q or that it induces an abelian congruence of Q if [X, X]Q = 1, while we say that  a normal subloop X of a loop Q is abelian if [X, X]X = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Thus, for instance, the phrase  “X is an abelian normal subloop of Q” means that X is a commutative group and X is a  normal subloop of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  Every normal subloop X of Q that induces an abelian congruence of Q is an abelian  normal subloop of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' The converse is also true in the variety of groups, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='9, but  there are numerous examples of loops Q with an abelian normal subloop X that does not  induce an abelian congruence of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' See [24] for examples of order 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Given a commutative group (X, +, 0), a loop (F, ·, 1) and mappings ϕ, ψ :  F × F → Aut(X) and θ : F × F → X, the loop defined on F × X by  (r, x)(s, y) = (rs, ϕr,s(x) + ψr,s(y) + θr,s)  is an abelian extension of X by F if ϕr,1 = idX = ψ1,r and θ1,r = 0 = θr,1 for every r ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  Central extensions are therefore those abelian extensions in which the automorphisms ϕr,s  and ψr,s are trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='8 ([25, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Let X be a normal subloop of a loop Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Then X is  abelian in Q (that is, [X, X]Q = 1) if and only if Q is isomorphic to an abelian extension of  X by Q/X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  The above external description of abelian extensions can be rewritten internally as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  Let (X, ·, 1) be an abelian normal subloop of a loop (Q, ·, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Let U be a (left) transversal  to X in Q such that 1 ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Then Q is an abelian extension of X by Q/X if there exist ϕ,  ψ : U × U → Aut(X) and θ : U × U → X satisfying ϕr,1 = idX = ψ1,r and θ1,r = 1 = θr,1 for  every r ∈ U, and  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='1)  rx · sy = ur,s · ϕr,s(x)ψr,s(y)θr,s  holds for every r, s ∈ U and x, y ∈ X, where ur,s is the unique element of U ∩ (rs)X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  There is a substantial difference between abelian normal subloops of Q and normal subloops  of Q that are abelian in Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' This can be illustrated by considering the multiplication table  of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  Let X be a commutative group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  For X to be an abelian normal subloop of Q,  nothing else is required but that Q is a disjoint union of {uX : u ∈ U} for some subset  8  U ⊆ Q, the multiplication table of X is reproduced in the subsquare X × X, and for every  r, s ∈ U the subsquare rX × sX is a latin square with entries running through ur,sX, where  ur,s ∈ U ∩(rs)X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' However, for X to induce an abelian congruence of Q, the structure of the  subsquare rX × sX must be much more rigid, conforming to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  Using the notion of abelian extension, it is easy to show that every abelian normal subgroup  of a group Q is abelian in Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Here is a more general result:  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Let Q be a loop and let X be an abelian normal subloop of Q such that X ≤  Nucm(Q) ∩ Nucr(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Then X induces an abelian congruence of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' By Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='8, it suffices to show that Q is an abelian extension of X by Q/X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  Let U be a transversal to X in Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' For r, s ∈ U, let ϕr,s be the restriction of T −1  s  to X, let  ψr,s = idX ∈ Aut(X), let ur,s ∈ U ∩ (rs)X and let θr,s = ur,s\\(rs) ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' By Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='1,  ϕr,s ∈ Aut(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  Let x, y ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Note that sT −1  s (x) = s(s\\(xs)) = xs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Using the fact that x, y, T −1  s (x) ∈  X ≤ Nucm(Q) ∩ Nucr(Q), we have  rx · sy = r(x · sy) = r(xs · y) = r(sT −1  s (x) · y) = r(s · T −1  s (x)y) = rs · T −1  s (x)y  = ur,sθr,s · T −1  s (x)y = ur,s · θr,s(T −1  s (x)y) = utr,s · T −1  s (x)yθr,s,  where the last step follows from the fact that θr,s, T −1  s (x) and y lie in the abelian group  X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' We have obtained an instance of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='1), proving that Q an abelian extension of X by  Q/X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  □  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Classical solvability and congruence solvability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' The history of the notion of  solvability in loop theory is convoluted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  Albert defined solvable loops in [1, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' 412] as loops whose composition factors have no  notrivial subloops, mimicking a result from groups that states that a finite group is solvable  if and only if each of its composition factors is isomorphic to a group of prime order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Albert’s  definition of solvability has been abandoned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  Bruck introduced the notion of a derived subloop in [3, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' 268].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' The derived subloop Q′  of a loop Q is the smallest normal subloop H of Q such that Q/H is a commutative group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  The derived series of Q is then the (possibly infinite) series  Q = Q0 ≥ Q1 ≥ · · · ≥ Qn ≥ · · ·  such that for every i ≥ 0, Qi+1 is the derived subloop of Qi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Bruck then defines solvable  loops as loops whose derived series reaches the trivial subloop 1 in finitely many steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  Another definition of solvability for loops was given by Glauberman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' A loop Q is said to  be solvable in [14, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' 397] if there exists a series  Q = Q0 ≥ Q1 ≥ · · · ≥ Qn = 1  such that for every 0 ≤ i < n, Qi+1 is a normal subloop of Qi and the factor Qi/Qi+1 is a  commutative group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' By adopting the standard proof from group theory, it is not difficult to  show that Bruck’s and Glauberman’s definitions of solvability for loops are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' In  particular, Glauberman’s definition of solvability will not be affected if we demand that for  every 0 ≤ i < n, Qi+1 is a normal subloop of Q, not just a normal subloop of Qi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  The commutator theory of Freese and McKenzie [11] naturally leads to a definition of  solvability in congruence modular varieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  In loops, their concept of solvability, called  9  congruence solvability in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='10, is strictly stronger than the equivalent solvabil-  ity concepts of Bruck and Glauberman, called classical solvability in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' The  terminology comes from [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' A classically solvable series for a loop Q is a series  Q = Q0 ≥ Q1 ≥ · · · ≥ Qn = 1,  such that for every 0 ≤ i < n, Qi+1 is a normal subloop of Q and the factor Qi/Qi+1 is  abelian (that is, a commutative group).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' A loop Q is classically solvable if it contains a  classically solvable series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  A congruence solvable series for a loop Q is a series  Q = Q0 ≥ Q1 ≥ · · · ≥ Qn = 1  such that for every 0 ≤ i < n, Qi+1 is a normal subloop of Q and the factor Qi/Qi+1 is  abelian in Q/Qi+1 (that is, Qi/Qi+1 induces an abelian congruence of Q/Qi+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' A loop Q is  congruence solvable if it contains a congruence solvable series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  Obviously, every congruence solvable loop is classically solvable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' The converse is true for  groups but not for loops, with small counterexamples easy to construct, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' [11, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' The  following problem is open:  Problem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Is every classically solvable Moufang loop congruence solvable?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  Towards a solution of Problem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='11, we prove in Sections 4 and 5 that if Q is a 6-divisible  classically solvable Moufang loop or a Moufang loop of odd order, then Q is congruence  solvable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  Since every central series is a congruence solvable series, we have:  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Centrally nilpotent loops are congruence solvable and classically solvable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  Call a loop Q an iterated abelian extension if either Q is a commutative group, or Q is  an abelian extension of a commutative group X by some loop that is an iterated abelian  extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' It was shown in [25, Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='1] that iterated abelian extensions of loops are  precisely congruence solvable loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' For the sake of completeness, let us give a short proof  here:  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' A loop is congruence solvable if and only if it is an iterated abelian  extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Let Q = Q0 > Q1 > · · · > Qn = 1 be a congruence solvable series for Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' We prove  by induction on n that Q is an iterated abelian extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' If n = 1 then the series becomes  Q > 1 and Q is a commutative group, hence a congruence solvable loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' If n > 1, let  X = Qn−1 ⊴ Q and note that Q > X > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Since X and 1 are adjacent terms in the original  series, X/1 = X is abelian in Q/1 = Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' By Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='8, Q is an abelian extension of X by  Q/X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' It remains to show that Q/X is an iterated abelian extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Consider the series  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='2)  Q/X = Q0/X > Q1/X > · · · > Qn−1/X = X/X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  Since Qi ⊴Q, we have Qi/X ⊴Q/X by the Correspondence Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' As Qi/Qi+1 is abelian  in Q/Qi+1, the factor loop (Qi/X)/(Qi+1/X) ∼= Qi/Qi+1 is abelian in (Q/X)/(Qi+1/X) ∼=  Q/Qi+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Hence (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='2) is a congruence solvable series, Q/X is congruence solvable and, by the  induction assumption, Q/X is an iterated abelian extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  10  Conversely, suppose that Q is an iterated abelian extension constructed in n steps, each  an abelian extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' We prove by induction on n that Q is congruence solvable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' If n = 0  then Q is a commutative group and we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Else let Q be an abelian extension of a  commutative group X by Q/X, where Q/X is constructed by n − 1 abelian extensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' By  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='8, X is abelian in Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' By the induction assumption, Q/X is congruence solvable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  Let Q/X = Q0/X > · · · > Qm/X = X/X be a congruence solvable series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Consider the  series  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='3)  Q = Q0 > Q1 > · · · > Qm−1 > Qm = X > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  Since Qi/X ⊴ Q/X, we have Qi ⊴ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Moreover, Qi/Qi+1 ∼= (Qi/X)/(Qi+1/X) is abelian  in (Q/X)/(Qi+1/X) ∼= Q/Qi+1 for every i < m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' The last inclusion in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='3) is X > 1, and  certainly X/1 = X is abelian in Q = Q/1 as we have already shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Therefore (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='3) is a  congruence solvable series and Q is congruence solvable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  □  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Moufang loops with an abelian normal subgroup that does not induce  an abelian congruence  By inspecting the library of small Moufang loops available in the GAP [13] package LOOPS  [20], it was observed in [24] that there exists a Moufang loop Q of order 16 with an abelian  normal subloop X (isomorphic to C2×C4) such that X does not induce an abelian congruence  of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  In this section we offer a general construction of Moufang loops Q containing an abelian  normal subloop X that does not induce an abelian congruence of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  Recall that for a vector space V over a field F, a mapping q : V → F is a quadratic  form if q(λu) = λ2u for every λ ∈ F, u ∈ V , and if h : V × V → F defined by h(u, v) =  q(u + v) − q(u) − q(v) is a bilinear form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' The form h is referred to as the associated bilinear  form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  Also recall that a loop Q is an extra loop if it satisfies the identity x(y · zx) = (xy · z)x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  Fenyves proved in [10] that extra loops are Moufang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Let W = (W, +) be a commutative group with subgroups F ≤ B ≤ W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  Suppose that F = {0, 1} and W = W/B is an elementary abelian 2-group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Let q : W → F  be a quadratic form with associated bilinear form h : W × W → F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Let q : W → F and h :  W × W → F be defined by q(u) = q(u) and h(u, v) = h(u, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Denote by Q = Q(F, B, W, q)  the magma defined on F × W by  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='1)  (i, u) · (j, v) = (i + j, u + v + jq(u) + ih(u, v)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  Then:  (i) Q is a centrally nilpotent loop, a central extension of the commutative group B by the  elementary abelian 2-group F × W,  (ii) Q is congruence solvable and hence classically solvable,  (iii) Q is an extra loop,  (iv) Q is a group if and only if the quadratic form q is linear,  (v) X = 0 × W is an abelian normal subloop of Q,  (vi) if Q is not a group, then the congruence of Q induced by X is not abelian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  11  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' (i) The commutative group W is (isomorphic to) a central extension of B by W, with  multiplication on W × B given by  (u, a)(v, b) = (u + v, a + b + θu,v),  where θ : W × W → B is some group cocycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' The multiplication formula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='1) on F × W  can then be seen as a multiplication on F × W × B, namely  (i, u, a)(j, v, b) = (i + j, u + v, a + b + θu,v + jq(u) + ih(u, v)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  Hence Q is isomorphic to the magma defined on (F × W) × B by  ((i, u), a)((j, v), b) = ((i, u) + (j, v), a + b + θ(i,u),(j,v)),  where θ(i,u),(j,v) = θu,v + jq(u) + ih(u, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Since θ(0,0),(j,v) = θ0,v + jq(0) = 0 and θ(i,u),(0,0) =  θu,0 + ih(u, 0) = 0, we see at once that θ is a loop cocycle and Q is a central extension of  the commutative group B by the elementary abelian 2-group F ×W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Hence Q is a centrally  nilpotent loop, finishing the proof of part (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Part (ii) then follows from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  (iii) Chein an Robinson characterized extra loops as Moufang loops in which squares  are in the nucleus [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  In the isomorphic copy of Q from part (i) we have ((i, u), a)2 =  ((0, 0), 2a+θ(i,u),(i,u)) ∈ 0×B ≤ Z(Q) ≤ Nuc(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' To prove that Q is extra, it therefore suffices  to check one of the Moufang identities, say (xy·x)z = x(y·xz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' By definition, q(b+v) = q(v)  and h(b+v, w) = h(v, w) for every b ∈ B and v, w ∈ W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Note that 2W ⊆ B since W = W/B  is an elementary abelian 2-group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Finally observe that h(u, u) = h(u, u) = q(0) + 2q(u) = 0  and h(u, u + v) = h(u, u) + h(u, v) = h(u, v) for every u, v ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Let x = (i, u), y = (j, v) and  z = (k, w) ∈ F × W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' We then calculate  ((i, u)(j, v) · (i, u))(k, w)  = (i + j, u + v + jq(u) + ih(u, v))(i, u) · (k, w)  = (j, 2u + v + jq(u) + ih(u, v) + iq(u + v) + (i + j)h(v, u)) · (k, w)  = (j, 2u + v + jq(u) + iq(u + v) + jh(u, v)) · (k, w)  = (j + k, 2u + v + w, jq(u) + iq(u + v) + jh(u, v) + kq(v) + jh(v, w)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  On the other hand  (i, u)((j, v) · (i, u)(k, w))  = (i, u) · (j, v)(i + k, u + w + kq(u) + ih(u, w))  = (i, u) · (i + j + k, u + v + w + kq(u) + ih(u, w) + (i + k)q(v) + jh(v, u + w))  = (j + k, 2u+v+w+kq(u)+ih(u, w)+(i+k)q(v)+jh(v, u+w)+(i+j+k)q(u)+ih(u, v+w))  = (j + k, 2u + v + w + (i + j)q(u) + ih(u, v) + (i + k)q(v) + jh(v, u + w)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  These two products agree in the first coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Upon canceling like terms in the second  coordinate (while taking advantage of bilinearity of h), only iq(u + v) remains in the first  product, while iq(u) + ih(u, v) + iq(v) remains in the second product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  Since h(u, v) =  q(u + v) + q(u) + q(v), we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  (iv) Suppose that q is linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Then h = 0 and the cocycle of (i) reduces to θ(i,u),(j,v) =  θu,v + jq(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Since θ is a group cocycle, it satisfies the group cocycle identity  θu,v + θu+v,w = θv,w + θu,v+w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  12  We then have  θ(i,u),(j,v) + θ(i,u)+(j,v),(k,w) = θu,v + θu+v,w + jq(u) + kq(u + v)  = θv,w + θu,v+w + kq(v) + (j + k)q(u)  = θ(j,v),(k,w) + θ(i,u),(j,v)+(k,w),  which is a group cocycle identity for θ, so Q is a group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  Conversely, if Q is a group then for all u, v ∈ W the product  (1, 0)(0, u) · (1, v) = (1, u)(1, v) = (0, u + v + q(u) + h(u, v))  is equal to  (1, 0) · (0, u)(1, v) = (1, 0)(1, u + v + q(u)) = (0, u + v + q(u)),  so h = 0 and q is linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  (v) On X = 0 × W, the multiplication formula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='1) reduces to  (0, u)(0, v) = (0, u + v),  and X is therefore isomorphic to the commutative group W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' The mapping f : Q → F,  (i, u) �→ i is clearly a homomorphism with kernel equal to X, which shows that X is a  normal subloop of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  (vi) Suppose that X induces an abelian congruence of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' By Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='8, Q is an abelian  extension of X = W by F, and there exist ϕ, ψ : F × F → Aut(X) and θ : F × F → X such  that ϕi,0 = ψ0,i = idX, θi,0 = θ0,i = 0 and  (i, u)(j, v) = (i + j, ϕi,j(u) + ψi,j(v) + θi,j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  Since (0, u)(1, v) = (1, u+v+q(u)) by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='1) and ϕ0,1(u)+ψ0,1(v)+θ0,1 = ϕ0,1(u)+v, we must  have ϕ0,1(u) = u + q(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' As ϕ0,1 is an automorphism of X, we deduce (u + v) + q(u + v) =  (u + q(u)) + (v + q(v)) for all u, v ∈ W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' This shows that q is linear and Q is a group by  (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  □  The nonassociative loops Q(F, B, W, q) afforded by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='1 demonstrate that it  is possible for a Moufang loop to be congruence solvable (even centrally nilpotent) and yet  posses an abelian normal subloop that does not induce an abelian congruence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Smallest examples of interest are obtained from Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='1 when q is  a nonlinear quadratic form on a vector space W/B of dimension two, which forces (up to  isomorphism) either F = B = 0 × 0 × C2 ≤ W = C2 × C2 × C2 or F = B = 0 × C2 ≤ W =  C2 × C4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' We then obtain a Moufang loop Q of order 16 with an abelian normal subloop X  (isomorphic to C2 × C2 × C2 or to C2 × C4) that does not induce an abelian congruence of  Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' This covers the example that was found in [24] by an exhaustive search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Moufang loops in which classical solvability and congruence solvability  coincide  We start with the following easy fact:  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Let X be a 2-divisible commutative group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Then every semiautomorphism of  X is an automorphism of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  13  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Let f be a semiautomorphism of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Recall that f(xn) = f(x)n for every x ∈ X  and n ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  Let x, y ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  By 2-divisibility, there is u ∈ X such that x = u2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  Then  f(xy) = f(u2y) = f(uyu) = f(u)f(y)f(u) = f(u)2f(y) = f(u2)f(y) = f(x)f(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  □  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Let Q be a Moufang loop and X a 2-divisible abelian normal subgroup of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  Then every inner mapping of Q restricts to an automorphism of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Let f ∈ Inn(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' By the introductory remarks in Subsection 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='3, f is a pseudoauto-  morphism of Q and hence a semiautomorphism of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='1, the restriction f|X of  f to X is an automorphism of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  □  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Let Q be a 3-divisible Moufang loop and let X be a 2-divisible normal subgroup  of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Then the congruence {aX : a ∈ Q} on Q induced by X is an abelian congruence of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Let U be a transversal to X containing 1, let r, s ∈ U and x, y ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' There are  uniquely determined u = ur,s ∈ U and z ∈ X such that rs = uz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' We wish to apply Theorem  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='8 and hence to find ϕr,s, ψr,s ∈ Aut(X) and θr,s ∈ X such that rx·sy = u·ϕr,s(x)ψr,s(y)θr,s  as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' In addition, we must verify ϕr,1 = idX = ψ1,r and θ1,r = θr,1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' We will build  the automorphisms in several steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Consider  f1 = (Ts)|X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='2, f1 ∈ Aut(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' We have f1(y)s = Ts(y)s = sys−1s = sy, so  rx · sy = rx · f1(y)s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  Let  f2 = (L−1  s−1rL−1  s Lr)|X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  Since f2(1) = (s−1r)−1(s−1r) = 1, f2 is a restriction of an inner mapping of Q to X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' By  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='2, f2 ∈ Aut(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Moreover, we have s · (s−1r)f2(x) = rx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Therefore  rx · sy = rx · f1(y)s = (s · (s−1r)f2(x))(f1(y)s) = s((s−1r)f2(x) · f1(y))s,  where we have used a Moufang identity (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='3) in the last step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' By the identity (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='5), we have  uv · w = u(u−1 · (uv · w)) = u(vu−1 · uw) for any u, v, w ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' In particular, with u = s−1r,  v = f2(x) ∈ X and w = f1(y) ∈ X, we obtain  rx · sy = s(uv · w)s = s(u(vu−1 · uw))s = su · (vu−1 · uw)s,  where we have again used a Moufang identity in the last step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Since Q is 3-divisible, there  is a ∈ Q such that u = a3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='9 then yields  vu−1 · uw = va−3 · a3w = T −1  a (Ta(v)Ta(w)) = vw,  where in the last step we used Ta|X ∈ Aut(X), by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' So far we showed  rx · sy = su · (vw)s = s(s−1r) · (vw)s = r · (vw)s = r · (f2(x)f1(y))s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  Now, sf −1  1 (x0) = ss−1x0s = x0s for any x0 ∈ Q and thus  rx · sy = r · (f2(x)f1(y))s = r · sf −1  1 (f2(x)f1(y)) = r · s(f −1  1 f2(x) · y),  taking advantage of f1 ∈ Aut(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Consider  f3 = (L−1  rs LrLs)|X  14  and note that f3(1) = (rs)−1(rs) = 1, which implies f3 ∈ Aut(X) as usual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Moreover,  rs · f3(x0) = r · sx0 for any x0 ∈ Q, and hence  rx · sy = r · s(f −1  1 f2(x) · y) = rs · f3(f −1  1 f2(x) · y) = rs · (f3f −1  1 f2(x) · f3(y)),  using f3 ∈ Aut(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Recall that rs = uz with u ∈ U, z ∈ X, and consider  f4 = (L−1  z L−1  u Lrs)|X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  We have f4(1) = z−1(u−1(rs)) = 1 (since rs = uz) and f4 ∈ Aut(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Moreover, u·zf4(x0) =  rs · x0 for any x0 ∈ Q, and thus  rx · sy=rs · (f3f −1  1 f2(x) · f3(y))=u · zf4(f3f −1  1 f2(x) · f3(y))=u · f4f3f −1  1 f2(x)f4f3(y)z,  where we have used f4 ∈ Aut(X) and commutativity of the group X in the last step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  Rewriting, we have  rx · sy = u · ϕr,s(x)ψr,s(y)θr,s,  where ϕr,s = f4f3f −1  1 f2 ∈ Aut(X), ψr,s = f4f3 ∈ Aut(X) and θr,s = z = u−1(rs) ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  If s = 1, we observe f1 = (T1)|X = idX, f2 = (L−1  r L−1  1 Lr)|X = idX, f3 = (L−1  r LrL1)|X =  idX, u ∈ (r1)X ∩ U = {r}, θr,1 = z = 1, f4 = (L−1  1 L−1  r Lr)|X = idX and therefore ϕr,1 = id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  If r = 1, we observe f3 = (L−1  s L1Ls)|X = idX, u ∈ (1s)X ∩ U = {s}, θ1,s = z = 1,  f4 = (L−1  1 L−1  s Ls)|X = idX and therefore ψ1,s = idX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  □  Recall the power mapping hd of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='2) and note that h6 = h3h2 = h2h3 in a power associative  loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Hence a power associative loop is 6-divisible if and only if it is 2-divisible and 3-divisible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  We therefore deduce from Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='3:  Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Let Q be a 6-divisible Moufang loop and let X be an abelian normal subloop  of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Then X induces an abelian congruence of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  Since in the two definitions of solvability (Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='10), the only difference is whether  the quotients Qi/Qi+1 are merely commutative groups or whether they induce an abelian  congruence, we have:  Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Let Q be a 6-divisible Moufang loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Then Q is solvable if and only if it is  congruence solvable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Moufang loops of odd order are congruence solvable  We proceed to show that Moufang loops of odd order are congruence solvable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  This  strengthens Glauberman’s Odd Order Theorem for Moufang loops [14, Theorem 16]:  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='1 (Glauberman).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' A Moufang loop of odd order is classically solvable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  Our proof is based on the recent, deep result of Cs¨org˝o [6, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='1]:  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='2 (Cs¨org˝o).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' A nontrivial Moufang loop of odd order has a nontrivial nucleus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  A subloop X of a loop Q is characteristic if f(X) = X for every f ∈ Aut(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Unlike in  groups, a characteristic subloop of Q is not necessarily a normal subloop of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' It is clear  from the definition of the nucleus that Nuc(Q) is a characteristic subloop of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' A Moufang loop of odd order is congruence solvable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  15  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' We proceed by induction on the order of the Moufang loop Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' There is nothing to  prove when Q = 1, so suppose that Q is nontrivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' In every Moufang loop, the nucleus is a  normal subloop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' By Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='2, 1 < N = Nuc(Q) ⊴ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Since N is a group of odd order,  it is solvable by the Odd Order Theorem for groups [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  Let X be a minimal characteristic subgroup of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Let f ∈ Inn(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' By Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='2, f  restricts to an automorphism of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Since X is characteristic in N, f(X) = X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Hence X ⊴Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  A variation on a standard group-theoretic argument, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' [23, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='24], now shows that  X is an abelian group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' (Consider the derived subgroup X′ of the solvable group X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=')  Altogether, we have shown that X is an abelian normal subgroup of Q and 1 < X ≤  Nuc(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='9, X induces an abelian congruence of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' By Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='8, Q is an  abelian extension of X by Q/X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' By the induction assumption, Q/X is congruence solvable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  By Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='13, Q/X is an iterated abelian extension, hence Q is an iterated abelian  extension, and Q is congruence solvable by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='13 again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  □  Note that in the finite case, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='3 implies Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Indeed, if Q is a finite  6-divisible Moufang loop then it is of odd order by Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='13, congruence solvable by  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='3 and thus also classically solvable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' The Cauchy property for p = 3 in Moufang loops  Here we give an elementary proof of the Cauchy property for p = 3 in Moufang loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' We  start with a lemma motivated by triality for Moufang loops, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Let  Com(Q) = {x ∈ Q : xy = yx for all y ∈ Q}  be the commutant of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  In addition to the already introduced bijections Lx, Rx and Tx = R−1  x Lx, consider also  Mx = RxLx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' In diassociative loops, we have Ln  x = Lxn, Mx = LxRx, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Let Q be a diassociative loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Then a mapping  σ : {Lx, Rx, L−1  x , R−1  x  : x ∈ Q} → Mlt(Q)  is well-defined by σ(Lx) = Rx, σ(L−1  x ) = R−1  x , σ(Rx) = M−1  x  and σ(R−1  x ) = Mx if and only  if x3 = 1 for every x ∈ Com(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Note that if Lx ∈ {Ly, Ry, L−1  y , R−1  y } then x ∈ {y, y−1}, and similarly for Rx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' To check  that σ is well-defined, we therefore need to establish the following implications: (a) Lx = L−1  x  implies Rx = R−1  x , (b) Rx = R−1  x  implies M−1  x  = Mx, (c) Lx = Rx implies Rx = M−1  x , (d)  Lx = R−1  x  implies Rx = Mx, (e) L−1  x  = Rx implies R−1  x  = M−1  x , and (f) L−1  x  = R−1  x  implies  R−1  x  = Mx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  The implication (a) is immediate, for if Lx = L−1  x  then x = x−1 and Rx = R−1  x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' The  argument for (b) is similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' The implications (c) and (f) are equivalent, and so are the  implications (d) and (e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  Concerning (c), suppose that Lx = Rx, that is, x ∈ Com(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' We will show that then  Rx = M−1  x  iff x3 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Certainly if Rx = M−1  x  then evaluating at 1 yields x = x−2, that is,  x3 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Conversely, if x3 = 1 then R3  x = idQ and Rx = R−2  x  = R−1  x L−1  x  = M−1  x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  Finally, for (d), suppose that Lx = R−1  x , that is, x2 = 1 and x ∈ Com(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' We will again  show that Rx = Mx iff x3 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Certainly if Rx = Mx then Rx = RxLx, Lx = idQ, x = 1 and  x3 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Conversely, if x3 = 1 then from x2 = 1 we deduce x = 1 and Rx = Mx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  □  16  Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' If Q is a Moufang loop with trivial nucleus then the mapping σ of Lemma  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='1 is well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Let x ∈ Com(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Then Tx = idQ, so in particular Tx is an automorphism of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Since  (x−3, Tx) ∈ Psaℓ(Q) by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='4), we have x−3Tx(y) · Tx(z) = x−3Tx(yz) = x−3(Tx(y)Tx(z)) for  all y, z ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Therefore x−3 ∈ Nuc(Q) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  □  Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Let Q be a Moufang loop with trivial nucleus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Then there exists a unique σ ∈  Aut(Mlt(Q)) such that σ(Lx) = Rx and σ(Rx) = M−1  x , for every x ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' This automorphism  satisfies σ3 = idMlt(Q), and if ϕ is an inner mapping of Q with companion c (when seen as a  pseudoautomorphism) then σ(ϕ) = R−1  c ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' The maping σ of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='1 is well-defned by Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Our first goal is to show  that it is possible to extend it into an automorphism of Mlt(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' For that it suffices to verify  that  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='1)  ψ1 · · · ψk = idQ  ⇔  σ(ψ1) · · ·σ(ψk) = idQ  whenever each ψi, 1 ≤ i ≤ k, belongs to {L±1  x , R±1  x  : x ∈ Q}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  The first Moufang identity of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='3) can be rewritten as (Lx, Rx, Mx) ∈ Atp(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Substi-  tuting yx−1 for y in the second identity of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='5) yields (zy)x−1 = zx · x−1(yx−1), which says  (Rx, M−1  x , R−1  x ) ∈ Atp(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Taking inverses into consideration, we see that all four triples  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='2)  (Lx, Rx, Mx), (L−1  x , R−1  x , M−1  x ), (Rx, M−1  x , R−1  x ) and (R−1  x , Mx, Rx),  are autotopisms of Q, for every x ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  Now, autotopisms may be chosen from this list in such a way that the first coordinate is  equal to ψi and the second coordinate is equal to σ(ψi), 1 ≤ i ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' By composing these  autotopisms we obtain an autotopism in which the first coordinate is equal to ψ1 · · · ψk and  the second coordinate is equal to σ(ψ1) · · ·σ(ψk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Since all the (equal) nuclei of Q are trivial,  Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='4 implies that the first coordinate is trivial if and only if the second coordinate  is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' We proved σ ∈ Aut(Mlt(Q)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  Let us establish σ3 = idMlt(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  We have σ(Lx) = Rx, σ(Rx) = M−1  x  = R−1  x L−1  x  and  σ(M−1  x ) = σ(R−1  x )σ(L−1  x ) = MxR−1  x  = LxRxR−1  x  = Lx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' This shows that the automorphism  σ3 is identical on a generating set of Mlt(Q), hence also on Mlt(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  Finally, let ϕ ∈ Inn(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Then ϕ is a pseudoautomorphism with some companion c ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  This can also be expressed as (Lcϕ, ϕ, Lcϕ) ∈ Atp(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Let us compose the autotopisms  of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='2) so that the first coordinate of the resulting autotopism (f, g, h) is equal to Lcϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  Certainly g = σ(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' By Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='4, (f, g, h) = (Lcϕ, ϕ, Lcϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Thus Rcσ(ϕ) = σ(Lc)σ(ϕ) =  σ(Lcϕ) = σ(f) = g = ϕ and σ(ϕ) = R−1  c ϕ follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  □  Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Let Q be a finite Moufang loop of order divisible by three.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' If the nucleus  of Q is trivial, then Q possesses a proper subloop of order divisible by three.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Let σ be as in Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='3, H = ⟨σ⟩, and let G = Mlt(Q) ⋉ H be the semidirect  product defined by the natural action of H on Mlt(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Hence λσi · ρσj = λσi(ρ) · σi+j for  all λ, ρ ∈ Mlt(Q) and i, j ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  By Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='3, σ3 = idMlt(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Should we have σ = idMlt(Q), then Rx = σ(Rx) = Mx =  RxLx would imply Lx = idQ, x = 1 and Q = 1, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Hence |H| = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Consider  a 3-Sylow subgroup S of G that contains H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Let P = Mlt(Q) ∩ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' As S contains H, the  underlying set of S is equal to P ×H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Since | Mlt(Q)| = |Q|·| Inn(Q)| in any loop, and since  17  3 divides |Q| here, it follows that 3 divides | Mlt(Q)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Then |S| > 3 and P > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' As Mlt(Q)  is normal in G, the group P = Mlt(Q) ∩ S is normal in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' The center Z(P) ̸= 1 of P is also  normal in S, being a characteristic subgroup of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' The automorphism σ acts by conjugation  on Z(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Since σ3 = idMlt(Q) and Z(P) ̸= 1 is a 3-group, the conjugation by σ also fixes  some 1 ̸= ψ ∈ Z(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Passing to a suitable power of ψ, we can assume that |ψ| = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Note  that ψσ = σψ = σ(ψ)σ implies σ(ψ) = ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  Write ψ = Lxϕ for some x ∈ Q and ϕ ∈ Inn(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Let c be the companion of ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' By  Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='3, Lxϕ = ψ = σ(ψ) = σ(Lxϕ) = σ(Lx)σ(ϕ) = RxR−1  c ϕ, hence Lx = RxR−1  c ,  c = 1 (so ϕ is an automorphism), Lx = Rx and x ∈ Com(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' By Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='2 and Lemma  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='1, x3 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' If x ̸= 1 then ⟨x⟩ is the sought-after subloop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Else x = 1 and ψ = ϕ is an  automorphism of order 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Since 3 divides |Q|, it then also divides the order of the proper  subloop Fix(ϕ) = {u ∈ Q : ϕ(u) = u}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  □  We are ready to prove the Cauchy property for p = 3 in Moufang loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Let Q be a  finite Moufang loop whose order is divisible by 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' We proceed by induction on |Q|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Let  N = Nuc(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' If N = 1 then Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='4 yields a proper subloop of Q whose order is  divisible by 3, and we are done by the induction assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Suppose from now on that  N ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' If 3 divides |N| then N contains an element of order 3 since N is a group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Else 3  divides |Q/N|, and by the induction assumption there is x ∈ Q such that xN is of order 3  in Q/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Since xN is the homomorphic image of x under the natural projection modulo N,  the order of xN divides the order of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' A suitable power of x is then of order 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  References  [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Albert, Quasigroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=', Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' 55 (1944), 401–419.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  [2] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
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+page_content=' Paige, A class of simple Moufang loops, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' 7 (1956), 471–482.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  [23] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Rotman, An introduction to the theory of groups, fourth edition, Graduate Texts in Mathematics,  Springer-Verlag, 1995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  [24] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Stanovsk´y and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Vojtˇechovsk´y, Commutator theory for loops, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Algebra 399 (2014), 290–322.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  [25] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Stanovsk´y and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' Vojtˇechovsk´y, Abelian extensions and solvable loops, Results Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' 66 (2014),  367–384.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='  (Dr´apal) Dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' of Mathematics, Charles University, Sokolovsk´a 83, 186 75 Praha 8, Czech  Republic  Email address, Dr´apal: drapal@karlin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='mff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='cuni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='cz  (Vojtˇechovsk´y) Dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' of Mathematics, University of Denver, 2390 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=' York St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content=', Denver, CO  80208, USA  Email address, Vojtˇechovsk´y: petr@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='du.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
+page_content='edu  19' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfFgYY/content/2301.03646v1.pdf'}
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+Frequency Band Analysis of Nonstationary Multivariate
+Time Series ∗†
+Raanju R. Sundararajan‡
+and
+Scott A. Bruce§
+‡Department of Statistical Science, Southern Methodist University, Dallas, Texas, USA
+§Department of Statistics, Texas A&M University, College Station, Texas, USA
+Abstract
+Information from frequency bands in biomedical time series provides useful summaries
+of the observed signal. Many existing methods consider summaries of the time se-
+ries obtained over a few well-known, pre-defined frequency bands of interest. However,
+these methods do not provide data-driven methods for identifying frequency bands that
+optimally summarize frequency-domain information in the time series. A new method
+to identify partition points in the frequency space of a multivariate locally stationary
+time series is proposed. These partition points signify changes across frequencies in
+the time-varying behavior of the signal and provide frequency band summary measures
+that best preserve the nonstationary dynamics of the observed series. An L2 norm-
+based discrepancy measure that finds differences in the time-varying spectral density
+matrix is constructed, and its asymptotic properties are derived. New nonparametric
+bootstrap tests are also provided to identify significant frequency partition points and
+∗AMS subject classification. Primary: 62M10. Secondary: 62M15.
+†Keywords and phrases: Multivariate time series, nonstationary, frequency domain, spectral matrix,
+electroencephalography (EEG)
+‡Email: rsundararajan@smu.edu (both authors contributed equally to this work)
+§Email: sabruce@tamu.edu
+1
+arXiv:2301.03664v1  [stat.ME]  9 Jan 2023
+
+to identify components and cross-components of the spectral matrix exhibiting changes
+over frequencies. Finite-sample performance of the proposed method is illustrated via
+simulations. The proposed method is used to develop optimal frequency band summary
+measures for characterizing time-varying behavior in resting-state electroencephalog-
+raphy (EEG) time series, as well as identifying components and cross-components
+associated with each frequency partition point.
+1
+Introduction
+Frequency-domain information contained in biomedical time series provides important sum-
+maries leading to meaningful physiological interpretations.
+Oscillatory behavior in non-
+stationary time series are often characterized by the time-varying spectral matrix, which
+is a time- and frequency-dependent matrix-valued function. Rather than analyzing the full
+time-varying spectral matrix, practitioners often partition frequencies into a few well-known,
+pre-defined bands that serve as a pseudo partition of the frequency space. Numerical sum-
+maries of the spectral matrix are then computed within these frequency bands, and these
+summaries have been shown to provide meaningful biological interpretations of the observed
+signal. Examples of biomedical time series where frequency bands are identified and asso-
+ciated with physiological characteristics include heart rate variability (HRV) (Hall et al.,
+2004), local field potential (LFP) (Liu and Newsome, 2006), resting-state functional mag-
+netic resonance imaging (rs-fMRI) (Yuen et al., 2019) and electroencephalography (EEG)
+(Klimesch, 1999).
+In analyzing neuroimaging data, several works have identified frequency bands of interest
+that carry meaningful biological interpretations. Biswal et al. (1995) analyze correlations in
+resting state fMRI that characterize functional connectivity in the brain. In their study, high
+2
+
+correlations in low frequency oscillations (< 0.1 Hz) are detected within the sensorimotor
+cortex of the brain and also with other regions of the brain that associate with motor func-
+tion. Lowe et al. (2000) is another work that computes cross-correlations in low frequency
+oscillations (< 0.08 Hz) of the fMRI signal between widely separated anatomic regions of the
+brain. To detect regions of the brain that exhibit strong cross-correlations, their work uses
+these low frequency oscillations to discriminate between different memory tasks. In Yuen
+et al. (2019), frequency partitions within the 0.01-0.25 Hz band were obtained for rs-fMRI
+signals and the oscillations from each of these partitions were associated with different bio-
+logical functions such as respiration, pulse, and vasomotor activity. In Henrie and Shapley
+(2005), spectral analysis is performed on a LFP signal gathered from the primary visual cor-
+tex of a macaque. The power of the spectral density of this LFP signal is computed at the
+low (< 10 Hz), gamma (25 − 240 Hz) and broad (8 − 240 Hz) frequency bands, and changes
+in power are analyzed in response to various stimuli. In analyzing EEG data, many methods
+resort to analyzing oscillations for known frequency bands, such as Alpha, Beta, Gamma
+and Delta, and associate these oscillations with various biological functions. As an example,
+Newson and Thiagarajan (2019) provides a review of several methods that utilize pre-defined
+and well-known frequency bands to analyze resting state EEG data from individuals with
+neurological disorders. In all of the above cited works, irrespective of the modality of choice
+(fMRI, LFP or EEG), the selection of frequency bands used to summarize oscillatory pat-
+terns is often predetermined either through manual inspection of the time series or through
+historical precedents. Doppelmayr et al. (1998) mention the possibility of differences in the
+endpoints of the Alpha frequency band in EEG data from different individuals, and Ghazi
+et al. (2021) is another example that discusses differences in the peak frequency in the Al-
+pha band of EEG data among individuals from different sexes. Analyses like these point to
+3
+
+the need for a data-driven method for automatically identifying frequency bands that best
+describe changes in the frequency space of neuroimaging time series data.
+Several methods in the signal processing and applied harmonic analysis literature pur-
+sue time-frequency analysis of univariate nonstationary time series. Time-frequency analysis
+helps understand the time-varying properties of different frequency components of the time
+series. For obtaining a time-frequency description of the observed signal, Flandrin (1998)
+describes a few solutions, such as the short-time Fourier transform and wavelet transform,
+along with their properties. While most methods aim at estimating the time-varying am-
+plitudes of the signal, very few methods discuss estimation of changes happening in the
+frequency space of nonstationary time series. An online algorithm for detecting a single
+prominent time-varying frequency band is proposed in Tiganj et al. (2012). In their work, at
+each local time window, the prominent frequency band is estimated by using a band-limited
+signal as the input. In Cohen et al. (2016), the observed signal is assumed to be composed
+of multiple, uncorrelated cyclostationary processes (Gardner et al., 2006), and they provide
+an algorithm for detecting multiple peaks in the spectral density of the signal.
+In the statistical literature, time-frequency analysis of nonstationary time series has been
+studied using evolutionary spectra (Priestley, 1965) and the time-varying spectral density
+of locally stationary processes (Dahlhaus, 1997).
+Most methods utilize the time-varying
+spectral matrix or its finite sample estimates for detecting changes in the time space, i.e.,
+temporal change point detection; see Adak (1998), Ombao et al. (2005), Kirch et al. (2015),
+Preuss et al. (2015) for examples in univariate and multivariate locally stationary time
+series.
+In Schr¨oder and Ombao (2019), temporal change points are obtained over user-
+specified frequency bands and the obvious limitation in their work is the need for the user
+to specify the frequency bands over which the temporal change points are found. Bruce
+4
+
+et al. (2020) is a recent work that considers univariate locally stationary time series and
+aims at finding partition points in the frequency space that best describe the nonstationary
+characteristics of the observed time series. Their work constructs an estimate of the time-
+varying spectral density by assuming a piecewise stationary approximation and applying a
+multitaper estimator of the spectral density. A CUSUM statistic designed to spot changes
+in the frequency space is formulated based on this estimator, and significant partition points
+are obtained.
+In this work, we propose a new method to identify partition points in the frequency space of a
+multivariate locally stationary time series. These partition points are detected such that they
+best preserve the nonstationary dynamics of the time-varying spectral density matrix. To
+detect these partition points, an L2 norm-based discrepancy measure is constructed using
+the time-varying spectral matrix. This measure computes differences in the time-varying
+spectral matrix over local neighborhoods of frequencies and its asymptotic properties are
+derived. With the discrepancy measure viewed as our test statistic, a new nonparametric
+bootstrap test is proposed to obtain the critical values. The proposed bootstrap procedure
+is also further utilized in identifying the components and cross-components contributing to
+the changes in the frequency space. The proposed method is motivated by an application in
+analyzing resting-state electroencephalography (EEG) time series from 14 individuals. The
+experiment involves a straightforward eyes-open/eyes-closed scheme with the signal being
+recorded from 16 electrodes (Cattan et al., 2018). After down-sampling the original signal
+to the rate of 64 Hz, the left plots in Figure 1 depicts the standardized EEG time series
+from the Oz occipital channel in participants 2 and 13. These illustrated signals cover a time
+segment of the entire experiment that includes a total of five consecutive blocks, with each
+block being an eyes-closed or eyes-open scenario. The right plots in the same figure show
+5
+
+the estimated time-varying spectral densities for this channel, and the time-varying behavior
+can be witnessed around the 10 Hz frequency.
+This work addresses several critical aspects of frequency-domain analysis of nonstationary
+multivariate signals. First, a data-driven estimator for frequency bands of interest for each
+individual is developed, as opposed to investigating commonly known pre-defined frequency
+bands that are assumed to be the same for all individuals. Second, the proposed method is
+used to better understand variability in estimated partition points across different individuals
+from a population of interest. Finally, it is typically not the case that all components of the
+spectral matrix exhibit changes across frequencies for each partition point. Accordingly, the
+proposed method allows for detection of the particular components and cross-components
+of the time series that are contributing to each estimated frequency partition point. With
+the proposed method being uniquely positioned to address such problems, in Section 4 we
+illustrate and discuss the application in detail.
+6
+
+−1
+0
+1
+0.00
+0.25
+0.50
+0.75
+1.00
+Time
+Participant 2 (Oz)
+Participant 2 (Oz) Standardized EEG
+(a)
+0.2
+0.4
+0.6
+0.8
+5
+10
+15
+20
+25
+30
+Participant 2 (Oz)
+Time
+Frequency (Hz)
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+(b)
+−2
+−1
+0
+1
+2
+0.00
+0.25
+0.50
+0.75
+1.00
+Time
+Participant 13 (Oz)
+Participant 13 (Oz) Standardized EEG
+(c)
+0.2
+0.4
+0.6
+0.8
+5
+10
+15
+20
+25
+30
+Participant 13 (Oz)
+Time
+Frequency (Hz)
+0.0
+0.5
+1.0
+1.5
+2.0
+(d)
+Figure 1: Oz channel EEG for participants 2 and 13 and corresponding estimated spectral
+densities.
+The rest of the paper is organized as follows. Section 2 describes the proposed method in
+detail along with the theoretical results. Section 3 discusses the finite sample performance of
+the method using a few simulation schemes. The application to modeling resting-state EEG
+time series data is presented in Section 4. The concluding remarks are given in Section 5.
+2
+Methodology
+In this section, we describe our proposed method to find frequency partition points in the
+time-varying spectral matrix of a locally stationary process. First, the model for the locally
+7
+
+stationary process and frequency-banded time-varying behavior is introduced. A discrep-
+ancy measure and estimator is then proposed for identifying frequency partition points that
+characterize the frequency band structure. Resampling-based testing procedures are then
+developed to determine the significance of potential frequency partition points and to identify
+components associated with significant frequency partition points.
+2.1
+Model
+We begin with the definition of a locally stationary process using a two-sided MA(∞) repre-
+sentation followed by the required assumptions on the coefficient matrices (Dahlhaus, 1997,
+Dahlhaus, 2000). Let Xt,T be a p-variate locally stationary process given by
+Xt,T =
+∞
+�
+j=−∞
+Φt,T,jεt−j,
+t = 1, 2, . . . , T,
+(1)
+where εt are i.i.d Gaussian with unit variance matrix. The time-varying coefficient matrices,
+Φt,T,j, are assumed to be temporally smooth in the following sense.
+Assumption 1 (Temporal smoothness). There exists temporally smooth functions Φ : [0, 1]×
+Z → Rp×p such that
+∞
+�
+j=−∞
+sup
+t=1,2,...,T
+||Φt,T,j − Φ (t/T, j)||∞ = O (1/T) ,
+where || · ||∞ denotes the infinity norm, and
+∞
+�
+j=−∞
+sup
+u∈[0,1]
+||Φ(u, j)||∞|j| < ∞,
+8
+
+∞
+�
+j=−∞
+sup
+u∈[0,1]
+||Φ
+′(u, j)||∞|j| < ∞,
+∞
+�
+j=−∞
+sup
+u∈[0,1]
+||Φ
+′′(u, j)||∞ < ∞.
+With the above model, the p × p time-varying spectral matrix of Xt,T is given by
+f(u, ω) = 1
+2π
+�
+r,s
+Φ(u, r)Φ(u, s)
+′e−i2πω(r−s),
+(2)
+for ω ∈ [−1/2, 1/2]. With the above temporal smoothness assumption, the series Xt,T
+can also be expressed using the Cram´er representation with a time-varying transfer function
+matrix that can be approximated by a temporally-smooth matrix-valued function A : [0, 1]×
+[−1/2, 1/2] → Cp×p. Then, the time-varying spectral matrix can be written as f(u, ω) =
+A(u, ω)A∗(u, ω), where A∗(u, ω) denotes the conjugate transpose.
+To characterize the nonstationary behavior of the spectral matrix beyond a simple level
+shift, we consider the demeaned time-varying spectral matrix given by
+g(u, ω) = f(u, ω) −
+� 1
+0
+f(u, ω)du.
+(3)
+We will further assume that g(u, ω) has the following partition in the frequency space.
+Assumption 2 (Frequency-banded time-varying behavior).
+g(u, ω) =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+g1(u)
+if ω ∈ [0, ω1)
+g2(u)
+if ω ∈ [ω1, ω2)
+...
+gK+1(u)
+if ω ∈ [ωK, 1/2],
+9
+
+where PK = {ω1, ω2, . . . , ωK} denotes the set of K frequency partition points in the interval
+(0, 1/2), with K and PK being fixed and unknown.
+2.2
+Comparing frequency-specific time-varying behavior
+To determine both K and PK, we consider the following L2 norm-based discrepancy measure
+on the demeaned spectral matrix g(u, ω). For any ω ∈ (0, 1/2) we have,
+D(ω) = 1
+δ
+� 1
+u=0
+� δ
+λ=0
+||g(u, ω − λ) − g(u, ω + λ)||2 dλ du,
+(4)
+where || · ||2 denotes the squared L2 norm, and δ > 0. Observe that at any frequency ω, the
+nonnegative measure D(·) represents the difference in the demeaned time-varying spectral
+matrix in a local neighborhood of frequencies surrounding ω.
+In order to estimate the discrepancy measure in (4), we use the local periodogram IN(u, ω)
+(Dahlhaus, 1997) given by
+IN(u, ω) = JN(u, ω)J∗
+N(u, ω), with JN(u, ω) =
+1
+√
+2πN
+N−1
+�
+s=0
+X⌊uT⌋−N/2+1+s,T e−i2πωs,
+(5)
+where N is the length of the local neighborhood around time point u and J∗
+N(u, ω) denotes
+the conjugate transpose. Viewing IN(u, ω) as a local estimate of the time-varying spectral
+matrix f(u, ω), we obtain the estimated version of our measure in (4) as
+�D(ω) = 1
+T
+T
+�
+t=1
+1
+W
+W
+�
+k=1
+���
+����g
+�
+t/T, ω − λk
+�
+− �g
+�
+t/T, ω + λk
+����
+���
+2
+,
+(6)
+where �g(t/T, ω − λk) = IN(t/T, ω − λk) − 1
+T
+�T
+t1=1 IN(t1/T, ω − λk), and λk = k/N, k =
+1, 2, . . . , W. W corresponds to the parameter δ in (4), and in finite sample situations, we
+10
+
+resort to a multiscale approach in which we consider a range of plausible values for W. This
+approach is discussed in Section 2.5.
+To better understand the behavior of the measure �D(·) in (6), we consider the following
+sets.
+CN =
+�W
+N , W + 1
+N
+, . . . , 1
+2 − W
+N
+�
+,
+CN,1 =
+K
+�
+j=1
+�
+ωj − W
+N , . . . , ωj + W
+N
+�
+, and
+CN,2 = CN \ CN,1.
+(7)
+Here, CN denotes the set of all candidate partition points minus a neighborhood of
+length W/N on the ends. The set CN,1 takes the union of the neighborhoods of all partition
+points, where each neighborhood around a partition point is of radius W/N. The set CN,2
+denotes the set of all points located at a distance of least W/N from all partition points. To
+establish large sample properties of the measure �D(·), we assume frequency partition points
+are reasonably well-separated as follows.
+Assumption 3 (Separated partition points). There exists a constant c ∈ (0, 1/2) such that
+ω1 > c, ωK < 1
+2 − c,
+min
+j=1,2,...,K−1 |ωj − ωj+1| ≥ c.
+With these assumptions, the following theorem describes the asymptotic behavior of the
+proposed estimator �D(ω).
+Theorem 2.1. Suppose that the conditions stated in Assumptions 1-3 hold. Let N → ∞
+and T → ∞ such that N/T → 0 and W/N → c. Then,
+(a) for ω ∈ CN,2, �D(ω)
+p
+−→ 0,
+11
+
+(b) for ω ∈ PK = {ω1, ω2, . . . , ωK},
+�D(ω)
+p
+−→
+1
+2πc
+� 1
+0
+� 2πc
+0
+p
+�
+a,b=1
+�
+ga,b(u, ω−λ)−ga,b(u, ω+λ)
+��
+ga,b(u, ω−λ)−ga,b(u, ω+ λ)
+�∗
+dλ du,
+where a, b = 1, 2, . . . , p, and
+p
+−→ denotes convergence in probability.
+Proof is made available in the Appendix and relies on computing the asymptotic mean
+and variance of �D(·). This result means that the discrepancy measure will approach zero
+for frequencies not near a partition point and approach a positive constant for frequencies
+representing partition points. The above result also implies that a hypothesis test using �D(·)
+as the test statistic leads to a consistent test. Such a test would be defined in the following
+manner. Let ηT,ω be the threshold satisfying lim
+T→∞ ηT,ω ≥ B > 0, for some positive constant
+B. Then, a candidate partition point ω ∈ (0, 1/2) is designated as a partition point if
+�D(ω) > ηT,ω.
+(8)
+With the discrepancy measure �D(ω) as the test statistic, the threshold ηT,ω can be de-
+termined by the bootstrap procedure described in Section 2.3.
+2.3
+Bootstrapping locally stationary processes
+Here we describe the resampling procedure to obtain the threshold ηT,ω given in (8). This
+threshold can be viewed as a critical value of the discrepancy measure �D(ω) from (6), under
+the null hypothesis that ω is not a partition point. We resort to a nonparametric bootstrap
+method that generates samples under the null hypothesis in order to approximate ηT,ω and
+provide a corresponding p-value.
+Under the null hypothesis, assume Xt,T = Yt + σ(t/T)Zt where Yt is a second-order
+12
+
+stationary p-variate process, σ(t/T) is a p × p time-varying matrix, Zt is i.i.d. N(0, Ip),
+and Yt is independent of Zt.
+In this case, the spectral matrix of Xt,T is fX(t/T, ω) =
+fY (ω) + σ(t/T)σ′(t/T), where fY (ω) is the spectral matrix of Yt.
+However, fY (ω) does
+not appear in the demeaned time-varying spectral matrix (3) since fY (ω) =
+� 1
+0 fY (ω)du.
+Therefore, we can simplify computation for our bootstrap procedure by avoiding estimation
+of fY (ω) and instead assuming Xt,T = σ(t/T)Zt in order to generate bootstrap samples. Let
+ω ∈ CN be a candidate frequency partition point. The bootstrap procedure is carried out
+through the following steps.
+Step 1. Assume Xt,T = σ(t/T)Zt, where Zt is i.i.d.
+N(0, Ip).
+Compute the time-varying
+variance matrix estimator as
+�ΓX,0(u) = 1
+T
+T
+�
+t=1
+XtX
+′
+tKh(u − t/T),
+(9)
+where Kh(u) =
+1
+hK( u
+h), h denotes the bandwidth, and K(·) is a symmetric kernel
+function that integrates to 1.
+Step 2. Compute the time-varying square root matrix �σ(u) =
+�
+�ΓX,0(u)
+�1/2
+.
+Step 3. Obtain R bootstrap resamples as X(r)
+t,T = �σ(t/T)Z(r)
+t , where Z(r)
+t
+∼ N(0, Ip), t =
+1, 2, . . . , T, and r = 1, 2, . . . R.
+Step 4. With each resample X(r)
+t,T, t = 1, 2, . . . , T, compute the discrepancy measure �D(r)(ω).
+Step 5. Obtain the p-value of this test as
+�R
+r=1 1 �
+D(r)(ω)> �
+D(ω)
+R
+, where �D(ω) is the observed value
+of the test statistic.
+Recall that the assumption in Step 1 follows from the behavior of the demeaned spectral
+matrix g(u, ω) under the null hypothesis that ω is not a partition point.
+In this case,
+13
+
+g(u, ω) = g(u) which does not depend on frequency ω. Under this assumption, Steps 2-5
+then describe the resampling procedure that produces the required p-value. In Step 1, in
+finite sample situations discussed in Sections 3 and 4, we utilize the triangular kernel for
+K(·) with a bandwidth h = T −0.3.
+2.4
+Finding the number and locations of partition points
+Here we describe the steps to detect the locations and number of frequency partition points.
+Recall from (7), the set CN represents the set of candidate frequency partition points to
+search over. Our iterative procedure to detect the locations of partition points involves the
+following steps.
+Step 0. Initialize the set �CN = CN and �P = ∅, where �P is the final set of partition points
+returned by the procedure.
+Step 1. Compute the measure �D(ω), for every ω ∈ CN.
+Step 2. Find the point ω∗, where
+ω∗ = arg. max
+λ∈ �CN
+�D(λ).
+(10)
+Step 3. Determine the p-value for testing if ω∗ is a partition point using the resampling pro-
+cedure described in Section 2.3. If found to be significant, set �P = �P �{ω∗}, and set
+�CN = �CN \ {ω∗ − W
+N , . . . , ω∗ + W
+N }.
+Step 4. Repeat Steps 2 and 3 until the significance test in Step 3 fails to return a significant
+frequency partition point.
+The above iterative procedure results in the set �P that contains the final set of frequency
+partition points, and the cardinality of this set provides an estimate �K of the number of
+14
+
+partition points. Next, we provide a large sample result on the estimate �K of the number of
+partition points obtained through this procedure.
+Theorem 2.2. Suppose that the conditions stated in Assumptions 1-3 hold. Let N → ∞
+and T → ∞ such that N/T → 0 and W/N → c. Then,
+P( �K ̸= K)
+T→∞
+−→ 0.
+(11)
+Proof. See Appendix for details of the proof.
+2.5
+Choice of W: length of neighborhood of frequencies
+The choice of W used to estimate the discrepancy measure in (6) depends on the nature
+and magnitude of changes in the frequency space. Smaller changes need larger values of
+W for detection while larger changes can be identified even with smaller values of W. In
+order to detect partition points associated with both small and large changes, we adopt a
+multiscale approach (Messer et al., 2014). In practice, a sequence of q choices for W given
+by Wmin < W1 < W2, ... < Wq < Wmax is considered. Let �Pi denote the set of partition
+points estimated using the iterative procedure from Section 2.4 with neighborhood length
+choice Wi. Set P = �P1, where P denotes the final set of estimated change points returned
+by our multiscale approach. For any point ω ∈ �P2, ω is added to the set P only if it does
+not belong to a W2-neighborhood of any of the existing points in the set P. The procedure
+is successively moved forward until all choices for W have been considered. Algorithm 1
+provides the pseudocode that illustrates the full implementation of the multiscale frequency
+band estimation procedure.
+It should be noted that Wmin should be selected small enough to ensure partition points
+associated with more subtle changes are detected, but not too small, which may lead to false
+15
+
+Algorithm 1: Multiscale Frequency Band Estimation
+�CN ← CN =
+� W1
+N , W1+1
+N
+, . . . , 1
+2 − W1
+N
+�
+�P ← ∅
+for W ∈ {W1, W2, . . . , Wq} do
+�CN ← �CN \ { W1
+N , . . . , W
+N , 1
+2 − W
+N , . . . , 1
+2 − W1
+N }
+for λ ∈ �P do �CN ← �CN \ {λ − W
+N , . . . , λ + W
+N }
+stop ← 0
+while stop = 0 and �CN ̸= ∅ do
+ω∗ ← arg maxλ∈ �CN �D(λ) where �D(λ) is calculated by (6) given W
+Determine p-value for �D(ω∗) using bootstrap procedure (see Section 2.3)
+if p−value is significant then
+�P ← �P �{ω∗}
+�CN ← �CN \ {ω∗ − W
+N , . . . , ω∗ + W
+N }
+else stop ← 1
+�K = | �P|
+return �P, �K
+positives. In finite sample cases in Section 3, we present the performance results for different
+choices of Wmin. Based on our simulation results, Wmin = N/8 provides the best estimation
+performance for the simulation settings considered herein.
+In scenarios where a single frequency neighborhood length choice W must be selected,
+one can take the largest value of W ∈ {W1, W2, ..., Wq} for which there is an addition of
+a partition point to the set P in the iterative procedure described above. More precisely
+W = Wi∗, where i∗ is the largest value in the set {1, 2, ..., q} for which the iterative procedure
+described above adds a point to the set P during iteration i∗ (i.e., with neighborhood length
+choice Wi∗). In case there is no point added to the set P for any choice Wi, i = 1, 2, ..., q, we
+set W = Wq.
+16
+
+2.6
+Finding components responsible for partition points
+In this section, we present a new technique to identify the components and cross-components
+of the multivariate series that significantly contribute to each of the partition points iden-
+tified in the frequency space. For every identified partition point, a resampling procedure
+for finding the components significantly contributing to the change characterized by the fre-
+quency partition point is discussed. The proposed approach, similar to the bootstrap method
+given in Section 2.3, generates samples under the null hypothesis, and results in p-values for
+every component and cross-component of the series Xt,T.
+Let ωc be a partition point detected by our method. With every component (a, b), where
+1 ≤ a ≤ b ≤ p, the goal is to estimate the p-value corresponding to the null hypothesis that
+component (a, b) of the series Xt,T does not have a significant contribution to the partition
+at frequency ωc. The component-specific test statistic �D(a,b) is then written as
+�D(a,b)(ωc) = 1
+T
+T
+�
+t=1
+1
+W
+W
+�
+k=1
+����ga,b
+�
+t/T, ωc − λk
+�
+− �ga,b
+�
+t/T, ωc + λk
+����
+2
+,
+(12)
+where �g(t/T, ωc − λk) = IN(t/T, ωc − λk) − 1
+T
+�T
+t1=1 IN( t1
+T , ωc − λk) and λk =
+k
+N , k =
+1, 2, . . . , W. Note that �ga,b is component (a, b) of the p × p matrix �g(·). The p-value is
+obtained through the following steps.
+Step 1. Assume Xt,T = σ(t/T)Zt, where Zt is i.i.d.
+N(0, Ip).
+Compute the time-varying
+variance matrix estimator as
+�ΓX,0(u) = 1
+T
+T
+�
+t=1
+XtX
+′
+tKh(u − t/T),
+(13)
+where Kh(u) =
+1
+hK( u
+h), h denotes the bandwidth, and K(·) is a symmetric kernel
+function which integrates to 1.
+17
+
+Step 2. Compute the time-varying square root matrix �σ(u) =
+�
+�ΓX,0(u)
+�1/2
+.
+Step 3. Obtain R bootstrap resamples as X(r)
+t,T = �σ(t/T)Z(r)
+t , where Z(r)
+t
+∼ N(0, Ip), t =
+1, 2, . . . , T, and r = 1, 2, . . . R.
+Step 4. With each resample X(r)
+t,T, t = 1, 2, . . . , T, compute the component-specific test statistic
+�D(r)
+(a,b)(ωc).
+Step 5. Obtain the p-value of this test as
+�R
+r=1 1 �
+D(r)
+(a,b)(ωc)> �
+D(a,b)(ωc)
+R
+, where �D(a,b)(ω) is the observed
+value of the test statistic.
+The above procedure is applied to every component and cross-component (a, b), 1 ≤ a ≤
+b ≤ p, of the series Xt,T. The components and cross-components that carry a significant
+p-value are deemed as the components responsible for the partition at frequency ωc. An
+illustration of this procedure can be seen in the application presented in Section 4. In order
+to account for simultaneous testing of multiple components, multiple testing adjustments
+can and should be used to control the experiment-wide error rate; for example, a Bonferroni
+adjustment is implemented for the application presented in Section 4.
+3
+Simulation study
+Performance of the proposed method is assessed through a few simulation examples. The
+five simulation schemes are described first followed by the presentation of the performance
+results.
+The first scheme (WN1B) is a multivariate white noise model with no partition points in
+the frequency space. This setting is considered to ensure that the method does not produce
+an unreasonable number of false positives. The second scheme (L3B) considers partition
+18
+
+points at frequencies 0.15 and 0.35, with spectral density f2(u, ω) exhibiting a linear trend
+in time u ∈ (0, 1). The third scheme (S3B) also considers partition points at frequencies
+0.15 and 0.35, but with spectral density f3(u, ω) exhibiting a non-linear trend in time. The
+second and third schemes illustrate performance of the method in capturing partition points
+associated with both linear and nonlinear time-varying dynamics. The fourth scheme (M3B-
+1), with partition points at frequencies 0.15 and 0.35, considers a mixture of time series
+exhibiting linear and non-linear trends from the models L3B and S3B, respectively. This
+setting considers performance of the method when components of the series have a similar
+frequency band structure, but differing time-varying dynamics across components. The fifth
+scheme (M3B-2) again considers a mixture of linear and non-linear trends in the spectral
+density, but assumes only 20% of the p components in Xt,T contribute to the partition at
+frequency 0.15, whereas the remaining 80% of the p components contribute to the partition
+at frequency 0.35. This setting is the most challenging and represents both differing time-
+varying dynamics and differing partition points across components.
+1. White noise (WN1B). Xk,t = z1,t+k−1 for k = 1, 2, . . . , p, and z1,t has time-varying
+spectral density f1(u, ω) given by
+f1(u, ω) = 1 for ω ∈ (0, 0.5)
+2. Linear, 3-Bands (L3B). Xk,t = z2,t+k−1 for k = 1, 2, . . . , p, and z2,t has time-varying
+spectral density f2(u, ω) given by
+19
+
+f2(u, ω) =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+10 − 9u for ω ∈ (0, 0.15)
+1 for ω ∈ [0.15, 0.35)
+1 + 9u for ω ∈ [0.35, 0.5)
+3. Sinusoidal, 3-Bands (S3B). Xk,t = z3,t+k−1 for k = 1, 2, . . . , p, and z3,t has time-
+varying spectral density f3(u, ω) given by
+f3(u, ω) =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+10 + 10 sin(4πu − π/2) for ω ∈ (0, 0.15]
+5 + 5 cos(4πu) for ω ∈ (0.15, 0.35]
+8.5 + 8.5 sin(3πu − π/16) for ω ∈ (0.35, 0.5)
+4. Linear and Sinusoidal, 3-Bands, Mixture (M3B-1). Xk,t = z2,t+k−1 for k =
+1, 2, . . . , ⌊p/2⌋, and Xk,t = z3,t+k−⌊p/2⌋−1 for k = ⌊p/2⌋ + 1, . . . , p. Here, the series z2,t
+and z3,t are given by the schemes L3B and S3B described above.
+5. Linear and Sinusoidal, 3-Bands, Differing Proportions (M3B-2).
+Xk,t =
+z4,t+k−1 for k = 1, 2, . . . , ⌊0.2p⌋, and Xk,t = z5,t for k = ⌊0.2p⌋ + 1 . . . , p. Here, z4,t and
+z5,t have time-varying spectral densities f4(u, ω) and f5(u, ω), respectively.
+f4(u, ω) =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+10 − 9u for ω ∈ (0, 0.15)
+1 for ω ∈ [0.15, 0.5)
+f5(u, ω) =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+5 + 5 cos(4πu) for ω ∈ (0, 0.35]
+8.5 + 8.5 sin(3πu − π/16) for ω ∈ (0.35, 0.5)
+20
+
+To assess the performance of the proposed method, we first present the results on esti-
+mating the true number of frequency bands, i.e., the quantity K + 1, where K is the true
+number of partition points defined in (2). Table 1 reports the estimated mean number of
+frequency bands for the five simulation schemes based on 100 replications. Note that the
+bootstrap procedure from Section 2.4 is used for estimating the number and locations of the
+partition points. In implementing this procedure, the triangular kernel is used as the kernel
+choice K(·), and the bandwidth is h = T −0.3. The choice for the frequency neighborhood
+length W involves the multiscale approach described in Section 2.5, and we consider an
+equally-spaced sequence Wmin = N
+8 < W1 < W2, ... < Wq < Wmax = N
+4 , with N = T 0.7,
+and the sample size T ∈ {200, 500, 1000}. The results in Table 1 show that as the sample
+size increases, accuracy in estimating the number of partition points increases for all five
+simulation schemes. Table 2 presents estimation results at a fixed sample size T = 1000, but
+for different choices of Wmin used in the multiscale procedure from Section 2.5. It is seen
+that in most schemes the number of frequency bands estimated increases as Wmin decreases,
+with the best results at parameter choice Wmin = N/8.
+p
+T
+WN1B
+L3B
+S3B
+M3B-1
+M3B-2
+10
+200
+1(0)
+2.24(0.55)
+2.15(0.36)
+2.09(0.35)
+2.14(0.64)
+500
+1(0)
+2.94(0.34)
+2.56(0.50)
+2.47(0.52)
+2.88(0.57)
+1000
+1(0)
+3.06(0.24)
+2.96(0.24)
+2.92(0.31)
+3.09(0.35)
+15
+200
+1(0)
+2.34(0.52)
+2.18(0.39)
+2.07(0.29)
+2.25(0.59)
+500
+1(0)
+2.99(0.30)
+2.61(0.55)
+2.53(0.52)
+2.92(0.51)
+1000
+1(0)
+3.05(0.22)
+3.00(0.20)
+2.98(0.20)
+3.12(0.41)
+Table 1: Mean(sd) for estimated number of frequency bands, ˆK + 1, for 100 replications
+(Wmin = N/8). True value K + 1 is 1 for WN1B, and 3 for all other schemes.
+Next, we present the proportion of the 100 replications that result in correct detection.
+At any given replication, a correct detection occurs when the proposed method identifies the
+correct number of frequency partition points and all estimated partition points are within a
+21
+
+p
+Wmin
+WN1B
+L3B
+S3B
+M3B-1
+M3B-2
+10
+N/8
+1(0)
+3.06(0.24)
+2.96(0.24)
+2.92(0.31)
+3.09(0.35)
+N/10
+1(0)
+3.64(0.64)
+3.43(0.52)
+3.13(0.34)
+3.77(0.47)
+N/12
+1.07(0.26)
+3.36(0.58)
+4.52(0.56)
+4.11(0.71)
+4.26(0.66)
+15
+N/8
+1(0)
+3.05(0.22)
+3.00(0.20)
+2.98(0.20)
+3.12(0.41)
+N/10
+1(0)
+3.69(0.66)
+3.50(0.54)
+3.22(0.42)
+3.87(0.42)
+N/12
+1.06(0.24)
+3.44(0.59)
+4.68(0.51)
+4.32(0.63)
+4.41(0.64)
+Table 2: Mean(sd) for estimated number of frequency bands, ˆK + 1, for 100 replications
+(T = 1000). True value K + 1 is 1 for WN1B, and 3 for all other schemes.
+distance of ζ from the true partition point. Table 3 presents the correct detection rate based
+on 100 replications for the four simulation schemes with more than one frequency band.
+We observe that in almost all cases, as the length of the time series increases, the correct
+detection rate increases.
+The scheme M3B-2 exhibits the lowest correct detection rate relative to other settings.
+Recall that under the M3B-2 setting, only 20% of the p components of the multivariate series
+Xt,T contribute to the partition at frequency 0.15, and the remaining 80% of the components
+contribute to the partition at frequency 0.35. With weaker contribution from components
+towards the frequency partition point 0.15, the proposed method requires much longer time
+series (T) to improve the correct detection rate.
+p
+T
+L3B
+S3B
+M3B-1
+M3B-2
+10
+200
+0.3
+0.15
+0.11
+0.17
+500
+0.88
+0.53
+0.45
+0.36
+1000
+0.94
+0.93
+0.90
+0.42
+15
+200
+0.36
+0.18
+0.08
+0.20
+500
+0.91
+0.52
+0.51
+0.41
+1000
+0.95
+0.96
+0.96
+0.39
+Table 3: Correct Detection. Proportion of 100 replications that correctly estimate the number
+of bands, K +1, and the distance between the estimated and true partition points is no more
+than ζ. Here ζ = 1/16 and Wmin = N/8.
+Finally, in Table 4, we provide the results on correct detection rates for different choices
+of ζ and fixed time series length T = 1000 for the four settings with more than one frequency
+22
+
+p
+ζ
+L3B
+S3B
+M3B-1
+M3B-2
+10
+1/12
+0.94
+0.93
+0.90
+0.73
+1/16
+0.94
+0.93
+0.90
+0.42
+1/24
+0.94
+0.93
+0.90
+0.27
+15
+1/12
+0.95
+0.96
+0.96
+0.67
+1/16
+0.95
+0.96
+0.96
+0.39
+1/24
+0.95
+0.96
+0.96
+0.25
+Table 4: Correct Detection. Proportion of 100 replications that correctly estimate the number
+of bands, K +1, and the distance between the estimated and true partition points is no more
+than ζ. Here T = 1000 and Wmin = N/8.
+band. We notice that for the first three simulation schemes, the correct detection rate is not
+sensitive to the choice of ζ. Here again, the scheme M3B-2 sees the lower correct detection
+rates. This can be attributed to the fact that fewer components (only 20%) are contributing
+to the partition at frequency 0.15, and hence a much larger sample size is needed to achieve
+higher correct detection rates.
+4
+Application
+To illustrate the usefulness of the proposed method in analyzing multivariate biomedical time
+series, we turn to frequency band analysis of EEG signals. Frequency bands are commonly
+used in the scientific literature to generate summary measures of the EEG signal, so a
+principled approach to frequency band estimation would be a welcomed development.
+Our analysis considers 16-channel EEG signals from 14 participants in a simple resting-
+state eyes-open, eyes-closed experimental protocol (Cattan et al., 2018). For computational
+efficiency, signals are standardized and downsampled to a sampling rate of 64 Hz and subset
+to include 5 consecutive blocks alternating between eyes-closed and eyes-open conditions
+lasting approximately 15 seconds per block. This produces, for each participant, a time series
+approximately 4800 observations in length. For illustration, Figure 1 displays standardized
+23
+
+EEG time series from the Oz occipital channel in participants 2 and 13. The time-varying
+behavior can be witnessed around the 10 Hz frequency, which corresponds to the traditional
+Alpha frequency band (8-12 Hz) and is associated with the alternating eyes-closed and eyes-
+open conditions.
+In practice, the detection of Alpha waves is a useful indicator of stress levels, concentra-
+tion, relaxation, or mental load (Banquet, 1973; Antonenko et al., 2010). However, Alpha
+power may vary in both its peak frequency and range of frequencies across participants (Dop-
+pelmayr et al., 1998; Ghazi et al., 2021), so a data-driven approach is essential for accurately
+characterizing Alpha power across different individuals. To illustrate this, Figure 2 displays
+the estimated frequency partition points using the proposed method applied on each partici-
+pant’s EEG signal separately. A triangular kernel is used as the kernel choice K(·) with band-
+width h = T −0.3, and the multiscale approach described in Section 2.5 was used considering
+a sequence of equally-spaced values Wmin = N
+12 < W1 < W2 < W3 < W4 < W5 < Wmax = N
+4 ,
+with N = T 0.7.
+14
+13
+12
+11
+10
+9
+8
+7
+6
+5
+4
+3
+2
+1
+0
+6
+12
+18
+24
+30
+Frequency
+Participant
+Estimated Frequency Partition Points by Participant
+Figure 2: Frequency partition points estimated using proposed methodology.
+24
+
+0.2
+0.4
+0.6
+0.8
+5
+10
+15
+20
+25
+30
+Participant 13 (Oz)
+Time
+Frequency
+0.0
+0.5
+1.0
+1.5
+2.0
+(a)
+0.2
+0.4
+0.6
+0.8
+5
+10
+15
+20
+25
+30
+Participant 13 (P7)
+Time
+Frequency
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+(b)
+Figure 3: Local periodogram for two components and estimated frequency bands.
+While frequency bands identified are similar across participants, some participants (e.g.
+Participant 13) exhibit Alpha power in a slightly higher band of frequencies than the con-
+ventional 8-12 Hz band used in practice, which demonstrates the advantage of the proposed
+data-driven frequency band estimator. For further illustration, Figure 3 displays the esti-
+mated spectral density for Participant 13 in two channels, one from the occipital region (Oz)
+and another from the parietal region (P7), along with the estimated frequency partition
+points (green lines). It is not surprising to find that the time-varying behavior in this band
+is prominent for these two channels, since the parietal and occipital brain regions are known
+to exhibit strong Alpha band power (Pfurtscheller et al., 1996).
+Next, in addition to identifying the frequency bands, the proposed method also provides
+a data-driven approach to identifying components and cross-components of the multivariate
+signal that are significantly associated with each identified frequency partition point. Ap-
+plying the bootstrap technique described in Section 2.6, we can identify components that
+significantly contribute to the upper and lower Alpha power frequency partition points iden-
+tified by the proposed method for Participants 2 and 13 (see Figure 4).
+Unsurprisingly, we find that channels from the parietal (P7, P3, Pz, P4, P8) and occipital
+(O1, Oz, O2) regions play a major role in establishing both the upper and lower bounds of the
+25
+
+FP1
+FP2
+FC5
+FC6
+FZ
+T7
+CZ
+T8
+P7
+P3
+PZ
+P4
+P8
+O1
+Oz
+O2
+FP1
+FP2
+FC5
+FC6
+FZ
+T7
+CZ
+T8
+P7
+P3
+PZ
+P4
+P8
+O1
+Oz
+O2
+Participant 2 Frequency 6 Hz
+FP1
+FP2
+FC5
+FC6
+FZ
+T7
+CZ
+T8
+P7
+P3
+PZ
+P4
+P8
+O1
+Oz
+O2
+FP1
+FP2
+FC5
+FC6
+FZ
+T7
+CZ
+T8
+P7
+P3
+PZ
+P4
+P8
+O1
+Oz
+O2
+Participant 2 Frequency 11.7 Hz
+FP1
+FP2
+FC5
+FC6
+FZ
+T7
+CZ
+T8
+P7
+P3
+PZ
+P4
+P8
+O1
+Oz
+O2
+FP1
+FP2
+FC5
+FC6
+FZ
+T7
+CZ
+T8
+P7
+P3
+PZ
+P4
+P8
+O1
+Oz
+O2
+Participant 13 Frequency 7.3 Hz
+FP1
+FP2
+FC5
+FC6
+FZ
+T7
+CZ
+T8
+P7
+P3
+PZ
+P4
+P8
+O1
+Oz
+O2
+FP1
+FP2
+FC5
+FC6
+FZ
+T7
+CZ
+T8
+P7
+P3
+PZ
+P4
+P8
+O1
+Oz
+O2
+Participant 13 Frequency 12.8 Hz
+Figure 4: Components significantly contributing to estimated frequency partition point (red)
+for two participants and two partition points.
+26
+
+Alpha band. However, there are two additional findings of interest. First, in differentiating
+the time-varying dynamics of lower frequency power vs. Alpha band power (i.e., components
+responsible for frequency partition points at 6 Hz and 7.3 Hz for Participants 2 and 13
+respectively), the prefrontal channels (FP1, FP2) are also involved. The prefrontal region
+has been shown to be the dominant source of power in lower frequencies (< 4 Hz) during both
+eyes-open and eyes-closed resting state conditions. Further, low frequency power has been
+shown to increase from the eyes-closed to eyes-open condition (Chen et al., 2008). Since the
+prefrontal region is not meaningfully involved in Alpha power fluctuations, the differences in
+the time-varying dynamics for lower frequencies vs. Alpha power frequencies in the prefrontal
+region are shown to be significant using the proposed method. More precisely, our method
+shows that the prefrontal region contributes towards the identification of the lower Alpha
+band frequency partition points (6 Hz and 7.3 Hz for Participants 2 and 13 respectively), but
+not the upper Alpha band frequency partition points (11.7 Hz and 12.8 Hz for Participants 2
+and 13 respectively). Second, the cross-components between the parietal and occipital region
+channels and the prefrontal, frontal, and central region channels contribute differently to the
+lower and upper Alpha band frequency partition points.
+This suggests that interaction
+components between the anterior and posterior brain regions also exhibit different time
+varying behavior in the lower and higher frequencies surrounding the Alpha band.
+Across all participants, Figure 5 displays the average component-wise p-values for test-
+ing the significance of the contribution towards the lower and upper Alpha band frequency
+partition points that were identified. Here we can see more clearly that the lower frequency
+partition point, which separates the lower frequencies from the Alpha band, can be attributed
+to the prefrontal components and the interaction components between the prefrontal and
+parietal/occipital regions. However, the upper frequency partition point separating the Al-
+27
+
+FP1
+FP2
+FC5
+FC6
+FZ
+T7
+CZ
+T8
+P7
+P3
+PZ
+P4
+P8
+O1
+Oz
+O2
+FP1 FP2 FC5 FC6 FZ
+T7
+CZ
+T8
+P7
+P3
+PZ
+P4
+P8
+O1
+Oz
+O2
+0.1
+0.2
+0.3
+0.4
+First Partition Point: Average P−values
+(a)
+FP1
+FP2
+FC5
+FC6
+FZ
+T7
+CZ
+T8
+P7
+P3
+PZ
+P4
+P8
+O1
+Oz
+O2
+FP1 FP2 FC5 FC6 FZ
+T7
+CZ
+T8
+P7
+P3
+PZ
+P4
+P8
+O1
+Oz
+O2
+0.2
+0.4
+0.6
+Second Partition Point: Average P−values
+(b)
+Figure 5: Component-wise significance of contribution towards partition points
+pha band from higher frequencies is associated with the parietal and occipital components
+and cross-components. These findings are made possible by the proposed method that is
+uniquely able to identify frequency partition points and also the corresponding sets of sig-
+nificant components and cross-components associated with each partition point.
+5
+Concluding remarks
+The frequency band analysis framework introduced in this article offers a quantitative ap-
+proach to identifying frequency bands that best preserve the nonstationary dynamics of the
+underlying multivariate time series. This framework allows for estimation of both the num-
+ber of frequency bands, their corresponding frequency partition points, and the components
+and cross-components of the multivariate signal associated with each of the partition points.
+This is made possible by the development of a sensible discrepancy measure and computa-
+tionally efficient bootstrap testing procedure within an iterative search algorithm. However,
+the proposed method is not without limitations. Motivated by the application to EEG fre-
+quency band analysis, it would be interesting to extend this framework to directly consider
+multiple subjects and produce a single set of frequency bands that jointly characterizes the
+28
+
+time-varying dynamics of the collection of signals in a data-driven manner. Second, in order
+to extend the proposed methodology for analyzing high-dimensional EEG signals (64 to 512
+channels), the bootstrap testing procedure would need to be modified to accommodate high-
+dimensional covariance structures. This would require appropriate simplifying assumptions
+on the covariance structure, such as factor-based, sparse, and block covariance structures.
+Finally, as seen in the last simulation setting (M3B-2) in Section 3, the current discrepancy
+measure requires large amounts of data to detect frequency partition points associated with
+changes in a few components of the multivariate signal. Modifications of the discrepancy
+measure that may be more powerful in detecting such changes (e.g. weighted squared L2
+norm or L∞ norm) are also worth further investigation and development.
+Funding Statement
+Research reported in this publication was supported by the National Institute Of General
+Medical Sciences of the National Institutes of Health under Award Number R01GM140476.
+The content is solely the responsibility of the authors and does not necessarily represent the
+official views of the National Institutes of Health. Portions of this research were conducted
+with the advanced computing resources provided by Texas A&M High Performance Research
+Computing.
+Data Availability Statement
+The data underlying this article are publicly available via Zenodo at https://doi.org/10.
+5281/zenodo.2348892.
+29
+
+Supplementary
+R code for “Frequency Band Analysis of Multivariate Time Series”.
+R code, a quick start demo, and descriptions of all functions and parameters needed to
+generate simulated data introduced in Section 3 and to implement the proposed method on
+data for use in practice can be downloaded from GitHub at this link: https://github.com/
+sbruce23/mEBA.
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+33
+
+A
+Proofs
+Proof of Theorem 2.1. We have from (6) the estimated discrepancy measure given by
+�D(ω) = 1
+T
+T
+�
+t=1
+1
+W
+W
+�
+k=1
+p
+�
+a,b=1
+�
+�ga,b
+� t
+T , ω − λk
+�
+− �ga,b
+� t
+T , ω + λk
+��
+×
+�
+�ga,b
+� t
+T , ω − λk
+�
+−
+�ga,b
+� t
+T , ω + λk
+��∗
+= 1
+T
+T
+�
+t=1
+1
+W
+W
+�
+k=1
+p
+�
+a,b=1
+�
+IN,a,b( t
+T , ω − λk) − 1
+T
+T
+�
+t1=1
+IN,a,b(t1
+T , ω − λk)−
+IN,a,b( t
+T , ω + λk) + 1
+T
+T
+�
+t2=1
+IN,a,b(t2
+T , ω + λk)
+�
+×
+�
+IN,a,b( t
+T , ω − λk) − 1
+T
+T
+�
+t1=1
+IN,a,b(t1
+T , ω − λk) − IN,a,b( t
+T , ω + λk) + 1
+T
+T
+�
+t2=1
+IN,a,b(t2
+T , ω + λk)
+�∗
+.
+where �ga,b and IN,a,b denote the entry (a, b) in the respective matrices. Expanding the inside
+term of the above expression involves several terms and we consider one of each kind and
+show that the expected value of the discrepancy measure tends to zero under H0.
+First, we consider the terms of the type IN( t
+T , ω ± λk)I∗
+N( t
+T , ω ± λk).
+We have, for
+34
+
+component (a, b),
+IN,a,b( t
+T , ω ± λk)I∗
+N,a,b( t
+T , ω ± λk) = JN,a( t
+T , ω ± λk)J∗
+N,b( t
+T , ω ± λk)JN,a( t
+T , ω ± λk)J∗
+N,b( t
+T , ω ± λk)
+=
+1
+(2πN)2
+� N−1
+�
+s1=0
+Xa,⌊utT⌋−N/2+1+s1,T e−is1θk,±
+�
+×
+� N−1
+�
+s2=0
+Xb,⌊utT⌋−N/2+1+s2,T eis2θk,±
+�
+×
+� N−1
+�
+s3=0
+Xa,⌊utT⌋−N/2+1+s3,T e−is3θk,±
+�
+×
+� N−1
+�
+s4=0
+Xb,⌊utT⌋−N/2+1+s4,T eis4θk,±
+�
+=
+1
+(2πN)2
+�
+s1,s2,s3,s4
+∞
+�
+l,m,n,o=−∞
+�
+Φa(uts1, l)εts1−l
+�
+×
+�
+Φb(uts2, m)εts2−m
+�
+�
+Φa(uts3, n)εts3−n
+�
+×
+�
+Φb(uts4, o)εts4−o
+�
+exp(−iθk,±(s1 − s2 + s3 − s4)) + O( 1
+T )
+=
+1
+(2πN)2
+�
+s1,s2,s3,s4
+∞
+�
+l,m,n,o=−∞
+�
+Za,ts1−lZb,ts2−mZa,ts3−nZb,ts4−o
+�
+exp(−iθk,±(s1 − s2 + s3 − s4)) + O( 1
+T )
+where θk,± = ω ± λk, ut =
+t
+T and tsj = ⌊utT⌋ − N/2 + 1 + sj for j = 1, 2, 3, 4.
+Also,
+Za,ts1−l =
+�
+Φ
+′
+a(uts1, l)εts1−l
+�
+where Φ
+′
+a(uts1, l) denotes the ath row of the coefficient matrix
+Φ(uts1, l). For the expectation above, we apply Theorem 2.3.2 of Brillinger (2001). Noting
+that the Z random variables above are Gaussian by the assumption in (1), the expected
+values simplifies to
+E
+�
+IN,a,b( t
+T , ω ± λk)I∗
+N,a,b( t
+T , ω ± λk)
+�
+= E(a,b)
+1,T + E(a,b)
+2,T + O( 1
+N ) + O( 1
+T ) + O(N 2
+T 2 ),
+35
+
+where
+E(a,b)
+1,T
+=
+1
+(2πN)2
+�
+s1,s2,s3,s4
+∞
+�
+l,m,n,o=−∞
+exp(−iθk,±(s1 − s2 + s3 − s4))E
+��
+Φ
+′
+a(ut, l)εts1−l
+�
+×
+�
+Φ
+′
+b(ut, m)εts2−m
+��
+×
+E
+��
+Φ
+′
+a(ut, n)εts3−n
+�
+×
+�
+Φ
+′
+b(ut, o)εts4−o
+��
+E(a,b)
+2,T
+=
+1
+(2πN)2
+�
+s1,s2,s3,s4
+∞
+�
+l,m,n,o=−∞
+exp(−iθk,±(s1 − s2 + s3 − s4))E
+��
+Φ
+′
+a(ut, l)εts1−l
+�
+×
+�
+Φ
+′
+b(ut, o)εts4−o
+��
+×
+E
+��
+Φ
+′
+a(ut, n)εts3−n
+�
+×
+�
+Φ
+′
+b(ut, m)εts3−m
+��
+For E(a,b)
+1,T
+it can be seen that the expectations are non-zero only when ts1 − l = ts2 − m
+and ts3 − n = ts4 − o. We hence have
+1
+T
+T
+�
+t=1
+1
+W
+W
+�
+k=1
+E(a,b)
+1,T
+= 1
+T
+T
+�
+t=1
+1
+W
+W
+�
+k=1
+fa,b( t
+T , θk,±)fa,b(ut, θk,±)∗ + o(1).
+(14)
+Similarly, for the term E(a,b)
+2,T
+it can be seen that the expectations are non-zero only when
+ts1 − l = ts4 − o and ts3 − n = ts2 − m. We get
+1
+T
+T
+�
+t=1
+1
+W
+W
+�
+k=1
+E(a,b)
+2,T
+= 1
+T
+T
+�
+t=1
+1
+W
+W
+�
+k=1
+fa,b( t
+T , θk,±)fa,b(ut, θk,±)∗ + o(1).
+(15)
+where θk,± = ω ± λk, ut = t/T, fa,b(ut, θk,±) denotes component (a, b) of the spectral matrix
+f(ut, θk,±) and fa,b(ut, θk,±)∗ is the conjugate transpose. The terms in (14) and (15) are
+approximated by
+1
+2πc
+� 1
+0
+� 2πc
+0
+fa,b(u, ω ± λ)fa,b(u, ω ± λ)∗dλ du + o(1).
+Next, we consider the terms of the type IN( t
+T , ω ± λk)I∗
+N( t
+T , ω ∓ λk).
+We have, for
+36
+
+component (a, b),
+IN,a,b( t
+T , ω ± λk)I∗
+N,a,b( t
+T , ω ∓ λk) =
+1
+(2πN)2
+� N−1
+�
+s1=0
+Xa,⌊utT⌋−N/2+1+s1,T e−is1θk,±
+�
+×
+� N−1
+�
+s2=0
+Xb,⌊utT⌋−N/2+1+s2,T eis2θk,±
+�
+×
+� N−1
+�
+s3=0
+Xa,⌊utT⌋−N/2+1+s3,T e−is3θk,∓
+�
+×
+� N−1
+�
+s4=0
+Xb,⌊utT⌋−N/2+1+s4,T eis4θk,∓
+�
+=
+1
+(2πN)2
+�
+s1,s2,s3,s4
+∞
+�
+l,m,n,o=−∞
+�
+Za,ts1−lZb,ts2−mZa,ts3−nZb,ts4−o
+�
+exp(−iθk,±(s1 − s2))exp(−iθk,∓(s3 − s4)) +
+O( 1
+T ),
+where θk,± = ω ± λk, θk,∓ = ω ∓ λk. The expected values simplifies to E
+�
+IN,a,b( t
+T , ω ±
+λk)I∗
+N,a,b( t
+T , ω ± λk)
+�
+= E(a,b)
+3,T + E(a,b)
+4,T + o(1), where
+E(a,b)
+3,T
+=
+1
+(2πN)2
+�
+s1,s2,s3,s4
+∞
+�
+l,m,n,o=−∞
+exp(−iθk,±(s1 − s2))exp(−iθk,∓(s3 − s4))E
+��
+Φ
+′
+a(ut, l)εts1−l
+�
+×
+�
+Φ
+′
+b(ut, m)εts2−m
+��
+× E
+��
+Φ
+′
+a(ut, n)εts3−n
+�
+×
+�
+Φ
+′
+b(ut, o)εts4−o
+��
+E(a,b)
+4,T
+=
+1
+(2πN)2
+�
+s1,s2,s3,s4
+∞
+�
+l,m,n,o=−∞
+exp(−iθk,±(s1 − s4))exp(−iθk,∓(s3 − s2))E
+��
+Φ
+′
+a(ut, l)εts1−l
+�
+×
+�
+Φ
+′
+b(ut, o)εts4−o
+��
+× E
+��
+Φ
+′
+a(ut, n)εts3−n
+�
+×
+�
+Φ
+′
+b(ut, m)εts3−m
+��
+.
+For E(a,b)
+3,T , the expectations are non-zero only when ts1 −l = ts2 −m and ts3 −n = ts4 −o.
+We hence have
+1
+T
+T
+�
+t=1
+1
+W
+W
+�
+k=1
+E(a,b)
+1,T
+= 1
+T
+T
+�
+t=1
+1
+W
+W
+�
+k=1
+fa,b( t
+T , θk,±)fa,b(ut, θk,∓)∗ + o(1).
+(16)
+For E(a,b)
+4,T , the expectations are non-zero only when ts1 − l = ts4 − o and ts3 − n = ts2 − m.
+37
+
+We get
+1
+T
+T
+�
+t=1
+1
+W
+W
+�
+k=1
+E(a,b)
+2,T
+= 1
+T
+T
+�
+t=1
+1
+W
+W
+�
+k=1
+fa,b( t
+T , θk,±)fa,b(ut, θk,∓)∗ + o(1).
+(17)
+The remaining types of terms considered are i). IN( t
+T , ω±λk)
+�
+1
+T
+�T
+t1=1 IN,a,b( t1
+T , ω±λk)∗�
+;
+ii). IN( t
+T , ω ± λk)
+�
+1
+T
+�T
+t1=1 IN,a,b( t1
+T , ω ∓ λk)∗�
+; iii).
+�
+1
+T
+�T
+t1=1 IN,a,b( t1
+T , ω ± λk)∗�
+IN( t
+T , ω ±
+λk)∗; iv).
+�
+1
+T
+�T
+t1=1 IN,a,b( t1
+T , ω ± λk)∗�
+IN( t
+T , ω ∓ λk)∗; v).
+�
+1
+T
+�T
+t1=1 IN,a,b( t1
+T , ω ± λk)
+�
+×
+�
+1
+T
+�T
+t1=1 IN,a,b( t1
+T , ω ± λk)∗�
+and vi).
+�
+1
+T
+�T
+t1=1 IN,a,b( t1
+T , ω ± λk)
+�
+×
+�
+1
+T
+�T
+t1=1 IN,a,b( t1
+T , ω ∓
+λk)∗�
+. The treatment of all these types of terms is similar to the approach that lead to
+(14)-(17) for the first two types of terms considered.
+When ω ∈ CN,2, it can be seen that by combining the 16 limiting expressions that we
+get from all the different types of terms, the expected value of �D(ω) tends to zero. With the
+same approach, when ω ∈ {ω1, ω2, . . . , ωK}, the expected value of �D(ω) is approximated by
+1
+2πc
+� 1
+0
+� 2πc
+0
+�p
+a,b=1
+�
+ga,b(u, ω−λ)−ga,b(u, ω+λ)
+��
+ga,b(u, ω−λ)−ga,b(u, ω+λ)
+�∗
+dλ du+o(1).
+Next, we consider the variance of the discrepancy measure. Here we look at the 2nd order
+38
+
+cumulant of discrepancy measure that is written as
+cum( �D(ω), �D(ω)) = 1
+T 2
+T
+�
+t1,t2=1
+1
+W 2
+W
+�
+k1,k2=1
+p
+�
+a,b,c,d=1
+cum
+��
+IN,a,b(t1
+T , ω − λk1) − 1
+T
+T
+�
+x1=1
+IN,a,b(x1
+T , ω − λk1)−
+IN,a,b(t1
+T , ω + λk1) + 1
+T
+T
+�
+x2=1
+IN,a,b(x2
+T , ω + λk1)
+�
+×
+�
+IN,a,b(t1
+T , ω − λk1) − 1
+T
+T
+�
+x3=1
+IN,a,b(x3
+T , ω − λk1) − IN,a,b(t1
+T , ω + λk1) + 1
+T
+T
+�
+x4=1
+IN,a,b(x4
+T , ω + λk1)
+�∗
+,
+�
+IN,c,d(t2
+T , ω − λk2) − 1
+T
+T
+�
+x1=1
+IN,c,d(x1
+T , ω − λk2)−
+IN,c,d(t2
+T , ω + λk2) + 1
+T
+T
+�
+x2=1
+IN,c,d(x2
+T , ω + λk2)
+�
+×
+�
+IN,c,d(t2
+T , ω − λk2) − 1
+T
+T
+�
+x3=1
+IN,c,d(x3
+T , ω − λk2) − IN,c,d(t2
+T , ω + λk2) + 1
+T
+T
+�
+x4=1
+IN,c,d(x4
+T , ω + λk2)
+�∗
+�
+.
+(18)
+We can now look at cumulant terms of different types and see the behavior as T → ∞.
+First, for components (a, b) and (c, d),we consider cumulant terms of the type
+cum
+�
+IN,a,b(t1
+T , ω ± λk1)I∗
+N,a,b(t1
+T , ω ± λk1), IN,c,d(t2
+T , ω ± λk2)I∗
+N,c,d(t2
+T , ω ± λk2)
+�
+=
+1
+(2πN)4
+N−1
+�
+r1,r2,r3,r4=0
+N−1
+�
+s1,s2,s3,s4=0
+∞
+�
+l1,m1,n1,o1=−∞
+∞
+�
+l2,m2,n2,o2=−∞
+cum
+��
+Φa(utr1, l)εtr1−l1
+�
+×
+�
+Φb(utr2, m1)εtr2−m1
+�
+×
+�
+Φa(utr3, n1)εtr3−n1
+�
+×
+�
+Φb(utr4, o1)εtr4−o1
+�
+,
+�
+Φc(uts1, l2)εts1−l2
+�
+×
+�
+Φd(uts2, m2)εts2−m2
+�
+×
+�
+Φc(uts3, n2)εts3−n2
+�
+×
+�
+Φd(uts4, o2)εts4−o2
+��
+×
+exp(−iθk1(r1 − r2 + r3 − r4)) × exp(−iθk2(s1 − s2 + s3 − s4)) + O( 1
+T ),
+(19)
+39
+
+where θk1,± = ω ± λk1 and θk2,± = ω ± λk2. Noting that εt is Gaussian, an application of
+Theorem 2.3.2 of Brillinger (2001) yield certain terms that are non-vanishing asymptotically.
+The term in (A) leads to
+1
+T 2
+T
+�
+t1,t2=1
+1
+W 2
+W
+�
+k1,k2=1
+cum
+�
+IN,a,b(t1
+T , θk1,±)I∗
+N,a,b(t1
+T , θk1,±), IN,c,d(t2
+T , θk2,±)I∗
+N,c,d(t2
+T , θk2,±)
+�
+=
+1
+T 2
+T
+�
+t=1
+1
+W 2
+W
+�
+k1,k2=1
+40
+(2π)4fa,b( t
+T , θk1,±)fa,b( t
+T , θk1,±)∗fc,d( t
+T , θk2,±)fc,d( t
+T , θk2,±)∗ + o(1). (20)
+The above term can be approximated by
+1
+T2πc
+� 1
+0
+� 2πc
+0
+fa,b(u, ω ± λ)fa,b(u, ω ± λ)∗fc,d(u, ω ±
+λ)fc,d(u, ω ± λ)∗dλ du + o(1). The treatment of the cumulant terms from the other term
+types follows similarly.
+Proof of Theorem 2.2. With the sets CN,1 and CN,2 defined in (7), the proof of this result
+follows from the following two results.
+(a). P
+� �
+ω∈CN,2 �D(ω) > ηT,ω
+�
+T→∞
+−→ 0,
+(b). P
+� �
+ω∈{ω1,ω2,...,ωK} �D(ω) > ηT,ω
+�
+T→∞
+−→ 1.
+For any ω ∈ CN,2, an application of Theorem 2.1(a) yields
+P(
+�
+ω∈CN,2
+�D(ω) > ηT,ω) ≤
+�
+ω∈CN,2
+P( �D(ω) > ηT,ω)
+(21)
+≤
+�
+ω∈CN,2
+E( �D(ω))
+ηT,ω
+T→∞
+−→ 0.
+(22)
+The result in (b) follows by an application of Theorem 2.1(b).
+40
+
diff --git a/gNE2T4oBgHgl3EQfHAau/content/tmp_files/load_file.txt b/gNE2T4oBgHgl3EQfHAau/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..1bf9f721a687b0512a95566ace1f0250441dc55f
--- /dev/null
+++ b/gNE2T4oBgHgl3EQfHAau/content/tmp_files/load_file.txt
@@ -0,0 +1,1288 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf,len=1287
+page_content='Frequency Band Analysis of Nonstationary Multivariate  Time Series ∗†  Raanju R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Sundararajan‡  and  Scott A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Bruce§  ‡Department of Statistical Science, Southern Methodist University, Dallas, Texas, USA  §Department of Statistics, Texas A&M University, College Station, Texas, USA  Abstract  Information from frequency bands in biomedical time series provides useful summaries  of the observed signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Many existing methods consider summaries of the time se-  ries obtained over a few well-known, pre-defined frequency bands of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' However,  these methods do not provide data-driven methods for identifying frequency bands that  optimally summarize frequency-domain information in the time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' A new method  to identify partition points in the frequency space of a multivariate locally stationary  time series is proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' These partition points signify changes across frequencies in  the time-varying behavior of the signal and provide frequency band summary measures  that best preserve the nonstationary dynamics of the observed series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' An L2 norm-  based discrepancy measure that finds differences in the time-varying spectral density  matrix is constructed, and its asymptotic properties are derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' New nonparametric  bootstrap tests are also provided to identify significant frequency partition points and  ∗AMS subject classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Primary: 62M10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Secondary: 62M15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  †Keywords and phrases: Multivariate time series, nonstationary, frequency domain, spectral matrix,  electroencephalography (EEG)  ‡Email: rsundararajan@smu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='edu (both authors contributed equally to this work)  §Email: sabruce@tamu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='edu  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='03664v1  [stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='ME]  9 Jan 2023  to identify components and cross-components of the spectral matrix exhibiting changes  over frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Finite-sample performance of the proposed method is illustrated via  simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' The proposed method is used to develop optimal frequency band summary  measures for characterizing time-varying behavior in resting-state electroencephalog-  raphy (EEG) time series, as well as identifying components and cross-components  associated with each frequency partition point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  1  Introduction  Frequency-domain information contained in biomedical time series provides important sum-  maries leading to meaningful physiological interpretations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  Oscillatory behavior in non-  stationary time series are often characterized by the time-varying spectral matrix, which  is a time- and frequency-dependent matrix-valued function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Rather than analyzing the full  time-varying spectral matrix, practitioners often partition frequencies into a few well-known,  pre-defined bands that serve as a pseudo partition of the frequency space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Numerical sum-  maries of the spectral matrix are then computed within these frequency bands, and these  summaries have been shown to provide meaningful biological interpretations of the observed  signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Examples of biomedical time series where frequency bands are identified and asso-  ciated with physiological characteristics include heart rate variability (HRV) (Hall et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=',  2004), local field potential (LFP) (Liu and Newsome, 2006), resting-state functional mag-  netic resonance imaging (rs-fMRI) (Yuen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=', 2019) and electroencephalography (EEG)  (Klimesch, 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  In analyzing neuroimaging data, several works have identified frequency bands of interest  that carry meaningful biological interpretations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Biswal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' (1995) analyze correlations in  resting state fMRI that characterize functional connectivity in the brain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' In their study, high  2  correlations in low frequency oscillations (< 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='1 Hz) are detected within the sensorimotor  cortex of the brain and also with other regions of the brain that associate with motor func-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Lowe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' (2000) is another work that computes cross-correlations in low frequency  oscillations (< 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='08 Hz) of the fMRI signal between widely separated anatomic regions of the  brain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' To detect regions of the brain that exhibit strong cross-correlations, their work uses  these low frequency oscillations to discriminate between different memory tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' In Yuen  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' (2019), frequency partitions within the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='01-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='25 Hz band were obtained for rs-fMRI  signals and the oscillations from each of these partitions were associated with different bio-  logical functions such as respiration, pulse, and vasomotor activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' In Henrie and Shapley  (2005), spectral analysis is performed on a LFP signal gathered from the primary visual cor-  tex of a macaque.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' The power of the spectral density of this LFP signal is computed at the  low (< 10 Hz), gamma (25 − 240 Hz) and broad (8 − 240 Hz) frequency bands, and changes  in power are analyzed in response to various stimuli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' In analyzing EEG data, many methods  resort to analyzing oscillations for known frequency bands, such as Alpha, Beta, Gamma  and Delta, and associate these oscillations with various biological functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' As an example,  Newson and Thiagarajan (2019) provides a review of several methods that utilize pre-defined  and well-known frequency bands to analyze resting state EEG data from individuals with  neurological disorders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' In all of the above cited works, irrespective of the modality of choice  (fMRI, LFP or EEG), the selection of frequency bands used to summarize oscillatory pat-  terns is often predetermined either through manual inspection of the time series or through  historical precedents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Doppelmayr et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' (1998) mention the possibility of differences in the  endpoints of the Alpha frequency band in EEG data from different individuals, and Ghazi  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' (2021) is another example that discusses differences in the peak frequency in the Al-  pha band of EEG data among individuals from different sexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Analyses like these point to  3  the need for a data-driven method for automatically identifying frequency bands that best  describe changes in the frequency space of neuroimaging time series data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  Several methods in the signal processing and applied harmonic analysis literature pur-  sue time-frequency analysis of univariate nonstationary time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Time-frequency analysis  helps understand the time-varying properties of different frequency components of the time  series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' For obtaining a time-frequency description of the observed signal, Flandrin (1998)  describes a few solutions, such as the short-time Fourier transform and wavelet transform,  along with their properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' While most methods aim at estimating the time-varying am-  plitudes of the signal, very few methods discuss estimation of changes happening in the  frequency space of nonstationary time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' An online algorithm for detecting a single  prominent time-varying frequency band is proposed in Tiganj et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' In their work, at  each local time window, the prominent frequency band is estimated by using a band-limited  signal as the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' In Cohen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' (2016), the observed signal is assumed to be composed  of multiple, uncorrelated cyclostationary processes (Gardner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=', 2006), and they provide  an algorithm for detecting multiple peaks in the spectral density of the signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  In the statistical literature, time-frequency analysis of nonstationary time series has been  studied using evolutionary spectra (Priestley, 1965) and the time-varying spectral density  of locally stationary processes (Dahlhaus, 1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  Most methods utilize the time-varying  spectral matrix or its finite sample estimates for detecting changes in the time space, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=',  temporal change point detection;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' see Adak (1998), Ombao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' (2005), Kirch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' (2015),  Preuss et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' (2015) for examples in univariate and multivariate locally stationary time  series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  In Schr¨oder and Ombao (2019), temporal change points are obtained over user-  specified frequency bands and the obvious limitation in their work is the need for the user  to specify the frequency bands over which the temporal change points are found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Bruce  4  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' (2020) is a recent work that considers univariate locally stationary time series and  aims at finding partition points in the frequency space that best describe the nonstationary  characteristics of the observed time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Their work constructs an estimate of the time-  varying spectral density by assuming a piecewise stationary approximation and applying a  multitaper estimator of the spectral density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' A CUSUM statistic designed to spot changes  in the frequency space is formulated based on this estimator, and significant partition points  are obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  In this work, we propose a new method to identify partition points in the frequency space of a  multivariate locally stationary time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' These partition points are detected such that they  best preserve the nonstationary dynamics of the time-varying spectral density matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' To  detect these partition points, an L2 norm-based discrepancy measure is constructed using  the time-varying spectral matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' This measure computes differences in the time-varying  spectral matrix over local neighborhoods of frequencies and its asymptotic properties are  derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' With the discrepancy measure viewed as our test statistic, a new nonparametric  bootstrap test is proposed to obtain the critical values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' The proposed bootstrap procedure  is also further utilized in identifying the components and cross-components contributing to  the changes in the frequency space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' The proposed method is motivated by an application in  analyzing resting-state electroencephalography (EEG) time series from 14 individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' The  experiment involves a straightforward eyes-open/eyes-closed scheme with the signal being  recorded from 16 electrodes (Cattan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' After down-sampling the original signal  to the rate of 64 Hz, the left plots in Figure 1 depicts the standardized EEG time series  from the Oz occipital channel in participants 2 and 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' These illustrated signals cover a time  segment of the entire experiment that includes a total of five consecutive blocks, with each  block being an eyes-closed or eyes-open scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' The right plots in the same figure show  5  the estimated time-varying spectral densities for this channel, and the time-varying behavior  can be witnessed around the 10 Hz frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  This work addresses several critical aspects of frequency-domain analysis of nonstationary  multivariate signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' First, a data-driven estimator for frequency bands of interest for each  individual is developed, as opposed to investigating commonly known pre-defined frequency  bands that are assumed to be the same for all individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Second, the proposed method is  used to better understand variability in estimated partition points across different individuals  from a population of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Finally, it is typically not the case that all components of the  spectral matrix exhibit changes across frequencies for each partition point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Accordingly, the  proposed method allows for detection of the particular components and cross-components  of the time series that are contributing to each estimated frequency partition point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' With  the proposed method being uniquely positioned to address such problems, in Section 4 we  illustrate and discuss the application in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  6  −1  0  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='75  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='00  Time  Participant 2 (Oz)  Participant 2 (Oz) Standardized EEG  (a)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='8  5  10  15  20  25  30  Participant 2 (Oz)  Time  Frequency (Hz)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='0  (b)  −2  −1  0  1  2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='75  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='00  Time  Participant 13 (Oz)  Participant 13 (Oz) Standardized EEG  (c)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='8  5  10  15  20  25  30  Participant 13 (Oz)  Time  Frequency (Hz)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='0  (d)  Figure 1: Oz channel EEG for participants 2 and 13 and corresponding estimated spectral  densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  The rest of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Section 2 describes the proposed method in  detail along with the theoretical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Section 3 discusses the finite sample performance of  the method using a few simulation schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' The application to modeling resting-state EEG  time series data is presented in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' The concluding remarks are given in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  2  Methodology  In this section, we describe our proposed method to find frequency partition points in the  time-varying spectral matrix of a locally stationary process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' First, the model for the locally  7  stationary process and frequency-banded time-varying behavior is introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' A discrep-  ancy measure and estimator is then proposed for identifying frequency partition points that  characterize the frequency band structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Resampling-based testing procedures are then  developed to determine the significance of potential frequency partition points and to identify  components associated with significant frequency partition points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='1  Model  We begin with the definition of a locally stationary process using a two-sided MA(∞) repre-  sentation followed by the required assumptions on the coefficient matrices (Dahlhaus, 1997,  Dahlhaus, 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Let Xt,T be a p-variate locally stationary process given by  Xt,T =  ∞  �  j=−∞  Φt,T,jεt−j,  t = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' , T,  (1)  where εt are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='d Gaussian with unit variance matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' The time-varying coefficient matrices,  Φt,T,j, are assumed to be temporally smooth in the following sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  Assumption 1 (Temporal smoothness).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' There exists temporally smooth functions Φ : [0, 1]×  Z → Rp×p such that  ∞  �  j=−∞  sup  t=1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=',T  ||Φt,T,j − Φ (t/T, j)||∞ = O (1/T) ,  where || · ||∞ denotes the infinity norm, and  ∞  �  j=−∞  sup  u∈[0,1]  ||Φ(u, j)||∞|j| < ∞,  8  ∞  �  j=−∞  sup  u∈[0,1]  ||Φ  ′(u, j)||∞|j| < ∞,  ∞  �  j=−∞  sup  u∈[0,1]  ||Φ  ′′(u, j)||∞ < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  With the above model, the p × p time-varying spectral matrix of Xt,T is given by  f(u, ω) = 1  2π  �  r,s  Φ(u, r)Φ(u, s)  ′e−i2πω(r−s),  (2)  for ω ∈ [−1/2, 1/2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' With the above temporal smoothness assumption, the series Xt,T  can also be expressed using the Cram´er representation with a time-varying transfer function  matrix that can be approximated by a temporally-smooth matrix-valued function A : [0, 1]×  [−1/2, 1/2] → Cp×p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Then, the time-varying spectral matrix can be written as f(u, ω) =  A(u, ω)A∗(u, ω), where A∗(u, ω) denotes the conjugate transpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  To characterize the nonstationary behavior of the spectral matrix beyond a simple level  shift, we consider the demeaned time-varying spectral matrix given by  g(u, ω) = f(u, ω) −  � 1  0  f(u, ω)du.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  (3)  We will further assume that g(u, ω) has the following partition in the frequency space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  Assumption 2 (Frequency-banded time-varying behavior).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  g(u, ω) =  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  g1(u)  if ω ∈ [0, ω1)  g2(u)  if ω ∈ [ω1, ω2)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  gK+1(u)  if ω ∈ [ωK, 1/2],  9  where PK = {ω1, ω2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' , ωK} denotes the set of K frequency partition points in the interval  (0, 1/2), with K and PK being fixed and unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='2  Comparing frequency-specific time-varying behavior  To determine both K and PK, we consider the following L2 norm-based discrepancy measure  on the demeaned spectral matrix g(u, ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' For any ω ∈ (0, 1/2) we have,  D(ω) = 1  δ  � 1  u=0  � δ  λ=0  ||g(u, ω − λ) − g(u, ω + λ)||2 dλ du,  (4)  where || · ||2 denotes the squared L2 norm, and δ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Observe that at any frequency ω, the  nonnegative measure D(·) represents the difference in the demeaned time-varying spectral  matrix in a local neighborhood of frequencies surrounding ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  In order to estimate the discrepancy measure in (4), we use the local periodogram IN(u, ω)  (Dahlhaus, 1997) given by  IN(u, ω) = JN(u, ω)J∗  N(u, ω), with JN(u, ω) =  1  √  2πN  N−1  �  s=0  X⌊uT⌋−N/2+1+s,T e−i2πωs,  (5)  where N is the length of the local neighborhood around time point u and J∗  N(u, ω) denotes  the conjugate transpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Viewing IN(u, ω) as a local estimate of the time-varying spectral  matrix f(u, ω), we obtain the estimated version of our measure in (4) as  �D(ω) = 1  T  T  �  t=1  1  W  W  �  k=1  ���  ����g  �  t/T, ω − λk  �  − �g  �  t/T, ω + λk  ����  ���  2  ,  (6)  where �g(t/T, ω − λk) = IN(t/T, ω − λk) − 1  T  �T  t1=1 IN(t1/T, ω − λk), and λk = k/N, k =  1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' , W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' W corresponds to the parameter δ in (4), and in finite sample situations, we  10  resort to a multiscale approach in which we consider a range of plausible values for W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' This  approach is discussed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  To better understand the behavior of the measure �D(·) in (6), we consider the following  sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  CN =  �W  N , W + 1  N  , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' , 1  2 − W  N  �  ,  CN,1 =  K  �  j=1  �  ωj − W  N , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' , ωj + W  N  �  , and  CN,2 = CN \\ CN,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  (7)  Here, CN denotes the set of all candidate partition points minus a neighborhood of  length W/N on the ends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' The set CN,1 takes the union of the neighborhoods of all partition  points, where each neighborhood around a partition point is of radius W/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' The set CN,2  denotes the set of all points located at a distance of least W/N from all partition points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' To  establish large sample properties of the measure �D(·), we assume frequency partition points  are reasonably well-separated as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  Assumption 3 (Separated partition points).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' There exists a constant c ∈ (0, 1/2) such that  ω1 > c, ωK < 1  2 − c,  min  j=1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=',K−1 |ωj − ωj+1| ≥ c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  With these assumptions, the following theorem describes the asymptotic behavior of the  proposed estimator �D(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Suppose that the conditions stated in Assumptions 1-3 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Let N → ∞  and T → ∞ such that N/T → 0 and W/N → c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Then,  (a) for ω ∈ CN,2, �D(ω)  p  −→ 0,  11  (b) for ω ∈ PK = {ω1, ω2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' , ωK},  �D(ω)  p  −→  1  2πc  � 1  0  � 2πc  0  p  �  a,b=1  �  ga,b(u, ω−λ)−ga,b(u, ω+λ)  ��  ga,b(u, ω−λ)−ga,b(u, ω+ λ)  �∗  dλ du,  where a, b = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' , p, and  p  −→ denotes convergence in probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  Proof is made available in the Appendix and relies on computing the asymptotic mean  and variance of �D(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' This result means that the discrepancy measure will approach zero  for frequencies not near a partition point and approach a positive constant for frequencies  representing partition points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' The above result also implies that a hypothesis test using �D(·)  as the test statistic leads to a consistent test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Such a test would be defined in the following  manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Let ηT,ω be the threshold satisfying lim  T→∞ ηT,ω ≥ B > 0, for some positive constant  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Then, a candidate partition point ω ∈ (0, 1/2) is designated as a partition point if  �D(ω) > ηT,ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  (8)  With the discrepancy measure �D(ω) as the test statistic, the threshold ηT,ω can be de-  termined by the bootstrap procedure described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='3  Bootstrapping locally stationary processes  Here we describe the resampling procedure to obtain the threshold ηT,ω given in (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' This  threshold can be viewed as a critical value of the discrepancy measure �D(ω) from (6), under  the null hypothesis that ω is not a partition point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' We resort to a nonparametric bootstrap  method that generates samples under the null hypothesis in order to approximate ηT,ω and  provide a corresponding p-value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  Under the null hypothesis, assume Xt,T = Yt + σ(t/T)Zt where Yt is a second-order  12  stationary p-variate process, σ(t/T) is a p × p time-varying matrix, Zt is i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' N(0, Ip),  and Yt is independent of Zt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  In this case, the spectral matrix of Xt,T is fX(t/T, ω) =  fY (ω) + σ(t/T)σ′(t/T), where fY (ω) is the spectral matrix of Yt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  However, fY (ω) does  not appear in the demeaned time-varying spectral matrix (3) since fY (ω) =  � 1  0 fY (ω)du.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  Therefore, we can simplify computation for our bootstrap procedure by avoiding estimation  of fY (ω) and instead assuming Xt,T = σ(t/T)Zt in order to generate bootstrap samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Let  ω ∈ CN be a candidate frequency partition point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' The bootstrap procedure is carried out  through the following steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Assume Xt,T = σ(t/T)Zt, where Zt is i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  N(0, Ip).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  Compute the time-varying  variance matrix estimator as  �ΓX,0(u) = 1  T  T  �  t=1  XtX  ′  tKh(u − t/T),  (9)  where Kh(u) =  1  hK( u  h), h denotes the bandwidth, and K(·) is a symmetric kernel  function that integrates to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Compute the time-varying square root matrix �σ(u) =  �  �ΓX,0(u)  �1/2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Obtain R bootstrap resamples as X(r)  t,T = �σ(t/T)Z(r)  t , where Z(r)  t  ∼ N(0, Ip), t =  1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' , T, and r = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  Step 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' With each resample X(r)  t,T, t = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' , T, compute the discrepancy measure �D(r)(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  Step 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Obtain the p-value of this test as  �R  r=1 1 �  D(r)(ω)> �  D(ω)  R  , where �D(ω) is the observed value  of the test statistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  Recall that the assumption in Step 1 follows from the behavior of the demeaned spectral  matrix g(u, ω) under the null hypothesis that ω is not a partition point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  In this case,  13  g(u, ω) = g(u) which does not depend on frequency ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Under this assumption, Steps 2-5  then describe the resampling procedure that produces the required p-value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' In Step 1, in  finite sample situations discussed in Sections 3 and 4, we utilize the triangular kernel for  K(·) with a bandwidth h = T −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='4  Finding the number and locations of partition points  Here we describe the steps to detect the locations and number of frequency partition points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  Recall from (7), the set CN represents the set of candidate frequency partition points to  search over.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Our iterative procedure to detect the locations of partition points involves the  following steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  Step 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Initialize the set �CN = CN and �P = ∅, where �P is the final set of partition points  returned by the procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Compute the measure �D(ω), for every ω ∈ CN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Find the point ω∗, where  ω∗ = arg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' max  λ∈ �CN  �D(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  (10)  Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Determine the p-value for testing if ω∗ is a partition point using the resampling pro-  cedure described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' If found to be significant, set �P = �P �{ω∗}, and set  �CN = �CN \\ {ω∗ − W  N , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' , ω∗ + W  N }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  Step 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Repeat Steps 2 and 3 until the significance test in Step 3 fails to return a significant  frequency partition point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  The above iterative procedure results in the set �P that contains the final set of frequency  partition points, and the cardinality of this set provides an estimate �K of the number of  14  partition points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Next, we provide a large sample result on the estimate �K of the number of  partition points obtained through this procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Suppose that the conditions stated in Assumptions 1-3 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Let N → ∞  and T → ∞ such that N/T → 0 and W/N → c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Then,  P( �K ̸= K)  T→∞  −→ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  (11)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' See Appendix for details of the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='5  Choice of W: length of neighborhood of frequencies  The choice of W used to estimate the discrepancy measure in (6) depends on the nature  and magnitude of changes in the frequency space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Smaller changes need larger values of  W for detection while larger changes can be identified even with smaller values of W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' In  order to detect partition points associated with both small and large changes, we adopt a  multiscale approach (Messer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=', 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' In practice, a sequence of q choices for W given  by Wmin < W1 < W2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' < Wq < Wmax is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Let �Pi denote the set of partition  points estimated using the iterative procedure from Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='4 with neighborhood length  choice Wi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Set P = �P1, where P denotes the final set of estimated change points returned  by our multiscale approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' For any point ω ∈ �P2, ω is added to the set P only if it does  not belong to a W2-neighborhood of any of the existing points in the set P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' The procedure  is successively moved forward until all choices for W have been considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Algorithm 1  provides the pseudocode that illustrates the full implementation of the multiscale frequency  band estimation procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  It should be noted that Wmin should be selected small enough to ensure partition points  associated with more subtle changes are detected, but not too small, which may lead to false  15  Algorithm 1: Multiscale Frequency Band Estimation  �CN ← CN =  � W1  N , W1+1  N  , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' , 1  2 − W1  N  �  �P ← ∅  for W ∈ {W1, W2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' , Wq} do  �CN ← �CN \\ { W1  N , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' , W  N , 1  2 − W  N , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' , 1  2 − W1  N }  for λ ∈ �P do �CN ← �CN \\ {λ − W  N , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' , λ + W  N }  stop ← 0  while stop = 0 and �CN ̸= ∅ do  ω∗ ← arg maxλ∈ �CN �D(λ) where �D(λ) is calculated by (6) given W  Determine p-value for �D(ω∗) using bootstrap procedure (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='3)  if p−value is significant then  �P ← �P �{ω∗}  �CN ← �CN \\ {ω∗ − W  N , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' , ω∗ + W  N }  else stop ← 1  �K = | �P|  return �P, �K  positives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' In finite sample cases in Section 3, we present the performance results for different  choices of Wmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Based on our simulation results, Wmin = N/8 provides the best estimation  performance for the simulation settings considered herein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  In scenarios where a single frequency neighborhood length choice W must be selected,  one can take the largest value of W ∈ {W1, W2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=', Wq} for which there is an addition of  a partition point to the set P in the iterative procedure described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' More precisely  W = Wi∗, where i∗ is the largest value in the set {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=', q} for which the iterative procedure  described above adds a point to the set P during iteration i∗ (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=', with neighborhood length  choice Wi∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' In case there is no point added to the set P for any choice Wi, i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=', q, we  set W = Wq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  16  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='6  Finding components responsible for partition points  In this section, we present a new technique to identify the components and cross-components  of the multivariate series that significantly contribute to each of the partition points iden-  tified in the frequency space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' For every identified partition point, a resampling procedure  for finding the components significantly contributing to the change characterized by the fre-  quency partition point is discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' The proposed approach, similar to the bootstrap method  given in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='3, generates samples under the null hypothesis, and results in p-values for  every component and cross-component of the series Xt,T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  Let ωc be a partition point detected by our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' With every component (a, b), where  1 ≤ a ≤ b ≤ p, the goal is to estimate the p-value corresponding to the null hypothesis that  component (a, b) of the series Xt,T does not have a significant contribution to the partition  at frequency ωc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' The component-specific test statistic �D(a,b) is then written as  �D(a,b)(ωc) = 1  T  T  �  t=1  1  W  W  �  k=1  ����ga,b  �  t/T, ωc − λk  �  − �ga,b  �  t/T, ωc + λk  ����  2  ,  (12)  where �g(t/T, ωc − λk) = IN(t/T, ωc − λk) − 1  T  �T  t1=1 IN( t1  T , ωc − λk) and λk =  k  N , k =  1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' , W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Note that �ga,b is component (a, b) of the p × p matrix �g(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' The p-value is  obtained through the following steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Assume Xt,T = σ(t/T)Zt, where Zt is i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  N(0, Ip).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  Compute the time-varying  variance matrix estimator as  �ΓX,0(u) = 1  T  T  �  t=1  XtX  ′  tKh(u − t/T),  (13)  where Kh(u) =  1  hK( u  h), h denotes the bandwidth, and K(·) is a symmetric kernel  function which integrates to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  17  Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Compute the time-varying square root matrix �σ(u) =  �  �ΓX,0(u)  �1/2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Obtain R bootstrap resamples as X(r)  t,T = �σ(t/T)Z(r)  t , where Z(r)  t  ∼ N(0, Ip), t =  1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' , T, and r = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  Step 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' With each resample X(r)  t,T, t = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' , T, compute the component-specific test statistic  �D(r)  (a,b)(ωc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  Step 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Obtain the p-value of this test as  �R  r=1 1 �  D(r)  (a,b)(ωc)> �  D(a,b)(ωc)  R  , where �D(a,b)(ω) is the observed  value of the test statistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  The above procedure is applied to every component and cross-component (a, b), 1 ≤ a ≤  b ≤ p, of the series Xt,T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' The components and cross-components that carry a significant  p-value are deemed as the components responsible for the partition at frequency ωc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' An  illustration of this procedure can be seen in the application presented in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' In order  to account for simultaneous testing of multiple components, multiple testing adjustments  can and should be used to control the experiment-wide error rate;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' for example, a Bonferroni  adjustment is implemented for the application presented in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  3  Simulation study  Performance of the proposed method is assessed through a few simulation examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' The  five simulation schemes are described first followed by the presentation of the performance  results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  The first scheme (WN1B) is a multivariate white noise model with no partition points in  the frequency space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' This setting is considered to ensure that the method does not produce  an unreasonable number of false positives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' The second scheme (L3B) considers partition  18  points at frequencies 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='15 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='35, with spectral density f2(u, ω) exhibiting a linear trend  in time u ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' The third scheme (S3B) also considers partition points at frequencies  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='15 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='35, but with spectral density f3(u, ω) exhibiting a non-linear trend in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' The  second and third schemes illustrate performance of the method in capturing partition points  associated with both linear and nonlinear time-varying dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' The fourth scheme (M3B-  1), with partition points at frequencies 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='15 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='35, considers a mixture of time series  exhibiting linear and non-linear trends from the models L3B and S3B, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' This  setting considers performance of the method when components of the series have a similar  frequency band structure, but differing time-varying dynamics across components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' The fifth  scheme (M3B-2) again considers a mixture of linear and non-linear trends in the spectral  density, but assumes only 20% of the p components in Xt,T contribute to the partition at  frequency 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='15, whereas the remaining 80% of the p components contribute to the partition  at frequency 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' This setting is the most challenging and represents both differing time-  varying dynamics and differing partition points across components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' White noise (WN1B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Xk,t = z1,t+k−1 for k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' , p, and z1,t has time-varying  spectral density f1(u, ω) given by  f1(u, ω) = 1 for ω ∈ (0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='5)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Linear, 3-Bands (L3B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Xk,t = z2,t+k−1 for k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' , p, and z2,t has time-varying  spectral density f2(u, ω) given by  19  f2(u, ω) =  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  10 − 9u for ω ∈ (0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='15)  1 for ω ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='15, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='35)  1 + 9u for ω ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='35, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='5)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Sinusoidal, 3-Bands (S3B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Xk,t = z3,t+k−1 for k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' , p, and z3,t has time-  varying spectral density f3(u, ω) given by  f3(u, ω) =  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  10 + 10 sin(4πu − π/2) for ω ∈ (0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='15]  5 + 5 cos(4πu) for ω ∈ (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='15, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='35]  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='5 + 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='5 sin(3πu − π/16) for ω ∈ (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='35, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='5)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Linear and Sinusoidal, 3-Bands, Mixture (M3B-1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Xk,t = z2,t+k−1 for k =  1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' , ⌊p/2⌋, and Xk,t = z3,t+k−⌊p/2⌋−1 for k = ⌊p/2⌋ + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' , p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Here, the series z2,t  and z3,t are given by the schemes L3B and S3B described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Linear and Sinusoidal, 3-Bands, Differing Proportions (M3B-2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  Xk,t =  z4,t+k−1 for k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' , ⌊0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='2p⌋, and Xk,t = z5,t for k = ⌊0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='2p⌋ + 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' , p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Here, z4,t and  z5,t have time-varying spectral densities f4(u, ω) and f5(u, ω), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  f4(u, ω) =  �  �  �  �  �  �  �  �  �  10 − 9u for ω ∈ (0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='15)  1 for ω ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='15, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='5)  f5(u, ω) =  �  �  �  �  �  �  �  �  �  5 + 5 cos(4πu) for ω ∈ (0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='35]  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='5 + 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='5 sin(3πu − π/16) for ω ∈ (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='35, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='5)  20  To assess the performance of the proposed method, we first present the results on esti-  mating the true number of frequency bands, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=', the quantity K + 1, where K is the true  number of partition points defined in (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Table 1 reports the estimated mean number of  frequency bands for the five simulation schemes based on 100 replications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Note that the  bootstrap procedure from Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='4 is used for estimating the number and locations of the  partition points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' In implementing this procedure, the triangular kernel is used as the kernel  choice K(·), and the bandwidth is h = T −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' The choice for the frequency neighborhood  length W involves the multiscale approach described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='5, and we consider an  equally-spaced sequence Wmin = N  8 < W1 < W2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' < Wq < Wmax = N  4 , with N = T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='7,  and the sample size T ∈ {200, 500, 1000}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' The results in Table 1 show that as the sample  size increases, accuracy in estimating the number of partition points increases for all five  simulation schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Table 2 presents estimation results at a fixed sample size T = 1000, but  for different choices of Wmin used in the multiscale procedure from Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' It is seen  that in most schemes the number of frequency bands estimated increases as Wmin decreases,  with the best results at parameter choice Wmin = N/8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  p  T  WN1B  L3B  S3B  M3B-1  M3B-2  10  200  1(0)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='24(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='55)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='15(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='36)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='09(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='35)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='14(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='64)  500  1(0)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='94(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='34)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='56(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='50)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='47(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='52)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='88(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='57)  1000  1(0)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='06(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='24)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='96(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='24)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='92(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='31)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='09(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='35)  15  200  1(0)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='34(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='52)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='18(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='39)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='07(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='29)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='25(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='59)  500  1(0)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='99(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='30)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='61(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='55)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='53(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='52)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='92(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='51)  1000  1(0)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='05(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='22)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='00(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='20)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='98(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='20)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='12(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='41)  Table 1: Mean(sd) for estimated number of frequency bands, ˆK + 1, for 100 replications  (Wmin = N/8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' True value K + 1 is 1 for WN1B, and 3 for all other schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  Next, we present the proportion of the 100 replications that result in correct detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  At any given replication, a correct detection occurs when the proposed method identifies the  correct number of frequency partition points and all estimated partition points are within a  21  p  Wmin  WN1B  L3B  S3B  M3B-1  M3B-2  10  N/8  1(0)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='06(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='24)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='96(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='24)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='92(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='31)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='09(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='35)  N/10  1(0)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='64(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='64)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='43(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='52)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='13(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='34)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='77(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='47)  N/12  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='07(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='26)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='36(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='58)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='52(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='56)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='11(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='71)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='26(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='66)  15  N/8  1(0)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='05(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='22)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='00(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='20)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='98(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='20)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='12(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='41)  N/10  1(0)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='69(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='66)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='50(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='54)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='22(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='42)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='87(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='42)  N/12  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='06(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='24)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='44(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='59)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='68(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='51)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='32(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='63)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='41(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='64)  Table 2: Mean(sd) for estimated number of frequency bands, ˆK + 1, for 100 replications  (T = 1000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' True value K + 1 is 1 for WN1B, and 3 for all other schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  distance of ζ from the true partition point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Table 3 presents the correct detection rate based  on 100 replications for the four simulation schemes with more than one frequency band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  We observe that in almost all cases, as the length of the time series increases, the correct  detection rate increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  The scheme M3B-2 exhibits the lowest correct detection rate relative to other settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  Recall that under the M3B-2 setting, only 20% of the p components of the multivariate series  Xt,T contribute to the partition at frequency 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='15, and the remaining 80% of the components  contribute to the partition at frequency 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' With weaker contribution from components  towards the frequency partition point 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='15, the proposed method requires much longer time  series (T) to improve the correct detection rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  p  T  L3B  S3B  M3B-1  M3B-2  10  200  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
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+page_content='17  500  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='88  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
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+page_content='42  15  200  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='36  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
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+page_content='20  500  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
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+page_content='52  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='51  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='41  1000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='95  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
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+page_content='39  Table 3: Correct Detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Proportion of 100 replications that correctly estimate the number  of bands, K +1, and the distance between the estimated and true partition points is no more  than ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Here ζ = 1/16 and Wmin = N/8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  Finally, in Table 4, we provide the results on correct detection rates for different choices  of ζ and fixed time series length T = 1000 for the four settings with more than one frequency  22  p  ζ  L3B  S3B  M3B-1  M3B-2  10  1/12  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
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+page_content='96  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='96  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='39  1/24  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='95  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
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+page_content='25  Table 4: Correct Detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Proportion of 100 replications that correctly estimate the number  of bands, K +1, and the distance between the estimated and true partition points is no more  than ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Here T = 1000 and Wmin = N/8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' We notice that for the first three simulation schemes, the correct detection rate is not  sensitive to the choice of ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Here again, the scheme M3B-2 sees the lower correct detection  rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' This can be attributed to the fact that fewer components (only 20%) are contributing  to the partition at frequency 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='15, and hence a much larger sample size is needed to achieve  higher correct detection rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  4  Application  To illustrate the usefulness of the proposed method in analyzing multivariate biomedical time  series, we turn to frequency band analysis of EEG signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Frequency bands are commonly  used in the scientific literature to generate summary measures of the EEG signal, so a  principled approach to frequency band estimation would be a welcomed development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  Our analysis considers 16-channel EEG signals from 14 participants in a simple resting-  state eyes-open, eyes-closed experimental protocol (Cattan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' For computational  efficiency, signals are standardized and downsampled to a sampling rate of 64 Hz and subset  to include 5 consecutive blocks alternating between eyes-closed and eyes-open conditions  lasting approximately 15 seconds per block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' This produces, for each participant, a time series  approximately 4800 observations in length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' For illustration, Figure 1 displays standardized  23  EEG time series from the Oz occipital channel in participants 2 and 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' The time-varying  behavior can be witnessed around the 10 Hz frequency, which corresponds to the traditional  Alpha frequency band (8-12 Hz) and is associated with the alternating eyes-closed and eyes-  open conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  In practice, the detection of Alpha waves is a useful indicator of stress levels, concentra-  tion, relaxation, or mental load (Banquet, 1973;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Antonenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=', 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' However, Alpha  power may vary in both its peak frequency and range of frequencies across participants (Dop-  pelmayr et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=', 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Ghazi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=', 2021), so a data-driven approach is essential for accurately  characterizing Alpha power across different individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' To illustrate this, Figure 2 displays  the estimated frequency partition points using the proposed method applied on each partici-  pant’s EEG signal separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' A triangular kernel is used as the kernel choice K(·) with band-  width h = T −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='3, and the multiscale approach described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='5 was used considering  a sequence of equally-spaced values Wmin = N  12 < W1 < W2 < W3 < W4 < W5 < Wmax = N  4 ,  with N = T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  14  13  12  11  10  9  8  7  6  5  4  3  2  1  0  6  12  18  24  30  Frequency  Participant  Estimated Frequency Partition Points by Participant  Figure 2: Frequency partition points estimated using proposed methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  24  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
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+page_content='0  (b)  Figure 3: Local periodogram for two components and estimated frequency bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  While frequency bands identified are similar across participants, some participants (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  Participant 13) exhibit Alpha power in a slightly higher band of frequencies than the con-  ventional 8-12 Hz band used in practice, which demonstrates the advantage of the proposed  data-driven frequency band estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' For further illustration, Figure 3 displays the esti-  mated spectral density for Participant 13 in two channels, one from the occipital region (Oz)  and another from the parietal region (P7), along with the estimated frequency partition  points (green lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' It is not surprising to find that the time-varying behavior in this band  is prominent for these two channels, since the parietal and occipital brain regions are known  to exhibit strong Alpha band power (Pfurtscheller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=', 1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  Next, in addition to identifying the frequency bands, the proposed method also provides  a data-driven approach to identifying components and cross-components of the multivariate  signal that are significantly associated with each identified frequency partition point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Ap-  plying the bootstrap technique described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='6, we can identify components that  significantly contribute to the upper and lower Alpha power frequency partition points iden-  tified by the proposed method for Participants 2 and 13 (see Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  Unsurprisingly,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' we find that channels from the parietal (P7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' P3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Pz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' P4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' P8) and occipital  (O1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Oz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' O2) regions play a major role in establishing both the upper and lower bounds of the  25  FP1  FP2  FC5  FC6  FZ  T7  CZ  T8  P7  P3  PZ  P4  P8  O1  Oz  O2  FP1  FP2  FC5  FC6  FZ  T7  CZ  T8  P7  P3  PZ  P4  P8  O1  Oz  O2  Participant 2 Frequency 6 Hz  FP1  FP2  FC5  FC6  FZ  T7  CZ  T8  P7  P3  PZ  P4  P8  O1  Oz  O2  FP1  FP2  FC5  FC6  FZ  T7  CZ  T8  P7  P3  PZ  P4  P8  O1  Oz  O2  Participant 2 Frequency 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='7 Hz  FP1  FP2  FC5  FC6  FZ  T7  CZ  T8  P7  P3  PZ  P4  P8  O1  Oz  O2  FP1  FP2  FC5  FC6  FZ  T7  CZ  T8  P7  P3  PZ  P4  P8  O1  Oz  O2  Participant 13 Frequency 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='3 Hz  FP1  FP2  FC5  FC6  FZ  T7  CZ  T8  P7  P3  PZ  P4  P8  O1  Oz  O2  FP1  FP2  FC5  FC6  FZ  T7  CZ  T8  P7  P3  PZ  P4  P8  O1  Oz  O2  Participant 13 Frequency 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='8 Hz  Figure 4: Components significantly contributing to estimated frequency partition point (red)  for two participants and two partition points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  26  Alpha band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' However, there are two additional findings of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' First, in differentiating  the time-varying dynamics of lower frequency power vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Alpha band power (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=', components  responsible for frequency partition points at 6 Hz and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='3 Hz for Participants 2 and 13  respectively), the prefrontal channels (FP1, FP2) are also involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' The prefrontal region  has been shown to be the dominant source of power in lower frequencies (< 4 Hz) during both  eyes-open and eyes-closed resting state conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Further, low frequency power has been  shown to increase from the eyes-closed to eyes-open condition (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=', 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Since the  prefrontal region is not meaningfully involved in Alpha power fluctuations, the differences in  the time-varying dynamics for lower frequencies vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Alpha power frequencies in the prefrontal  region are shown to be significant using the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' More precisely, our method  shows that the prefrontal region contributes towards the identification of the lower Alpha  band frequency partition points (6 Hz and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='3 Hz for Participants 2 and 13 respectively), but  not the upper Alpha band frequency partition points (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='7 Hz and 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='8 Hz for Participants 2  and 13 respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Second, the cross-components between the parietal and occipital region  channels and the prefrontal, frontal, and central region channels contribute differently to the  lower and upper Alpha band frequency partition points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  This suggests that interaction  components between the anterior and posterior brain regions also exhibit different time  varying behavior in the lower and higher frequencies surrounding the Alpha band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  Across all participants, Figure 5 displays the average component-wise p-values for test-  ing the significance of the contribution towards the lower and upper Alpha band frequency  partition points that were identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Here we can see more clearly that the lower frequency  partition point, which separates the lower frequencies from the Alpha band, can be attributed  to the prefrontal components and the interaction components between the prefrontal and  parietal/occipital regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' However, the upper frequency partition point separating the Al-  27  FP1  FP2  FC5  FC6  FZ  T7  CZ  T8  P7  P3  PZ  P4  P8  O1  Oz  O2  FP1 FP2 FC5 FC6 FZ  T7  CZ  T8  P7  P3  PZ  P4  P8  O1  Oz  O2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='4  First Partition Point: Average P−values  (a)  FP1  FP2  FC5  FC6  FZ  T7  CZ  T8  P7  P3  PZ  P4  P8  O1  Oz  O2  FP1 FP2 FC5 FC6 FZ  T7  CZ  T8  P7  P3  PZ  P4  P8  O1  Oz  O2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='6  Second Partition Point: Average P−values  (b)  Figure 5: Component-wise significance of contribution towards partition points  pha band from higher frequencies is associated with the parietal and occipital components  and cross-components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' These findings are made possible by the proposed method that is  uniquely able to identify frequency partition points and also the corresponding sets of sig-  nificant components and cross-components associated with each partition point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  5  Concluding remarks  The frequency band analysis framework introduced in this article offers a quantitative ap-  proach to identifying frequency bands that best preserve the nonstationary dynamics of the  underlying multivariate time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' This framework allows for estimation of both the num-  ber of frequency bands, their corresponding frequency partition points, and the components  and cross-components of the multivariate signal associated with each of the partition points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  This is made possible by the development of a sensible discrepancy measure and computa-  tionally efficient bootstrap testing procedure within an iterative search algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' However,  the proposed method is not without limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Motivated by the application to EEG fre-  quency band analysis, it would be interesting to extend this framework to directly consider  multiple subjects and produce a single set of frequency bands that jointly characterizes the  28  time-varying dynamics of the collection of signals in a data-driven manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Second, in order  to extend the proposed methodology for analyzing high-dimensional EEG signals (64 to 512  channels), the bootstrap testing procedure would need to be modified to accommodate high-  dimensional covariance structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' This would require appropriate simplifying assumptions  on the covariance structure, such as factor-based, sparse, and block covariance structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  Finally, as seen in the last simulation setting (M3B-2) in Section 3, the current discrepancy  measure requires large amounts of data to detect frequency partition points associated with  changes in a few components of the multivariate signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Modifications of the discrepancy  measure that may be more powerful in detecting such changes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' weighted squared L2  norm or L∞ norm) are also worth further investigation and development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  Funding Statement  Research reported in this publication was supported by the National Institute Of General  Medical Sciences of the National Institutes of Health under Award Number R01GM140476.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  The content is solely the responsibility of the authors and does not necessarily represent the  official views of the National Institutes of Health.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Portions of this research were conducted  with the advanced computing resources provided by Texas A&M High Performance Research  Computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  Data Availability Statement  The data underlying this article are publicly available via Zenodo at https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  5281/zenodo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='2348892.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  29  Supplementary  R code for “Frequency Band Analysis of Multivariate Time Series”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  R code, a quick start demo, and descriptions of all functions and parameters needed to  generate simulated data introduced in Section 3 and to implement the proposed method on  data for use in practice can be downloaded from GitHub at this link: https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='com/  sbruce23/mEBA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
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+page_content=' Society for Industrial and Applied Mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  Bruce, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=', C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Tang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Hall, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Krafty (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Empirical frequency band anal-  ysis of nonstationary time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Journal of the American Statistical Association 115(532),  1933–1945.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  30  Cattan, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=', P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
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+page_content=' Congedo (2018, December).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  EEG alpha waves  dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
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+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
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+page_content='2348892.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  Chen, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=', W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Feng, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Zhao, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Yin, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
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+page_content=' Neuroimage 41(2), 561–574.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
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+page_content=' Pollak, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Eldar (2016, March).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Carrier frequency and bandwidth  estimation of cyclostationary multiband signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' In 2016 IEEE International Conference  on Acoustics, Speech and Signal Processing (ICASSP), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
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+page_content='  Dahlhaus, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
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+page_content='  Dahlhaus, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
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+page_content='  Statist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
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+page_content=' Biological Cybernetics 79(1), 49-57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
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+page_content=' Napolitano, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
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+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
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+page_content='  Frontiers in Human Neuroscience 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
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+page_content=' Ombao, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Chen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
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+page_content=' Kupfer, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
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+page_content=' Psychosomatic  Medicine 66(1), 56-62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
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+page_content=' Lfp power spectra in v1 cortex: The graded effect of  stimulus contrast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Journal of Neurophysiology 94(1), 479–490.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
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+page_content=', B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Muhsal, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
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+page_content=' Detection of changes in multivariate time series  with application to EEG data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
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+page_content=' EEG alpha and theta oscillations reflect cognitive and memory per-  formance: A review and analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
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+page_content=' Local field potential in cortical area mt: Stimulus tuning  and behavioral correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
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+page_content=' Lurito, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
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+page_content=' Mathews, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Phillips (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Cor-  relations in low-frequency bold fluctuations reflect cortico-cortical connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' NeuroIm-  age 12(5), 582–587.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  Messer, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=', M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Kirchner, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Schiemann, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Roeper, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Neininger, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Schneider (2014,  12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' A multiple filter test for the detection of rate changes in renewal processes with  varying variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' 8(4), 2027–2067.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  Newson, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Thiagarajan (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Eeg frequency bands in psychiatric disorders:  A review of resting state studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Frontiers in Human Neuroscience 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  Ombao, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=', R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' von Sachs, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Guo (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' SLEX analysis of multivariate nonstationary  time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Journal of the American Statistical Association 100(470), 519–531.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  32  Pfurtscheller, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=', A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Stancak Jr, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Neuper (1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Event-related synchronization (ERS)  in the alpha band—an electrophysiological correlate of cortical idling: A review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Interna-  tional Journal of Psychophysiology 24(1-2), 39–46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  Preuss, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=', R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Puchstein, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Dette (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Detection of multiple structural breaks in  multivariate time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Journal of the American Statistical Association 110(510), 654–  668.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  Priestley, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' (1965).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Evolutionary spectra and non-stationary processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Journal of the  Royal Statistical Society, Series B 28, 228-240.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  Schr¨oder, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Ombao (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' FreSpeD: Frequency-specific change-point detection  in epileptic seizure multi-channel EEG data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Journal of the American Statistical Associ-  ation 114(525), 115–128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  Tiganj, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=', M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Mboup, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Chevallier, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Kalunga (2012, Aug).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Online frequency band  estimation and change-point detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' In 2012 1st International Conference on Systems  and Computer Science (ICSCS), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' 1-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  Yuen, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=', N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Osachoff, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Chen (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Intrinsic frequencies of the resting-state  fmri signal: The frequency dependence of functional connectivity and the effect of mode  mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Frontiers in Neuroscience 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  33  A  Proofs  Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' We have from (6) the estimated discrepancy measure given by  �D(ω) = 1  T  T  �  t=1  1  W  W  �  k=1  p  �  a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='b=1  �  �ga,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='b  � t  T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' ω − λk  �  − �ga,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='b  � t  T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' ω + λk  ��  ×  �  �ga,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='b  � t  T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' ω − λk  �  −  �ga,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='b  � t  T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' ω + λk  ��∗  = 1  T  T  �  t=1  1  W  W  �  k=1  p  �  a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='b=1  �  IN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='b( t  T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' ω − λk) − 1  T  T  �  t1=1  IN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='b(t1  T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' ω − λk)−  IN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='b( t  T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' ω + λk) + 1  T  T  �  t2=1  IN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='b(t2  T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' ω + λk)  �  ×  �  IN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='b( t  T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' ω − λk) − 1  T  T  �  t1=1  IN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='b(t1  T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' ω − λk) − IN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='b( t  T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' ω + λk) + 1  T  T  �  t2=1  IN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='b(t2  T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' ω + λk)  �∗  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  where �ga,b and IN,a,b denote the entry (a, b) in the respective matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Expanding the inside  term of the above expression involves several terms and we consider one of each kind and  show that the expected value of the discrepancy measure tends to zero under H0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  First, we consider the terms of the type IN( t  T , ω ± λk)I∗  N( t  T , ω ± λk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  We have,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' for  34  component (a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' b),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  IN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='b( t  T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' ω ± λk)I∗  N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='b( t  T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' ω ± λk) = JN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='a( t  T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' ω ± λk)J∗  N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='b( t  T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' ω ± λk)JN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='a( t  T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' ω ± λk)J∗  N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='b( t  T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' ω ± λk)  =  1  (2πN)2  � N−1  �  s1=0  Xa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='⌊utT⌋−N/2+1+s1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='T e−is1θk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='±  �  ×  � N−1  �  s2=0  Xb,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='⌊utT⌋−N/2+1+s2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='T eis2θk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='±  �  ×  � N−1  �  s3=0  Xa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='⌊utT⌋−N/2+1+s3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='T e−is3θk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='±  �  ×  � N−1  �  s4=0  Xb,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='⌊utT⌋−N/2+1+s4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='T eis4θk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='±  �  =  1  (2πN)2  �  s1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='s2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='s3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='s4  ∞  �  l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='o=−∞  �  Φa(uts1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' l)εts1−l  �  ×  �  Φb(uts2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' m)εts2−m  �  �  Φa(uts3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' n)εts3−n  �  ×  �  Φb(uts4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' o)εts4−o  �  exp(−iθk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='±(s1 − s2 + s3 − s4)) + O( 1  T )  =  1  (2πN)2  �  s1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='s2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='s3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='s4  ∞  �  l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='o=−∞  �  Za,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='ts1−lZb,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='ts2−mZa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='ts3−nZb,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='ts4−o  �  exp(−iθk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='±(s1 − s2 + s3 − s4)) + O( 1  T )  where θk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='± = ω ± λk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' ut =  t  T and tsj = ⌊utT⌋ − N/2 + 1 + sj for j = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  Also,  Za,ts1−l =  �  Φ  ′  a(uts1, l)εts1−l  �  where Φ  ′  a(uts1, l) denotes the ath row of the coefficient matrix  Φ(uts1, l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' For the expectation above, we apply Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='2 of Brillinger (2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Noting  that the Z random variables above are Gaussian by the assumption in (1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' the expected  values simplifies to  E  �  IN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='b( t  T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' ω ± λk)I∗  N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='b( t  T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' ω ± λk)  �  = E(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='b)  1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='T + E(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='b)  2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='T + O( 1  N ) + O( 1  T ) + O(N 2  T 2 ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  35  where  E(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='b)  1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='T  =  1  (2πN)2  �  s1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='s2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='s3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='s4  ∞  �  l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='o=−∞  exp(−iθk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='±(s1 − s2 + s3 − s4))E  ��  Φ  ′  a(ut,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' l)εts1−l  �  ×  �  Φ  ′  b(ut,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' m)εts2−m  ��  ×  E  ��  Φ  ′  a(ut,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' n)εts3−n  �  ×  �  Φ  ′  b(ut,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' o)εts4−o  ��  E(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='b)  2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='T  =  1  (2πN)2  �  s1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='s2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='s3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='s4  ∞  �  l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='o=−∞  exp(−iθk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='±(s1 − s2 + s3 − s4))E  ��  Φ  ′  a(ut,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' l)εts1−l  �  ×  �  Φ  ′  b(ut,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' o)εts4−o  ��  ×  E  ��  Φ  ′  a(ut,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' n)εts3−n  �  ×  �  Φ  ′  b(ut,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' m)εts3−m  ��  For E(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='b)  1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='T  it can be seen that the expectations are non-zero only when ts1 − l = ts2 − m  and ts3 − n = ts4 − o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' We hence have  1  T  T  �  t=1  1  W  W  �  k=1  E(a,b)  1,T  = 1  T  T  �  t=1  1  W  W  �  k=1  fa,b( t  T , θk,±)fa,b(ut, θk,±)∗ + o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  (14)  Similarly, for the term E(a,b)  2,T  it can be seen that the expectations are non-zero only when  ts1 − l = ts4 − o and ts3 − n = ts2 − m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' We get  1  T  T  �  t=1  1  W  W  �  k=1  E(a,b)  2,T  = 1  T  T  �  t=1  1  W  W  �  k=1  fa,b( t  T , θk,±)fa,b(ut, θk,±)∗ + o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  (15)  where θk,± = ω ± λk, ut = t/T, fa,b(ut, θk,±) denotes component (a, b) of the spectral matrix  f(ut, θk,±) and fa,b(ut, θk,±)∗ is the conjugate transpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' The terms in (14) and (15) are  approximated by  1  2πc  � 1  0  � 2πc  0  fa,b(u, ω ± λ)fa,b(u, ω ± λ)∗dλ du + o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  Next, we consider the terms of the type IN( t  T , ω ± λk)I∗  N( t  T , ω ∓ λk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  We have, for  36  component (a, b),  IN,a,b( t  T , ω ± λk)I∗  N,a,b( t  T , ω ∓ λk) =  1  (2πN)2  � N−1  �  s1=0  Xa,⌊utT⌋−N/2+1+s1,T e−is1θk,±  �  ×  � N−1  �  s2=0  Xb,⌊utT⌋−N/2+1+s2,T eis2θk,±  �  ×  � N−1  �  s3=0  Xa,⌊utT⌋−N/2+1+s3,T e−is3θk,∓  �  ×  � N−1  �  s4=0  Xb,⌊utT⌋−N/2+1+s4,T eis4θk,∓  �  =  1  (2πN)2  �  s1,s2,s3,s4  ∞  �  l,m,n,o=−∞  �  Za,ts1−lZb,ts2−mZa,ts3−nZb,ts4−o  �  exp(−iθk,±(s1 − s2))exp(−iθk,∓(s3 − s4)) +  O( 1  T ),  where θk,± = ω ± λk, θk,∓ = ω ∓ λk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' The expected values simplifies to E  �  IN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='b( t  T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' ω ±  λk)I∗  N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='b( t  T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' ω ± λk)  �  = E(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='b)  3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='T + E(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='b)  4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='T + o(1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' where  E(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='b)  3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='T  =  1  (2πN)2  �  s1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='s2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='s3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='s4  ∞  �  l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='o=−∞  exp(−iθk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='±(s1 − s2))exp(−iθk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='∓(s3 − s4))E  ��  Φ  ′  a(ut,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' l)εts1−l  �  ×  �  Φ  ′  b(ut,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' m)εts2−m  ��  × E  ��  Φ  ′  a(ut,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' n)εts3−n  �  ×  �  Φ  ′  b(ut,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' o)εts4−o  ��  E(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='b)  4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='T  =  1  (2πN)2  �  s1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='s2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='s3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='s4  ∞  �  l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='o=−∞  exp(−iθk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='±(s1 − s4))exp(−iθk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='∓(s3 − s2))E  ��  Φ  ′  a(ut,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' l)εts1−l  �  ×  �  Φ  ′  b(ut,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' o)εts4−o  ��  × E  ��  Φ  ′  a(ut,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' n)εts3−n  �  ×  �  Φ  ′  b(ut,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' m)εts3−m  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  For E(a,b)  3,T , the expectations are non-zero only when ts1 −l = ts2 −m and ts3 −n = ts4 −o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  We hence have  1  T  T  �  t=1  1  W  W  �  k=1  E(a,b)  1,T  = 1  T  T  �  t=1  1  W  W  �  k=1  fa,b( t  T , θk,±)fa,b(ut, θk,∓)∗ + o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  (16)  For E(a,b)  4,T , the expectations are non-zero only when ts1 − l = ts4 − o and ts3 − n = ts2 − m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  37  We get  1  T  T  �  t=1  1  W  W  �  k=1  E(a,b)  2,T  = 1  T  T  �  t=1  1  W  W  �  k=1  fa,b( t  T , θk,±)fa,b(ut, θk,∓)∗ + o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  (17)  The remaining types of terms considered are i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' IN( t  T , ω±λk)  �  1  T  �T  t1=1 IN,a,b( t1  T , ω±λk)∗�  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' IN( t  T , ω ± λk)  �  1  T  �T  t1=1 IN,a,b( t1  T , ω ∓ λk)∗�  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  �  1  T  �T  t1=1 IN,a,b( t1  T , ω ± λk)∗�  IN( t  T , ω ±  λk)∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' iv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  �  1  T  �T  t1=1 IN,a,b( t1  T , ω ± λk)∗�  IN( t  T , ω ∓ λk)∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  �  1  T  �T  t1=1 IN,a,b( t1  T , ω ± λk)  �  ×  �  1  T  �T  t1=1 IN,a,b( t1  T , ω ± λk)∗�  and vi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  �  1  T  �T  t1=1 IN,a,b( t1  T , ω ± λk)  �  ×  �  1  T  �T  t1=1 IN,a,b( t1  T , ω ∓  λk)∗�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' The treatment of all these types of terms is similar to the approach that lead to  (14)-(17) for the first two types of terms considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  When ω ∈ CN,2, it can be seen that by combining the 16 limiting expressions that we  get from all the different types of terms, the expected value of �D(ω) tends to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' With the  same approach, when ω ∈ {ω1, ω2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' , ωK}, the expected value of �D(ω) is approximated by  1  2πc  � 1  0  � 2πc  0  �p  a,b=1  �  ga,b(u, ω−λ)−ga,b(u, ω+λ)  ��  ga,b(u, ω−λ)−ga,b(u, ω+λ)  �∗  dλ du+o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  Next, we consider the variance of the discrepancy measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Here we look at the 2nd order  38  cumulant of discrepancy measure that is written as  cum( �D(ω),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' �D(ω)) = 1  T 2  T  �  t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='t2=1  1  W 2  W  �  k1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='k2=1  p  �  a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='d=1  cum  ��  IN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='b(t1  T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' ω − λk1) − 1  T  T  �  x1=1  IN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='b(x1  T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' ω − λk1)−  IN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='b(t1  T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' ω + λk1) + 1  T  T  �  x2=1  IN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='b(x2  T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' ω + λk1)  �  ×  �  IN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='b(t1  T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' ω − λk1) − 1  T  T  �  x3=1  IN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='b(x3  T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' ω − λk1) − IN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='b(t1  T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' ω + λk1) + 1  T  T  �  x4=1  IN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='b(x4  T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' ω + λk1)  �∗  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  �  IN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='d(t2  T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' ω − λk2) − 1  T  T  �  x1=1  IN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='d(x1  T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' ω − λk2)−  IN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='d(t2  T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' ω + λk2) + 1  T  T  �  x2=1  IN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='d(x2  T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' ω + λk2)  �  ×  �  IN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='d(t2  T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' ω − λk2) − 1  T  T  �  x3=1  IN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='d(x3  T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' ω − λk2) − IN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='d(t2  T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' ω + λk2) + 1  T  T  �  x4=1  IN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='d(x4  T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' ω + λk2)  �∗  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  (18)  We can now look at cumulant terms of different types and see the behavior as T → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  First,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' for components (a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' b) and (c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' d),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='we consider cumulant terms of the type  cum  �  IN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='b(t1  T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' ω ± λk1)I∗  N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='b(t1  T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' ω ± λk1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' IN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='d(t2  T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' ω ± λk2)I∗  N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='d(t2  T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' ω ± λk2)  �  =  1  (2πN)4  N−1  �  r1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='r2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='r3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='r4=0  N−1  �  s1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='s2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='s3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='s4=0  ∞  �  l1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='m1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='n1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='o1=−∞  ∞  �  l2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='m2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='n2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='o2=−∞  cum  ��  Φa(utr1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' l)εtr1−l1  �  ×  �  Φb(utr2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' m1)εtr2−m1  �  ×  �  Φa(utr3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' n1)εtr3−n1  �  ×  �  Φb(utr4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' o1)εtr4−o1  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  �  Φc(uts1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' l2)εts1−l2  �  ×  �  Φd(uts2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' m2)εts2−m2  �  ×  �  Φc(uts3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' n2)εts3−n2  �  ×  �  Φd(uts4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' o2)εts4−o2  ��  ×  exp(−iθk1(r1 − r2 + r3 − r4)) × exp(−iθk2(s1 − s2 + s3 − s4)) + O( 1  T ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  (19)  39  where θk1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='± = ω ± λk1 and θk2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='± = ω ± λk2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' Noting that εt is Gaussian, an application of  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='2 of Brillinger (2001) yield certain terms that are non-vanishing asymptotically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  The term in (A) leads to  1  T 2  T  �  t1,t2=1  1  W 2  W  �  k1,k2=1  cum  �  IN,a,b(t1  T , θk1,±)I∗  N,a,b(t1  T , θk1,±), IN,c,d(t2  T , θk2,±)I∗  N,c,d(t2  T , θk2,±)  �  =  1  T 2  T  �  t=1  1  W 2  W  �  k1,k2=1  40  (2π)4fa,b( t  T , θk1,±)fa,b( t  T , θk1,±)∗fc,d( t  T , θk2,±)fc,d( t  T , θk2,±)∗ + o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' (20)  The above term can be approximated by  1  T2πc  � 1  0  � 2πc  0  fa,b(u, ω ± λ)fa,b(u, ω ± λ)∗fc,d(u, ω ±  λ)fc,d(u, ω ± λ)∗dλ du + o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' The treatment of the cumulant terms from the other term  types follows similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' With the sets CN,1 and CN,2 defined in (7), the proof of this result  follows from the following two results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' P  � �  ω∈CN,2 �D(ω) > ηT,ω  �  T→∞  −→ 0,  (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=' P  � �  ω∈{ω1,ω2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content=',ωK} �D(ω) > ηT,ω  �  T→∞  −→ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  For any ω ∈ CN,2, an application of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='1(a) yields  P(  �  ω∈CN,2  �D(ω) > ηT,ω) ≤  �  ω∈CN,2  P( �D(ω) > ηT,ω)  (21)  ≤  �  ω∈CN,2  E( �D(ω))  ηT,ω  T→∞  −→ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  (22)  The result in (b) follows by an application of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
+page_content='  40' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfHAau/content/2301.03664v1.pdf'}
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+arXiv:2301.11515v1  [cs.IT]  27 Jan 2023
+1
+Secrecy Analysis for MISO Broadcast Systems with
+Regularized Zero-Forcing Precoding
+Xin Zhang, Graduate Student Member, IEEE, Shenghui Song, Senior Member, IEEE, and Yonina C.
+Eldar, Fellow, IEEE
+Abstract—As an effective way to enhance the physical layer
+security (PLS) for the broadcast channel (BC), regularized zero-
+forcing (RZF) precoding has attracted much attention. However,
+the reliability performance, i.e., secrecy outage probability (SOP),
+of RZF is not well investigated in the literature. In this paper,
+we characterize the secrecy performance of RZF precoding in
+the large multiple-input single-output (MISO) broadcast system.
+For this purpose, we first consider a central limit theorem (CLT)
+for the joint distribution of the users’ signal-to-interference-plus-
+noise ratio (SINR) and the eavesdropper’s (Eve’s) signal-to-noise
+ratio (ESNR) by leveraging random matrix theory (RMT). The
+result is then utilized to obtain a closed-form approximation
+for the ergodic secrecy rate (ESR) and SOP of three typical
+scenarios: the case with only external Eves, the case with only
+internal Eves, and that with both. The derived results are then
+used to evaluate the percentage of users in secrecy outage and
+the required number of transmit antennas to achieve a positive
+secrecy rate. It is shown that, with equally-capable Eves, the
+secrecy loss caused by external Eves is higher than that caused
+by internal Eves. Numerical simulations validate the accuracy of
+the theoretical results.
+Index Terms—Physical layer security (PLS), regularized zero-
+forcing (RZF) precoding, multiple-input single-output (MISO),
+central limit theorem (CLT), random matrix theory (RMT).
+I. INTRODUCTION
+A pivotal problem for wireless communications is eaves-
+dropping due to the broadcast nature of wireless channels [1],
+[2]. To ensure security, traditional network-layer key-based
+cryptography has been widely used. However, the dynamics
+of wireless environments, e.g., fading, lead to issues in key
+distribution and management, and cause high computational
+complexity [3]. To provide additional security enhancements,
+physical layer security (PLS) [4], which utilizes the random-
+ness of the noise and dynamics of the fading channel to
+limit information leakage to potential eavesdroppers, has been
+considered as an appealing low-complexity alternative [5].
+PLS has become an active area of research, especially in
+the broadcast channel (BC), where the transmitter tries to
+broadcast messages to users, with minimum leakage to the
+eavesdroppers. To this end, linear precoding schemes have
+been used in confidential transmission of multiple antenna
+systems due to its simple implementation and effectiveness in
+interference controlling [6]–[8]. In [3], [9], the authors utilized
+regularized zero-forcing (RZF) precoding, also referred to as
+Xin Zhang and Shenghui Song are with the Department of Electronic and
+Computer Engineering, The Hong Kong University of Science and Technol-
+ogy, Hong Kong (e-mail: xzhangfe@connect.ust.hk; eeshsong@ust.hk).
+Yonina C. Eldar is with the Faculty of Math and CS, Weizmann Institute
+of Science, Rehovot 7610001, Israel (e-mail: yonina.eldar@weizmann.ac.il).
+regularized channel inversion (RCI), to mitigate the signal
+leaked to the undesired users and maximize the secrecy sum
+rate. Despite the wide usage of RZF, its performance analysis
+is still in the infant stage. Existing works on RZF in MISO
+BC can be categorized into two types according to the nature
+of the eavesdroppers, referred to as Eves.
+1. Internal Eve: In a MISO BC, users can act maliciously
+as Eves for other users [10], [11]. Given the Eves are within
+the BC system, we refer to this scenario as the internal Eve
+case. Under such circumstances, for a given user, other users
+are regarded as single-antenna Eves. In [3], [9], the secrecy
+sum rate of the MISO BC system with internal Eves over
+independent and identically distributed (i.i.d.) channels was
+given in closed form by large random matrix theory (RMT).
+Considering the unequal path loss and correlations of the
+transmit antennas, the authors of [12] evaluated the secrecy
+sum rate of RZF using RMT.
+2. External Eve: In a practical scenario, external devices can
+also act as Eves, which are referred to as external Eves. The
+major difference between internal and external Eves is that the
+channels of internal Eves are typically correlated with those
+of the users while the channels of external Eves are typically
+independent of the users. The impact of external Eves on the
+secure connectivity was investigated by stochastic geometry
+(SG) [13]–[15]. In [16], assuming that the locations of the Eves
+are distributed as a Poisson point process (PPP), the authors
+derived the sum rate and the secrecy outage probability (SOP)
+with RZF over uncorrelated MISO channels by utilizing RMT
+and SG. In [17], assuming both users’ and Eves’ locations are
+distributed as a PPP, the authors derived approximate closed-
+form results for the SOP and ergodic secrecy rate (ESR) with
+RZF precoding over uncorrelated MISO channels.
+Most existing works focus on the secrecy sum rate of
+the system [3], [12], [16]. However, the secure reliability
+performance, i.e., SOP, is not well studied. In [17], the SOP
+of RZF precoding was derived by SG, considering only large-
+scale fading, where the impact of small-scale fading on the
+SOP is ignored. Although the SG based approach is powerful
+in modeling the dynamics of the network and performing
+coverage analysis [18], it may not be able to capture long-term
+behavior of the system where one BS serves a fixed group of
+users. In this case, the large-scale fading is deterministic and
+thus the PPP model is not suitable. In particular, the PPP model
+will dominate the performance analysis such that the impact of
+small-scale fading can not be fully understood. In this paper,
+we focus on the impact of small-scale fading on the secrecy
+performance of MISO BC with RZF precoding and derive the
+
+2
+SOP, which is not available in the literature.
+A closed-form expression for the SOP is essential for the
+performance evaluation and will be very beneficial for system
+design. However, characterization of the secrecy performance,
+especially the SOP, turns out to be a difficult problem due
+to the fractional structure of the signal-to-interference-plus-
+noise ratio (SINR) and the power normalization factor of RZF.
+In particular, these challenges make it extremely difficult to
+obtain the closed-form results for the joint distribution of the
+users’ SINR and Eve’s signal-to-noise ratio (ESNR) in finite
+dimensions. To overcome this difficulty, asymptotic RMT will
+be adopted to derive the asymptotic distribution of the SINRs
+and ESNRs by assuming that the number of transmit antennas,
+the number of users, and the number of Eves go to infinity
+with the same pace. In fact, asymptotic RMT has been widely
+used in large-system analysis of multiple-antenna systems to
+obtain strikingly simple results [19]–[21], which are shown to
+be effective even for small dimensions.
+The contributions of this paper are summarized as follows:
+1. We consider a CLT for the joint distribution of the SINRs
+and ESNRs of a finite number of users and derive explicit
+expressions for the asymptotic mean and variance. Specifically,
+we utilize the CLT for the quadratic form and compute the
+asymptotic second-order attributes by Gaussian tools [20],
+[22]. The derived results can be extended to the case with
+imperfect CSI and the analysis of the channel-inversion based
+precoding method, e.g., secure RZF [23].
+2. Based on the CLT, we derive closed-form approxima-
+tions for the SOP of RZF precoding including three sce-
+narios, namely the External-Eve-Only, Internal-Eve-Only, and
+External+Internal-Eve cases.
+3. The derived results are then used to evaluate how many
+transmit antennas are needed to guarantee a positive secrecy
+rate and to estimate the percentage of users in outage. We
+further compare the performance of RZF for the Internal-Eve-
+Only and External-Eve-Only cases and derive the theoretical
+gap between the two secrecy rates, which indicates that the
+external Eves incur more information leakage than internal
+ones. Numerical results validate the accuracy of the theoretical
+results.
+The rest of this paper is organized as follows. In Section II,
+the system model is introduced and the problem is formulated.
+In Section III, some preliminary results on RMT, which
+are critical for the derivations in this paper, are given. In
+Section IV, a CLT for the joint distribution of SINRs and
+ESNRs and the approximation for the SOP are derived. The
+derived results are used in Section V to evaluate the percentage
+of users in outage and the required number of antennas for
+a positive secrecy rate. Simulation results are provided in
+Section VI. Section VII concludes the paper.
+Notations: Bold, upper case letters and bold, lower case
+letters represent matrices and vectors, respectively. The prob-
+ability operator and expectation operator are denoted by P(·)
+and E(·), respectively. The circularly complex Gaussian distri-
+bution and real Gaussian distribution are denoted by CN and
+N, respectively. The N-dimensional complex vector space and
+real vector space are represented CN and RN, respectively.
+The M-by-N complex and real matrix space, are represented
+by CM×N and RM×N, respectively. The centered version of
+the random variable x is denoted by x = x−Ex. AH denotes
+the Hermitian transpose of A and [A]i,j or Aij denotes
+the (i, j)-th entry. The conjugate of a complex number is
+represented by (·)∗. The spectral norm of A is given by ∥A∥.
+Tr A is the trace of A, IN is the N by N identity matrix, and
+Φ(x) represents the cumulative distribution function (CDF) of
+the standard Gaussian distribution. Almost sure convergence,
+convergence in probability, and convergence in distribution,
+are represented by
+a.s.
+−−−−→
+N→∞ ,
+i.p.
+−−−−→
+N→∞ , and
+d
+−−−−→
+N→∞ , respectively.
+The set {1, 2, ..., N} is denoted by [N] and
+1(·) represents the
+indicator function.
+II. SYSTEM MODEL AND PROBLEM FORMULATION
+A. System Model
+We consider the downlink MISO system, where N single-
+antenna users are served by a base station (BS) with M trans-
+mit antennas. We assume that perfect channel state information
+(CSI) of users is available at the BS and RZF is adopted to
+suppress information leakage.
+RZF is a linear precoding scheme proposed to serve
+multiple users in the MISO downlink channel, which often
+demonstrates better performance than ZF, especially with low
+SNR [6]. With RZF, the received signal at the k-th user is
+given by
+yk =
+�
+ξhH
+k
+N
+�
+i=1
+√piwisi + vk,
+(1)
+where si ∼ N(0, 1) and vk ∼ N(0, σ2) denote the transmitted
+signal for the i-th user and the additive white Gaussian noise
+(AWGN) at the k-th user, respectively. Here hk
+∈ CM
+and wi ∈ CM represent the channel vector of the k-th
+user and the precoding vector for the i-th user, respectively,
+pi denotes the power of the i-th message and ξ represents
+the power normalization factor. The precoding matrix W =
+[w1, w2, ..., wN] ∈ CM×N of the RZF precoder is given
+by [6]
+W =
+�
+HHH + zIN
+�−1 H,
+(2)
+where H = [h1, h2, ..., hN] ∈ CM×N and z > 0 is the
+regularization parameter of the RZF precoder. The power
+normalization factor satisfies
+ξ Tr PWHW
+N
+= 1, where P =
+diag(p1, p2, ..., pN). The k-th user’s SINR is given by
+SINRk =
+pk|hH
+k wk|2
+�
+j̸=k
+pj|hH
+k wj|2 + σ2
+ξ
+.
+(3)
+Here we consider L single antenna Eves. The channel vector
+between the BS and the l-th Eve is denoted by he,l ∈ CM,
+l ∈ [L]. The received signal of the l-th Eve, ul, is given by
+ul =
+�
+ξhH
+e,l
+N
+�
+i=1
+√piwisi + wl,
+(4)
+where wl ∼ N(0, ρ2) represents the AWGN at the l-th Eve
+with ρ2 denoting the power of the noise.
+It is not straightforward to directly characterize the infor-
+mation leakage, i.e., the information received by the undesired
+
+3
+users. Here we consider the worst-case scenario, where all
+Eves are assumed to work together and be able to cancel
+out the multiuser interference. This model is widely used in
+secrecy analysis [3], [12], [24]. Under such circumstances, the
+ESNR for all Eves to collaboratively overhear the message sk
+is given by
+ESNRk = ξ∥HH
+e wk∥2
+ρ2
+,
+(5)
+where He = [he,1, he,2, ..., he,L]. The corresponding informa-
+tion leakage rate is given by Ce,n = log(1 + ESNRk). As a
+result, the secrecy rate of the k-th user is
+Rk = ⌈log(1 + SINRk) − log(1 + ESNRk)⌉+.
+(6)
+The SOP of the k-th user for a given rate requirement R is
+given by
+Pout,k = P(Rk ≤ R).
+(7)
+Note that the Eves can be both internal and external. For the
+cases with only internal Eves, the channel matrix for the Eves
+with respect to the k-th user can be given by He = Hk, where
+Hk is obtained by removing hk from H. In this paper, we
+will evaluate the ESR in (6) and SOP in (7) in a large system
+setting where the number of transmit antennas and numbers
+of users and Eves go to infinity with the same space.
+B. Channel Model
+We consider a correlated Rayleigh fading channel for users.
+Accordingly, the channel vector for the k-th user can be rep-
+resented by hk = t
+1
+2
+k R
+1
+2 xk, where tk is the large-scale fading
+(pathloss) from the BS to the k-th user, xk ∼ CN(0, IM), and
+R denotes the correlation matrix of the BS towards the users.
+In this paper, we assume a common transmit correlation matrix
+for different users, which is widely used in [12], [25]–[27]
+for tractability of the problem. With the common correlation
+matrix, the channel matrix H = [h1, ..., hN] is given by
+H = R
+1
+2 XT
+1
+2 ,
+(8)
+where T = diag(t1, t2, ..., tN) and X = [x1, ..., xN].
+Assume that there are L external single-antenna Eves and
+denote the channel vector for the l-th Eve as gl, l ∈ [L].
+Similar to the channel matrix for the users, the channel matrix
+of the external Eves, G = [g1, g2, ..., gL], is given by
+G = C
+1
+2 YD
+1
+2 ,
+(9)
+where C ∈ CM×M and D ∈ CL×L = diag(d1, d2, ..., dL)
+represent the correlation matrix and the large-scale fading for
+the Eves, respectively. Here Y ∈ CM×L is a random matrix
+with i.i.d. circularly Gaussian entries, i.e., Ym,l ∼ N(0, 1
+M ).
+Notice that the external Eves overhear the message passively.
+Thus, we can assume that hk, k ∈ [N] and gl, l ∈ [L] are
+independent, i.e., X and Y are independent. Due to the same
+reason, the CSI of the Eves is not available at the BS. If there
+are only external Eves, we have He = G.
+In this paper, we consider three typical scenarios.
+The External-Eve-Only case. According to (5), we can
+obtain the ESNR for overhearing message sk as
+ESNRex,k = ξwH
+k GGHwk
+ρ2
+.
+(10)
+The achievable secrecy rate for the k-th user is given by
+Rex,k = ⌈log(1 + SINRk) − log(1 + ESNRex,k)⌉+.
+(11)
+The Internal-Eve-Only case. In this case, the unintended
+users within the MISO BC behave maliciously so that the
+crosstalk among users causes information leakage [12]. For
+each user, the other N − 1 users behave as single-antenna
+Eves. Specifically, users in Mk = [N] \ k can cooperate to
+jointly eavesdrop the message sk. In the worst case scenario,
+they can be regarded as a single Eve with N − 1 antennas.
+The ESNR for overhearing message sk is then given by [3],
+[9]
+ESNRin,k = ξwH
+k HkHH
+k wk
+σ2
+.
+(12)
+The achievable secrecy rate for the k-th user is given by
+Rin,k = ⌈log(1 + SINRk) − log(1 + ESNRin,k)⌉+.
+(13)
+In fact, (11) and (13) can be obtained by taking He = G and
+He = Hk in (6), respectively.
+The External+Internal-Eve case. In this case, both the
+internal and external Eves exist and they work in a non-
+colluding mode. The secrecy rate of the k-th user is given
+by [28], [29]
+Rk = min(Rex,k, Rin,k).
+(14)
+Note that the first two cases are differentiated by whether
+the channels of the Eves are independent with those of the
+intended users. The secrecy sum rate of the Intern-Eve-Only
+case has been investigated in [3], [9], [12] but the SOP is not
+available in the literature. In the following, we investigate the
+ESR and SOP for the three cases.
+III. PRELIMINARIES
+In this section, we provide some preliminary results for the
+evaluation of SINR and ESNR, and introduce the RMT results
+and notations which will be frequently used in the following
+derivations. Necessarily, we first introduce the resolvent matrix
+Q = (zIN + HHH)−1, which will be extensively used in the
+following analysis. Further define Qk = (zIN + HkHH
+k )−1,
+where Hk is obtained by removing the k-th column from H.
+By utilizing the identity hH
+k Q =
+hH
+k Qk
+1+hH
+k Qkhk , the SINR for
+k-th user can be rewritten as
+SINRk =
+pkA2
+k
+Bk + σ2(1 + Ak)2C ,
+(15)
+where Ak
+=
+hH
+k Qkhk, Bk
+=
+hH
+k QkHkPkHH
+k Qkhk,
+and C
+=
+1
+M Tr PHH(HHH + zIN)−2H. Here Pk is
+obtained by removing the k-th row and column of P =
+diag(p1, p2, ..., pN). Similarly, ESNRex,k and ESNRin,k can
+be rewritten as
+ESNRex,k =
+pkEk
+ρ2C(1 + Ak)2 ,
+(16)
+and
+ESNRin,k =
+pkFk
+σ2C(1 + Ak)2 ,
+(17)
+
+4
+respectively, where Ek
+= hH
+k QkGGHQkhk and Fk
+=
+hH
+k QkHkHH
+k Qkhk. The formulation above resorts the inves-
+tigation of SINRk and ESNRk to that of the Ak, Bk, Dk, and
+Ek, which can be regarded as the quadratic forms of hk when
+Hk is given. We next introduce three assumptions based on
+which the analysis in this paper is performed.
+Assumption A.1. The dimensions N, L, and M go to
+infinity at the same pace, i.e., when M → ∞,
+0 < lim inf M
+N ≤ M
+N ≤ lim sup M
+N < ∞,
+0 < lim inf L
+M ≤ L
+M ≤ lim sup L
+M < ∞.
+(18)
+Assumption A.2. ∥R∥ ≤ ∞, ∥T∥ ≤ ∞, ∥P∥ ≤ ∞ [30],
+[31].
+Assumption
+A.3.
+inf
+M≤1
+Tr R
+M
+>
+0,
+inf
+M≤1
+Tr T
+M
+>
+0,
+inf
+M≤1
+Tr C
+M
+> 0, inf
+M≤1
+Tr D
+M
+> 0 [20], [32].
+Assumption A.1 indicates the asymptotic regime where the
+number of transmit antennas, the number of users, and the
+number of external Eves are in the same order. Assump-
+tion A.2 and A.3 eliminate the extremely low-rank case of
+R and C, i.e., the ranks of R and C do not increase with
+the number of antennas. The bound ∥P∥ ≤ ∞ guarantees the
+power for each message is finite. We further define two ratios
+τ =
+M
+N−1 and θ = L
+M .
+Lemma 1. Define matrices GR =
+�
+zIM + �δR
+�−1
+and
+GT =
+�
+IN + �δT
+�−1
+, where (δ, �δ) is the unique positive
+solution for the following system of equations
+�
+δ
+=
+1
+M Tr RGR,
+�δ
+=
+1
+M Tr TGT .
+(19)
+Given assumptions A.1-A.3, for the random matrix H defined
+in (8), it holds true that for any deterministic matrix A with
+bounded norm [30, Theorem 1],
+1
+M Tr AQ
+a.s.
+−−−−→
+N→∞
+1
+M Tr AGR,
+(20)
+and [20, Step B in Section IV]
+1
+M E Tr AQ = 1
+M Tr AGR + O(M −2),
+(21)
+Lemma 1 indicates that the normalized trace of the resolvent
+converges almost surely to its deterministic equivalent, which
+is a good approximation for the expectation of the trace of the
+resolvent. The computation of the variances for the SINR and
+ESNR is highly related to the trace of the resolvent.
+We summarize some frequently used notations in Table I,
+which are all derived from δ and �δ. In the following analysis,
+some of the symbols have the subscript k. They are derived
+similarly as those in Table I, but are parameterized by R and
+Tk with
+�
+δk
+=
+1
+M Tr RGR,k,
+�δk
+=
+1
+M Tr TkGT,k,
+(22)
+where Tk is obtained by removing the k-th row and k-
+th column in T, GR,k =
+�
+zIM + �δkR
+�−1
+and GT,k =
+(IN−1 + δkTk)−1. These k-related quantities are used to
+approximate the Qk related statistics.
+TABLE I: List of Notations.
+Symbols
+Expression
+Symbols
+Expression
+γ
+1
+M Tr R2G2
+R
+γ(A)
+1
+M Tr AG2
+RR
+�γ
+1
+M Tr T2G2
+T
+�γ(B)
+1
+M Tr BG2
+T T
+η
+1
+M Tr R3G3
+R
+η(A)
+1
+M Tr AG3
+RR2
+�η
+1
+M Tr T3G3
+T
+�η(B)
+1
+M Tr BG3
+T T2
+ζ
+1
+M Tr R4G4
+R
+ζ(A)
+1
+M Tr AG4
+RR3
+�ζ
+1
+M Tr T4G4
+T
+�ζ(B)
+1
+M Tr BG4
+T T3
+IV. MAIN RESULTS
+In this section, we develop theoretical results on the asymp-
+totic distribution of SINR, ESNRex, and ESNRin, and then
+derive the ESR and SOP of the MISO BC.
+A. First-order Analysis of the SINR, ESNR, and Secrecy Rate
+In this part, we derive approximations for the mean of SINR
+and ESNR, which will be used to obtain the ESR.
+1) SINR and ESNR: According to (15)-(17), an approxima-
+tion for SINR, ESNRex, and ESNRin can be obtained by the
+continuous mapping theorem [33], where we need to obtain
+the deterministic equivalents of Ak, Bk, Ek, Fk, and C.
+By Lemma 1 and [34, Lemma 14.2], we have
+Ak
+a.s.
+−−−−→
+N→∞
+tkE Tr RQk
+M
+N→∞
+−−−−→ tkδk := Ak,
+(23)
+and
+Bk
+a.s.
+−−−−→
+N→∞
+tkE Tr RQkHkPkHH
+k Qk
+M
+= tk�γk(Pk)γk
+∆k
++ O(M −2) := Bk + O(M −2),
+(24)
+where �γk(Pk) and γk are given in Table I. Here ∆k = 1 −
+γk�γk,
+C
+a.s.
+−−−−→
+N→∞
+�γ(P)γ(IM)
+∆
+:= C,
+(25)
+Dk
+a.s.
+−−−−→
+N→∞
+E Tr RQkGGHQk
+M
+= θdγk(C)
+∆k
++ O(M −2) := Dk + O(M −2),
+(26)
+and
+Ek
+a.s.
+−−−−→
+N→∞
+E Tr QkHkHH
+k QkR
+M
+= �γk(IN−1)γk
+∆k
++ O(M −2) := Ek + O(M −2).
+(27)
+We define Ck = Tr QkHkPkHH
+k Qk
+M
+and its deterministic equiv-
+alent is Ck = �γk(Pk)γk(IM)
+∆k
+. In fact, we have
+√
+M(C − Ck) =
+√
+M[
+pkhH
+k Q2
+khk
+M(1 + hH
+k Qkhk)2
++ hH
+k QkHPkHHQkhkhH
+k Qkhk
+M(1 + hH
+k Qkhk)2
+− 2hH
+k QkHPkHHQ2
+khk
+M(1 + hH
+k Qkhk)
+]
+a.s.
+−−−−→
+N→∞ O(M − 1
+2 ),
+(28)
+
+5
+which indicates that C can be replaced by Ck in the asymptotic
+regime A.1. To keep a concise expression for the deterministic
+equivalent, we will use Ck instead of C.
+By the continuous mapping theorem [33], the approxima-
+tion for SINRk, ESNRex,k, and ESNRin,k can be obtained
+by replacing the random quantities with their deterministic
+equivalents in (23)-(27). Thus, we obtain
+SINRk =
+pkt2
+kδ2
+k∆k
+γk�γk(Pk) + σ2(1 + tkδk)2γk(IM)�γk(Pk),
+ESNRex,k =
+pkθdγk(C)
+ρ2(1 + tkδk)2γk(IM)�γk(Pk),
+ESNRin,k =
+pk�γk(IN−1)γk
+σ2(1 + tkδk)2�γk(Pk)γk(IM),
+(29)
+where d = Tr D
+L
+represents the averaged large-scale fading of
+the external Eves.
+2) Ergodic Secrecy Rate (ESR): We approximate the ESR
+(expectation of the secrecy rate) by replacing the instanta-
+neous SINR and ESNR with their deterministic equivalents
+in (29) [35]. Specifically, we plug (29) into (11) and (13),
+respectively, to obtain
+ERin,k
+M→∞
+−−−−→ ⌈Cin,k⌉+,
+ERex,k
+M→∞
+−−−−→ ⌈Cex,k⌉+,
+Cin,k = log(1 + SINRk) − log(1 + ESNRin,k),
+Cex,k = log(1 + SINRk) − log(1 + ESNRex,k).
+(30)
+Note that, the results in this paper are more general than the
+existing ones, because the spatial correlation at the transmitter
+and the power allocation scheme are considered. In particular,
+the above results include the first-order results of previous
+works [12, Eq. (48)-(50)(K = 1)], [9, Eq. (31)], and [3,
+Theorem 1] as special cases, i.e., by taking R = IM, T = IN,
+and P = IN.
+B. Second-order Analysis of the SINR, ESNR, and Secrecy
+Rate
+In this part, we investigate the fluctuations of SINR and
+ESNR to further investigate the distribution of the secrecy
+rate. The concerned random quantities, i.e., the SINRs and
+ESNRs, converge almost surely to a constant as shown in Sec-
+tion IV-A1, which means that their variances vanish when
+M grows to infinity. In order to investigate the second-order
+performance, in this section, we show that
+√
+MSINRk and
+√
+MESNRk converge to a Gaussian distribution when M goes
+to infinity.
+For a finite number K, define SINR, ESNRex, and ESNRin
+for K users as the vector gK ∈ R3K, given by
+gK = (SINR1, SINR2, ..., SINRK, ESNRex,1, ESNRex,2,
+..., ESNRex,K, ESNRin,1, ESNRin,2, ..., ESNRin,K)T .
+In the following, we first derive the joint distribution for the
+SINRs and ESNRs and then utilize the result to evaluate the
+SOP.
+1) Joint distribution for SINRs and ESNRs: The following
+theorem presents the asymptotic joint distribution of gK.
+Theorem 1. (CLT for the joint distribution of SINRs and
+ESNRs) For a finite number K, the distribution of the vector
+√
+MgK converges to a Gaussian distribution
+√
+M (gK − gK)
+d
+−−−−→
+M→∞ N(0, VK),
+(31)
+where
+gK = (SINR1, SINR2, ..., SINRK, ESNRex,1, ESNRex,2,
+..., ESNRex,K, ESNRin,1, ESNRin,2, ..., ESNRin,K)T .
+(32)
+Here, SINRk, ESNRex,k, and ESNRin,1 are given in (29). The
+covariance matrix VK is given by
+VK =
+
+
+UK
+FK
+JK
+FK
+WK
+OK
+JK
+OK
+ZK
+
+ ,
+(33)
+with
+UK
+=
+diag([U1, U2, ..., UK]),
+WK
+=
+diag([W1, W2, ..., WK]),
+ZK
+=
+diag([Z1, Z2, ..., ZK]),
+FK = diag([F1, F2, ..., FK]), JK = diag([J1, J2, ..., JK]),
+and OK = diag([O1, O2, ..., OK]). The parameters Uk, Wk,
+Vk, Fk, Jk and Ok are given by
+Uk =
+a2
+k,1t2
+kγk
+∆k
++ a2
+k,2t2
+kΠk(Pk) + 2ak,1ak,2t2
+kκk(Pk), (34)
+Wk =
+a2
+k,3t2
+kγk
+∆k
++ a2
+k,4t2
+k[θd2γk(C)2
+∆2
+k
++ θ2d
+2χk(C, C)] + 2ak,3ak,4t2
+kθdΓk(C),
+(35)
+Zk =
+a2
+k,5t2
+kγk
+∆
++ a2
+k,6t2
+kΠk(IN) + 2t2
+kak,5ak,6κk(IN), (36)
+Fk = ak,1ak,3t2
+kγk
+∆k
++ ak,1ak,4t2
+kθdΓk(C)
++ ak,2ak,3t2
+kκk(Pk) + ak,2ak,4t2
+kθdβk(Pk, C),
+(37)
+Jk = ak,1ak,6t2
+kκk(IM) + ak,2ak,5t2
+kκk(Pk)+
+ak,1ak,5t2
+kγk
+∆k
++ ak,2ak,6t2
+k[κk(Pk) − zβk(Pk, IM)],
+(38)
+Ok = ak,3ak,5t2
+kγk
+∆k
++ ak,3ak,6t2
+kκk(IN−1)
++ ak,4ak,5t2
+kθdΓk(C) + ak,4ak,6t2
+kθdβk(IN−1, C),
+(39)
+where Πk(·), κk(·), χk(·), βk(·), and Γk(·) are given by (90)
+to (94) in Appendix B. Here, ak,i, i = 1, 2, ..., 6 are given
+in (40) at the top of the next page.
+The terms χ(C, C), Γ(C), Π(P), and κ(P) are the
+deterministic approximations for
+1
+M E Tr QCQRQCQR,
+1
+M E Tr QRQRQC,
+1
+M E Tr(QRQHPHH)2,
+and
+1
+M E Tr RQHPHHQRQ, respectively, which are proved by
+Lemma 3 in Appendix B.
+Proof. The proof of Theorem 1 is given in Appendix A.
+Theorem 1 indicates that for a finite number of users, i.e.,
+K ≪ M, the joint distribution of the SINRs and ESNRs
+converges to a joint Gaussian distribution when M, L, and
+
+6
+ak,1 = 2pk[tkδ∆k(tkγk + σ2(1 + tkδk)γ(IM))]
+(tkγk + σ2(1 + tkδk)2γk(IM))2�γk(Pk) , ak,2 =
+−pkt2
+kδ2
+k∆2
+k
+(tkγk + σ2(1 + tkδk)2γk(IM))�γk(Pk)2 , ak,3 =
+−2pktkdγk(C)
+ρ2�γk(Pk)γk(IM)(1 + tkδk)3 ,
+ak,4 =
+pk∆k
+ρ2�γk(Pk)γk(IM)(1 + tkδk)2 , ak,5 =
+−2pktk�γ(IN)γk
+σ2�γk(Pk)γk(IM)(1 + tkδk)3 , ak,6 =
+pk∆k
+σ2�γ(Pk)γk(IM)(1 + tkδk)2 .
+(40)
+N go to infinity with the same pace. From the structure of the
+covariance matrix VK, we observe that the SINRs of different
+users are asymptotically independent. Similar results can also
+be obtained for the ESNR. However, SINRk and ESNRex,k
+are not independent, and their covariances are characterized
+by the diagonal entries of FK. Similarly, the covariance
+of the pairs (SINRk, ESNRin,k) and (ESNRex,k, ESNRin,k)
+is characterized by the diagonal entries of JK and OK,
+respectively. It follows from Theorem 1 that for a large system,
+the SINR and ESNR can be approximated by a Gaussian
+distribution, which will be utilized to evaluate the SOP in the
+following.
+Theorem 1 is obtained by assuming a common correlation
+matrix as mentioned in Section II-B. When different correla-
+tion matrices are assumed, the system of equations in (19) will
+become a system of N equations parameterized by N different
+correlation matrices, i.e., R1, R2,...,RN [35, Eq. (11)] and
+there will be N different δs instead of only one in (19). In this
+case, the characterization of the higher order trace of resolvents
+is more complex. For example, E Tr QRQR
+M
+M→∞
+−−−−→ γ
+∆ will be-
+come E Tr QRiQRj
+M
+M→∞
+−−−−→ [(IM −S)−1v]i, where S ∈ RN×N
+and v ∈ RN are determined by R1, R2,...,RN. The inverse
+matrix (IM − S)−1 will appear frequently and its order will
+increase in the computation of high-order resolvents like
+1
+∆
+in (90) to (94). Although the simplified common correlation
+matrix is considered, Theorem 1 sets up a framework for
+the second-order analysis of SINRs and ESNRs in the MISO
+system with RZF, which can also be used to analyze the
+scenario with imperfect CSI and other channel-inversion based
+precoding schemes like the secure RZF proposed recently
+in [23].
+2) SOP Evaluation: Based on Theorem 1, the SOP of the
+three cases can be obtained by Propositions 1 to 3 as follows,
+which are not yet available in the literature.
+Proposition 1. (External-Eve-Only) Given a rate threshold R,
+the SOP of user k with only external Eves can be approximated
+by
+Pex,out,k(R) ≈ φ
+�√
+M(R − Cex,k)
+�Vex,k
+�
+,
+(41)
+where Vex,k =
+Uk
+(1+gk)2 +
+Wk
+(1+gex,k)2 −
+2Fk
+(1+gk)(1+gex,k).
+Proposition 2. (Internal-Eve-Only) Given a rate threshold R,
+the SOP of user k with only internal Eves can be approximated
+by
+Pin,out,k(R) = φ
+�√
+M(R − Cin,k)
+�
+Vin,k
+�
+,
+(42)
+where Vin,k =
+Uk
+(1+gk)2 +
+Zk
+(1+gin,k)2 −
+2Jk
+(1+gk)(1+gin,k).
+Proposition 3. (External+Internal-Eve) Given a rate thresh-
+old R, the SOP of user k with both external and internal Eves
+can be approximated by
+Pco,out,k(R) ≈ 1 −
+� ∞
+R
+� ∞
+R
+(2π)−1[det(Cco,k)]− 1
+2 ×
+exp
+�
+−1
+2(x − µco,k)T C−1
+co,k(x − µco,k)
+�
+dx1dx2,
+(43)
+where x = [x1, x2]T , µco,k = [Cex,k, Cin,k]T , and Cco,k =
+�
+Vex,k
+Vco,k
+Vco,k
+Vin,k
+�
+, Vco,k
+=
+Uk
+(1+gk)2 +
+Ok
+(1+gex,k)(1+gin,k) −
+Jk
+(1+gk)(1+gin,k) −
+Fk
+(1+gk)(1+gex,k).
+Propositions 1-3 follow from Theorem 1 by utilizing the delta-
+method [36] and indicate that the SOP can be approximated
+by a Gaussian distribution in the large system setting A.1-A.3.
+V. LARGE SYSTEM ANALYSIS
+In this section, we utilize the derived results in Section IV to
+analyze the impact of the system dimensions. To achieve this
+goal, we ignore the correlation, pathloss, and power allocation
+and consider the i.i.d. channel, i.e., R = IM, T = IN, C =
+IM, D = IN, and P = IN. In this case, the subscript used to
+differentiate users can be removed and the system of equations
+in (22) degenerate to the quadratic equation zτδ2 + (τz − τ +
+1)δ − τ = 0, whose positive solution is given by
+δ = τ − 1 − τz +
+�
+(τ − 1)2 + 2(1 + τ)τz + z2τ 2
+2zτ
+.
+(44)
+In addition, δ(1) = dδ
+dz, δ(2) = d2δ
+dz2 , and δ(3) = d3δ
+dz3 represent
+the first to third derivatives of δ with respect to z. With the
+setting above, we only need to consider the joint distribution
+of gi.i.d = (SINR, ESNRex, ESNRin)T , and Theorem 1 can
+be simplified as follows.
+Theorem 2. Given assumption A.1, we have
+√
+M (giid − giid)
+d
+−−−−→
+M→∞ N(0, Viid),
+(45)
+where giid = (giid, giid,ex, giid,in)T ∈ R3 is given by
+giid =
+�δ[1 + zτ(1 + δ)2]
+1 + σ2(1 + δ)2 , τθ
+ρ2 ,
+1
+σ2(1 + δ)2
+�T
+.
+(46)
+The covariance matrix Viid ∈ C3×3 is given by
+Viid =
+
+
+Uiid
+Fiid
+Jiid
+Fiid
+Wiid
+Oiid
+Jiid
+Oiid
+Ziid
+
+ , with
+(47)
+Uiid = −[(a1 + a2)2δ(1) + (a2
+2 + a1a2)zδ(2) + a2
+2z2δ(3)
+6
+],
+Wiid = −[a2
+3δ(1) − a2
+4θδ2
+(1) + θ2a2
+4δ(3)
+6
+− a3a4θδ(2)],
+Ziid = −[(a5 + a6)2δ(1) + a5a6zδ(2) + a2
+6z2δ(3)
+6
+],
+
+7
+Fiid = −[(a1 + a2)a3δ(1) − [(a1 + a2)a4θ − a2a3z]δ(2)
+2
+− a2a4θzδ(3)
+6
+],
+Jiid = −[(a1 + a2)(a5 + a6)δ(1)
++ z(a2a5 + a1a6 + 2a2a6)δ(2)
+2
++ za2a6δ(3)
+6
+],
+Oiid = −[a3(a5 + a6)δ(1) + (za3a6 − θa4(a5 + a6))δ(2)
+2
+− za4a6θδ(3)
+6
+].
+(48)
+The parameters ai, i = 1, 2, ..., 6, are given as
+a1 =
+2δ[1 + σ2(1 + δ)]
+[1 + σ2(1 + δ)2]2(δ + zδ(1)), a4 =
+1
+ρ2(1 + δ)2(δ + zδ(1)),
+a2 =
+−δ2
+[1 + σ2(1 + δ)2]2(δ + zδ(1))2 , a5 = −
+2
+σ2(1 + δ)3 ,
+a3 =
+2θδ(1)
+ρ2(1 + δ)3(δ + zδ(1)), a6 =
+1
+σ2(1 + δ)2(δ + zδ(1)).
+The SINR distribution of the MISO BC with RZF was
+investigated in [37], where the i.i.d channel and equal power
+allocation are assumed. If we discard ESNRex and ESNRin,
+the results in Theorem 2 degenerate to [37, Theorem 3]. The
+outage probability for the i.i.d. case can be obtained by a
+similar approach as Propositions 1 to 3. In this paper, the joint
+distribution of SINR and ESNR is investigated with correlated
+transmit antennas and unequal power allocation.
+A. How many users are in secrecy outage?
+Theorem 2 can be utilized to evaluate how many users are in
+outage. For that purpose, we define the empirical distribution
+αN,case(R) = 1
+N
+N
+�
+i=1
+1{Rcase≤R},
+(49)
+where case can be either External-Eve-Only or Internal-Eve-
+Only. Here we consider the External-Eve-Only case and the
+results for the Internal-Eve-Only case can be obtained simi-
+larly. The following proposition is an application of Theorem 2
+to compute the quantiles of the users in outage.
+Proposition
+4.
+Given
+assumptions
+A.1
+to
+A.3,
+define
+Rex(σ2, ρ2, q) and Rin(σ2, q) as
+Rex(σ2, ρ2, q) = log(1 + giid,ex) +
+�
+Vex,iid
+M
+φ−1(q),
+Rin(σ2, q) = log(1 + giid,in) +
+�
+Vin,iid
+M
+φ−1(q).
+(50)
+Then, we have
+αN,ex(Rex(σ2, ρ2, q))
+a.s.
+−−−−→
+M→∞ q,
+αN,in(Rin(σ2, q))
+a.s.
+−−−−→
+M→∞ q.
+(51)
+Proof. The result follow from the law of large numbers and
+Theorem 2.
+Proposition 4 indicates that for large N and M, given
+a target secrecy rate Rcase and the noise level σ2, ρ2, the
+proportion of the users in outage is approximated by q, where
+q is obtained by solving equation (50). This means that almost
+qN users are in outage. Given ρ2 and σ2, q is an increasing
+function of Rcase, which indicates that a larger threshold
+causes more users in outage.
+B. The Secrecy Loss of the External-Eve-Only and Internal-
+Eve-Only Cases
+Next, we investigate the secrecy loss induced by external
+and internal Eves. To make a fair comparison, we consider the
+case where the internal and external Eves are equally powerful,
+i.e., L = N −1 and ρ2 = σ2, and the precoders have the same
+z. Under such circumstances, giid,in
+M→∞
+−−−−→
+giid,ex
+(1+δ)2 , which
+indicates that the information leakage due to external Eves is
+larger than that due to internal Eves. This is because, with
+only internal Eves, the information received by other users
+is mitigated by RZF due to its ability to cancel interference.
+With external Eves, RZF precoder will not be able to cancel
+the interference since the channels of Eves are independent of
+those of the users.
+It can also be observed from (46) that ESNRex is constant
+when the ratio L
+N is a constant. Moreover, ESNRex does not
+depend on the regularization parameter z, indicating that the
+optimal regularization parameter for maximizing the secrecy
+sum rate is equivalent to that without considering the Eves,
+i.e., R = �N
+i=1 log(1 + SINRi). The optimal value of the
+regularization parameter is z = σ2
+τ [38].
+C. How many transmit antennas do we need to achieve a
+positive secrecy rate?
+For the External-Eve-Only case with the optimal regular-
+ization parameter z = σ2
+τ , the inequality µ = δ > µex must
+hold true, i.e.,
+τ − 1 − σ2 +
+�
+(τ − 1)2 + 2(1 + τ)σ2 + σ4
+2σ2
+≥ τθ
+ρ2 ,
+(52)
+in order to obtain a positive secrecy rate. From (52), the
+minimum τ required for a positive secrecy rate is
+τ ∗ =
+L
+Nρ2 (1 + σ2) + L2σ2
+N 2ρ4
+L
+Nρ2 + 1
+.
+(53)
+This indicates that if M > Nτ ∗, a positive secrecy rate will be
+guaranteed. The estimation of M is accurate in the high SNR
+regime because only the term 4σ2 is omitted in the relaxation.
+When
+1
+ρ2 → ∞, M = Nτ ≈
+Lσ2
+ρ2 , we can obtain that M
+grows with the order O(ρ−2) and a larger L requires a higher
+increasing rate of M.
+VI. NUMERICAL RESULTS
+In this section, the theoretical results derived in Sections IV
+and V are validated by numerical results. Specifically, the
+accuracy of the SOP approximation and the evaluation of the
+percentage of users in secure outage are validated by Monte-
+Carlo simulations. The impact of the regularization factor
+
+8
+z and number of transmit antennas required for a positive
+secrecy rate are also investigated.
+A. Simulation Settings: In the simulation, we consider
+a uniform linear array of antennas at the BS. According to
+the model for conventional linear antenna arrays [39], the
+correlation matrix at the BS can be obtained by
+[L(dr, α, ν, N)]m,n
+=
+� 180
+−180
+1
+√
+2πδ2 e 2π
+λ dr(m−n) sin( πφ
+180 )− (φ−α)2
+2ν2
+dφ,
+(54)
+where m and n represent the indices of antennas. Here, dr and
+N denote the relative antenna spacing (in wavelengths) and the
+dimension of the matrix, respectively, and α and ν2 represent
+the mean angle and the mean-square angle spreads, whose
+units are degree. We adopt the setting dr = λ. In the following
+simulations, the correlation matrices are generated according
+to (54), i.e., R = L(1, αR, νR, M), C = L(1, αC, νC, M)
+with αR = νR = 10 and αC = νC = 5. Without loss of
+generality, we consider the first user, i.e., k = 1.
+Following [12], a simple model for the large-scale fading
+of different users ti, i = 1, 2, ..., N is used, with ti = a−η
+i
+,
+where ai represents the distance between the BS and i-th user.
+The path loss exponent η = 3 is used to model a shadowed
+urban area [40]. Here we divide the users into Ga groups and
+users in the same group have a common distance. We choose
+ai = a⌊ i−1
+Ga ⌋ so that ti = a−η⌊ i−1
+Ga ⌋. Similarly, di is generated
+by di = b−η⌊ i−1
+Gb ⌋. The power of each message is given by
+pi = c⌊ i−1
+Gc ⌋ and the total power is normalized to be N. In
+the following, we use the setting a = 1.0772, b = 1.1262,
+c = 0.9, and Ga = Gb = Gc = 4. In the figures, we use the
+notations “Ana.” and “Sim.” to represent the theoretical and
+the simulation results, respectively.
+B. The Approximation Accuracy for SOP: In Fig. 1a, the
+SOP for three cases are given. The dimensions of the system
+are set as M = 64, N = 32, and L = 16. The SNRs at
+the user and external Eves, i.e.,
+1
+σ2 and
+1
+ρ2 , are 10 dB and
+4.5 dB, respectively. The regularization parameter z is set
+as z = 0.2. The number of the Monte-Carlo realizations is
+5 × 105. It can be observed that the approximations for the
+SOP in Proposition 1 to 3 are accurate. The External+Internal-
+Eve case has the highest SOP and the gap between the
+External+Internal-Eve case and the other two represents the
+performance loss induced by different types of Eves.
+In Fig. 1b, the SOP of the i.i.d. system is given with the
+setting M = 64, N = 32, L = 16,
+1
+σ2 = 6 dB, and
+1
+ρ2 = 2
+dB. This result validates the accuracy of Theorem 2.
+C. Optimal z Control: The theoretical results of this paper
+can be used to investigate the optimal z for enhancing secure
+reliability. Figs. 2a and 2b investigate the impact of z on
+ESR and SOP for the External-Eve-Only and Internal-Eve-
+Only case, respectively. Here the i.i.d channel is considered
+with equal power allocation. The SNR for internal Eves and
+external Eves are 5 dB and 2 dB, respectively. The rate thresh-
+olds for the two cases are set as 0.8/ log(2) and 1.55/ log(2),
+respectively. The optimal z that maximizes the ESR can be
+obtained by z =
+σ2
+τ
+(External-Eve-Only) and [3, Eq. (12)]
+(Internal-Eve-Only). The optimal values are determined as
+0.6
+0.8
+1
+1.2
+1.4
+1.6
+1.8
+2
+2.2
+10-4
+10-3
+10-2
+10-1
+100
+(a) M = 64.
+0.5
+1
+1.5
+2
+10-4
+10-3
+10-2
+10-1
+100
+(b) i.i.d. channel.
+Fig. 1: Secrecy outage probability.
+0
+0.05
+0.1
+0.15
+0.2
+0.8
+0.9
+1
+1.1
+1.2
+1.3
+1.4
+1.5
+1.6
+(a) ESR
+0
+0.05
+0.1
+0.15
+0.2
+0
+0.2
+0.4
+0.6
+0.8
+1
+(b) SOP
+Fig. 2: The impact of z
+z = 0.1532 and z = 0.0664, respectively, which agree with the
+simulation results shown in Fig. 2a. It can be observed from
+Fig. 2 that the optimal z that minimizes the SOP is different
+from the one that maximizes the ESR.
+D. Percentage of Users in Outage: Fig. 3a and Fig. 3b
+depict percentages of users in outage for the External-Eve-
+Only and Internal-Eve-Only case, respectively. The parameters
+are set as M = 256, N = 128, L = 64, z = 0.1, and
+1
+ρ2 = 4 dB. It can be observed that the evaluation in (50)
+matches the empirical result well, which validates the accuracy
+of Propostion 4 .
+E. The Number of Transmit Antennas Required for a
+Positive Secrecy: With only the external Eves, Fig. 4 shows
+the number antennas required to achieve a positive secrecy rate
+for a given SNR at the Eves. The transmit SNR is set to be
+10 dB and N = 128. It can be observed that as the ability of
+Eves increases, the required number of antennas for a positive
+secrecy rate increases. The increasing rate grows larger as L
+increases, which agrees with the analysis in Section V-C.
+VII. CONCLUSION
+In this paper, the secrecy performance of RZF in the
+MISO broadcasting system was investigated. In the asymptotic
+regime that the numbers of transmit antennas, users, and
+1.5
+2
+2.5
+3
+0
+0.2
+0.4
+0.6
+0.8
+1
+(a) External-Eve-Only.
+2.4
+2.6
+2.8
+3
+3.2
+3.4
+3.6
+3.8
+4
+0
+0.2
+0.4
+0.6
+0.8
+1
+(b) Internal-Eve-Only.
+Fig. 3: The percentage of users in secrecy outage.
+
+9
+2
+4
+6
+8
+10
+12
+0
+100
+200
+300
+400
+500
+600
+700
+800
+Fig. 4: Required number of antennas for a positive secrecy
+rate.
+Eves go to infinity with the same pace, a CLT for the
+joint distribution of SINRs and ESNRs was derived. The
+CLT was then used to obtain a closed-form approximation
+for the SOP of three cases, i.e., External-Eve-Only, Internal-
+Eve-Only, and External+Internal-Eve. Based on the derived
+results, the required number of transmit antennas for a positive
+secrecy rate and the percentage of user in secrecy outage were
+evaluated in a closed form. The secrecy loss caused by the
+external Eves and internal Eves were compared, showing that
+the loss caused by the internal Eves is less than that caused
+by the external Eves. The derived results were validated by
+numerical simulations. The methods used in this paper can be
+applied to analyze the performance of RZF with imperfect CSI
+and investigate the channel-inversion based precoding scheme
+like the recently proposed secure RZF [23].
+APPENDIX A
+PROOF OF THEOREM 1
+Proof. The proof can be summarized by three steps including:
+1. Show the Gaussianity of the SINRk; 2. Show the Gaus-
+sianity of the ESNRk; and 3. Derive the covariances. In the
+first two steps, without loss of generosity, we consider user
+1, i.e., k = 1, and the derivation for other users k = 2, ..., K
+are the same. The proof of the Gaussianity relies on the CLT
+for the quadratic forms shown in Lemma 2 of Appendix B.
+Specifically, we will first show that the SINRs and ESNRs
+can be approximated by a linear combination of the quadratic
+forms Ak, Bk, Ek, and Gk, and the approximation is tight in
+probability. The asymptotic variances obtained in this step are
+related to the high order resolvents, so we need to determine
+the variance by their deterministic equivalent. This process
+involves computations for the high-order resolvents, which are
+summarized by Lemma 3 in Appendix B. The results will also
+be utilized to evaluate the covariances. Now we turn to the first
+step.
+A. The Asymptotic Gaussianity of the SINR1
+In this step, the fluctuation of
+√
+M(SINR1 − SINR1) will
+be investigated. To achieve this goal, we first rewrite SINR1 −
+SINR1 as
+SINR1 − SINR1 = 2p1A1(A1 − A1)
+D1
+− p1A
+2
+1(D1 − D1)
+D
+2
+1
++
+1
+√
+M
+εs.
+(55)
+where εs =
+√
+Mp1D1( A1
+D1 − A1
+D1 )2. For any given ǫ, by the
+Markov inequality, we have
+P(|εs| > ǫ) ≤
+√
+M
+ǫ
+E|D1|| A1
+D1
+− A1
+D1
+|2
+≤ 2
+√
+M
+ǫ
+(E|A1 − A1|2
+|D1|
++ |A1|2
+|D1|2 E|D1 − D1|2
+|D1|
+).
+(56)
+Since |D1| is bounded away from zero almost surely, E|A1 −
+A1|2
+= O( 1
+M2 ), E|A1 − A1|2
+= O( 1
+M ), and E|D1 −
+D1|2 = O( 1
+M ), we have P(|εs| > ǫ)
+M→∞
+−−−−→ 0. Therefore,
+√
+M(SINR1 − SINR1) can be approximated by the first two
+terms at the right hand side of (55) and the approximation is
+tight in probability. D1 can be written as
+D1 − D1 = B1 − B1 + 2σ2(1 + A1)C(A1 − A1) + εD, (57)
+where εD = σ2(A1 − A1)2C + σ2(1 + A1)2(C − C). By
+the variance control in [20], [41], we can show that E|A1 −
+A1|4 = O( 1
+M2 ) and E|C − C|2 = O( 1
+M2 ) so that we can
+prove P(|εD| > ǫ)
+M→∞
+−−−−→ 0 for a positive ǫ by the same
+approach in (56). Then, by substituting (57) into (55), we have
+the following linear approximation
+√
+M(SINR1 − SINR1)
+i.p.
+−−−−→
+M→∞
+√
+M[a1,1(A1 − A1) + a1,2(B1 − B1)],
+(58)
+where a1,1 and a1,2 are given in (40).
+Now we turn to determine the asymptotic distribution of
+the random process at the right hand side of (58). Given H1,
+√
+M(A1 − A1) and
+√
+M(B1 − B1) are both in quadratic
+forms. According to Lemma2, we can show that
+√
+MA1
+and
+√
+MB1 both converge to a Gaussian distribution when
+H1 is given so that the linear combination of them also
+converges to a Gaussian distribution. Thus, we only need
+to determine the asymptotic variances for
+√
+MA1,
+√
+MB1
+and their covariance, which can be obtained by Lemma 3 in
+Appendix B as
+VA1 = t2
+1 Tr Q1RQ1RQ1
+M
+i.p.
+−−−−→
+M→∞
+t2
+1γ1
+∆1
+,
+(59)
+VB1 = t2
+1 Tr(Q1RQ1H1P1HH
+1 )2
+M
+i.p.
+−−−−→
+M→∞ t2
+1Π1(P1). (60)
+The covariance of A1 and B1 can be evaluated by
+VA1,B1 = t2
+1 Tr RQ1RQ1H1P1HH
+1 Q1
+M
+i.p.
+−−−−→
+M→∞ t2
+1κ1(P1).
+(61)
+Therefore, we can obtain the asymptotic variance of SINR1 as
+U1 = a2
+1,1VA1 + a2
+1,2VB1 + 2a1,1a1,2VA1,B1.
+(62)
+
+10
+B. The Asymptotic Gaussianity of ESNRex,1
+In this part, we will show the Gaussianity of ESNRex,1 by
+the same approach as Appendix A-A. We first replace C by
+C in (16) to obtain
+ESNRex,1 =
+p1E1
+ρ2C(1 + A1)2 + εC,
+(63)
+where εC =
+√
+M(
+E1
+C(1+A1)2 −
+E1
+C(1+A1)2 ). By the variance
+control in [22, Lemma 3], we have Var(C) = O( 1
+M2 ) to show
+εC
+i.p.
+−−−−→
+M→∞ 0 so that we only need to consider the asymptotic
+distribution of
+√
+ME1
+(1+A1)2 . To achieve this goal, we first perform
+the following decomposition
+E1
+(1 + A1)2 −
+E1
+(1 + A1)2 = −2E1(A1 − A1)
+(1 + A1)3
++ E1 − E1
+(1 + A1)2 + e1 + e2 + e3,
+(64)
+where e1 =
+2E1(A1−A1)2
+(1+A1)(1+A1)3 , e2 =
+E1(A1−A1)2
+(1+A1)2(1+A1)2 , and e3 =
+2(E1−E1)(A1−A1)
+(1+A1)3
+. We can then show that e1, e2, and e3 vanish
+in probability and here we take e1 as an example. E
+√
+M|e1|
+can be upper bounded by
+E
+√
+M|e1| ≤ 2
+√
+M(E|E1 − E1|(A1 − A1)2
++ E1E|A1 − A1|2) ≤ 2
+√
+M(E
+1
+2 (E1 − E1)2×
+E
+1
+2 (A1 − A1)4 + E1E|A1 − A1|2).
+(65)
+The term (E1 − E1)2 can be upper bounded by
+(E1 − E1)2 ≤ 2[(E1 − t1 Tr RQ1GHGQ1
+M
+)2
++ (t1 Tr RQ1GHGQ1
+M
+− E1)2].
+(66)
+The first term of the RHS in (66) is bounded by
+E|E1 − t1 Tr RQ1GHGQ1
+M
+|2
+= t2
+1E Tr(RQ1GHGQ1)2
+M 2
+≤ KE∥G∥4
+M
+(a)
+≤ K′
+M ,
+(67)
+where the inequality (a) holds true due to the boundness of
+E∥G∥4, which was shown in [22, Lemma 2]. We also have
+the bound for the second term in the RHS of (66) as
+E|t1 Tr RQ1GHGQ1
+M
+− E1|2 ≤ 2Var(t1 Tr RQ1GHGQ1
+M
+)
++ 2|Et1 Tr RQ1GHGQ1
+M
+− E1|2 ≤ K′′
+M 2 ,
+(68)
+and
+E|A1 − A1|4 ≤ K′′′
+M 2 ,
+(69)
+where K′, K′′, and K′′′ are constants which are independent
+of M, L, and N. By substituting (67)-(68) into (66) and
+then (65), we can obtain that
+√
+ME|e1| = O(M − 1
+2 ) so that
+P(
+√
+M|e1| > ε) ≤
+√
+ME|e1|
+ε
+M→∞
+−−−−→ 0.
+(70)
+Similarly, we can show that e2
+i.p.
+−−−−→
+M→∞ 0 and e3
+i.p.
+−−−−→
+M→∞ 0.
+Therefore, we have
+√
+M(ESNRex,1 − ESNRex,1)
+i.p.
+−−−−→
+M→∞
+√
+Ma1,3(A1 − A1) +
+√
+Ma1,4(E1 − E1),
+(71)
+VE1 = t2
+1 Tr(Q1GGHQ1R)2
+M
+i.p.
+−−−−→
+M→∞
+t2
+1E Tr(Q1GGHQ1R)2
+M
+,
+(72)
+where the last step follows from Var(Tr(Q1GGHQ1R)2) =
+O(1), which can be obtained by the Nash-Poincar´e Inequality
+in [20], [42].
+E Tr(Q1GGHQ1R)2
+M
+can be evaluated by the
+integration by parts formula [20], [42],
+E Tr Q1RQ1GGHQ1RQ1GGH
+M
+= 1
+M
+�
+i,j
+EY ∗
+i,jd
+1
+2
+j [C
+1
+2 Q1RQ1GGHQRQ1G]i,j
+= (E Tr CQ1RQ1)2 Tr D2
+M 3
++ (Tr D)2E Tr CQ1RQ1CQ1RQ1
+M 3
++ O(M −2).
+(73)
+By the evaluations in Lemma 3, we have
+VE1
+i.p.
+−−−−→
+M→∞ t2
+1[θd2γ1(C)
+∆1
++ θ2d
+2χ1(C, C)],
+(74)
+VA1,E1 = t2
+1 Tr Q1GGHQ1RQ1R
+M
+i.p.
+−−−−→
+M→∞ t2
+1θdΓ1(C). (75)
+Therefore, we can obtain the asymptotic variance of ESNRex,1
+as
+W1 = a2
+1,3VA1 + a2
+1,4VE1 + 2a1,3a1,4VA1,E1.
+(76)
+C. The Asymptotic Gaussianity of ESNRin,1
+Similar to the manipulation of SINR1 and ESNRex,1,
+ESNRin,1 can be approximated by
+√
+M(ESNRin,1 − ESNRin,1)
+i.p.
+−−−−→
+M→∞
+√
+Mp1
+C
+[−2F 1(A1 − A1)
+(1 + A1)3
++ F1 − F 1
+(1 + A1)2 ]
+i.p.
+−−−−→
+M→∞
+√
+M[a1,5(A1 − A1) + a1,6(F1 − F 1)].
+(77)
+The asymptotic variance of F1 and the covariance between F1
+and A1 can be obtained by
+VF1 = t2
+1 Tr(Q1H1HH
+1 Q1R)2
+M
+i.p.
+−−−−→
+M→∞ t2
+1Π1(IN−1),
+VA1,F1 = t2
+1 Tr RQ1RQ1H1P1HH
+1 Q1
+M
+i.p.
+−−−−→
+M→∞ t2
+1κ1(IN−1).
+Therefore, we can obtain the asymptotic variance of ESNRin,1
+as
+Z1 = a2
+1,5VA1 + a2
+1,6VF1 + 2a1,5a1,6VF1,F1.
+
+11
+D. The Evaluation of Covariances
+We will first give the closed-form expression for the asymp-
+totic covariances between SINR1, ESNRin,1, and ESNRex,1.
+By (58), (71) and (88) in Lemma (3), we can obtain
+MESINR1ESNRex,1
+M→∞
+−−−−→ E(a1a3VA1+
+a2a3VA1,B1 + a1a4VA1,E1 + a3a3VB1,E1),
+(78)
+EVB1,E1 = E Tr Q1H1P1HH
+1 Q1RQ1GGHQ1R
+M
+= θdE Tr Q1H1P1HH
+1 Q1RQ1CQ1R
+M
+= θdβ1(P1, C) + O(M −2).
+(79)
+The evaluations of VA1, VA1,B1, and VA1,E1 can be found
+in (59), (61), and (75), respectively. The covariances of the
+pairs (SINR1, ESNRin,1) and (ESNRex,1, ESNRin,1) can be
+given by
+MESINR1ESNRin,1
+M→∞
+−−−−→ E(a1a5VA1 + a2a5VA1,B1
++ a1a4VA1,G1 + a3a3VB1,G1),
+MEESNRex,1ESNRin,1
+M→∞
+−−−−→ E(a3a5VA1+
+a4a5VA1,E1 + a3a6VA1,G1 + a4a6VE1,G1),
+where
+EVB1,G1 = t2
+1E Tr Q1H1P1HH
+1 Q1RQ1H1HH
+1 Q1R
+M
+= t2
+1E Tr Q1H1P1HH
+1 Q1RQ1R
+M
+− t2
+1zE Tr Q1H1P1HH
+1 Q1RQ2
+1R
+M
+= t2
+1[κ1(P1) − zβ1(IM, P1)] + O(M −2),
+(80)
+and
+EVE1,G1 = t2
+1E Tr Q1H1HH
+1 Q1RQ1GGHQ1R
+M
+= t2
+1θdE Tr Q1H1HH
+1 Q1RQ1CQ1R
+M
+= t2
+1θdβ1(IN−1, C) + O(M −2).
+(81)
+By far, we have shown that SINR1, ESNRex,1, and ESNRin,1
+converge to a Gaussian distribution when M, K, L go to
+infinity with the same pace. Next, we need to investigate
+the covariance between SINR and ESNR and the covariance
+between different users. It has been proved in [43, Eq.(2.9)-
+(2.11)] that
+√
+MAi and
+√
+MAj are asymptotically indepen-
+dent. By a similar approach, we can also prove the asymptotic
+independence between
+√
+MBi,
+√
+MBj,
+√
+MEi,
+√
+MEj, and
+√
+MGi,
+√
+MGj when i ̸= j so that the asymptotic covari-
+ances are all zero.
+APPENDIX B
+USEFUL RESULTS
+Lemma 2. (CLT for quadratic forms) Given assumptions A.1,
+Hk, G, and P, there holds true that
+√
+M(Ak − Ak)
+d
+−−−−→
+M→∞ N(0, VAk),
+(82)
+where VAk = t2
+k Tr QkRQkR
+M
+. Similarly, we also have
+√
+M(Bk − Bk)
+d
+−−−−→
+M→∞ N(0, VBk),
+√
+M(Ek − Ek)
+d
+−−−−→
+M→∞ N(0, VEk),
+(83)
+where
+VBk
+=
+t2
+k Tr RQkHkPkHH
+k Qk
+M
+and
+VEk
+=
+t2
+k Tr RQkGGHQk
+M
+.
+Proof. The first CLT is the separable case in [21] when the
+resolvent matrix is given. The other two CLTs can be proved
+by the same approach and the proof is omitted here.
+Lemma 3. (Computation results about the high order re-
+solvents) Given assumptions A.1-A.3 and any deterministic
+matrices C, P with bounded norm, the following evaluations
+hold true
+E Tr QRQC
+M
+= γ(C)
+∆
++ O(M −2),
+(84)
+E Tr QRQRQC
+M
+= Γ(C) + O(M −2),
+(85)
+E Tr RQHPHHQRQ
+M
+= κ(P) + O(M −2),
+(86)
+E Tr QCQRQCQR
+M
+= χ(C, C) + O(M −2),
+(87)
+E Tr QHPHHQRQCQR
+M
+= β(P, C) + O(M −2),
+(88)
+E Tr(QRQHPHH)2
+M
+= Π(P) + O(M −2),
+(89)
+Γ(C) = η(C) − (η(C)γ − ηγ(C))�γ − �ηγ2γ(C)
+∆3
+,
+(93)
+κ(P) = �γ(P)(η − �ηγ3)
+∆3
+− γ2�η(P)
+∆2
+.
+(94)
+where χ(C, C), β(P, C), and Π(P) are given in (90) to (92)
+at the top of the next page. γ(C) and ∆ are given in Table (I)
+Proof. The proof of Lemma 3 is given in Appendix C.
+The evaluation of the high order resolvent is considered
+in [22], which is used to set up a CLT for the signal-to-
+noise ratio (SNR) of minimum variance distortionless response
+(MVDR) filter. (85) is equivalent to [22, Proposition 3].
+However, more complex forms for the fourth order resolvents
+related to multiple system parameters, e.g. C, P, need to be
+evaluated while in [22], only one undetermined parameter Θ
+is considered. Lemma (3) is more general than those in [22].
+If we take the first C to be R, (87) is equivalent to the result
+in [22, Proposition 4] .
+
+12
+χ(C, C) = ζ(C, C)
+∆2
++ 2ζ(C)γ(C)�γ
+∆3
++ 2η(C)2�γ
+∆3
++ [4�γ2γ(C)η − 3�ηγγ(C) − �η�γγ2γ(C)]η(C)
+∆4
++ γ(C)2(ζ�γ2 + �ζγ2 − η�η)
+∆4
++ γ(C)2(2η2�γ3 + 2�η2γ3 − 3η�ηγ�γ − η�ηγ2�γ2)
+∆5
+.
+(90)
+Π(P) = γ2(�η(P2) − δ�ζ(P2))
+∆2
++ 2γ3�η(P)2 + 2γ3�ζ(P)�γ(P)
+∆3
++ (ζ + γ4�ζ)�γ(P)2 − 4(η − γ3�η)γ�η(P)�γ(P)
+∆4
++ 2�γ(P)2(�γη2 + γ5�η2 − 2γ2η�η)
+∆5
+.
+(91)
+β(P, C) = 2�γ(P)(�γη − γ2�η)η(C) + (�γζ + γ3�ζ)γ(C)�γ(P) − 2�η(P)γ[η(C) − (η(C)γ − ηγ(C))�γ − �ηγ2γ(C)]
+∆4
++ ζ(C)�γ(P) + γ2γ(C)�ζ(P)
+∆3
++ 2[�γ2η2 + γ4�η2 − γ(1 + γ�γ)η�η]γ(C)�γ(P)
+∆5
+.
+(92)
+APPENDIX C
+PROOF OF LEMMA 3
+Proof. The main idea is to evaluate the high-order resolvents
+based on the lower-order results. As a result, we will perform
+the computation from low order to high order iteratively.
+We first turn to the evaluation of the second-order resolvent
+in (84). For any deterministic matrices A, B, C, and D with
+bounded norm, we have
+E Tr CQHAHHDQ
+M
+= Tr ATGT
+M
+E Tr CQRDQ
+M
+− E Tr HATGT HHDQ
+M
+E Tr CQRQ
+M
++ O(M −2)
+= Tr ATGT
+M
+Tr CGRRDGR
+M
++ 1
+∆(Tr ATGT
+M
+�γγ(C)
+× γ(RD) − Tr AT2G2
+T Tr DRGRγ(C)
+M 2
+) + O(M −2),
+(95)
+by integration by parts formula. Then we turn to evaluate
+the third-order resolvent. By the resolvent identity and the
+integration by parts formula, we have
+zE[QCQDQ]i,j = E[QCQD]i,j − E[QCQDQHHH]i,j
+= E{[QCQD]i,j − �δ[QCQDQR]i,j
++ Tr RQCQDQ
+M
+[QHTGT HH]i,j
++ Tr RQDQ
+M
+[QCQHTGT HH]i,j}.
+(96)
+By solving E[QCQDQ]i,j in (96), taking the trace operation,
+and using variance control in [20], [22], we can obtain
+E Tr QRQCQD
+M
+= 1
+∆(E Tr QCQDRGR
+M
++ E Tr QDQR
+M
+E Tr QCQHTGT HHRGR
+M
+) + O(M −2)
+= 1
+∆[η(C, D) + 1
+∆(γ(C)η(D)�γ + γ(D)�γη(C))
++ 1
+∆2 (γ(D)�γ2γ(C)η − �ηγγ(C))] + O(M −2).
+(97)
+By letting D = R in (97), we can obtain (85). Then, by the
+integration by parts formula [20], we can obtain the evaluation
+for E Tr RQHAHHDQCQ
+M
+by plugging (95) into (98) below
+E Tr RQHAHHDQCQ
+M
+= E Tr RQRDQCQ
+M 2
+× Tr ATGT − E Tr RQCQE Tr TGT HHDQRQHA
+M 2
+− E Tr RQRQCQE Tr TGT HHDQHA
+M 2
++ O(M −2).
+(98)
+By replacing the trace of the third-order resolvents by (97)
+and letting D = IM, A = P, C = R in (98), we can
+obtain (86). The fourth-order resolvent in (87) can be evaluated
+by the similar approach, which is given in (99) at the top
+of the next page, where step (a) in (99) is obtained by
+plugging (97) and (98) into (99) to replace E Tr RQRQCQ
+M
+and
+E Tr QCQRQHTGT HHGRR
+M
+, respectively. To obtain (88), we
+first follow similar steps as in (96) to obtain the evaluation for
+E Tr QRQCQRQR
+M
+. Then by replacing the third-order terms in
+E Tr QRQCQRQHAHH
+M
+= −E Tr QRQR
+M
+E Tr TGT HHQCQRQHA
+M
+− E Tr QRQCQR
+M
+E Tr TGT HHQRQHA
+M
++ E Tr QRQCQRQR
+M
+Tr ATG2
+T
+M
++ O(M −2),
+(100)
+by (97) and (98), we can conclude (88). The tedious compu-
+tation is omitted here. Then we turn to evaluate (89). By the
+integration by parts formula, we can obtain
+E Tr QRQHAHHQRQHBHH
+M
+=
+− E Tr QRQR
+M
+E Tr TGT HHQHAHHQRQHB
+M
++ E Tr QRQRE Tr TGT AHHQRQHB
+M 2
+− E Tr RQRQHAHHQ
+M
+E Tr TGT HHQRQHB
+M
+− E Tr RQRQHAHHQRQ
+M
+E Tr TGT HHQHB
+M
++ E Tr RQRQHAHHQRQ
+M
+Tr TGT B
+M
++ O(M −2).
+(101)
+The third-order term
+E Tr TGT HHQHAHHQRQHB
+M
+in (101)
+can be obtained by
+E Tr QHAHHQRQHBHH
+M
+=
+E Tr QRE Tr TGT AHHQRQHB
+M 2
+− E Tr RQHAHHQE Tr TGT HHQRQHB
+M 2
+− E Tr RQHAHHQRQ Tr TGT HHQHB
+M 2
++ E Tr RQHAHHQRQ Tr TGT B
+M 2
++ O(M −2).
+(102)
+
+13
+E Tr QCQRQCQR
+M
+= 1
+∆[E Tr RQRQCQ
+M
+E Tr QCQHTGT HHGRR
+M
++ E Tr RQCQ
+M
+E Tr QCQRQHTGT HHGRR
+M
+]+O(M −2)
+(a)
+=
+1
+∆4 {γ(C)[ζ(C)∆ + ζ�γγ(C) + ζ(η(C)�γ∆ + �γ2ηγ(C) − γ(C)γ�η) + [η(C) − (η(C)γ − ηγ(C))�γ − �ηγ2γ(C)]]} + O(M −2),
+(99)
+The evaluation of (102) can be obtained by plugging the
+evaluations in (95) and (98) to replace the expectations for the
+trace of the lower order resolvents. With these evaluations, all
+the terms in (101) can be obtained so that (89) is concluded.
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diff --git a/htFJT4oBgHgl3EQfWCwj/content/tmp_files/load_file.txt b/htFJT4oBgHgl3EQfWCwj/content/tmp_files/load_file.txt
new file mode 100644
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@@ -0,0 +1,1247 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf,len=1246
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='11515v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='IT]  27 Jan 2023  1  Secrecy Analysis for MISO Broadcast Systems with  Regularized Zero-Forcing Precoding  Xin Zhang, Graduate Student Member, IEEE, Shenghui Song, Senior Member, IEEE, and Yonina C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  Eldar, Fellow, IEEE  Abstract—As an effective way to enhance the physical layer  security (PLS) for the broadcast channel (BC), regularized zero-  forcing (RZF) precoding has attracted much attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' However,  the reliability performance, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=', secrecy outage probability (SOP),  of RZF is not well investigated in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' In this paper,  we characterize the secrecy performance of RZF precoding in  the large multiple-input single-output (MISO) broadcast system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  For this purpose, we first consider a central limit theorem (CLT)  for the joint distribution of the users’ signal-to-interference-plus-  noise ratio (SINR) and the eavesdropper’s (Eve’s) signal-to-noise  ratio (ESNR) by leveraging random matrix theory (RMT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' The  result is then utilized to obtain a closed-form approximation  for the ergodic secrecy rate (ESR) and SOP of three typical  scenarios: the case with only external Eves, the case with only  internal Eves, and that with both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' The derived results are then  used to evaluate the percentage of users in secrecy outage and  the required number of transmit antennas to achieve a positive  secrecy rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' It is shown that, with equally-capable Eves, the  secrecy loss caused by external Eves is higher than that caused  by internal Eves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' Numerical simulations validate the accuracy of  the theoretical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  Index Terms—Physical layer security (PLS), regularized zero-  forcing (RZF) precoding, multiple-input single-output (MISO),  central limit theorem (CLT), random matrix theory (RMT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' INTRODUCTION  A pivotal problem for wireless communications is eaves-  dropping due to the broadcast nature of wireless channels [1],  [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' To ensure security, traditional network-layer key-based  cryptography has been widely used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' However, the dynamics  of wireless environments, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=', fading, lead to issues in key  distribution and management, and cause high computational  complexity [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' To provide additional security enhancements,  physical layer security (PLS) [4], which utilizes the random-  ness of the noise and dynamics of the fading channel to  limit information leakage to potential eavesdroppers, has been  considered as an appealing low-complexity alternative [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  PLS has become an active area of research, especially in  the broadcast channel (BC), where the transmitter tries to  broadcast messages to users, with minimum leakage to the  eavesdroppers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' To this end, linear precoding schemes have  been used in confidential transmission of multiple antenna  systems due to its simple implementation and effectiveness in  interference controlling [6]–[8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' In [3], [9], the authors utilized  regularized zero-forcing (RZF) precoding, also referred to as  Xin Zhang and Shenghui Song are with the Department of Electronic and  Computer Engineering, The Hong Kong University of Science and Technol-  ogy, Hong Kong (e-mail: xzhangfe@connect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='ust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='hk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' eeshsong@ust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='hk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  Yonina C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' Eldar is with the Faculty of Math and CS, Weizmann Institute  of Science, Rehovot 7610001, Israel (e-mail: yonina.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='eldar@weizmann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='il).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  regularized channel inversion (RCI), to mitigate the signal  leaked to the undesired users and maximize the secrecy sum  rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' Despite the wide usage of RZF, its performance analysis  is still in the infant stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' Existing works on RZF in MISO  BC can be categorized into two types according to the nature  of the eavesdroppers, referred to as Eves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' Internal Eve: In a MISO BC, users can act maliciously  as Eves for other users [10], [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' Given the Eves are within  the BC system, we refer to this scenario as the internal Eve  case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' Under such circumstances, for a given user, other users  are regarded as single-antenna Eves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' In [3], [9], the secrecy  sum rate of the MISO BC system with internal Eves over  independent and identically distributed (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=') channels was  given in closed form by large random matrix theory (RMT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  Considering the unequal path loss and correlations of the  transmit antennas, the authors of [12] evaluated the secrecy  sum rate of RZF using RMT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' External Eve: In a practical scenario, external devices can  also act as Eves, which are referred to as external Eves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' The  major difference between internal and external Eves is that the  channels of internal Eves are typically correlated with those  of the users while the channels of external Eves are typically  independent of the users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' The impact of external Eves on the  secure connectivity was investigated by stochastic geometry  (SG) [13]–[15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' In [16], assuming that the locations of the Eves  are distributed as a Poisson point process (PPP), the authors  derived the sum rate and the secrecy outage probability (SOP)  with RZF over uncorrelated MISO channels by utilizing RMT  and SG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' In [17], assuming both users’ and Eves’ locations are  distributed as a PPP, the authors derived approximate closed-  form results for the SOP and ergodic secrecy rate (ESR) with  RZF precoding over uncorrelated MISO channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  Most existing works focus on the secrecy sum rate of  the system [3], [12], [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' However, the secure reliability  performance, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=', SOP, is not well studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' In [17], the SOP  of RZF precoding was derived by SG, considering only large-  scale fading, where the impact of small-scale fading on the  SOP is ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' Although the SG based approach is powerful  in modeling the dynamics of the network and performing  coverage analysis [18], it may not be able to capture long-term  behavior of the system where one BS serves a fixed group of  users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' In this case, the large-scale fading is deterministic and  thus the PPP model is not suitable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' In particular, the PPP model  will dominate the performance analysis such that the impact of  small-scale fading can not be fully understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' In this paper,  we focus on the impact of small-scale fading on the secrecy  performance of MISO BC with RZF precoding and derive the  2  SOP, which is not available in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  A closed-form expression for the SOP is essential for the  performance evaluation and will be very beneficial for system  design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' However, characterization of the secrecy performance,  especially the SOP, turns out to be a difficult problem due  to the fractional structure of the signal-to-interference-plus-  noise ratio (SINR) and the power normalization factor of RZF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  In particular, these challenges make it extremely difficult to  obtain the closed-form results for the joint distribution of the  users’ SINR and Eve’s signal-to-noise ratio (ESNR) in finite  dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' To overcome this difficulty, asymptotic RMT will  be adopted to derive the asymptotic distribution of the SINRs  and ESNRs by assuming that the number of transmit antennas,  the number of users, and the number of Eves go to infinity  with the same pace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' In fact, asymptotic RMT has been widely  used in large-system analysis of multiple-antenna systems to  obtain strikingly simple results [19]–[21], which are shown to  be effective even for small dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  The contributions of this paper are summarized as follows:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' We consider a CLT for the joint distribution of the SINRs  and ESNRs of a finite number of users and derive explicit  expressions for the asymptotic mean and variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' Specifically,  we utilize the CLT for the quadratic form and compute the  asymptotic second-order attributes by Gaussian tools [20],  [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' The derived results can be extended to the case with  imperfect CSI and the analysis of the channel-inversion based  precoding method, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=', secure RZF [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' Based on the CLT, we derive closed-form approxima-  tions for the SOP of RZF precoding including three sce-  narios, namely the External-Eve-Only, Internal-Eve-Only, and  External+Internal-Eve cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' The derived results are then used to evaluate how many  transmit antennas are needed to guarantee a positive secrecy  rate and to estimate the percentage of users in outage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' We  further compare the performance of RZF for the Internal-Eve-  Only and External-Eve-Only cases and derive the theoretical  gap between the two secrecy rates, which indicates that the  external Eves incur more information leakage than internal  ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' Numerical results validate the accuracy of the theoretical  results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  The rest of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' In Section II,  the system model is introduced and the problem is formulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  In Section III, some preliminary results on RMT, which  are critical for the derivations in this paper, are given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' In  Section IV, a CLT for the joint distribution of SINRs and  ESNRs and the approximation for the SOP are derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' The  derived results are used in Section V to evaluate the percentage  of users in outage and the required number of antennas for  a positive secrecy rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' Simulation results are provided in  Section VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' Section VII concludes the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  Notations: Bold, upper case letters and bold, lower case  letters represent matrices and vectors, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' The prob-  ability operator and expectation operator are denoted by P(·)  and E(·), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' The circularly complex Gaussian distri-  bution and real Gaussian distribution are denoted by CN and  N, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' The N-dimensional complex vector space and  real vector space are represented CN and RN, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  The M-by-N complex and real matrix space, are represented  by CM×N and RM×N, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' The centered version of  the random variable x is denoted by x = x−Ex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' AH denotes  the Hermitian transpose of A and [A]i,j or Aij denotes  the (i, j)-th entry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' The conjugate of a complex number is  represented by (·)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' The spectral norm of A is given by ∥A∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  Tr A is the trace of A, IN is the N by N identity matrix, and  Φ(x) represents the cumulative distribution function (CDF) of  the standard Gaussian distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' Almost sure convergence,  convergence in probability, and convergence in distribution,  are represented by  a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  −−−−→  N→∞ ,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  −−−−→  N→∞ , and  d  −−−−→  N→∞ , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  The set {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=', N} is denoted by [N] and  1(·) represents the  indicator function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' SYSTEM MODEL AND PROBLEM FORMULATION  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' System Model  We consider the downlink MISO system, where N single-  antenna users are served by a base station (BS) with M trans-  mit antennas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' We assume that perfect channel state information  (CSI) of users is available at the BS and RZF is adopted to  suppress information leakage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  RZF is a linear precoding scheme proposed to serve  multiple users in the MISO downlink channel, which often  demonstrates better performance than ZF, especially with low  SNR [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' With RZF, the received signal at the k-th user is  given by  yk =  �  ξhH  k  N  �  i=1  √piwisi + vk,  (1)  where si ∼ N(0, 1) and vk ∼ N(0, σ2) denote the transmitted  signal for the i-th user and the additive white Gaussian noise  (AWGN) at the k-th user, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' Here hk  ∈ CM  and wi ∈ CM represent the channel vector of the k-th  user and the precoding vector for the i-th user, respectively,  pi denotes the power of the i-th message and ξ represents  the power normalization factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' The precoding matrix W =  [w1, w2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=', wN] ∈ CM×N of the RZF precoder is given  by [6]  W =  �  HHH + zIN  �−1 H,  (2)  where H = [h1, h2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=', hN] ∈ CM×N and z > 0 is the  regularization parameter of the RZF precoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' The power  normalization factor satisfies  ξ Tr PWHW  N  = 1, where P =  diag(p1, p2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=', pN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' The k-th user’s SINR is given by  SINRk =  pk|hH  k wk|2  �  j̸=k  pj|hH  k wj|2 + σ2  ξ  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  (3)  Here we consider L single antenna Eves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' The channel vector  between the BS and the l-th Eve is denoted by he,l ∈ CM,  l ∈ [L].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' The received signal of the l-th Eve, ul, is given by  ul =  �  ξhH  e,l  N  �  i=1  √piwisi + wl,  (4)  where wl ∼ N(0, ρ2) represents the AWGN at the l-th Eve  with ρ2 denoting the power of the noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  It is not straightforward to directly characterize the infor-  mation leakage, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=', the information received by the undesired  3  users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' Here we consider the worst-case scenario, where all  Eves are assumed to work together and be able to cancel  out the multiuser interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' This model is widely used in  secrecy analysis [3], [12], [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' Under such circumstances, the  ESNR for all Eves to collaboratively overhear the message sk  is given by  ESNRk = ξ∥HH  e wk∥2  ρ2  ,  (5)  where He = [he,1, he,2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=', he,L].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' The corresponding informa-  tion leakage rate is given by Ce,n = log(1 + ESNRk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' As a  result, the secrecy rate of the k-th user is  Rk = ⌈log(1 + SINRk) − log(1 + ESNRk)⌉+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  (6)  The SOP of the k-th user for a given rate requirement R is  given by  Pout,k = P(Rk ≤ R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  (7)  Note that the Eves can be both internal and external.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' For the  cases with only internal Eves, the channel matrix for the Eves  with respect to the k-th user can be given by He = Hk, where  Hk is obtained by removing hk from H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' In this paper, we  will evaluate the ESR in (6) and SOP in (7) in a large system  setting where the number of transmit antennas and numbers  of users and Eves go to infinity with the same space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' Channel Model  We consider a correlated Rayleigh fading channel for users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  Accordingly, the channel vector for the k-th user can be rep-  resented by hk = t  1  2  k R  1  2 xk, where tk is the large-scale fading  (pathloss) from the BS to the k-th user, xk ∼ CN(0, IM), and  R denotes the correlation matrix of the BS towards the users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  In this paper, we assume a common transmit correlation matrix  for different users, which is widely used in [12], [25]–[27]  for tractability of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' With the common correlation  matrix, the channel matrix H = [h1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=', hN] is given by  H = R  1  2 XT  1  2 ,  (8)  where T = diag(t1, t2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=', tN) and X = [x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=', xN].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  Assume that there are L external single-antenna Eves and  denote the channel vector for the l-th Eve as gl, l ∈ [L].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  Similar to the channel matrix for the users, the channel matrix  of the external Eves, G = [g1, g2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=', gL], is given by  G = C  1  2 YD  1  2 ,  (9)  where C ∈ CM×M and D ∈ CL×L = diag(d1, d2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=', dL)  represent the correlation matrix and the large-scale fading for  the Eves, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' Here Y ∈ CM×L is a random matrix  with i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' circularly Gaussian entries, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=', Ym,l ∼ N(0, 1  M ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  Notice that the external Eves overhear the message passively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  Thus, we can assume that hk, k ∈ [N] and gl, l ∈ [L] are  independent, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=', X and Y are independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' Due to the same  reason, the CSI of the Eves is not available at the BS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' If there  are only external Eves, we have He = G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  In this paper, we consider three typical scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  The External-Eve-Only case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' According to (5), we can  obtain the ESNR for overhearing message sk as  ESNRex,k = ξwH  k GGHwk  ρ2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  (10)  The achievable secrecy rate for the k-th user is given by  Rex,k = ⌈log(1 + SINRk) − log(1 + ESNRex,k)⌉+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  (11)  The Internal-Eve-Only case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' In this case, the unintended  users within the MISO BC behave maliciously so that the  crosstalk among users causes information leakage [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' For  each user, the other N − 1 users behave as single-antenna  Eves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' Specifically, users in Mk = [N] \\ k can cooperate to  jointly eavesdrop the message sk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' In the worst case scenario,  they can be regarded as a single Eve with N − 1 antennas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  The ESNR for overhearing message sk is then given by [3],  [9]  ESNRin,k = ξwH  k HkHH  k wk  σ2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  (12)  The achievable secrecy rate for the k-th user is given by  Rin,k = ⌈log(1 + SINRk) − log(1 + ESNRin,k)⌉+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  (13)  In fact, (11) and (13) can be obtained by taking He = G and  He = Hk in (6), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  The External+Internal-Eve case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' In this case, both the  internal and external Eves exist and they work in a non-  colluding mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' The secrecy rate of the k-th user is given  by [28], [29]  Rk = min(Rex,k, Rin,k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  (14)  Note that the first two cases are differentiated by whether  the channels of the Eves are independent with those of the  intended users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' The secrecy sum rate of the Intern-Eve-Only  case has been investigated in [3], [9], [12] but the SOP is not  available in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' In the following, we investigate the  ESR and SOP for the three cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' PRELIMINARIES  In this section, we provide some preliminary results for the  evaluation of SINR and ESNR, and introduce the RMT results  and notations which will be frequently used in the following  derivations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' Necessarily, we first introduce the resolvent matrix  Q = (zIN + HHH)−1, which will be extensively used in the  following analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' Further define Qk = (zIN + HkHH  k )−1,  where Hk is obtained by removing the k-th column from H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  By utilizing the identity hH  k Q =  hH  k Qk  1+hH  k Qkhk , the SINR for  k-th user can be rewritten as  SINRk =  pkA2  k  Bk + σ2(1 + Ak)2C ,  (15)  where Ak  =  hH  k Qkhk, Bk  =  hH  k QkHkPkHH  k Qkhk,  and C  =  1  M Tr PHH(HHH + zIN)−2H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' Here Pk is  obtained by removing the k-th row and column of P =  diag(p1, p2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=', pN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' Similarly, ESNRex,k and ESNRin,k can  be rewritten as  ESNRex,k =  pkEk  ρ2C(1 + Ak)2 ,  (16)  and  ESNRin,k =  pkFk  σ2C(1 + Ak)2 ,  (17)  4  respectively, where Ek  = hH  k QkGGHQkhk and Fk  =  hH  k QkHkHH  k Qkhk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' The formulation above resorts the inves-  tigation of SINRk and ESNRk to that of the Ak, Bk, Dk, and  Ek, which can be regarded as the quadratic forms of hk when  Hk is given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' We next introduce three assumptions based on  which the analysis in this paper is performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  Assumption A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' The dimensions N, L, and M go to  infinity at the same pace, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=', when M → ∞,  0 < lim inf M  N ≤ M  N ≤ lim sup M  N < ∞,  0 < lim inf L  M ≤ L  M ≤ lim sup L  M < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  (18)  Assumption A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' ∥R∥ ≤ ∞, ∥T∥ ≤ ∞, ∥P∥ ≤ ∞ [30],  [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  Assumption  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  inf  M≤1  Tr R  M  >  0,  inf  M≤1  Tr T  M  >  0,  inf  M≤1  Tr C  M  > 0, inf  M≤1  Tr D  M  > 0 [20], [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  Assumption A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='1 indicates the asymptotic regime where the  number of transmit antennas, the number of users, and the  number of external Eves are in the same order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' Assump-  tion A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='2 and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='3 eliminate the extremely low-rank case of  R and C, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=', the ranks of R and C do not increase with  the number of antennas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' The bound ∥P∥ ≤ ∞ guarantees the  power for each message is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' We further define two ratios  τ =  M  N−1 and θ = L  M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' Define matrices GR =  �  zIM + �δR  �−1  and  GT =  �  IN + �δT  �−1  , where (δ, �δ) is the unique positive  solution for the following system of equations  �  δ  =  1  M Tr RGR,  �δ  =  1  M Tr TGT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  (19)  Given assumptions A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='1-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='3, for the random matrix H defined  in (8), it holds true that for any deterministic matrix A with  bounded norm [30, Theorem 1],  1  M Tr AQ  a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  −−−−→  N→∞  1  M Tr AGR,  (20)  and [20, Step B in Section IV]  1  M E Tr AQ = 1  M Tr AGR + O(M −2),  (21)  Lemma 1 indicates that the normalized trace of the resolvent  converges almost surely to its deterministic equivalent, which  is a good approximation for the expectation of the trace of the  resolvent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' The computation of the variances for the SINR and  ESNR is highly related to the trace of the resolvent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  We summarize some frequently used notations in Table I,  which are all derived from δ and �δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' In the following analysis,  some of the symbols have the subscript k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' They are derived  similarly as those in Table I, but are parameterized by R and  Tk with  �  δk  =  1  M Tr RGR,k,  �δk  =  1  M Tr TkGT,k,  (22)  where Tk is obtained by removing the k-th row and k-  th column in T, GR,k =  �  zIM + �δkR  �−1  and GT,k =  (IN−1 + δkTk)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' These k-related quantities are used to  approximate the Qk related statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  TABLE I: List of Notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  Symbols  Expression  Symbols  Expression  γ  1  M Tr R2G2  R  γ(A)  1  M Tr AG2  RR  �γ  1  M Tr T2G2  T  �γ(B)  1  M Tr BG2  T T  η  1  M Tr R3G3  R  η(A)  1  M Tr AG3  RR2  �η  1  M Tr T3G3  T  �η(B)  1  M Tr BG3  T T2  ζ  1  M Tr R4G4  R  ζ(A)  1  M Tr AG4  RR3  �ζ  1  M Tr T4G4  T  �ζ(B)  1  M Tr BG4  T T3  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' MAIN RESULTS  In this section, we develop theoretical results on the asymp-  totic distribution of SINR, ESNRex, and ESNRin, and then  derive the ESR and SOP of the MISO BC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' First-order Analysis of the SINR, ESNR, and Secrecy Rate  In this part, we derive approximations for the mean of SINR  and ESNR, which will be used to obtain the ESR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  1) SINR and ESNR: According to (15)-(17), an approxima-  tion for SINR, ESNRex, and ESNRin can be obtained by the  continuous mapping theorem [33], where we need to obtain  the deterministic equivalents of Ak, Bk, Ek, Fk, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  By Lemma 1 and [34, Lemma 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='2], we have  Ak  a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  −−−−→  N→∞  tkE Tr RQk  M  N→∞  −−−−→ tkδk := Ak,  (23)  and  Bk  a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  −−−−→  N→∞  tkE Tr RQkHkPkHH  k Qk  M  = tk�γk(Pk)γk  ∆k  + O(M −2) := Bk + O(M −2),  (24)  where �γk(Pk) and γk are given in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' Here ∆k = 1 −  γk�γk,  C  a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  −−−−→  N→∞  �γ(P)γ(IM)  ∆  := C,  (25)  Dk  a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  −−−−→  N→∞  E Tr RQkGGHQk  M  = θdγk(C)  ∆k  + O(M −2) := Dk + O(M −2),  (26)  and  Ek  a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  −−−−→  N→∞  E Tr QkHkHH  k QkR  M  = �γk(IN−1)γk  ∆k  + O(M −2) := Ek + O(M −2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  (27)  We define Ck = Tr QkHkPkHH  k Qk  M  and its deterministic equiv-  alent is Ck = �γk(Pk)γk(IM)  ∆k  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' In fact, we have  √  M(C − Ck) =  √  M[  pkhH  k Q2  khk  M(1 + hH  k Qkhk)2  + hH  k QkHPkHHQkhkhH  k Qkhk  M(1 + hH  k Qkhk)2  − 2hH  k QkHPkHHQ2  khk  M(1 + hH  k Qkhk)  ]  a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  −−−−→  N→∞ O(M − 1  2 ),  (28)  5  which indicates that C can be replaced by Ck in the asymptotic  regime A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' To keep a concise expression for the deterministic  equivalent, we will use Ck instead of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  By the continuous mapping theorem [33], the approxima-  tion for SINRk, ESNRex,k, and ESNRin,k can be obtained  by replacing the random quantities with their deterministic  equivalents in (23)-(27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' Thus, we obtain  SINRk =  pkt2  kδ2  k∆k  γk�γk(Pk) + σ2(1 + tkδk)2γk(IM)�γk(Pk),  ESNRex,k =  pkθdγk(C)  ρ2(1 + tkδk)2γk(IM)�γk(Pk),  ESNRin,k =  pk�γk(IN−1)γk  σ2(1 + tkδk)2�γk(Pk)γk(IM),  (29)  where d = Tr D  L  represents the averaged large-scale fading of  the external Eves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  2) Ergodic Secrecy Rate (ESR): We approximate the ESR  (expectation of the secrecy rate) by replacing the instanta-  neous SINR and ESNR with their deterministic equivalents  in (29) [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' Specifically, we plug (29) into (11) and (13),  respectively, to obtain  ERin,k  M→∞  −−−−→ ⌈Cin,k⌉+,  ERex,k  M→∞  −−−−→ ⌈Cex,k⌉+,  Cin,k = log(1 + SINRk) − log(1 + ESNRin,k),  Cex,k = log(1 + SINRk) − log(1 + ESNRex,k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  (30)  Note that, the results in this paper are more general than the  existing ones, because the spatial correlation at the transmitter  and the power allocation scheme are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' In particular,  the above results include the first-order results of previous  works [12, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' (48)-(50)(K = 1)], [9, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' (31)], and [3,  Theorem 1] as special cases, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=', by taking R = IM, T = IN,  and P = IN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' Second-order Analysis of the SINR, ESNR, and Secrecy  Rate  In this part, we investigate the fluctuations of SINR and  ESNR to further investigate the distribution of the secrecy  rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' The concerned random quantities, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=', the SINRs and  ESNRs, converge almost surely to a constant as shown in Sec-  tion IV-A1, which means that their variances vanish when  M grows to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' In order to investigate the second-order  performance, in this section, we show that  √  MSINRk and  √  MESNRk converge to a Gaussian distribution when M goes  to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  For a finite number K, define SINR, ESNRex, and ESNRin  for K users as the vector gK ∈ R3K, given by  gK = (SINR1, SINR2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=', SINRK, ESNRex,1, ESNRex,2,  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=', ESNRex,K, ESNRin,1, ESNRin,2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=', ESNRin,K)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  In the following, we first derive the joint distribution for the  SINRs and ESNRs and then utilize the result to evaluate the  SOP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  1) Joint distribution for SINRs and ESNRs: The following  theorem presents the asymptotic joint distribution of gK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' (CLT for the joint distribution of SINRs and  ESNRs) For a finite number K, the distribution of the vector  √  MgK converges to a Gaussian distribution  √  M (gK − gK)  d  −−−−→  M→∞ N(0, VK),  (31)  where  gK = (SINR1, SINR2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=', SINRK, ESNRex,1, ESNRex,2,  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=', ESNRex,K, ESNRin,1, ESNRin,2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=', ESNRin,K)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  (32)  Here, SINRk, ESNRex,k, and ESNRin,1 are given in (29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' The  covariance matrix VK is given by  VK =  \uf8ee  \uf8f0  UK  FK  JK  FK  WK  OK  JK  OK  ZK  \uf8f9  \uf8fb ,  (33)  with  UK  =  diag([U1, U2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=', UK]),  WK  =  diag([W1, W2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=', WK]),  ZK  =  diag([Z1, Z2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=', ZK]),  FK = diag([F1, F2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=', FK]), JK = diag([J1, J2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=', JK]),  and OK = diag([O1, O2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=', OK]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' The parameters Uk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' Wk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  Vk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' Fk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' Jk and Ok are given by  Uk =  a2  k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='1t2  kγk  ∆k  + a2  k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='2t2  kΠk(Pk) + 2ak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='1ak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='2t2  kκk(Pk),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' (34)  Wk =  a2  k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='3t2  kγk  ∆k  + a2  k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='4t2  k[θd2γk(C)2  ∆2  k  + θ2d  2χk(C,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' C)] + 2ak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='3ak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='4t2  kθdΓk(C),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  (35)  Zk =  a2  k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='5t2  kγk  ∆  + a2  k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='6t2  kΠk(IN) + 2t2  kak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='5ak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='6κk(IN),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' (36)  Fk = ak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='1ak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='3t2  kγk  ∆k  + ak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='1ak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='4t2  kθdΓk(C)  + ak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='2ak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='3t2  kκk(Pk) + ak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='2ak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='4t2  kθdβk(Pk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' C),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  (37)  Jk = ak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='1ak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='6t2  kκk(IM) + ak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='2ak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='5t2  kκk(Pk)+  ak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='1ak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='5t2  kγk  ∆k  + ak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='2ak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='6t2  k[κk(Pk) − zβk(Pk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' IM)],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  (38)  Ok = ak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='3ak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='5t2  kγk  ∆k  + ak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='3ak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='6t2  kκk(IN−1)  + ak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='4ak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='5t2  kθdΓk(C) + ak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='4ak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='6t2  kθdβk(IN−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' C),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  (39)  where Πk(·),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' κk(·),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' χk(·),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' βk(·),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' and Γk(·) are given by (90)  to (94) in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' Here, ak,i, i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=', 6 are given  in (40) at the top of the next page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  The terms χ(C, C), Γ(C), Π(P), and κ(P) are the  deterministic approximations for  1  M E Tr QCQRQCQR,  1  M E Tr QRQRQC,  1  M E Tr(QRQHPHH)2,  and  1  M E Tr RQHPHHQRQ, respectively, which are proved by  Lemma 3 in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' The proof of Theorem 1 is given in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  Theorem 1 indicates that for a finite number of users, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=',  K ≪ M, the joint distribution of the SINRs and ESNRs  converges to a joint Gaussian distribution when M, L, and  6  ak,1 = 2pk[tkδ∆k(tkγk + σ2(1 + tkδk)γ(IM))]  (tkγk + σ2(1 + tkδk)2γk(IM))2�γk(Pk) , ak,2 =  −pkt2  kδ2  k∆2  k  (tkγk + σ2(1 + tkδk)2γk(IM))�γk(Pk)2 , ak,3 =  −2pktkdγk(C)  ρ2�γk(Pk)γk(IM)(1 + tkδk)3 ,  ak,4 =  pk∆k  ρ2�γk(Pk)γk(IM)(1 + tkδk)2 , ak,5 =  −2pktk�γ(IN)γk  σ2�γk(Pk)γk(IM)(1 + tkδk)3 , ak,6 =  pk∆k  σ2�γ(Pk)γk(IM)(1 + tkδk)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  (40)  N go to infinity with the same pace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' From the structure of the  covariance matrix VK, we observe that the SINRs of different  users are asymptotically independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' Similar results can also  be obtained for the ESNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' However, SINRk and ESNRex,k  are not independent, and their covariances are characterized  by the diagonal entries of FK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' Similarly, the covariance  of the pairs (SINRk, ESNRin,k) and (ESNRex,k, ESNRin,k)  is characterized by the diagonal entries of JK and OK,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' It follows from Theorem 1 that for a large system,  the SINR and ESNR can be approximated by a Gaussian  distribution, which will be utilized to evaluate the SOP in the  following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  Theorem 1 is obtained by assuming a common correlation  matrix as mentioned in Section II-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' When different correla-  tion matrices are assumed, the system of equations in (19) will  become a system of N equations parameterized by N different  correlation matrices, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=', R1, R2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=',RN [35, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' (11)] and  there will be N different δs instead of only one in (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' In this  case, the characterization of the higher order trace of resolvents  is more complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' For example, E Tr QRQR  M  M→∞  −−−−→ γ  ∆ will be-  come E Tr QRiQRj  M  M→∞  −−−−→ [(IM −S)−1v]i, where S ∈ RN×N  and v ∈ RN are determined by R1, R2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=',RN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' The inverse  matrix (IM − S)−1 will appear frequently and its order will  increase in the computation of high-order resolvents like  1  ∆  in (90) to (94).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' Although the simplified common correlation  matrix is considered, Theorem 1 sets up a framework for  the second-order analysis of SINRs and ESNRs in the MISO  system with RZF, which can also be used to analyze the  scenario with imperfect CSI and other channel-inversion based  precoding schemes like the secure RZF proposed recently  in [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  2) SOP Evaluation: Based on Theorem 1, the SOP of the  three cases can be obtained by Propositions 1 to 3 as follows,  which are not yet available in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' (External-Eve-Only) Given a rate threshold R,  the SOP of user k with only external Eves can be approximated  by  Pex,out,k(R) ≈ φ  �√  M(R − Cex,k)  �Vex,k  �  ,  (41)  where Vex,k =  Uk  (1+gk)2 +  Wk  (1+gex,k)2 −  2Fk  (1+gk)(1+gex,k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' (Internal-Eve-Only) Given a rate threshold R,  the SOP of user k with only internal Eves can be approximated  by  Pin,out,k(R) = φ  �√  M(R − Cin,k)  �  Vin,k  �  ,  (42)  where Vin,k =  Uk  (1+gk)2 +  Zk  (1+gin,k)2 −  2Jk  (1+gk)(1+gin,k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' (External+Internal-Eve) Given a rate thresh-  old R, the SOP of user k with both external and internal Eves  can be approximated by  Pco,out,k(R) ≈ 1 −  � ∞  R  � ∞  R  (2π)−1[det(Cco,k)]− 1  2 ×  exp  �  −1  2(x − µco,k)T C−1  co,k(x − µco,k)  �  dx1dx2,  (43)  where x = [x1, x2]T , µco,k = [Cex,k, Cin,k]T , and Cco,k =  �  Vex,k  Vco,k  Vco,k  Vin,k  �  , Vco,k  =  Uk  (1+gk)2 +  Ok  (1+gex,k)(1+gin,k) −  Jk  (1+gk)(1+gin,k) −  Fk  (1+gk)(1+gex,k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  Propositions 1-3 follow from Theorem 1 by utilizing the delta-  method [36] and indicate that the SOP can be approximated  by a Gaussian distribution in the large system setting A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='1-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' LARGE SYSTEM ANALYSIS  In this section, we utilize the derived results in Section IV to  analyze the impact of the system dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' To achieve this  goal, we ignore the correlation, pathloss, and power allocation  and consider the i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' channel, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=', R = IM, T = IN, C =  IM, D = IN, and P = IN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' In this case, the subscript used to  differentiate users can be removed and the system of equations  in (22) degenerate to the quadratic equation zτδ2 + (τz − τ +  1)δ − τ = 0, whose positive solution is given by  δ = τ − 1 − τz +  �  (τ − 1)2 + 2(1 + τ)τz + z2τ 2  2zτ  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  (44)  In addition, δ(1) = dδ  dz, δ(2) = d2δ  dz2 , and δ(3) = d3δ  dz3 represent  the first to third derivatives of δ with respect to z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' With the  setting above, we only need to consider the joint distribution  of gi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='d = (SINR, ESNRex, ESNRin)T , and Theorem 1 can  be simplified as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' Given assumption A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='1, we have  √  M (giid − giid)  d  −−−−→  M→∞ N(0, Viid),  (45)  where giid = (giid, giid,ex, giid,in)T ∈ R3 is given by  giid =  �δ[1 + zτ(1 + δ)2]  1 + σ2(1 + δ)2 , τθ  ρ2 ,  1  σ2(1 + δ)2  �T  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  (46)  The covariance matrix Viid ∈ C3×3 is given by  Viid =  \uf8ee  \uf8f0  Uiid  Fiid  Jiid  Fiid  Wiid  Oiid  Jiid  Oiid  Ziid  \uf8f9  \uf8fb ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' with  (47)  Uiid = −[(a1 + a2)2δ(1) + (a2  2 + a1a2)zδ(2) + a2  2z2δ(3)  6  ],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  Wiid = −[a2  3δ(1) − a2  4θδ2  (1) + θ2a2  4δ(3)  6  − a3a4θδ(2)],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  Ziid = −[(a5 + a6)2δ(1) + a5a6zδ(2) + a2  6z2δ(3)  6  ],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  7  Fiid = −[(a1 + a2)a3δ(1) − [(a1 + a2)a4θ − a2a3z]δ(2)  2  − a2a4θzδ(3)  6  ],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  Jiid = −[(a1 + a2)(a5 + a6)δ(1)  + z(a2a5 + a1a6 + 2a2a6)δ(2)  2  + za2a6δ(3)  6  ],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  Oiid = −[a3(a5 + a6)δ(1) + (za3a6 − θa4(a5 + a6))δ(2)  2  − za4a6θδ(3)  6  ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  (48)  The parameters ai, i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=', 6, are given as  a1 =  2δ[1 + σ2(1 + δ)]  [1 + σ2(1 + δ)2]2(δ + zδ(1)), a4 =  1  ρ2(1 + δ)2(δ + zδ(1)),  a2 =  −δ2  [1 + σ2(1 + δ)2]2(δ + zδ(1))2 , a5 = −  2  σ2(1 + δ)3 ,  a3 =  2θδ(1)  ρ2(1 + δ)3(δ + zδ(1)), a6 =  1  σ2(1 + δ)2(δ + zδ(1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  The SINR distribution of the MISO BC with RZF was  investigated in [37], where the i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='d channel and equal power  allocation are assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' If we discard ESNRex and ESNRin,  the results in Theorem 2 degenerate to [37, Theorem 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' The  outage probability for the i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' case can be obtained by a  similar approach as Propositions 1 to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' In this paper, the joint  distribution of SINR and ESNR is investigated with correlated  transmit antennas and unequal power allocation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' How many users are in secrecy outage?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  Theorem 2 can be utilized to evaluate how many users are in  outage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' For that purpose, we define the empirical distribution  αN,case(R) = 1  N  N  �  i=1  1{Rcase≤R},  (49)  where case can be either External-Eve-Only or Internal-Eve-  Only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' Here we consider the External-Eve-Only case and the  results for the Internal-Eve-Only case can be obtained simi-  larly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' The following proposition is an application of Theorem 2  to compute the quantiles of the users in outage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  Proposition  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  Given  assumptions  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='1  to  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='3,  define  Rex(σ2, ρ2, q) and Rin(σ2, q) as  Rex(σ2, ρ2, q) = log(1 + giid,ex) +  �  Vex,iid  M  φ−1(q),  Rin(σ2, q) = log(1 + giid,in) +  �  Vin,iid  M  φ−1(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  (50)  Then, we have  αN,ex(Rex(σ2, ρ2, q))  a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  −−−−→  M→∞ q,  αN,in(Rin(σ2, q))  a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  −−−−→  M→∞ q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  (51)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' The result follow from the law of large numbers and  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  Proposition 4 indicates that for large N and M, given  a target secrecy rate Rcase and the noise level σ2, ρ2, the  proportion of the users in outage is approximated by q, where  q is obtained by solving equation (50).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' This means that almost  qN users are in outage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' Given ρ2 and σ2, q is an increasing  function of Rcase, which indicates that a larger threshold  causes more users in outage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' The Secrecy Loss of the External-Eve-Only and Internal-  Eve-Only Cases  Next, we investigate the secrecy loss induced by external  and internal Eves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' To make a fair comparison, we consider the  case where the internal and external Eves are equally powerful,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=', L = N −1 and ρ2 = σ2, and the precoders have the same  z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' Under such circumstances, giid,in  M→∞  −−−−→  giid,ex  (1+δ)2 , which  indicates that the information leakage due to external Eves is  larger than that due to internal Eves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' This is because, with  only internal Eves, the information received by other users  is mitigated by RZF due to its ability to cancel interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  With external Eves, RZF precoder will not be able to cancel  the interference since the channels of Eves are independent of  those of the users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  It can also be observed from (46) that ESNRex is constant  when the ratio L  N is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' Moreover, ESNRex does not  depend on the regularization parameter z, indicating that the  optimal regularization parameter for maximizing the secrecy  sum rate is equivalent to that without considering the Eves,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=', R = �N  i=1 log(1 + SINRi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' The optimal value of the  regularization parameter is z = σ2  τ [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' How many transmit antennas do we need to achieve a  positive secrecy rate?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  For the External-Eve-Only case with the optimal regular-  ization parameter z = σ2  τ , the inequality µ = δ > µex must  hold true, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=',  τ − 1 − σ2 +  �  (τ − 1)2 + 2(1 + τ)σ2 + σ4  2σ2  ≥ τθ  ρ2 ,  (52)  in order to obtain a positive secrecy rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' From (52), the  minimum τ required for a positive secrecy rate is  τ ∗ =  L  Nρ2 (1 + σ2) + L2σ2  N 2ρ4  L  Nρ2 + 1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  (53)  This indicates that if M > Nτ ∗, a positive secrecy rate will be  guaranteed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' The estimation of M is accurate in the high SNR  regime because only the term 4σ2 is omitted in the relaxation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  When  1  ρ2 → ∞, M = Nτ ≈  Lσ2  ρ2 , we can obtain that M  grows with the order O(ρ−2) and a larger L requires a higher  increasing rate of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' NUMERICAL RESULTS  In this section, the theoretical results derived in Sections IV  and V are validated by numerical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' Specifically, the  accuracy of the SOP approximation and the evaluation of the  percentage of users in secure outage are validated by Monte-  Carlo simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' The impact of the regularization factor  8  z and number of transmit antennas required for a positive  secrecy rate are also investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' Simulation Settings: In the simulation, we consider  a uniform linear array of antennas at the BS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' According to  the model for conventional linear antenna arrays [39], the  correlation matrix at the BS can be obtained by  [L(dr, α, ν, N)]m,n  =  � 180  −180  1  √  2πδ2 e\uf6be 2π  λ dr(m−n) sin( πφ  180 )− (φ−α)2  2ν2  dφ,  (54)  where m and n represent the indices of antennas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' Here, dr and  N denote the relative antenna spacing (in wavelengths) and the  dimension of the matrix, respectively, and α and ν2 represent  the mean angle and the mean-square angle spreads, whose  units are degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' We adopt the setting dr = λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' In the following  simulations, the correlation matrices are generated according  to (54), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=', R = L(1, αR, νR, M), C = L(1, αC, νC, M)  with αR = νR = 10 and αC = νC = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' Without loss of  generality, we consider the first user, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=', k = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  Following [12], a simple model for the large-scale fading  of different users ti, i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=', N is used, with ti = a−η  i  ,  where ai represents the distance between the BS and i-th user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  The path loss exponent η = 3 is used to model a shadowed  urban area [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' Here we divide the users into Ga groups and  users in the same group have a common distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' We choose  ai = a⌊ i−1  Ga ⌋ so that ti = a−η⌊ i−1  Ga ⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' Similarly, di is generated  by di = b−η⌊ i−1  Gb ⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' The power of each message is given by  pi = c⌊ i−1  Gc ⌋ and the total power is normalized to be N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' In  the following, we use the setting a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='0772, b = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='1262,  c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='9, and Ga = Gb = Gc = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' In the figures, we use the  notations “Ana.” and “Sim.” to represent the theoretical and  the simulation results, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' The Approximation Accuracy for SOP: In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' 1a, the  SOP for three cases are given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' The dimensions of the system  are set as M = 64, N = 32, and L = 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' The SNRs at  the user and external Eves, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=',  1  σ2 and  1  ρ2 , are 10 dB and  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='5 dB, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' The regularization parameter z is set  as z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' The number of the Monte-Carlo realizations is  5 × 105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' It can be observed that the approximations for the  SOP in Proposition 1 to 3 are accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' The External+Internal-  Eve case has the highest SOP and the gap between the  External+Internal-Eve case and the other two represents the  performance loss induced by different types of Eves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' 1b, the SOP of the i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' system is given with the  setting M = 64, N = 32, L = 16,  1  σ2 = 6 dB, and  1  ρ2 = 2  dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' This result validates the accuracy of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' Optimal z Control: The theoretical results of this paper  can be used to investigate the optimal z for enhancing secure  reliability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' 2a and 2b investigate the impact of z on  ESR and SOP for the External-Eve-Only and Internal-Eve-  Only case, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' Here the i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='d channel is considered  with equal power allocation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' The SNR for internal Eves and  external Eves are 5 dB and 2 dB, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' The rate thresh-  olds for the two cases are set as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='8/ log(2) and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='55/ log(2),  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' The optimal z that maximizes the ESR can be  obtained by z =  σ2  τ  (External-Eve-Only) and [3, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' (12)]  (Internal-Eve-Only).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' The optimal values are determined as  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='8  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='8  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='2  10-4  10-3  10-2  10-1  100  (a) M = 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='5  2  10-4  10-3  10-2  10-1  100  (b) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' 1: Secrecy outage probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='9  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='6  (a) ESR  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='2  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='8  1  (b) SOP  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' 2: The impact of z  z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='1532 and z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='0664, respectively, which agree with the  simulation results shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' 2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' It can be observed from  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' 2 that the optimal z that minimizes the SOP is different  from the one that maximizes the ESR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' Percentage of Users in Outage: Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' 3a and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' 3b  depict percentages of users in outage for the External-Eve-  Only and Internal-Eve-Only case, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' The parameters  are set as M = 256, N = 128, L = 64, z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='1, and  1  ρ2 = 4 dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' It can be observed that the evaluation in (50)  matches the empirical result well, which validates the accuracy  of Propostion 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' The Number of Transmit Antennas Required for a  Positive Secrecy: With only the external Eves, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' 4 shows  the number antennas required to achieve a positive secrecy rate  for a given SNR at the Eves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' The transmit SNR is set to be  10 dB and N = 128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' It can be observed that as the ability of  Eves increases, the required number of antennas for a positive  secrecy rate increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' The increasing rate grows larger as L  increases, which agrees with the analysis in Section V-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' CONCLUSION  In this paper, the secrecy performance of RZF in the  MISO broadcasting system was investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' In the asymptotic  regime that the numbers of transmit antennas, users, and  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='5  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='5  3  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='8  1  (a) External-Eve-Only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='4  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='6  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='8  3  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='2  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='4  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='6  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='8  4  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='8  1  (b) Internal-Eve-Only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' 3: The percentage of users in secrecy outage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  9  2  4  6  8  10  12  0  100  200  300  400  500  600  700  800  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' 4: Required number of antennas for a positive secrecy  rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  Eves go to infinity with the same pace, a CLT for the  joint distribution of SINRs and ESNRs was derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' The  CLT was then used to obtain a closed-form approximation  for the SOP of three cases, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=', External-Eve-Only, Internal-  Eve-Only, and External+Internal-Eve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' Based on the derived  results, the required number of transmit antennas for a positive  secrecy rate and the percentage of user in secrecy outage were  evaluated in a closed form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' The secrecy loss caused by the  external Eves and internal Eves were compared, showing that  the loss caused by the internal Eves is less than that caused  by the external Eves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' The derived results were validated by  numerical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' The methods used in this paper can be  applied to analyze the performance of RZF with imperfect CSI  and investigate the channel-inversion based precoding scheme  like the recently proposed secure RZF [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  APPENDIX A  PROOF OF THEOREM 1  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' The proof can be summarized by three steps including:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' Show the Gaussianity of the SINRk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' Show the Gaus-  sianity of the ESNRk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' Derive the covariances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' In the  first two steps, without loss of generosity, we consider user  1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=', k = 1, and the derivation for other users k = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=', K  are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' The proof of the Gaussianity relies on the CLT  for the quadratic forms shown in Lemma 2 of Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  Specifically, we will first show that the SINRs and ESNRs  can be approximated by a linear combination of the quadratic  forms Ak, Bk, Ek, and Gk, and the approximation is tight in  probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' The asymptotic variances obtained in this step are  related to the high order resolvents, so we need to determine  the variance by their deterministic equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' This process  involves computations for the high-order resolvents, which are  summarized by Lemma 3 in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' The results will also  be utilized to evaluate the covariances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' Now we turn to the first  step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' The Asymptotic Gaussianity of the SINR1  In this step, the fluctuation of  √  M(SINR1 − SINR1) will  be investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' To achieve this goal, we first rewrite SINR1 −  SINR1 as  SINR1 − SINR1 = 2p1A1(A1 − A1)  D1  − p1A  2  1(D1 − D1)  D  2  1  +  1  √  M  εs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  (55)  where εs =  √  Mp1D1( A1  D1 − A1  D1 )2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' For any given ǫ, by the  Markov inequality, we have  P(|εs| > ǫ) ≤  √  M  ǫ  E|D1|| A1  D1  − A1  D1  |2  ≤ 2  √  M  ǫ  (E|A1 − A1|2  |D1|  + |A1|2  |D1|2 E|D1 − D1|2  |D1|  ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  (56)  Since |D1| is bounded away from zero almost surely, E|A1 −  A1|2  = O( 1  M2 ), E|A1 − A1|2  = O( 1  M ), and E|D1 −  D1|2 = O( 1  M ), we have P(|εs| > ǫ)  M→∞  −−−−→ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' Therefore,  √  M(SINR1 − SINR1) can be approximated by the first two  terms at the right hand side of (55) and the approximation is  tight in probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' D1 can be written as  D1 − D1 = B1 − B1 + 2σ2(1 + A1)C(A1 − A1) + εD, (57)  where εD = σ2(A1 − A1)2C + σ2(1 + A1)2(C − C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' By  the variance control in [20], [41], we can show that E|A1 −  A1|4 = O( 1  M2 ) and E|C − C|2 = O( 1  M2 ) so that we can  prove P(|εD| > ǫ)  M→∞  −−−−→ 0 for a positive ǫ by the same  approach in (56).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' Then, by substituting (57) into (55), we have  the following linear approximation  √  M(SINR1 − SINR1)  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  −−−−→  M→∞  √  M[a1,1(A1 − A1) + a1,2(B1 − B1)],  (58)  where a1,1 and a1,2 are given in (40).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  Now we turn to determine the asymptotic distribution of  the random process at the right hand side of (58).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' Given H1,  √  M(A1 − A1) and  √  M(B1 − B1) are both in quadratic  forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' According to Lemma2, we can show that  √  MA1  and  √  MB1 both converge to a Gaussian distribution when  H1 is given so that the linear combination of them also  converges to a Gaussian distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' Thus, we only need  to determine the asymptotic variances for  √  MA1,  √  MB1  and their covariance, which can be obtained by Lemma 3 in  Appendix B as  VA1 = t2  1 Tr Q1RQ1RQ1  M  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  −−−−→  M→∞  t2  1γ1  ∆1  ,  (59)  VB1 = t2  1 Tr(Q1RQ1H1P1HH  1 )2  M  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  −−−−→  M→∞ t2  1Π1(P1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' (60)  The covariance of A1 and B1 can be evaluated by  VA1,B1 = t2  1 Tr RQ1RQ1H1P1HH  1 Q1  M  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  −−−−→  M→∞ t2  1κ1(P1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  (61)  Therefore, we can obtain the asymptotic variance of SINR1 as  U1 = a2  1,1VA1 + a2  1,2VB1 + 2a1,1a1,2VA1,B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  (62)  10  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' The Asymptotic Gaussianity of ESNRex,1  In this part, we will show the Gaussianity of ESNRex,1 by  the same approach as Appendix A-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' We first replace C by  C in (16) to obtain  ESNRex,1 =  p1E1  ρ2C(1 + A1)2 + εC,  (63)  where εC =  √  M(  E1  C(1+A1)2 −  E1  C(1+A1)2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' By the variance  control in [22, Lemma 3], we have Var(C) = O( 1  M2 ) to show  εC  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  −−−−→  M→∞ 0 so that we only need to consider the asymptotic  distribution of  √  ME1  (1+A1)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' To achieve this goal, we first perform  the following decomposition  E1  (1 + A1)2 −  E1  (1 + A1)2 = −2E1(A1 − A1)  (1 + A1)3  + E1 − E1  (1 + A1)2 + e1 + e2 + e3,  (64)  where e1 =  2E1(A1−A1)2  (1+A1)(1+A1)3 , e2 =  E1(A1−A1)2  (1+A1)2(1+A1)2 , and e3 =  2(E1−E1)(A1−A1)  (1+A1)3  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' We can then show that e1, e2, and e3 vanish  in probability and here we take e1 as an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' E  √  M|e1|  can be upper bounded by  E  √  M|e1| ≤ 2  √  M(E|E1 − E1|(A1 − A1)2  + E1E|A1 − A1|2) ≤ 2  √  M(E  1  2 (E1 − E1)2×  E  1  2 (A1 − A1)4 + E1E|A1 − A1|2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  (65)  The term (E1 − E1)2 can be upper bounded by  (E1 − E1)2 ≤ 2[(E1 − t1 Tr RQ1GHGQ1  M  )2  + (t1 Tr RQ1GHGQ1  M  − E1)2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  (66)  The first term of the RHS in (66) is bounded by  E|E1 − t1 Tr RQ1GHGQ1  M  |2  = t2  1E Tr(RQ1GHGQ1)2  M 2  ≤ KE∥G∥4  M  (a)  ≤ K′  M ,  (67)  where the inequality (a) holds true due to the boundness of  E∥G∥4, which was shown in [22, Lemma 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' We also have  the bound for the second term in the RHS of (66) as  E|t1 Tr RQ1GHGQ1  M  − E1|2 ≤ 2Var(t1 Tr RQ1GHGQ1  M  )  + 2|Et1 Tr RQ1GHGQ1  M  − E1|2 ≤ K′′  M 2 ,  (68)  and  E|A1 − A1|4 ≤ K′′′  M 2 ,  (69)  where K′, K′′, and K′′′ are constants which are independent  of M, L, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' By substituting (67)-(68) into (66) and  then (65), we can obtain that  √  ME|e1| = O(M − 1  2 ) so that  P(  √  M|e1| > ε) ≤  √  ME|e1|  ε  M→∞  −−−−→ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  (70)  Similarly, we can show that e2  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  −−−−→  M→∞ 0 and e3  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  −−−−→  M→∞ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  Therefore, we have  √  M(ESNRex,1 − ESNRex,1)  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  −−−−→  M→∞  √  Ma1,3(A1 − A1) +  √  Ma1,4(E1 − E1),  (71)  VE1 = t2  1 Tr(Q1GGHQ1R)2  M  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  −−−−→  M→∞  t2  1E Tr(Q1GGHQ1R)2  M  ,  (72)  where the last step follows from Var(Tr(Q1GGHQ1R)2) =  O(1), which can be obtained by the Nash-Poincar´e Inequality  in [20], [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  E Tr(Q1GGHQ1R)2  M  can be evaluated by the  integration by parts formula [20], [42],  E Tr Q1RQ1GGHQ1RQ1GGH  M  = 1  M  �  i,j  EY ∗  i,jd  1  2  j [C  1  2 Q1RQ1GGHQRQ1G]i,j  = (E Tr CQ1RQ1)2 Tr D2  M 3  + (Tr D)2E Tr CQ1RQ1CQ1RQ1  M 3  + O(M −2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  (73)  By the evaluations in Lemma 3, we have  VE1  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  −−−−→  M→∞ t2  1[θd2γ1(C)  ∆1  + θ2d  2χ1(C, C)],  (74)  VA1,E1 = t2  1 Tr Q1GGHQ1RQ1R  M  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  −−−−→  M→∞ t2  1θdΓ1(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' (75)  Therefore, we can obtain the asymptotic variance of ESNRex,1  as  W1 = a2  1,3VA1 + a2  1,4VE1 + 2a1,3a1,4VA1,E1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  (76)  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' The Asymptotic Gaussianity of ESNRin,1  Similar to the manipulation of SINR1 and ESNRex,1,  ESNRin,1 can be approximated by  √  M(ESNRin,1 − ESNRin,1)  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  −−−−→  M→∞  √  Mp1  C  [−2F 1(A1 − A1)  (1 + A1)3  + F1 − F 1  (1 + A1)2 ]  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  −−−−→  M→∞  √  M[a1,5(A1 − A1) + a1,6(F1 − F 1)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  (77)  The asymptotic variance of F1 and the covariance between F1  and A1 can be obtained by  VF1 = t2  1 Tr(Q1H1HH  1 Q1R)2  M  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  −−−−→  M→∞ t2  1Π1(IN−1),  VA1,F1 = t2  1 Tr RQ1RQ1H1P1HH  1 Q1  M  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  −−−−→  M→∞ t2  1κ1(IN−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  Therefore, we can obtain the asymptotic variance of ESNRin,1  as  Z1 = a2  1,5VA1 + a2  1,6VF1 + 2a1,5a1,6VF1,F1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  11  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' The Evaluation of Covariances  We will first give the closed-form expression for the asymp-  totic covariances between SINR1, ESNRin,1, and ESNRex,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  By (58), (71) and (88) in Lemma (3), we can obtain  MESINR1ESNRex,1  M→∞  −−−−→ E(a1a3VA1+  a2a3VA1,B1 + a1a4VA1,E1 + a3a3VB1,E1),  (78)  EVB1,E1 = E Tr Q1H1P1HH  1 Q1RQ1GGHQ1R  M  = θdE Tr Q1H1P1HH  1 Q1RQ1CQ1R  M  = θdβ1(P1, C) + O(M −2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  (79)  The evaluations of VA1, VA1,B1, and VA1,E1 can be found  in (59), (61), and (75), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' The covariances of the  pairs (SINR1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' ESNRin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='1) and (ESNRex,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' ESNRin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='1) can be  given by  MESINR1ESNRin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='1  M→∞  −−−−→ E(a1a5VA1 + a2a5VA1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='B1  + a1a4VA1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='G1 + a3a3VB1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='G1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  MEESNRex,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='1ESNRin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='1  M→∞  −−−−→ E(a3a5VA1+  a4a5VA1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='E1 + a3a6VA1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='G1 + a4a6VE1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='G1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  where  EVB1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='G1 = t2  1E Tr Q1H1P1HH  1 Q1RQ1H1HH  1 Q1R  M  = t2  1E Tr Q1H1P1HH  1 Q1RQ1R  M  − t2  1zE Tr Q1H1P1HH  1 Q1RQ2  1R  M  = t2  1[κ1(P1) − zβ1(IM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' P1)] + O(M −2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  (80)  and  EVE1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='G1 = t2  1E Tr Q1H1HH  1 Q1RQ1GGHQ1R  M  = t2  1θdE Tr Q1H1HH  1 Q1RQ1CQ1R  M  = t2  1θdβ1(IN−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' C) + O(M −2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  (81)  By far, we have shown that SINR1, ESNRex,1, and ESNRin,1  converge to a Gaussian distribution when M, K, L go to  infinity with the same pace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' Next, we need to investigate  the covariance between SINR and ESNR and the covariance  between different users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' It has been proved in [43, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='9)-  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='11)] that  √  MAi and  √  MAj are asymptotically indepen-  dent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' By a similar approach, we can also prove the asymptotic  independence between  √  MBi,  √  MBj,  √  MEi,  √  MEj, and  √  MGi,  √  MGj when i ̸= j so that the asymptotic covari-  ances are all zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  APPENDIX B  USEFUL RESULTS  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' (CLT for quadratic forms) Given assumptions A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='1,  Hk, G, and P, there holds true that  √  M(Ak − Ak)  d  −−−−→  M→∞ N(0, VAk),  (82)  where VAk = t2  k Tr QkRQkR  M  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' Similarly, we also have  √  M(Bk − Bk)  d  −−−−→  M→∞ N(0, VBk),  √  M(Ek − Ek)  d  −−−−→  M→∞ N(0, VEk),  (83)  where  VBk  =  t2  k Tr RQkHkPkHH  k Qk  M  and  VEk  =  t2  k Tr RQkGGHQk  M  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' The first CLT is the separable case in [21] when the  resolvent matrix is given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' The other two CLTs can be proved  by the same approach and the proof is omitted here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' (Computation results about the high order re-  solvents) Given assumptions A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='1-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='3 and any deterministic  matrices C, P with bounded norm, the following evaluations  hold true  E Tr QRQC  M  = γ(C)  ∆  + O(M −2),  (84)  E Tr QRQRQC  M  = Γ(C) + O(M −2),  (85)  E Tr RQHPHHQRQ  M  = κ(P) + O(M −2),  (86)  E Tr QCQRQCQR  M  = χ(C, C) + O(M −2),  (87)  E Tr QHPHHQRQCQR  M  = β(P, C) + O(M −2),  (88)  E Tr(QRQHPHH)2  M  = Π(P) + O(M −2),  (89)  Γ(C) = η(C) − (η(C)γ − ηγ(C))�γ − �ηγ2γ(C)  ∆3  ,  (93)  κ(P) = �γ(P)(η − �ηγ3)  ∆3  − γ2�η(P)  ∆2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  (94)  where χ(C, C), β(P, C), and Π(P) are given in (90) to (92)  at the top of the next page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' γ(C) and ∆ are given in Table (I)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' The proof of Lemma 3 is given in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  The evaluation of the high order resolvent is considered  in [22], which is used to set up a CLT for the signal-to-  noise ratio (SNR) of minimum variance distortionless response  (MVDR) filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' (85) is equivalent to [22, Proposition 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  However, more complex forms for the fourth order resolvents  related to multiple system parameters, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' C, P, need to be  evaluated while in [22], only one undetermined parameter Θ  is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' Lemma (3) is more general than those in [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  If we take the first C to be R, (87) is equivalent to the result  in [22, Proposition 4] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  12  χ(C, C) = ζ(C, C)  ∆2  + 2ζ(C)γ(C)�γ  ∆3  + 2η(C)2�γ  ∆3  + [4�γ2γ(C)η − 3�ηγγ(C) − �η�γγ2γ(C)]η(C)  ∆4  + γ(C)2(ζ�γ2 + �ζγ2 − η�η)  ∆4  + γ(C)2(2η2�γ3 + 2�η2γ3 − 3η�ηγ�γ − η�ηγ2�γ2)  ∆5  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  (90)  Π(P) = γ2(�η(P2) − δ�ζ(P2))  ∆2  + 2γ3�η(P)2 + 2γ3�ζ(P)�γ(P)  ∆3  + (ζ + γ4�ζ)�γ(P)2 − 4(η − γ3�η)γ�η(P)�γ(P)  ∆4  + 2�γ(P)2(�γη2 + γ5�η2 − 2γ2η�η)  ∆5  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  (91)  β(P, C) = 2�γ(P)(�γη − γ2�η)η(C) + (�γζ + γ3�ζ)γ(C)�γ(P) − 2�η(P)γ[η(C) − (η(C)γ − ηγ(C))�γ − �ηγ2γ(C)]  ∆4  + ζ(C)�γ(P) + γ2γ(C)�ζ(P)  ∆3  + 2[�γ2η2 + γ4�η2 − γ(1 + γ�γ)η�η]γ(C)�γ(P)  ∆5  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  (92)  APPENDIX C  PROOF OF LEMMA 3  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' The main idea is to evaluate the high-order resolvents  based on the lower-order results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' As a result, we will perform  the computation from low order to high order iteratively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  We first turn to the evaluation of the second-order resolvent  in (84).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' For any deterministic matrices A, B, C, and D with  bounded norm, we have  E Tr CQHAHHDQ  M  = Tr ATGT  M  E Tr CQRDQ  M  − E Tr HATGT HHDQ  M  E Tr CQRQ  M  + O(M −2)  = Tr ATGT  M  Tr CGRRDGR  M  + 1  ∆(Tr ATGT  M  �γγ(C)  × γ(RD) − Tr AT2G2  T Tr DRGRγ(C)  M 2  ) + O(M −2),  (95)  by integration by parts formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' Then we turn to evaluate  the third-order resolvent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' By the resolvent identity and the  integration by parts formula, we have  zE[QCQDQ]i,j = E[QCQD]i,j − E[QCQDQHHH]i,j  = E{[QCQD]i,j − �δ[QCQDQR]i,j  + Tr RQCQDQ  M  [QHTGT HH]i,j  + Tr RQDQ  M  [QCQHTGT HH]i,j}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  (96)  By solving E[QCQDQ]i,j in (96), taking the trace operation,  and using variance control in [20], [22], we can obtain  E Tr QRQCQD  M  = 1  ∆(E Tr QCQDRGR  M  + E Tr QDQR  M  E Tr QCQHTGT HHRGR  M  ) + O(M −2)  = 1  ∆[η(C, D) + 1  ∆(γ(C)η(D)�γ + γ(D)�γη(C))  + 1  ∆2 (γ(D)�γ2γ(C)η − �ηγγ(C))] + O(M −2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  (97)  By letting D = R in (97), we can obtain (85).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' Then, by the  integration by parts formula [20], we can obtain the evaluation  for E Tr RQHAHHDQCQ  M  by plugging (95) into (98) below  E Tr RQHAHHDQCQ  M  = E Tr RQRDQCQ  M 2  × Tr ATGT − E Tr RQCQE Tr TGT HHDQRQHA  M 2  − E Tr RQRQCQE Tr TGT HHDQHA  M 2  + O(M −2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  (98)  By replacing the trace of the third-order resolvents by (97)  and letting D = IM, A = P, C = R in (98), we can  obtain (86).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' The fourth-order resolvent in (87) can be evaluated  by the similar approach, which is given in (99) at the top  of the next page, where step (a) in (99) is obtained by  plugging (97) and (98) into (99) to replace E Tr RQRQCQ  M  and  E Tr QCQRQHTGT HHGRR  M  , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' To obtain (88), we  first follow similar steps as in (96) to obtain the evaluation for  E Tr QRQCQRQR  M  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' Then by replacing the third-order terms in  E Tr QRQCQRQHAHH  M  = −E Tr QRQR  M  E Tr TGT HHQCQRQHA  M  − E Tr QRQCQR  M  E Tr TGT HHQRQHA  M  + E Tr QRQCQRQR  M  Tr ATG2  T  M  + O(M −2),  (100)  by (97) and (98), we can conclude (88).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' The tedious compu-  tation is omitted here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' Then we turn to evaluate (89).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' By the  integration by parts formula, we can obtain  E Tr QRQHAHHQRQHBHH  M  =  − E Tr QRQR  M  E Tr TGT HHQHAHHQRQHB  M  + E Tr QRQRE Tr TGT AHHQRQHB  M 2  − E Tr RQRQHAHHQ  M  E Tr TGT HHQRQHB  M  − E Tr RQRQHAHHQRQ  M  E Tr TGT HHQHB  M  + E Tr RQRQHAHHQRQ  M  Tr TGT B  M  + O(M −2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  (101)  The third-order term  E Tr TGT HHQHAHHQRQHB  M  in (101)  can be obtained by  E Tr QHAHHQRQHBHH  M  =  E Tr QRE Tr TGT AHHQRQHB  M 2  − E Tr RQHAHHQE Tr TGT HHQRQHB  M 2  − E Tr RQHAHHQRQ Tr TGT HHQHB  M 2  + E Tr RQHAHHQRQ Tr TGT B  M 2  + O(M −2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='  (102)  13  E Tr QCQRQCQR  M  = 1  ∆[E Tr RQRQCQ  M  E Tr QCQHTGT HHGRR  M  + E Tr RQCQ  M  E Tr QCQRQHTGT HHGRR  M  ]+O(M −2)  (a)  =  1  ∆4 {γ(C)[ζ(C)∆ + ζ�γγ(C) + ζ(η(C)�γ∆ + �γ2ηγ(C) − γ(C)γ�η) + [η(C) − (η(C)γ − ηγ(C))�γ − �ηγ2γ(C)]]} + O(M −2),  (99)  The evaluation of (102) can be obtained by plugging the  evaluations in (95) and (98) to replace the expectations for the  trace of the lower order resolvents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' With these evaluations, all  the terms in (101) can be obtained so that (89) is concluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
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+page_content=' Shiu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' Chang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' Wu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
+page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
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+page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
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+page_content=' Mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFJT4oBgHgl3EQfWCwj/content/2301.11515v1.pdf'}
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+Uplink Scheduling in Federated Learning: an
+Importance-Aware Approach via Graph
+Representation Learning
+Marco Skocaj1, Pedro Enrique Iturria Rivera2, Roberto Verdone1 and Melike Erol-Kantarci2
+1University of Bologna, DEI & WiLab - National Laboratory for Wireless Communications, CNIT, Italy
+2University of Ottawa, School of Electrical Engineering and Computer Science, Canada
+Abstract—Federated Learning (FL) has emerged as a promis-
+ing framework for distributed training of AI-based services,
+applications, and network procedures in 6G. One of the major
+challenges affecting the performance and efficiency of 6G wireless
+FL systems is the massive scheduling of user devices over
+resource-constrained channels. In this work, we argue that the
+uplink scheduling of FL client devices is a problem with a rich
+relational structure. To address this challenge, we propose a
+novel, energy-efficient, and importance-aware metric for client
+scheduling in FL applications by leveraging Unsupervised Graph
+Representation Learning (UGRL). Our proposed approach intro-
+duces a relational inductive bias in the scheduling process and
+does not require the collection of training feedback information
+from client devices, unlike state-of-the-art importance-aware
+mechanisms. We evaluate our proposed solution against baseline
+scheduling algorithms based on recently proposed metrics in the
+literature. Results show that, when considering scenarios of nodes
+exhibiting spatial relations, our approach can achieve an average
+gain of up to 10% in model accuracy and up to 17 times in energy
+efficiency compared to state-of-the-art importance-aware policies.
+Index Terms—Federated Learning, Graph Representation
+Learning,
+Scheduling,
+Communication-efficient
+FL,
+Energy-
+efficient FL, 6G, Spatial Correlation
+I. INTRODUCTION & MOTIVATION
+Federated
+Learning
+(FL)
+[1]
+recently
+emerged
+as
+a
+new privacy-preserving paradigm for distributed training of
+Machine Learning (ML) algorithms without the need for
+explicit data sharing between users and a centralized com-
+putational unity. This framework is particularly appealing for
+next-generation 6G systems, which are foreseen to support
+ubiquitous Artificial Intelligence (AI) services and AI-native
+design of users’ network procedures [2]. Indeed, users of a
+6G network will naturally benefit from decentralized training
+that bypasses sharing and storing data in a centralized location.
+This will lead to the support of a new kind of AI-related traffic
+over wireless networks: frequent exchange of ML models
+introduces significant communication overhead, which raises a
+series of interesting novel challenges. This paved the way to a
+recent research area on communication-efficient FL for 6G [3].
+Wireless FL is an example of goal-oriented communication
+[4], for which traditional Radio Resource Management (RRM)
+methods are typically inadequate, and customized protocols
+must be developed [5].
+Arguably, one of the major challenges towards scalable and
+efficient 6G wireless FL systems is the massive scheduling
+of user devices, from now on referred to as clients. In fact,
+the central aggregation unity, namely the Parameter Server
+(PS), generally needs to link a vast number of User Equipment
+(UE)s through a resource-constrained spectrum and thus can
+allow only a limited number of UEs to send their trained
+weights via unreliable channels for global aggregation [6]. To
+this end, the concept of data importance, or importance-aware
+communications in FL has taken hold in the recent literature
+[7]–[13]. The main idea is that by prioritizing users with high
+data importance, the distributed ML training is accelerated
+[13]. Because explicit information on clients’ data is infeasible
+due to privacy concerns, state-of-the-art importance-aware
+approaches rely on feedback information from the training of
+local clients’ models. This approach, even though proven to be
+effective, has the major disadvantage of being energy and com-
+putationally inefficient. In fact, feedback-based importance-
+aware methods lead to the training of local models on all
+clients, regardless of the number of scheduled transmissions
+in the next communication round (Fig. 1).
+Fig. 1: Comparison between importance-aware scheduling protocols in FL.
+On the left side (a), classic energy-inefficient approaches require training
+feedback from all network clients every communication round. As the
+number of clients in a network grows, this becomes highly inefficient from
+the computational and energetic point of view. On the right side (b),
+importance-aware scheduling via UGRL requires only scheduled users to
+perform local training.
+In
+this
+work,
+we
+propose
+a
+novel
+energy-efficient,
+importance-aware FL metric based on graph representation
+learning, which leads to effective scheduling of client devices
+without any need for collecting training feedback information.
+arXiv:2301.11903v1  [cs.NI]  27 Jan 2023
+
+PS
+C1C2
+CM
+Global model
+Ops,t
+broadcast
+Local model
+update
+V.Qit
+Training
+feedback
+Scheduling
+Qit
+Local model
+transmissionPS
+C1 C2
+CM
+Global model
+broadcast &
+Ops,t
+scheduling
+Local model
+update
+Local model
+transmissionA. State of the Art & Contributions
+Over the last years, different metrics for FL client
+scheduling have been proposed and discussed. In [14], by
+leveraging the concept of Age of Information (AoI), a metric
+termed age of update (AoU) is introduced, which takes into
+account the staleness of the received parameters. In [10],
+[11], [14], [15] channel conditions experienced by different
+clients are considered during the scheduling decision. In [15],
+a channel prediction algorithm based on Gaussian process
+regression is incorporated into the scheduling process when
+dealing with imperfect channel state information. Authors
+from [11], instead, exploit both diversity in multiuser channels
+and diversity in the importance of the edge devices’ local
+learning updates. Importance, in this case, is measured by the
+local parameter update’s gradient divergence, which must be
+reported to the PS downstream of the training of all client
+devices. A similar training-feedback metric of importance
+is introduced in [10], where the significance of the model
+updates at the devices is captured by the L2-norm of the
+model update.
+Under the frequentist setting, training data constitute a
+fundamental part of the inductive bias of a model. In
+this context, in line with the idea of importance-aware
+communications, knowledge about data distribution among
+devices would suffice for driving proper FL scheduling
+decisions. However, as previously stated, this approach
+is unfeasible in FL systems. Nevertheless, clients exhibit
+relations and correlations in a network setting, especially
+in the context of massive Internet-of-Things (IoT) and
+ultra-dense 6G networks. Here, we argue that the scheduling
+of FL client devices is a problem with a rich relational
+structure and, as a consequence, there is a need to tackle this
+problem effectively by taking node correlations into account.
+Relations among clients, which relate to local data distribution
+too, can be learned and inferred by encompassing network
+geometry and relational representation learning, while at the
+same time preserving users’ privacy. Graphs, generally, are
+a representation that supports arbitrary relational structures,
+and computations over graphs afford a strong relational
+inductive bias [16]. By considering networks of clients as
+graphs, we introduce this bias in the clients’ scheduling
+process by leveraging Unsupervised Graph Representation
+Learning (UGRL). As results show, this effectively makes
+up for the impossibility of selecting users based on their
+data, while at the same time aiming for an energy and
+computationally-efficient scheduling protocol.
+The main contributions of this work are listed hereafter:
+• We consider network geometry in the form of node
+embeddings obtained via UGRL as a fundamental new
+metric for driving efficient FL scheduling decisions in
+the context of non-i.i.d. and spatially correlated data. We
+aim to show it is possible to make up for the absence
+of explicit knowledge information about clients’ data by
+introducing a relational inductive bias into the scheduling
+process.
+• We compare the performance of the proposed scheduling
+metric with respect to baseline metrics recently proposed
+in the literature.
+• We discuss the range of applicability of our proposed so-
+lution with respect to different kinds of data distributions
+with application to IoT and 6G networks.
+II. SYSTEM MODEL
+A. Network Scenario and Propagation Channel
+Let us consider a system comprised of one Access Point
+(AP) co-located with a PS and multiple client devices with
+local data and computation capabilities, as depicted in Fig. 2.
+Physically running on the PS, there are two software entities
+responsible for model aggregation and radio resource man-
+agement: the Federated Aggregator and the Graph Scheduler,
+respectively described in the following subsections.
+Fig. 2: System Model
+Each client m
+∈
+M holds a local data set Dm
+=
+�
+xm ∈ Rd, ym ∈ R
+�
+with
+cardinality
+|Dm|,
+such
+that
+�
+m∈M |Dm| = |D|, and is equipped with a single isotropic
+antenna. On the PS side, we consider an antenna with a
+directive gain of 15 dBi. All clients are randomly uniformly
+distributed within a radius R of the PS. The PS broadcasts
+the global model to the selected clients with a transmit power
+of 15 dBm, while the latter send their local updates with a
+transmit power of 10 dBm.
+As further detailed in the next subsections, we consider the
+two cases of non-i.i.d. spatially correlated data and spatially
+correlated/uncorrelated non-i.i.d clusters of i.i.d. data. Both
+cases are artificially reproduced in our experiments by spatially
+distributing MNIST digit labels among neighbor clients.
+Within the network area, we consider an Orthogonal Fre-
+quency Division Multiple Access (OFDMA) scheme with
+perfect equalization, where M = |M| clients share the same
+spectrum and can be assigned one of the K < M set of
+orthogonal sub-channels for model parameters transmission.
+Moreover, we assume a slow fading propagation model, where
+each model transmission from device m ∈ M to the PS is
+shorter than the channel coherence time Tc:
+TD,m < Tc
+for m ∈ M,
+(1)
+where TD,m is the model transmission time for client m. With
+the aforementioned assumptions, the channel impulse response
+hm,P S(f, t) from device m to the PS loses its time and
+
+ Graph Scheduler
+Nt()
+Ut+1
+Federated
+目
+Aggregator
+fM
+DM
+口
+目
+f1
+口
+D1
+f M-1
+目
+口
+DM-1
+f2
+目
+D2
+目frequency dependency within a block transmission duration
+TD,m (2):
+hm,P S(f, t) → hm,P S ∈ HM+1,
+(2)
+where HM+1 is the channel matrix of dimension M + 1.
+Finally, we consider the Okumura-Hata model for the median
+path loss, and the Nakagami-m distribution for the fading
+propagation model, as it provides a flexible formulation to
+characterize Rician and Rayleigh fading.
+B. Federated Learning Framework
+The goal of the Federated Aggregator is to learn a ML
+model by offloading and aggregating the training to the set
+M of distributed clients with local data. The federated training
+process involves a number of iterations, namely communica-
+tion rounds, until convergence. Each client m ∈ M, upon
+receiving a global model θP S from the PS at the beginning of
+a new round, executes multiple Stochastic Gradient Descent
+(SGD) updates to minimize the model’s loss function with
+respect to its local dataset (3):
+Fm(θ, Dm) =
+1
+|Dm|
+�
+{xi,yi}∈Dm
+f(θ, {xi, yi}),
+(3)
+where Fm is the m-th client loss function and f(θ, {xi, yi})
+indicates the task-dependent loss (e.g., Mean Square Error
+(MSE), categorical cross-entropy, etc.) for every training ex-
+ample {xi, yi}. At every j-th local SGD iteration, each client
+m updates its local model according to (4):
+θj+1
+m (t) = θj
+P S(t) − α(t) · gm,
+(4)
+where α(t) denotes the learning rate scheduled by the PS for
+communication round t and gm := ∇(Fm(θm, Dm) is the
+m-th client’s gradient of the local model’s weights. Once the
+training is terminated, each client selected for scheduling must
+forward its local model θm(t) to the PS, which will update
+the global model upon aggregation of all received clients’
+models. Here, we refer to FedAvg algorithm [1], for which
+the aggregation is a weighted sum described by (5):
+θP S(t + 1) = 1
+K
+�
+m∈K
+|Dm|
+�
+m∈K |Dm| · θm(t),
+(5)
+where we denote by K (|K| = K) the set of scheduled
+clients for communication. For model evaluation, we consider
+a separate centralized test set Dtest,P S locally residing on the
+PS.
+C. Graph Scheduler
+The Graph Scheduler is the entity responsible for the
+procedures depicted in Fig. 3: dynamic graph creation, UGRL,
+and distance maximization.
+• Sensing phase and reporting: a sensing phase is per-
+formed by all client devices, and it should be repeated
+with a periodicity that depends on environment dynam-
+icity. Following, all nodes v report a list of sensed
+neighbors Nt(v) (Fig. 2) to the graph scheduler.
+• Dynamic graph formation: the scheduler builds a dynamic
+graph from each node’s list of sensed neighbors. In
+Fig. 3: Graph scheduler block scheme. The scheduling sequence Ut+1 is
+computed downstream Distance Maximization scheduler and the UGRL
+block. The UGRL block is responsible for the transformation f from a
+graph space G|M| to the vectorial space RT x|M| of node embeddings.
+our approach, we retain the strongest-K neighbors, as it
+allows to adapt to the density of nodes’ deployment. Note
+that alternative approaches, such as retaining adjacencies
+based on a received power threshold, might also be used.
+• UGRL: graph representation learning involves the trans-
+formation via an encoding function f
+: (G|M|) →
+RT x|M| from the graph-structured node representation
+G|M|, to a vectorial space of T-dimensional node embed-
+dings. In this procedure, node embeddings from G|M|
+are efficiently computed in an unsupervised way with
+the use of random walks procedures. In our scenario, we
+make use of Node2Vec algorithm [17]. Further details are
+discussed in section IV.
+• Distance maximization: the final step involves the com-
+putation of the scheduling sequence Ut+1. This is based
+on the distance maximization between node embeddings
+retained in a context window of tunable dimension.
+Additional details are provided in section IV.
+D. Data Distribution
+The intuition behind the proposed approach is that the
+clients’ data distribution reflects the structural relation of nodes
+in a graph. Consequently, this method does not apply to
+the trivial case of i.i.d. data. Vice versa, it is possible to
+think of a plethora of applications and AI-driven network
+procedures foreseen for 6G in which this condition holds:
+localization, tracking, integrated sensing and communication,
+channel estimation and measures of a physical quantity from
+a sensors network is just a non-exhaustive list of examples
+where clients’ data are non-i.i.d. and spatially correlated.
+Another vertical of great importance for future 6G networks
+is Industrial Internet-of-Things (IIoT), where typically nodes
+are arranged into clusters, and nodes within a cluster might
+hold similar kinds of measurements (e.g., monitoring sensors
+inside automatic machines in a warehouse). To this end, we
+consider in this work typical kinds of client data distributions
+that find use in many real-world 6G applications:
+• Non-i.i.d. and spatially correlated data distribution (Fig.
+4a).
+• Spatially correlated/uncorrelated clusters of i.i.d. data
+(Fig. 4b, 4c).
+For benchmarking purposes, we reproduced the three sce-
+narios described above in our simulations by distributing
+MNIST data arranged by labels to a set of randomly distributed
+clients. According to the considered scenario, clients are
+distributed, at the beginning of every new simulation, a random
+
+Dynamic
+Sensing phase
+Distance
+→Ut+1
+graph
+UGRL
+& reporting
+maximization
+formation
+f : (GIMI) → RTa|MI(a)
+(b)
+(c)
+Fig. 4: Typical kinds of clients data distribution in AI-native network
+procedures and IoT: (a) Non-i.i.d, spatially correlated data distribution, (b)
+Non-i.i.d, spatially uncorrelated clusters of i.i.d. data and (c) Non-i.i.d,
+spatially correlated clusters of i.i.d. data
+number of examples according to: a) their respective position
+to the PS (for the case depicted in Fig. 4a), or b) their
+belonging cluster (for the case depicted in Fig. 4b, 4c).
+III. PROBLEM FORMULATION
+The observation space can be represented as a graph com-
+posed of nodes (client devices), edges (adjacency matrix),
+and node features F (FL metrics). In the most general for-
+mulation, each edge can also be associated with a weight
+|hi,j|, corresponding to the module of the complex channel
+impulse response hi,j between client i and j, drawn from
+a (M + 1)-dimensional channel matrix HM+1. Nevertheless,
+considering the case of orthogonal resources assignment (i.e.,
+no inter-users interference), and assuming link reciprocity,
+allows for a simplification of the problem formulation, since
+matrix HM+1 reduces to an M-dimensional vector H =
+[|h1,P S|, . . . , |hM,P S|]. Hence, its elements can be represented
+as node features instead of edges.
+St =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+F =
+�
+(F1, |h1,P S|), ...,
+(FM, |hM,P S|)
+�
+,
+with |hi,P S| ∈ R
+A =
+�
+���
+a11
+. . .
+a1M
+...
+...
+...
+aM1
+. . .
+aMM
+�
+���
+M
+,
+with aij ∈ {0, 1}
+(6)
+In (6), F is the feature space of every node, which includes the
+FL-metrics F used for decision-making during the scheduling
+procedure, and A is the adjacency matrix, which is obtained
+downstream of the dynamic graph formation block of Fig. 3.
+The feature space is composed of the following metrics:
+• Age of Information (AoI): a scalar indicator, introduced
+in [14], describing the number of rounds that elapsed
+since the client was last scheduled for model transmis-
+sion.
+• Path loss |hi,j|: the path loss value in dB related to the
+client-PS link, assuming link reciprocity.
+• L2-norm of model update ||θi,t+1 − θP S,t||2: an impor-
+tance metric indicating the L2 norm of the i-th client
+model update.
+Notably, no explicit informative content can be collected about
+the data of the clients, as this would violate the privacy-
+preserving nature of FL.
+IV. PROPOSED ALGORITHM
+In the formulation above, nodes connected by edges have
+similar data, but not necessarily similar FL metric features. In
+fact, AoI, for instance, does not show any spatial correlation
+property among nodes, as it purely depends on the schedul-
+ing mechanism. In turn, this, together with the inability to
+have explicit features about nodes’ data, depicts a situation
+where client features don’t necessarily reflect the structure of
+the graph. Moreover, the nature of the problem is naturally
+unsupervised, as nodes don’t have any label. To this end, we
+make use of UGRL via random walks in place of common
+supervised methods with Graph Neural Network (GNN)s to
+incorporate information about the structure of the graph into
+the decision-making process. By employing Node2Vec [17],
+we are able to compute the graph encoding function f(G)
+without the assistance of any node labels and features, but
+rather by maximizing the log-likelihood of the 2nd order
+biased random walks Ns(u) conditioned by f(u), for u ∈ G
+as per (7) [17].
+arg max
+f
+�
+u∈G(V )
+log (Prob(Ns(u)|f(u))
+(7)
+A. Distance Maximization via UGRL
+Once f(G) is obtained, clients’ scheduling relies on the
+distance maximization of the nodes in the embedding space.
+This has the effect of increasing data heterogeneity of the
+client devices during consecutive communication rounds. The
+distance among a pair of nodes (v, u) in the graph G is
+evaluated as the normalized dot product (cosine similarity)
+s(v, u) of their node embeddings zv, zu (8):
+s(v, u) =
+zT
+v · zu
+∥zv∥ · ∥zu∥.
+(8)
+To introduce memory of the past actions in the scheduling
+process, nodes scheduled in previous communication rounds
+are stored in a context window WL of tunable length L =
+max(|WL|). Accordingly, the similarity scores of the nodes
+v ∈ WL are computed and collected in a matrix Sv,u of
+dimension |WL| × (M − |WL|):
+Sv,u =
+�
+��
+s(v1, uL+1)
+. . .
+s(v1, uM)
+...
+...
+...
+s(vL, uL+1)
+. . .
+s(vL, uM)
+�
+�� .
+(9)
+A scheduling decision is finally determined by equation (10).
+arg
+min
+u∈G\{v∈WL}
+�
+v∈WL
+s(v, u).
+(10)
+The latter is equivalent to summing all elements of the matrix
+Sv,u by column, and selecting the next scheduled client as the
+argument corresponding to the column holding the minimum
+sum value, i.e., the node with maximum distance with respect
+to all previously scheduled nodes in WL.
+V. SIMULATION METHODOLOGY
+Experiments and evaluation were conducted on a simulator
+based on Tensorflow Federated Core API. The logical steps
+of the designed FL framework, namely ”FederatedEnv”, are
+reported in Algorithm 1.
+
+0
+1hAfter the initialization of clients’ positions (rm, φm)M and
+datasets Dm, the algorithm loops over a fixed number of
+communication rounds. Each round can be subdivided into
+7 logical steps:
+1) The module of the client-PS Downlink (DL) channel
+impulse response |hP S,m| is computed, assuming perfect
+CSI, by summing fading and shadowing contributions
+(f ∼ F, s ∼ N) to the median path loss PL. The signal-
+to-noise ratio γ is thereby computed with a noise floor
+NdB of -115 [dBm] and an AP gain Gtx of 15 [dBi].
+2) The PS model θP S,t is broadcasted to the clients. At
+the receiver side, Gaussian noise with standard deviation
+σDL =
+�
+E[N2] =
+�
+E[θ2]/γDL,m is added to the
+PS’s model weights, where θ is the discrete signal of
+weights of every model’s layer.
+3) Each client performs a local update of its model weights,
+controlled by the learning rate α(t), using stochastic
+gradient descent for Nep epochs, as per (4).
+4) After local model updates, a new round of Uplink (UL)
+channel estimation is performed in the same way as for
+step 1, and the corresponding UL signal-to-noise ratio
+γUL is computed.
+5) A binary mask of the scheduled users U = [u1, . . . , uM]
+is applied to the vector of updated model weights
+Θ′
+M = [θ′
+1, . . . , θ′
+M]T . During the aggregation phase,
+only models belonging to scheduled clients will be
+retained and included in the aggregation process.
+6) The noisy models from the scheduled clients are aggre-
+gated as per (5).
+7) The model is evaluated on a test set. We denote by
+f(Dtest|θP S(t + 1)) any metric computed over Dtest.
+Algorithm 1 FederatedEnv: logical steps
+for each episode do
+for m ∈ M do
+(rm, φm) = (rm ∼ U(0, R), φm ∼ U(0, 2π))
+|Dm| = |D| · (um ∼ U(0, 1)/
+�
+um)
+Dm = sample data(D, |Dm|, φm)
+end for
+for each round t in T and for every m ∈ M do
+Step 1: DL Channel State Estimation
+|hP S,m|dB = −PLm + s ∼ N(0, σs) + f ∼ F(k, ω)
+γDL,m = Ptx,DL + Gtx + |hP S,m|dB − NdB
+Step 2: Model Broadcast
+θm(t) ← θP S(t) + n ∼ N(0, σ∝γDL,m)
+Step 3: Client Update
+θj+1
+m
+(t) = θj
+m(t) − α(t) · gm
+Step 4: UL Channel State Estimation
+|hm,P S|dB = −PLm + s ∼ N(0, σs) + f ∼ F(k, ω)
+γUL,m = Ptx,UL + Gtx + |hm,P S|dB − NdB
+Step 5: Client Scheduling
+Θ′
+M = Θ′
+M · UT = [θ′
+1, . . . , θ′
+M]T · [u1, . . . , uM]
+Step 6: Model update
+θP S(t + 1) =
+1
+K
+�
+m∈K
+|Dm|
+�
+m∈K |Dm| · θm(t) + nm
+Step 7: Performance Evaluation
+f(Dtest|θP S(t + 1))
+end for
+end for
+VI. RESULTS
+In this section, we present and comment on the obtained
+results. We consider a scenario where M = 50 clients are
+randomly distributed among equally populated clusters (Fig.
+4b, 4c). The distance maximization scheduling was tested
+against baseline policies (Table I) based on the feature nodes
+metrics introduced in section III. All policies have been tested
+on MNIST classification when transmitting a shallow neural
+network with 1 hidden layer, achieving 0.91 accuracy in a
+centralized setting.
+POLICY
+DESCRIPTION.
+Max Age of Information (AoI)
+nodes with max AoI
+Random (RND)
+nodes chosen randomly
+Round robin (RR)
+nodes chosen in a round-trip fashion
+Best Channel (BC)
+nodes with the best channel condition
+Oracle (OCL)
+explicit data knowledge information -
+maximize label heterogeneity
+Max L2-norm (L2N)
+nodes with maximum L2-norm of
+their local model update
+Distance maximization (DM)
+our proposed scheduling policy
+TABLE I: Baseline policies
+Fig. 5 and Fig. 6 show performance comparison in terms of
+training accuracy and energy (Fig. 6a) vs. communication effi-
+ciency (Fig. 6b), respectively. The three metrics are evaluated
+as follows:
+• Accuracy: the sparse categorical cross accuracy on a
+centralized test set.
+• Communication efficiency: the number of communication
+rounds r to achieve an accuracy of 0.8.
+• Energy efficiency: the number of rounds r, multiplied
+by the number of training devices per round (K, or M,
+depending on the policy).
+In all three metrics, the proposed solution outperforms the
+baselines and approaches the performance of the oracle, which
+is an empirical upper bound. In particular, the performance
+gap between the proposed solutions increases as the number
+of schedulable users (K) decreases. It is of particular interest
+to compare our proposed DM policy with respect to the L2N
+importance-aware policy. Since the former does not require
+any form of training feedback (Fig. 1), it is much more
+energy efficient, as the number of clients training per round
+is reduced from M to K (i.e., only the scheduled users
+perform local training). Moreover, results show that, in the
+proposed scenario, DM outperforms L2N even in terms of
+model accuracy and communication efficiency (Fig. 5 and 6b).
+In fact, even though the L2N policy is successful in scheduling
+the users holding the most significant models every round
+(i.e., those contributing more significantly to the global model,
+according to the L2-norm of the local updates [10]), when
+nodes show spatial relation, this may result in the selection of
+nearby clients in space holding similar data. On the opposite,
+our proposed policy aims to achieve data heterogeneity by
+maximizing nodes’ distance in the graph space, making it
+more suitable for the considered scenario. For the same reason,
+policies such as AoI and random scheduling achieve better
+performance than round-robin, since they inherently increase
+
+0
+5 10 15 20 25 30 35 40 45 50 55 60 65 70 75
+Communication rounds
+0.2
+0.4
+0.6
+0.8
+Number of scheduled clients K=5
+AoI
+DM
+RND
+RR
+Oracle
+BC
+L2N
+0
+5 10 15 20 25 30 35 40 45 50 55 60 65 70 75
+Communication rounds
+0.2
+0.4
+0.6
+0.8
+Number of scheduled clients K=10
+AoI
+DM
+RND
+RR
+Oracle
+BC
+L2N
+0
+5 10 15 20 25 30 35 40 45 50 55 60 65 70 75
+Communication rounds
+0.2
+0.4
+0.6
+0.8
+Number of scheduled clients K=25
+AoI
+DM
+RND
+RR
+Oracle
+BC
+L2N
+Fig. 5: Training results: sparse categorical cross accuracy vs number of communication rounds for different number of schedulable users per round: (a)
+K = 5, M = 50, (b) K = 10, M = 50 and K = 25, M = 50
+data heterogeneity among the clients scheduled every round.
+Results show that DM can achieve an average 10% gain in
+model accuracy with respect to L2N at the end of the training
+while increasing energy efficiency by 17 times for K = 5.
+Moreover, we register an average gain of 4% accuracy at the
+end of the training and 31% in communication and energy
+efficiency with respect to the second-best performing policy
+(AoI). Finally, it is interesting to notice how the choice of
+K generates a tradeoff between communication and energy
+efficiency. Indeed, for our considered model and simulation
+parameters, if the aim of the designer is to maximize the
+system’s communication efficiency, then K = 10 yields a gain
+of 27, 9% in communication efficiency, but a loss of 30, 7%
+in energy efficiency with respect to K = 5. Therefore, this
+is an indicator that for energy-sensitive applications, like IoT,
+the maximization of the number of schedulable clients is not
+always the best design choice.
+AoI DM OCLRND RR BC L2N
+200
+300
+400
+500
+700
+3000
+4000
+Energy efficiency 
+ (num trainings on devices)
+Number of scheduled clients K=5
+AoI DM OCLRND RR BC L2N
+Number of scheduled clients K=10
+(a) Energy efficiency
+AoI DM OCLRND RR BC L2N
+0
+10
+20
+30
+40
+50
+60
+70
+Communication efficiency 
+ (number of rounds)
+Number of scheduled clients K=5
+AoI DM OCLRND RR BC L2N
+Number of scheduled clients K=10
+(b) Communication efficiency
+Fig. 6: Training results: energy (a) vs communication efficiency (b).
+VII. CONCLUSION
+In this study, we present a novel metric for the scheduling of
+client devices in FL applications leveraging the use of UGRL.
+With respect to state-of-the-art importance-aware scheduling
+methods, our solution does not require any training feedback
+from client devices. Hence, it provides a much more com-
+putationally and energy-efficient solution. Our results indicate
+that, when tested against baseline importance-aware policies,
+our solution achieves a gain of up to 10% in model accuracy,
+while requiring up to 17 times fewer local training phases on
+client devices.
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+Joint client scheduling and resource allocation,” in Proc. IEEE Wireless
+Commun. and Networking Conf. (WCNC), 2020, pp. 1–6.
+[16] P.W. Battaglia et al., “Relational inductive biases, deep learning, and
+graph networks,” arXiv preprint arXiv:1806.01261, 2018.
+[17] A. Grover et al., “node2vec: Scalable feature learning for networks,”
+Proc. of the 22nd ACM SIGKDD Int. Conf. on Knowledge Discovery
+and Data Mining, 2016.
+
diff --git a/kdFKT4oBgHgl3EQfxi4K/content/tmp_files/load_file.txt b/kdFKT4oBgHgl3EQfxi4K/content/tmp_files/load_file.txt
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf,len=450
+page_content='Uplink Scheduling in Federated Learning: an  Importance-Aware Approach via Graph  Representation Learning  Marco Skocaj1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' Pedro Enrique Iturria Rivera2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' Roberto Verdone1 and Melike Erol-Kantarci2  1University of Bologna,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' DEI & WiLab - National Laboratory for Wireless Communications,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' CNIT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' Italy  2University of Ottawa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' School of Electrical Engineering and Computer Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' Canada  Abstract—Federated Learning (FL) has emerged as a promis-  ing framework for distributed training of AI-based services,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='  applications,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' and network procedures in 6G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' One of the major  challenges affecting the performance and efficiency of 6G wireless  FL systems is the massive scheduling of user devices over  resource-constrained channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' In this work, we argue that the  uplink scheduling of FL client devices is a problem with a rich  relational structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' To address this challenge, we propose a  novel, energy-efficient, and importance-aware metric for client  scheduling in FL applications by leveraging Unsupervised Graph  Representation Learning (UGRL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' Our proposed approach intro-  duces a relational inductive bias in the scheduling process and  does not require the collection of training feedback information  from client devices, unlike state-of-the-art importance-aware  mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' We evaluate our proposed solution against baseline  scheduling algorithms based on recently proposed metrics in the  literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' Results show that, when considering scenarios of nodes  exhibiting spatial relations, our approach can achieve an average  gain of up to 10% in model accuracy and up to 17 times in energy  efficiency compared to state-of-the-art importance-aware policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='  Index Terms—Federated Learning, Graph Representation  Learning,  Scheduling,  Communication-efficient  FL,  Energy-  efficient FL, 6G, Spatial Correlation  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' INTRODUCTION & MOTIVATION  Federated  Learning  (FL)  [1]  recently  emerged  as  a  new privacy-preserving paradigm for distributed training of  Machine Learning (ML) algorithms without the need for  explicit data sharing between users and a centralized com-  putational unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' This framework is particularly appealing for  next-generation 6G systems, which are foreseen to support  ubiquitous Artificial Intelligence (AI) services and AI-native  design of users’ network procedures [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' Indeed, users of a  6G network will naturally benefit from decentralized training  that bypasses sharing and storing data in a centralized location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='  This will lead to the support of a new kind of AI-related traffic  over wireless networks: frequent exchange of ML models  introduces significant communication overhead, which raises a  series of interesting novel challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' This paved the way to a  recent research area on communication-efficient FL for 6G [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='  Wireless FL is an example of goal-oriented communication  [4], for which traditional Radio Resource Management (RRM)  methods are typically inadequate, and customized protocols  must be developed [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='  Arguably, one of the major challenges towards scalable and  efficient 6G wireless FL systems is the massive scheduling  of user devices, from now on referred to as clients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' In fact,  the central aggregation unity, namely the Parameter Server  (PS), generally needs to link a vast number of User Equipment  (UE)s through a resource-constrained spectrum and thus can  allow only a limited number of UEs to send their trained  weights via unreliable channels for global aggregation [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' To  this end, the concept of data importance, or importance-aware  communications in FL has taken hold in the recent literature  [7]–[13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' The main idea is that by prioritizing users with high  data importance, the distributed ML training is accelerated  [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' Because explicit information on clients’ data is infeasible  due to privacy concerns, state-of-the-art importance-aware  approaches rely on feedback information from the training of  local clients’ models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' This approach, even though proven to be  effective, has the major disadvantage of being energy and com-  putationally inefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' In fact, feedback-based importance-  aware methods lead to the training of local models on all  clients, regardless of the number of scheduled transmissions  in the next communication round (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' 1: Comparison between importance-aware scheduling protocols in FL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='  On the left side (a), classic energy-inefficient approaches require training  feedback from all network clients every communication round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' As the  number of clients in a network grows, this becomes highly inefficient from  the computational and energetic point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' On the right side (b),  importance-aware scheduling via UGRL requires only scheduled users to  perform local training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='  In  this  work,  we  propose  a  novel  energy-efficient,  importance-aware FL metric based on graph representation  learning, which leads to effective scheduling of client devices  without any need for collecting training feedback information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='11903v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='NI]  27 Jan 2023  PS  C1C2  CM  Global model  Ops,t  broadcast  Local model  update  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='Qit  Training  feedback  Scheduling  Qit  Local model  transmissionPS  C1 C2  CM  Global model  broadcast &  Ops,t  scheduling  Local model  update  Local model  transmissionA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' State of the Art & Contributions  Over the last years, different metrics for FL client  scheduling have been proposed and discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' In [14], by  leveraging the concept of Age of Information (AoI), a metric  termed age of update (AoU) is introduced, which takes into  account the staleness of the received parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' In [10],  [11], [14], [15] channel conditions experienced by different  clients are considered during the scheduling decision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' In [15],  a channel prediction algorithm based on Gaussian process  regression is incorporated into the scheduling process when  dealing with imperfect channel state information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' Authors  from [11], instead, exploit both diversity in multiuser channels  and diversity in the importance of the edge devices’ local  learning updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' Importance, in this case, is measured by the  local parameter update’s gradient divergence, which must be  reported to the PS downstream of the training of all client  devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' A similar training-feedback metric of importance  is introduced in [10], where the significance of the model  updates at the devices is captured by the L2-norm of the  model update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='  Under the frequentist setting, training data constitute a  fundamental part of the inductive bias of a model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' In  this context, in line with the idea of importance-aware  communications, knowledge about data distribution among  devices would suffice for driving proper FL scheduling  decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' However, as previously stated, this approach  is unfeasible in FL systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' Nevertheless, clients exhibit  relations and correlations in a network setting, especially  in the context of massive Internet-of-Things (IoT) and  ultra-dense 6G networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' Here, we argue that the scheduling  of FL client devices is a problem with a rich relational  structure and, as a consequence, there is a need to tackle this  problem effectively by taking node correlations into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='  Relations among clients, which relate to local data distribution  too, can be learned and inferred by encompassing network  geometry and relational representation learning, while at the  same time preserving users’ privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' Graphs, generally, are  a representation that supports arbitrary relational structures,  and computations over graphs afford a strong relational  inductive bias [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' By considering networks of clients as  graphs, we introduce this bias in the clients’ scheduling  process by leveraging Unsupervised Graph Representation  Learning (UGRL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' As results show, this effectively makes  up for the impossibility of selecting users based on their  data, while at the same time aiming for an energy and  computationally-efficient scheduling protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='  The main contributions of this work are listed hereafter:  We consider network geometry in the form of node  embeddings obtained via UGRL as a fundamental new  metric for driving efficient FL scheduling decisions in  the context of non-i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' and spatially correlated data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' We  aim to show it is possible to make up for the absence  of explicit knowledge information about clients’ data by  introducing a relational inductive bias into the scheduling  process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='  We compare the performance of the proposed scheduling  metric with respect to baseline metrics recently proposed  in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='  We discuss the range of applicability of our proposed so-  lution with respect to different kinds of data distributions  with application to IoT and 6G networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' SYSTEM MODEL  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' Network Scenario and Propagation Channel  Let us consider a system comprised of one Access Point  (AP) co-located with a PS and multiple client devices with  local data and computation capabilities, as depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='  Physically running on the PS, there are two software entities  responsible for model aggregation and radio resource man-  agement: the Federated Aggregator and the Graph Scheduler,  respectively described in the following subsections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' 2: System Model  Each client m  ∈  M holds a local data set Dm  =  �  xm ∈ Rd, ym ∈ R  �  with  cardinality  |Dm|,  such  that  �  m∈M |Dm| = |D|, and is equipped with a single isotropic  antenna.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' On the PS side, we consider an antenna with a  directive gain of 15 dBi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' All clients are randomly uniformly  distributed within a radius R of the PS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' The PS broadcasts  the global model to the selected clients with a transmit power  of 15 dBm, while the latter send their local updates with a  transmit power of 10 dBm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='  As further detailed in the next subsections, we consider the  two cases of non-i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' spatially correlated data and spatially  correlated/uncorrelated non-i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='d clusters of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' Both  cases are artificially reproduced in our experiments by spatially  distributing MNIST digit labels among neighbor clients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='  Within the network area, we consider an Orthogonal Fre-  quency Division Multiple Access (OFDMA) scheme with  perfect equalization, where M = |M| clients share the same  spectrum and can be assigned one of the K < M set of  orthogonal sub-channels for model parameters transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='  Moreover, we assume a slow fading propagation model, where  each model transmission from device m ∈ M to the PS is  shorter than the channel coherence time Tc:  TD,m < Tc  for m ∈ M,  (1)  where TD,m is the model transmission time for client m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' With  the aforementioned assumptions, the channel impulse response  hm,P S(f, t) from device m to the PS loses its time and  Graph Scheduler  Nt()  Ut+1  Federated  目  Aggregator  fM  DM  口  目  f1  口  D1  f M-1  目  口  DM-1  f2  目  D2  目frequency dependency within a block transmission duration  TD,m (2):  hm,P S(f, t) → hm,P S ∈ HM+1,  (2)  where HM+1 is the channel matrix of dimension M + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='  Finally, we consider the Okumura-Hata model for the median  path loss, and the Nakagami-m distribution for the fading  propagation model, as it provides a flexible formulation to  characterize Rician and Rayleigh fading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' Federated Learning Framework  The goal of the Federated Aggregator is to learn a ML  model by offloading and aggregating the training to the set  M of distributed clients with local data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' The federated training  process involves a number of iterations, namely communica-  tion rounds, until convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' Each client m ∈ M, upon  receiving a global model θP S from the PS at the beginning of  a new round, executes multiple Stochastic Gradient Descent  (SGD) updates to minimize the model’s loss function with  respect to its local dataset (3):  Fm(θ, Dm) =  1  |Dm|  �  {xi,yi}∈Dm  f(θ, {xi, yi}),  (3)  where Fm is the m-th client loss function and f(θ, {xi, yi})  indicates the task-dependent loss (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=', Mean Square Error  (MSE), categorical cross-entropy, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=') for every training ex-  ample {xi, yi}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' At every j-th local SGD iteration, each client  m updates its local model according to (4):  θj+1  m (t) = θj  P S(t) − α(t) · gm,  (4)  where α(t) denotes the learning rate scheduled by the PS for  communication round t and gm := ∇(Fm(θm, Dm) is the  m-th client’s gradient of the local model’s weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' Once the  training is terminated, each client selected for scheduling must  forward its local model θm(t) to the PS, which will update  the global model upon aggregation of all received clients’  models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' Here, we refer to FedAvg algorithm [1], for which  the aggregation is a weighted sum described by (5):  θP S(t + 1) = 1  K  �  m∈K  |Dm|  �  m∈K |Dm| · θm(t),  (5)  where we denote by K (|K| = K) the set of scheduled  clients for communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' For model evaluation, we consider  a separate centralized test set Dtest,P S locally residing on the  PS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' Graph Scheduler  The Graph Scheduler is the entity responsible for the  procedures depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' 3: dynamic graph creation, UGRL,  and distance maximization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='  Sensing phase and reporting: a sensing phase is per-  formed by all client devices, and it should be repeated  with a periodicity that depends on environment dynam-  icity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' Following, all nodes v report a list of sensed  neighbors Nt(v) (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' 2) to the graph scheduler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='  Dynamic graph formation: the scheduler builds a dynamic  graph from each node’s list of sensed neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' In  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' 3: Graph scheduler block scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' The scheduling sequence Ut+1 is  computed downstream Distance Maximization scheduler and the UGRL  block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' The UGRL block is responsible for the transformation f from a  graph space G|M| to the vectorial space RT x|M| of node embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='  our approach, we retain the strongest-K neighbors, as it  allows to adapt to the density of nodes’ deployment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' Note  that alternative approaches, such as retaining adjacencies  based on a received power threshold, might also be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='  UGRL: graph representation learning involves the trans-  formation via an encoding function f  : (G|M|) →  RT x|M| from the graph-structured node representation  G|M|, to a vectorial space of T-dimensional node embed-  dings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' In this procedure, node embeddings from G|M|  are efficiently computed in an unsupervised way with  the use of random walks procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' In our scenario, we  make use of Node2Vec algorithm [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' Further details are  discussed in section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='  Distance maximization: the final step involves the com-  putation of the scheduling sequence Ut+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' This is based  on the distance maximization between node embeddings  retained in a context window of tunable dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='  Additional details are provided in section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' Data Distribution  The intuition behind the proposed approach is that the  clients’ data distribution reflects the structural relation of nodes  in a graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' Consequently, this method does not apply to  the trivial case of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' Vice versa, it is possible to  think of a plethora of applications and AI-driven network  procedures foreseen for 6G in which this condition holds:  localization, tracking, integrated sensing and communication,  channel estimation and measures of a physical quantity from  a sensors network is just a non-exhaustive list of examples  where clients’ data are non-i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' and spatially correlated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='  Another vertical of great importance for future 6G networks  is Industrial Internet-of-Things (IIoT), where typically nodes  are arranged into clusters, and nodes within a cluster might  hold similar kinds of measurements (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=', monitoring sensors  inside automatic machines in a warehouse).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' To this end, we  consider in this work typical kinds of client data distributions  that find use in many real-world 6G applications:  Non-i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' and spatially correlated data distribution (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='  4a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='  Spatially correlated/uncorrelated clusters of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' data  (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' 4b, 4c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='  For benchmarking purposes, we reproduced the three sce-  narios described above in our simulations by distributing  MNIST data arranged by labels to a set of randomly distributed  clients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' According to the considered scenario, clients are  distributed, at the beginning of every new simulation, a random  Dynamic  Sensing phase  Distance  →Ut+1  graph  UGRL  & reporting  maximization  formation  f : (GIMI) → RTa|MI(a)  (b)  (c)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' 4: Typical kinds of clients data distribution in AI-native network  procedures and IoT: (a) Non-i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='d, spatially correlated data distribution, (b)  Non-i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='d, spatially uncorrelated clusters of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' data and (c) Non-i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='d,  spatially correlated clusters of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' data  number of examples according to: a) their respective position  to the PS (for the case depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' 4a), or b) their  belonging cluster (for the case depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' 4b, 4c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' PROBLEM FORMULATION  The observation space can be represented as a graph com-  posed of nodes (client devices), edges (adjacency matrix),  and node features F (FL metrics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' In the most general for-  mulation, each edge can also be associated with a weight  |hi,j|, corresponding to the module of the complex channel  impulse response hi,j between client i and j, drawn from  a (M + 1)-dimensional channel matrix HM+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' Nevertheless,  considering the case of orthogonal resources assignment (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=',  no inter-users interference), and assuming link reciprocity,  allows for a simplification of the problem formulation, since  matrix HM+1 reduces to an M-dimensional vector H =  [|h1,P S|, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' , |hM,P S|].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' Hence, its elements can be represented  as node features instead of edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='  St =  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  F =  �  (F1, |h1,P S|), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=',  (FM, |hM,P S|)  �  ,  with |hi,P S| ∈ R  A =  �  ���  a11  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='  a1M  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='  aM1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='  aMM  �  ���  M  ,  with aij ∈ {0, 1}  (6)  In (6), F is the feature space of every node, which includes the  FL-metrics F used for decision-making during the scheduling  procedure, and A is the adjacency matrix, which is obtained  downstream of the dynamic graph formation block of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='  The feature space is composed of the following metrics:  Age of Information (AoI): a scalar indicator, introduced  in [14], describing the number of rounds that elapsed  since the client was last scheduled for model transmis-  sion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='  Path loss |hi,j|: the path loss value in dB related to the  client-PS link, assuming link reciprocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='  L2-norm of model update ||θi,t+1 − θP S,t||2: an impor-  tance metric indicating the L2 norm of the i-th client  model update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='  Notably, no explicit informative content can be collected about  the data of the clients, as this would violate the privacy-  preserving nature of FL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' PROPOSED ALGORITHM  In the formulation above, nodes connected by edges have  similar data, but not necessarily similar FL metric features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' In  fact, AoI, for instance, does not show any spatial correlation  property among nodes, as it purely depends on the schedul-  ing mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' In turn, this, together with the inability to  have explicit features about nodes’ data, depicts a situation  where client features don’t necessarily reflect the structure of  the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' Moreover, the nature of the problem is naturally  unsupervised, as nodes don’t have any label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' To this end, we  make use of UGRL via random walks in place of common  supervised methods with Graph Neural Network (GNN)s to  incorporate information about the structure of the graph into  the decision-making process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' By employing Node2Vec [17],  we are able to compute the graph encoding function f(G)  without the assistance of any node labels and features, but  rather by maximizing the log-likelihood of the 2nd order  biased random walks Ns(u) conditioned by f(u), for u ∈ G  as per (7) [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='  arg max  f  �  u∈G(V )  log (Prob(Ns(u)|f(u))  (7)  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' Distance Maximization via UGRL  Once f(G) is obtained, clients’ scheduling relies on the  distance maximization of the nodes in the embedding space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='  This has the effect of increasing data heterogeneity of the  client devices during consecutive communication rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' The  distance among a pair of nodes (v, u) in the graph G is  evaluated as the normalized dot product (cosine similarity)  s(v, u) of their node embeddings zv, zu (8):  s(v, u) =  zT  v · zu  ∥zv∥ · ∥zu∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='  (8)  To introduce memory of the past actions in the scheduling  process, nodes scheduled in previous communication rounds  are stored in a context window WL of tunable length L =  max(|WL|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' Accordingly, the similarity scores of the nodes  v ∈ WL are computed and collected in a matrix Sv,u of  dimension |WL| × (M − |WL|):  Sv,u =  �  ��  s(v1, uL+1)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='  s(v1, uM)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='  s(vL, uL+1)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='  s(vL, uM)  �  �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='  (9)  A scheduling decision is finally determined by equation (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='  arg  min  u∈G\\{v∈WL}  �  v∈WL  s(v, u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='  (10)  The latter is equivalent to summing all elements of the matrix  Sv,u by column, and selecting the next scheduled client as the  argument corresponding to the column holding the minimum  sum value, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=', the node with maximum distance with respect  to all previously scheduled nodes in WL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' SIMULATION METHODOLOGY  Experiments and evaluation were conducted on a simulator  based on Tensorflow Federated Core API.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' The logical steps  of the designed FL framework, namely ”FederatedEnv”, are  reported in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='  0  1hAfter the initialization of clients’ positions (rm, φm)M and  datasets Dm, the algorithm loops over a fixed number of  communication rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' Each round can be subdivided into  7 logical steps:  1) The module of the client-PS Downlink (DL) channel  impulse response |hP S,m| is computed, assuming perfect  CSI, by summing fading and shadowing contributions  (f ∼ F, s ∼ N) to the median path loss PL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' The signal-  to-noise ratio γ is thereby computed with a noise floor  NdB of -115 [dBm] and an AP gain Gtx of 15 [dBi].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='  2) The PS model θP S,t is broadcasted to the clients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' At  the receiver side, Gaussian noise with standard deviation  σDL =  �  E[N2] =  �  E[θ2]/γDL,m is added to the  PS’s model weights, where θ is the discrete signal of  weights of every model’s layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='  3) Each client performs a local update of its model weights,  controlled by the learning rate α(t), using stochastic  gradient descent for Nep epochs, as per (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='  4) After local model updates, a new round of Uplink (UL)  channel estimation is performed in the same way as for  step 1, and the corresponding UL signal-to-noise ratio  γUL is computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='  5) A binary mask of the scheduled users U = [u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' , uM]  is applied to the vector of updated model weights  Θ′  M = [θ′  1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' , θ′  M]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' During the aggregation phase,  only models belonging to scheduled clients will be  retained and included in the aggregation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='  6) The noisy models from the scheduled clients are aggre-  gated as per (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='  7) The model is evaluated on a test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' We denote by  f(Dtest|θP S(t + 1)) any metric computed over Dtest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='  Algorithm 1 FederatedEnv: logical steps  for each episode do  for m ∈ M do  (rm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' φm) = (rm ∼ U(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' R),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' φm ∼ U(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' 2π))  |Dm| = |D| · (um ∼ U(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' 1)/  �  um)  Dm = sample data(D,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' |Dm|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' φm)  end for  for each round t in T and for every m ∈ M do  Step 1: DL Channel State Estimation  |hP S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='m|dB = −PLm + s ∼ N(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' σs) + f ∼ F(k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' ω)  γDL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='m = Ptx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='DL + Gtx + |hP S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='m|dB − NdB  Step 2: Model Broadcast  θm(t) ← θP S(t) + n ∼ N(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' σ∝γDL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='m)  Step 3: Client Update  θj+1  m  (t) = θj  m(t) − α(t) · gm  Step 4: UL Channel State Estimation  |hm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='P S|dB = −PLm + s ∼ N(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' σs) + f ∼ F(k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' ω)  γUL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='m = Ptx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='UL + Gtx + |hm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='P S|dB − NdB  Step 5: Client Scheduling  Θ′  M = Θ′  M · UT = [θ′  1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' , θ′  M]T · [u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' , uM]  Step 6: Model update  θP S(t + 1) =  1  K  �  m∈K  |Dm|  �  m∈K |Dm| · θm(t) + nm  Step 7: Performance Evaluation  f(Dtest|θP S(t + 1))  end for  end for  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' RESULTS  In this section, we present and comment on the obtained  results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' We consider a scenario where M = 50 clients are  randomly distributed among equally populated clusters (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='  4b, 4c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' The distance maximization scheduling was tested  against baseline policies (Table I) based on the feature nodes  metrics introduced in section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' All policies have been tested  on MNIST classification when transmitting a shallow neural  network with 1 hidden layer, achieving 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='91 accuracy in a  centralized setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='  POLICY  DESCRIPTION.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='  Max Age of Information (AoI)  nodes with max AoI  Random (RND)  nodes chosen randomly  Round robin (RR)  nodes chosen in a round-trip fashion  Best Channel (BC)  nodes with the best channel condition  Oracle (OCL)  explicit data knowledge information -  maximize label heterogeneity  Max L2-norm (L2N)  nodes with maximum L2-norm of  their local model update  Distance maximization (DM)  our proposed scheduling policy  TABLE I: Baseline policies  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' 5 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' 6 show performance comparison in terms of  training accuracy and energy (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' 6a) vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' communication effi-  ciency (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' 6b), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' The three metrics are evaluated  as follows:  Accuracy: the sparse categorical cross accuracy on a  centralized test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='  Communication efficiency: the number of communication  rounds r to achieve an accuracy of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='  Energy efficiency: the number of rounds r, multiplied  by the number of training devices per round (K, or M,  depending on the policy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='  In all three metrics, the proposed solution outperforms the  baselines and approaches the performance of the oracle, which  is an empirical upper bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' In particular, the performance  gap between the proposed solutions increases as the number  of schedulable users (K) decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' It is of particular interest  to compare our proposed DM policy with respect to the L2N  importance-aware policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' Since the former does not require  any form of training feedback (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' 1), it is much more  energy efficient, as the number of clients training per round  is reduced from M to K (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=', only the scheduled users  perform local training).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' Moreover, results show that, in the  proposed scenario, DM outperforms L2N even in terms of  model accuracy and communication efficiency (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' 5 and 6b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='  In fact, even though the L2N policy is successful in scheduling  the users holding the most significant models every round  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=', those contributing more significantly to the global model,  according to the L2-norm of the local updates [10]), when  nodes show spatial relation, this may result in the selection of  nearby clients in space holding similar data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' On the opposite,  our proposed policy aims to achieve data heterogeneity by  maximizing nodes’ distance in the graph space, making it  more suitable for the considered scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' For the same reason,  policies such as AoI and random scheduling achieve better  performance than round-robin, since they inherently increase  0  5 10 15 20 25 30 35 40 45 50 55 60 65 70 75  Communication rounds  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='8  Number of scheduled clients K=5  AoI  DM  RND  RR  Oracle  BC  L2N  0  5 10 15 20 25 30 35 40 45 50 55 60 65 70 75  Communication rounds  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='8  Number of scheduled clients K=10  AoI  DM  RND  RR  Oracle  BC  L2N  0  5 10 15 20 25 30 35 40 45 50 55 60 65 70 75  Communication rounds  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='8  Number of scheduled clients K=25  AoI  DM  RND  RR  Oracle  BC  L2N  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' 5: Training results: sparse categorical cross accuracy vs number of communication rounds for different number of schedulable users per round: (a)  K = 5, M = 50, (b) K = 10, M = 50 and K = 25, M = 50  data heterogeneity among the clients scheduled every round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='  Results show that DM can achieve an average 10% gain in  model accuracy with respect to L2N at the end of the training  while increasing energy efficiency by 17 times for K = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='  Moreover, we register an average gain of 4% accuracy at the  end of the training and 31% in communication and energy  efficiency with respect to the second-best performing policy  (AoI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' Finally, it is interesting to notice how the choice of  K generates a tradeoff between communication and energy  efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' Indeed, for our considered model and simulation  parameters, if the aim of the designer is to maximize the  system’s communication efficiency, then K = 10 yields a gain  of 27, 9% in communication efficiency, but a loss of 30, 7%  in energy efficiency with respect to K = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' Therefore, this  is an indicator that for energy-sensitive applications, like IoT,  the maximization of the number of schedulable clients is not  always the best design choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='  AoI DM OCLRND RR BC L2N  200  300  400  500  700  3000  4000  Energy efficiency  (num trainings on devices)  Number of scheduled clients K=5  AoI DM OCLRND RR BC L2N  Number of scheduled clients K=10  (a) Energy efficiency  AoI DM OCLRND RR BC L2N  0  10  20  30  40  50  60  70  Communication efficiency  (number of rounds)  Number of scheduled clients K=5  AoI DM OCLRND RR BC L2N  Number of scheduled clients K=10  (b) Communication efficiency  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' 6: Training results: energy (a) vs communication efficiency (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='  VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' CONCLUSION  In this study, we present a novel metric for the scheduling of  client devices in FL applications leveraging the use of UGRL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content='  With respect to state-of-the-art importance-aware scheduling  methods, our solution does not require any training feedback  from client devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' Hence, it provides a much more com-  putationally and energy-efficient solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
+page_content=' Our results indicate  that, when tested against baseline importance-aware policies,  our solution achieves a gain of up to 10% in model accuracy,  while requiring up to 17 times fewer local training phases on  client devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFKT4oBgHgl3EQfxi4K/content/2301.11903v1.pdf'}
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+Exact quantum ground state of a two-dimensional quasicrystalline antiferromagnet
+Pratyay Ghosh1, ∗
+1Institut f¨ur Theoretische Physik und Astrophysik and W¨urzburg-Dresden Cluster of Excellence ct.qmat,
+Universit¨at W¨urzburg, Am Hubland Campus S¨ud, W¨urzburg 97074, Germany
+We present the exact dimer ground state of a quantum antiferromagnet defined on a quasicrystal
+constructed from the Bronze-mean hexagonal quasicrystal. A coupling isotropy on the first and
+second-neighbor bonds is sufficient to stabilize a product state of singlets on the third-neighbor
+bonds. We also provide a systematic approach for constructing additional crystals, quasicrystals,
+and amorphous structures that can sustain an exact dimer ground state.
+Introduction. The quantum antiferromagnets on two-
+dimensional frustrated lattices have been a focus of con-
+densed matter research for decades [1, 2]. Owing to the
+non-commutativity of quantum spin operators and frus-
+trated magnetic interactions, it is often impossible to
+have an analytic solution to these systems’ ground states.
+So, while dealing with these systems, one usually ana-
+lyzes the point- and space-group symmetries of the un-
+derlying lattices, which serve as a constraint for the pos-
+sible ground states, and gain qualitative understandings
+or perform further numerical investigations [3–5]. With
+the appearance and increasing number of quasicrystalline
+materials [6–10], however, it became apparent that the
+spin systems may not always be periodic, and also can
+lack real-space symmetries [11], leading the frustrated
+quasicrystals to often realize exotic ground states, such
+as spin-glass [8, 12–14], which are hard to understand.
+From our experience with crystalline quantum mag-
+nets, we may infer that the leading-edge knowledge for
+understanding complex spin systems is often provided
+by exactly solvable models. Shastry and Sutherland pro-
+posed the first such exactly solvable model in 2D [15].
+The Shastry-Sutherland model (SSM), defined on a lat-
+tice of uniform tiling, was found to exhibit an exact dimer
+singlet ground state. Though such exact dimer ground
+states are limited only to two 2D lattices with uniform
+tiling [16] (maple-leaf model (MLM) is the other one),
+they have widely served as crystallization seeds for sev-
+eral theoretical and experimental advancements. Despite
+the importance, such exact solutions in quasicrystals are
+very limited [17, 18], although there has been an exten-
+sive number of works on quasicrystalline spin systems
+[19–24], due to the complexity of the aperiodicity.
+In this letter, we propose a model defined on a qua-
+sicrystal that admits an exact dimer ground state. We,
+first, construct the quasicrystal from the bronze-mean
+hexagonal quasicrystal (BMHQ) [25], via site and bond
+depletion. Next, we include a subset of third-neighbor
+couplings. By demonstrating that this model has an ex-
+act dimer ground state, a product state of dimer singlets,
+we, then, assess its stability. In the end, we introduce
+a generic method, that equally applies to crystals, qua-
+sicrystals, and amorphous systems, which creates further
+models bearing exact dimer ground states.
+Model.
+To construct our model, which hosts an ex-
+act dimer state, we start with the BMHQ [25] (see
+Fig.1), which is closely related to the celebrated Pen-
+rose tiling [26] and the Ammann-Beenker tiling [27], in
+a sense, as their inflation ratios are all metallic ratios –
+for the former it is the bronze ratio, and for the oth-
+ers, it is the golden, and silver ratio, respectively. The
+BMHQ consists of three different tiles: small equilateral
+triangles (edge length s), big equilateral triangles (edge
+length l), and s×l rectangles. A six-fold-symmetric irreg-
+ular dodecagon constructed of six copies of each tile is the
+elementary motif of BMHQ. The discussion in Ref. [16]
+makes it apparent that the BMHQ is incapable of sup-
+porting an exact dimer eigenstate if we place a spin on
+every vertex (site) and couple them pair-wise via first-
+and second-neighbor Heisenberg exchange interactions
+(bonds). To host an exact dimer state, the system must
+satisfy a necessary, but not sufficient, condition of hav-
+ing an odd coordination number, which is not the case for
+BMHQ. Therefore, we perform the following. We deplete
+all sites shared by six adjacent large triangles, and all
+bonds shared by two triangles. As a result, we get a dif-
+ferent quasicrystal (depicted in Fig.1), consisting of the
+four tiles: a hexagon, a small rhombus, a large rhombus,
+and a rectangle. Lastly, we add half of the third-neighbor
+couplings, i.e. one diagonal of each rectangle, such that
+every site is only part of one diagonal interaction. Note
+that there are two sets of diagonal bonds one can choose
+from; both are adequate for our purpose.
+We now define a Heisenberg spin Hamiltonian on this
+quasicrystal as
+H = Js
+�
+⟨kl⟩
+⃗Sk · ⃗Sl + Jl
+�
+⟨⟨km⟩⟩
+⃗Sk · ⃗Sm + Jd
+�
+(lm)
+⃗Sl · ⃗Sm. (1)
+Here, ⃗Si denotes the su(2) spin-S operator at site i, ⟨kl⟩,
+⟨⟨km⟩⟩, and (lm) runs over orange dashed, blue dotted,
+and thick green bonds, respectively (Fig. 1). Here, we
+only discuss the special case Jd = J and Js = Jl = J′
+of (1),
+HX = J′ �
+⟨kl⟩
+⃗Sk · ⃗Sl + J′ �
+⟨⟨km⟩⟩
+⃗Sk · ⃗Sm + J
+�
+(lm)
+⃗Sl · ⃗Sm, (2)
+(henceforth dubbed as Model-X), which can admit an
+exact dimer ground state.
+arXiv:2301.11331v1  [cond-mat.str-el]  26 Jan 2023
+
+2
+Bronze-mean hexagonal quasicrystal
+Model-X
+FIG. 1. The construction of Model-X from Bronze-mean hexagonal quasicrystal: First, we deplete all vertices that are shared
+by six big triangles to generate the hexagonal tiles. Next, we deplete the connections shared by two triangles, constructing the
+two rhombus tiles. On the resulting tiling, we place a spin on each vertex, and the connection between the vertices, which now
+connects a pair of spins, acts as the bonds. The bonds are of two different lengths, namely l and s, shown by dashed orange
+and dotted blue lines. We assume an equal Heisenberg coupling of strength J′ on both these bonds. We further add a subset of
+the third neighbor interactions, shown by thick green lines, with an exchange interaction of strength of J. We call the resulting
+spin-model Model-X. The singlets (light red ellipses) reside on J bonds. As the J′ couplings do not contribute to the energy,
+this product state of singlets is an eigenstate of (2).
+Quantum Ground State. The exact dimer ground state
+of Model-X can be obtained by following the same proce-
+dure as in Refs. [15, 16]. First, we recast (2) into a sum
+over the interacting spins on the right-angled triangles as
+HX =
+� �
++
++
++
++
++
+�
+. (3)
+For this, we have distributed the interaction J ⃗Sl · ⃗Sm
+on the thick green bonds in Fig. 1 equally between the
+two triangles that share this bond. Thus, these triangles
+now have three different colored bond interactions, the
+Hamiltonian for each, in general, reads as
+h△ = J′⃗Sk · ⃗Sl + J′⃗Sk · ⃗Sm + J
+2
+⃗Sl · ⃗Sm.
+(4)
+The triangular decomposition of (2), in a sense, sculpts
+the model to a frustration-free form [15, 28–30], as the
+ground state minimizes the energy of each h△. Note that
+if there are N spins in our model, there will be N such
+right-angled triangles, and N/2 thick green bonds.
+When J/2 > J′, the ground state of h△ is a spin-
+singlet forming on the green bond, which we denote by
+��[lm]
+�
+. Now, the parity of the singlet causes the first two
+terms in (4) to cancel each other’s contribution to the
+energy of a triangle, i.e.
+(J′⃗Sk · ⃗Sl + J′⃗Sk · ⃗Sm)
+��[lm]
+�
+= 0.
+Therefore, we can construct a product state of
+��[lm]
+�
+which covers the entire system as
+|ψ⟩ = ⊗
+�
+(lm)
+��[lm]
+�
+,
+(5)
+(see Fig.1 where we overlay |ψ⟩ on Model-X). Again, due
+to the parity of the
+��[lm]
+�
+s, the first two terms of (2)
+do not contribute to the energy of the full system or
+renormalize (5), thus, making (5) an exact eigenstate of
+Model-X with an energy density, which is independent of
+J′, given by
+E/N = −αS(S + 1)
+2
+J.
+(6)
+If the ground state energy of (4) is e△, then (6) sets
+an upper bound for the ground state energy of the entire
+system, i.e., Eg/N ≥ e△, the equality of which holds
+when J is greater than a lower bound Jb and |ψ⟩ is the
+ground state. For spin-1/2, Jb can be easily computed to
+be 2J′, and can likewise be obtained for other spins [31].
+Note that the variational principle-based analysis that we
+have performed thus far, does not prevent |ψ⟩ from being
+the ground state for J < Jb, only it cannot be shown
+analytically. The critical Jc, such that for Jb > J ≥ Jc
+the exact dimer state is the ground state of the system,
+can only be obtained numerically [16, 32–34], which we
+attempt next.
+Stability of the Exact Dimer Phase. Due to the aperi-
+odicity and frustration, Model-X is impregnable by most
+numerical techniques in their status quo. Therefore, we
+resort to the robust density matrix renormalization group
+(DMRG) approach to obtain an estimate of Jc/J′ [35].
+We perform our DMRG calculations on a 144-site spin-
+1/2 cluster [36] using the ITensor library [37]. The criti-
+cal Jc/J′ found from our calculations is shown in Tab. I.
+The phase transitions out of the dimer phase in both SSM
+and MLM have been found to be first-order in nature [32–
+34]. One may wonder if the aperiodicity for Model-X can
+change this nature or if the system can form domains of
+
+3
+Model-Y
+FIG. 2. The Model-Y, defined on a lattice, is made out of
+the same tiles as the Model-X. It contains three symmetry in-
+equivalent bonds marked by dashed orange, dotted blue, and
+thick green lines. In Model-Y, the dashed orange and dotted
+blue bonds have the same exchange interaction of strength J′,
+and the green bonds bare an exchange interaction of strength
+J. Similar to Model-X, Model-Y can also admit an exact sin-
+glet eigenstate with singlets on the J bonds, which becomes
+a ground state of the system for J ≳ 1.45J.
+exact dimer and non-dimer states. This seems implau-
+sible because when a local dimer switches from being a
+dimer to a non-dimer state, the system’s structure causes
+the dimers nearby to experience local non-uniform fields
+that force them to also become non-dimer states. As a
+result, a chain reaction spreads across the whole system,
+thereby, making the product singlet state become kaput
+as a whole. Thus, we do not expect a second-order phase
+transition out of the exact dimer phase in Model-X, which
+is confirmed by our DMRG calculations [36]. To gain fur-
+ther insights, we also compare our current dimer phase’s
+stability with the same in SSM and MLM in Tab. I, with
+intra- and inter-dimer dimer coupling being J and J′, re-
+spectively. We find Model-X to have a comparatively less
+stable exact dimer ground phase. To make a further com-
+parison, we introduce a lattice made out of the same tiles
+as Model-X, shown in Fig. 2 (henceforth called Model-Y).
+Model-Y also admits a product singlet ground state on
+the J dimers. Here, we again take a 144-site spin cluster
+and perform DMRG calculations to study the stability of
+its exact dimer phase [36]. The result is shown in Tab. I.
+The magnetic frustrations in Model-X and Model-Y are
+more than SSM but less than MLM. As the frustration
+impacts the dimer ground phase’s stability [16], one can
+anticipate that Jc for Model-X and Model-Y both will fall
+somewhere between the same for SSM and MLM. Model-
+Y exhibits this, but Model-X does not. The most likely
+explanation could be that the DMRG results on a finite
+section of quasicrystal, where the local structure might
+severely affect the exact dimer state, are very different
+from the results at the thermodynamic limit, where such
+local effects are averaged out. However, we can not ex-
+clude the possibility that the system’s aperiodicity might
+Jc/J′
+Refs.
+MLM
+1.35
+Ref. [16]
+Model-Y
+1.45(1)
+This work
+SSM
+1.48
+Ref. [32, 33]
+Model-X
+1.49(1)
+This work
+TABLE I. The comparison of the stability of the exact dimer
+ground state in different models. For J > Jc, the exact dimer
+states of the corresponding models become their ground state.
+For MLM, Ref. [34] finds a Jc/J′ ≈ 1.45.
+play a role in destabilizing the exact dimer state, even at
+the thermodynamic limit. A thorough study of the sta-
+bility of the exact dimer state is beyond the scope of
+this letter. Via our DMRG calculations, we only like to
+demonstrate that the exact dimer state on this quasicrys-
+tal can be stable even for J < 2J′.
+Though we do not investigate the models in detail
+beyond their exact dimer ground state, we can antici-
+pate the possibility that Model-X and Model-Y can have
+other novel phases in the high frustration regime J < Jc.
+This makes both our models worthy of further investiga-
+tions, e.g. the nature of these additional phases and the
+phase transitions which might feature the exotic decon-
+fined criticality [38].
+Further Possibilities.
+We have introduced Model-X,
+defined on one quasicrystal, which admits an exact dimer
+ground state. However, the question remains: Are there
+other quasicrystals with exact dimer ground states? The
+answer is yes, and we are now going to outline how to
+construct such systems. Since the prescription we present
+here applies to graphs, it is suitable for all types of sys-
+tems – crystalline, quasicrystalline, and amorphous. We
+begin with a graph where each vertex is connected to four
+other vertices (see Fig. 3). In the next step, the connec-
+tions are all decorated by adding a new vertex (the filled
+circles in Fig. 3). These new vertices will be our actual
+sites carrying the spins. After that, one applies a gen-
+eralized version of star-triangle-type transformation (we
+deem it as ×-□ transformation) to decimate the original
+vertices and form connections between the new vertices.
+In our spin model these new connections act as inter-
+dimer bonds. At this point, we have produced a graph
+where each vertex is part of two generalized quadran-
+gles. In our final step, we connect one diagonal of each
+quadrangle, which serves as our dimer bonds, while en-
+suring that any two dimers do not share a site. Thus, we
+construct a system that can host an exact dimer ground
+state, which is the product state of singlets on the dimer
+bonds, and the proof is similar to SSM [15], MLM [16],
+and Model-X. In the simplest case, one assigns equal
+strength to all inter-dimer couplings. All the intra-dimer
+couplings also have equal strength but are different from
+the inter-dimer couplings. However, more complex mod-
+els can also be defined on such a graph that allow exact
+dimer states. Note that the last step can result in two
+
+4
+FIG. 3.
+A general scheme for obtaining a model with an exact dimer ground state from a graph where each vertex has
+four connections.
+First, we decorate the connections with new vertices (marked by filled circles), which is followed by a
+generalized star-triangle-type transformation to decimate all the old vertices. From there, one can generate two possible graphs
+by connecting a subset of the diagonals of the newly generated quadrangles (keeping in mind that no site can be part of two
+diagonals), both of which can host an exact dimer state.
+independent graphs (see the last panel of Fig. 3), both
+ideal for our purpose.
+Taking the square and the kagome lattice as exam-
+ples of four coordinated lattices, and then following our
+procedure one obtains the SSM and the MLM, respec-
+tively. To construct such quasicrystals and amorphous
+systems, however, the initial difficulty is to acquire a sys-
+tem with coordination number 4 on which our prescrip-
+tion can be implemented. This is simple for amorphous
+systems. One can draw random straight lines on a plane
+and, in general, this would result in a system that has a
+coordination number of 4, when one considers the inter-
+sections as vertices and the line segments between them
+as edges [36]. For quasicrystals, one can start with an
+existing quasicrystal made up of quadrangles, e.g. the
+Penrose rhomb tiling [26, 27], place a vertex in the mid-
+dle of each tile, and then connect all pairs of vertices if
+their corresponding tiles share an edge, and thus, one can
+obtain a 4-coordinated quasicrystal.
+Conclusion and Outlook. We have introduced Model-
+X, which is defined on a quasicrystal made out of
+hexagons, rhombi, and rectangles, and studied its exact
+dimer ground state. To the best of our knowledge, no
+such model on quasicrystals has been reported so far to
+exhibit such a property. We also create a crystal using the
+same tiles and do similar studies on that as well. Finally,
+we lay out a general scheme for constructing crystals,
+quasicrystals, and amorphous systems, that can admit
+an exact dimer ground state.
+The Model-X opens up several questions which require
+further investigations.
+First, one needs to understand
+how the aperiodicity influences the stability of the dimer
+state (5). Second would be the study of the nature of
+the other phases, and the possible phase transitions in
+Model-X, also with the bond anisotropy Js ̸= Jl. The
+third is the investigation of Model-X in a finite magnetic
+field.
+The lattice versions of exact dimer models, e.g.
+the SSM and the MLM, show a series of spin-density
+wave and multi-triplet bound-state crystal-based magne-
+tization plateaux [39–44], behind all of which the lattice
+periodicity plays a pivotal role. One can still speculate
+the formation of two and three-triplet bound states in
+Model-X. However, how the aperiodicity of the model
+would affect the magnetization process in this system is
+an extremely tempting question. Lastly, a material re-
+alization of Model-X will be highly sought out for, in
+general, similar to how the SSM and its experimental
+realizations have played a central role in numerous the-
+oretical and experimental developments.
+Additionally,
+our scheme for creating systems with exact dimer ground
+states will significantly advance the study of amorphous
+spin systems, a subject in which the exact solution has
+just lately begun to emerge [45].
+Acknowledgments.
+The author acknowledges Tobias
+M¨uller and Ronny Thomale for useful discussions. The
+work is supported by the Deutsche Forschungsgemein-
+schaft (DFG, German Research Foundation) through
+Project-ID 258499086-SFB 1170 and the W¨urzburg-
+Dresden Cluster of Excellence on Complexity and Topol-
+ogy in Quantum Matter – ct.qmat Project-ID 390858490-
+EXC 2147.
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+and J. Knolle, 2208.08246v2.
+
+Supplemental Materials: Exact ground state of a two-dimensional quasicrystal
+quantum antiferromagnet
+Pratyay Ghosh1, ∗
+1Institut f¨ur Theoretische Physik und Astrophysik and W¨urzburg-Dresden Cluster of Excellence ct.qmat,
+Universit¨at W¨urzburg, Am Hubland Campus S¨ud, W¨urzburg 97074, Germany
+I.
+DETAILS OF DMRG CALCULATIONS
+The supplement contains further details about our DMRG calculations which are performed using iTENSOR
+library [1]. For both model-X and Y we choose a 144 site cluster (see model-X in Fig. S1 and model-Y in Fig. S2)
+with open boundary and perform 24 sweeps with a maximum bond dimension of 1024. The results, i.e. the ground
+state energy and the spin-spin correlations on a few selected bonds, are shown in Fig S3, which shows a clear first
+order transition out of the exact dimer phase.
+Figure S1. The 144-site Model-X cluster used to perform the DMRG calculations mentioned in the main text.
+∗ pratyay.ghosh@physik.uni-wuerzburg.de
+arXiv:2301.11331v1  [cond-mat.str-el]  26 Jan 2023
+
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+Figure S2. The 144-site Model-Y cluster used to perform the DMRG calculations mentioned in the main text.
+0.5
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+(a)
+𝐽′/𝐽
+𝑒𝑔
+(b)
+𝐽′/𝐽
+(c)
+𝐽′/𝐽
+𝑒𝑔
+(d)
+𝐽′/𝐽
+Figure S3. The DMRG results for Model-X: (a) the ground state energy per site and (b) the spin-spin correlations on selected
+bonds. (c) and (d) are respectively the same for Model-Y. The spin systems used for our calculations are depicted in Figs. S1
+and S1. In both calculations we have used a open boundary condition for a fair comparison.
+
+3
+Figure S4. An example of a amorphous system with exact ground state constructed via the prescription put forward in the
+main text. The dashed lines are placed randomly on the plane to create an amorphous system with coordination number 4.
+II.
+EXAMPLE OF A AMORPHOUS SYSTEM WITH EXACT GROUND STATE
+Fig. S4 shows a amorphous system with a exact dimer ground state which a product of singlets on the thick green
+bonds. This is created by using the procedure explained in the main text.
+[1] M. Fishman, S. R. White, and E. M. Stoudenmire, The itensor software library for tensor network calculations (2020),
+arXiv:2007.14822 [cs.MS].
+
diff --git a/ktFIT4oBgHgl3EQfrSul/content/tmp_files/load_file.txt b/ktFIT4oBgHgl3EQfrSul/content/tmp_files/load_file.txt
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@@ -0,0 +1,691 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf,len=690
+page_content='Exact quantum ground state of a two-dimensional quasicrystalline antiferromagnet  Pratyay Ghosh1, ∗  1Institut f¨ur Theoretische Physik und Astrophysik and W¨urzburg-Dresden Cluster of Excellence ct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content='qmat,  Universit¨at W¨urzburg, Am Hubland Campus S¨ud, W¨urzburg 97074, Germany  We present the exact dimer ground state of a quantum antiferromagnet defined on a quasicrystal  constructed from the Bronze-mean hexagonal quasicrystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' A coupling isotropy on the first and  second-neighbor bonds is sufficient to stabilize a product state of singlets on the third-neighbor  bonds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' We also provide a systematic approach for constructing additional crystals, quasicrystals,  and amorphous structures that can sustain an exact dimer ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content='  Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' The quantum antiferromagnets on two-  dimensional frustrated lattices have been a focus of con-  densed matter research for decades [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' Owing to the  non-commutativity of quantum spin operators and frus-  trated magnetic interactions, it is often impossible to  have an analytic solution to these systems’ ground states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content='  So, while dealing with these systems, one usually ana-  lyzes the point- and space-group symmetries of the un-  derlying lattices, which serve as a constraint for the pos-  sible ground states, and gain qualitative understandings  or perform further numerical investigations [3–5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' With  the appearance and increasing number of quasicrystalline  materials [6–10], however, it became apparent that the  spin systems may not always be periodic, and also can  lack real-space symmetries [11], leading the frustrated  quasicrystals to often realize exotic ground states, such  as spin-glass [8, 12–14], which are hard to understand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content='  From our experience with crystalline quantum mag-  nets, we may infer that the leading-edge knowledge for  understanding complex spin systems is often provided  by exactly solvable models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' Shastry and Sutherland pro-  posed the first such exactly solvable model in 2D [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content='  The Shastry-Sutherland model (SSM), defined on a lat-  tice of uniform tiling, was found to exhibit an exact dimer  singlet ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' Though such exact dimer ground  states are limited only to two 2D lattices with uniform  tiling [16] (maple-leaf model (MLM) is the other one),  they have widely served as crystallization seeds for sev-  eral theoretical and experimental advancements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' Despite  the importance, such exact solutions in quasicrystals are  very limited [17, 18], although there has been an exten-  sive number of works on quasicrystalline spin systems  [19–24], due to the complexity of the aperiodicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content='  In this letter, we propose a model defined on a qua-  sicrystal that admits an exact dimer ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' We,  first, construct the quasicrystal from the bronze-mean  hexagonal quasicrystal (BMHQ) [25], via site and bond  depletion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' Next, we include a subset of third-neighbor  couplings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' By demonstrating that this model has an ex-  act dimer ground state, a product state of dimer singlets,  we, then, assess its stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' In the end, we introduce  a generic method, that equally applies to crystals, qua-  sicrystals, and amorphous systems, which creates further  models bearing exact dimer ground states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content='  Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content='  To construct our model, which hosts an ex-  act dimer state, we start with the BMHQ [25] (see  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content='1), which is closely related to the celebrated Pen-  rose tiling [26] and the Ammann-Beenker tiling [27], in  a sense, as their inflation ratios are all metallic ratios –  for the former it is the bronze ratio, and for the oth-  ers, it is the golden, and silver ratio, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' The  BMHQ consists of three different tiles: small equilateral  triangles (edge length s), big equilateral triangles (edge  length l), and s×l rectangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' A six-fold-symmetric irreg-  ular dodecagon constructed of six copies of each tile is the  elementary motif of BMHQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' The discussion in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' [16]  makes it apparent that the BMHQ is incapable of sup-  porting an exact dimer eigenstate if we place a spin on  every vertex (site) and couple them pair-wise via first-  and second-neighbor Heisenberg exchange interactions  (bonds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' To host an exact dimer state, the system must  satisfy a necessary, but not sufficient, condition of hav-  ing an odd coordination number, which is not the case for  BMHQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' Therefore, we perform the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' We deplete  all sites shared by six adjacent large triangles, and all  bonds shared by two triangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' As a result, we get a dif-  ferent quasicrystal (depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content='1), consisting of the  four tiles: a hexagon, a small rhombus, a large rhombus,  and a rectangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' Lastly, we add half of the third-neighbor  couplings, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' one diagonal of each rectangle, such that  every site is only part of one diagonal interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' Note  that there are two sets of diagonal bonds one can choose  from;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' both are adequate for our purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content='  We now define a Heisenberg spin Hamiltonian on this  quasicrystal as  H = Js  �  ⟨kl⟩  ⃗Sk · ⃗Sl + Jl  �  ⟨⟨km⟩⟩  ⃗Sk · ⃗Sm + Jd  �  (lm)  ⃗Sl · ⃗Sm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' (1)  Here, ⃗Si denotes the su(2) spin-S operator at site i, ⟨kl⟩,  ⟨⟨km⟩⟩, and (lm) runs over orange dashed, blue dotted,  and thick green bonds, respectively (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' Here, we  only discuss the special case Jd = J and Js = Jl = J′  of (1),  HX = J′ �  ⟨kl⟩  ⃗Sk · ⃗Sl + J′ �  ⟨⟨km⟩⟩  ⃗Sk · ⃗Sm + J  �  (lm)  ⃗Sl · ⃗Sm, (2)  (henceforth dubbed as Model-X), which can admit an  exact dimer ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content='11331v1  [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content='str-el]  26 Jan 2023  2  Bronze-mean hexagonal quasicrystal  Model-X  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' The construction of Model-X from Bronze-mean hexagonal quasicrystal: First, we deplete all vertices that are shared  by six big triangles to generate the hexagonal tiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' Next, we deplete the connections shared by two triangles, constructing the  two rhombus tiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' On the resulting tiling, we place a spin on each vertex, and the connection between the vertices, which now  connects a pair of spins, acts as the bonds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' The bonds are of two different lengths, namely l and s, shown by dashed orange  and dotted blue lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' We assume an equal Heisenberg coupling of strength J′ on both these bonds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' We further add a subset of  the third neighbor interactions, shown by thick green lines, with an exchange interaction of strength of J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' We call the resulting  spin-model Model-X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' The singlets (light red ellipses) reside on J bonds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' As the J′ couplings do not contribute to the energy,  this product state of singlets is an eigenstate of (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content='  Quantum Ground State.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' The exact dimer ground state  of Model-X can be obtained by following the same proce-  dure as in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' [15, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' First, we recast (2) into a sum  over the interacting spins on the right-angled triangles as  HX =  � �  +  +  +  +  +  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' (3)  For this, we have distributed the interaction J ⃗Sl · ⃗Sm  on the thick green bonds in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' 1 equally between the  two triangles that share this bond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' Thus, these triangles  now have three different colored bond interactions, the  Hamiltonian for each, in general, reads as  h△ = J′⃗Sk · ⃗Sl + J′⃗Sk · ⃗Sm + J  2  ⃗Sl · ⃗Sm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content='  (4)  The triangular decomposition of (2), in a sense, sculpts  the model to a frustration-free form [15, 28–30], as the  ground state minimizes the energy of each h△.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' Note that  if there are N spins in our model, there will be N such  right-angled triangles, and N/2 thick green bonds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content='  When J/2 > J′, the ground state of h△ is a spin-  singlet forming on the green bond, which we denote by  ��[lm]  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' Now, the parity of the singlet causes the first two  terms in (4) to cancel each other’s contribution to the  energy of a triangle, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content='  (J′⃗Sk · ⃗Sl + J′⃗Sk · ⃗Sm)  ��[lm]  �  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content='  Therefore, we can construct a product state of  ��[lm]  �  which covers the entire system as  |ψ⟩ = ⊗  �  (lm)  ��[lm]  �  ,  (5)  (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content='1 where we overlay |ψ⟩ on Model-X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' Again, due  to the parity of the  ��[lm]  �  s, the first two terms of (2)  do not contribute to the energy of the full system or  renormalize (5), thus, making (5) an exact eigenstate of  Model-X with an energy density, which is independent of  J′, given by  E/N = −αS(S + 1)  2  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content='  (6)  If the ground state energy of (4) is e△, then (6) sets  an upper bound for the ground state energy of the entire  system, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=', Eg/N ≥ e△, the equality of which holds  when J is greater than a lower bound Jb and |ψ⟩ is the  ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' For spin-1/2, Jb can be easily computed to  be 2J′, and can likewise be obtained for other spins [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content='  Note that the variational principle-based analysis that we  have performed thus far, does not prevent |ψ⟩ from being  the ground state for J < Jb, only it cannot be shown  analytically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' The critical Jc, such that for Jb > J ≥ Jc  the exact dimer state is the ground state of the system,  can only be obtained numerically [16, 32–34], which we  attempt next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content='  Stability of the Exact Dimer Phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' Due to the aperi-  odicity and frustration, Model-X is impregnable by most  numerical techniques in their status quo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' Therefore, we  resort to the robust density matrix renormalization group  (DMRG) approach to obtain an estimate of Jc/J′ [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content='  We perform our DMRG calculations on a 144-site spin-  1/2 cluster [36] using the ITensor library [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' The criti-  cal Jc/J′ found from our calculations is shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content='  The phase transitions out of the dimer phase in both SSM  and MLM have been found to be first-order in nature [32–  34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' One may wonder if the aperiodicity for Model-X can  change this nature or if the system can form domains of  3  Model-Y  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' The Model-Y, defined on a lattice, is made out of  the same tiles as the Model-X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' It contains three symmetry in-  equivalent bonds marked by dashed orange, dotted blue, and  thick green lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' In Model-Y, the dashed orange and dotted  blue bonds have the same exchange interaction of strength J′,  and the green bonds bare an exchange interaction of strength  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' Similar to Model-X, Model-Y can also admit an exact sin-  glet eigenstate with singlets on the J bonds, which becomes  a ground state of the system for J ≳ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content='45J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content='  exact dimer and non-dimer states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' This seems implau-  sible because when a local dimer switches from being a  dimer to a non-dimer state, the system’s structure causes  the dimers nearby to experience local non-uniform fields  that force them to also become non-dimer states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' As a  result, a chain reaction spreads across the whole system,  thereby, making the product singlet state become kaput  as a whole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' Thus, we do not expect a second-order phase  transition out of the exact dimer phase in Model-X, which  is confirmed by our DMRG calculations [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' To gain fur-  ther insights, we also compare our current dimer phase’s  stability with the same in SSM and MLM in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' I, with  intra- and inter-dimer dimer coupling being J and J′, re-  spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' We find Model-X to have a comparatively less  stable exact dimer ground phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' To make a further com-  parison, we introduce a lattice made out of the same tiles  as Model-X, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' 2 (henceforth called Model-Y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content='  Model-Y also admits a product singlet ground state on  the J dimers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' Here, we again take a 144-site spin cluster  and perform DMRG calculations to study the stability of  its exact dimer phase [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' The result is shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content='  The magnetic frustrations in Model-X and Model-Y are  more than SSM but less than MLM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' As the frustration  impacts the dimer ground phase’s stability [16], one can  anticipate that Jc for Model-X and Model-Y both will fall  somewhere between the same for SSM and MLM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' Model-  Y exhibits this, but Model-X does not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' The most likely  explanation could be that the DMRG results on a finite  section of quasicrystal, where the local structure might  severely affect the exact dimer state, are very different  from the results at the thermodynamic limit, where such  local effects are averaged out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' However, we can not ex-  clude the possibility that the system’s aperiodicity might  Jc/J′  Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content='  MLM  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content='35  Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' [16]  Model-Y  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content='45(1)  This work  SSM  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content='48  Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' [32, 33]  Model-X  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content='49(1)  This work  TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' The comparison of the stability of the exact dimer  ground state in different models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' For J > Jc, the exact dimer  states of the corresponding models become their ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content='  For MLM, Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' [34] finds a Jc/J′ ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content='45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content='  play a role in destabilizing the exact dimer state, even at  the thermodynamic limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' A thorough study of the sta-  bility of the exact dimer state is beyond the scope of  this letter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' Via our DMRG calculations, we only like to  demonstrate that the exact dimer state on this quasicrys-  tal can be stable even for J < 2J′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content='  Though we do not investigate the models in detail  beyond their exact dimer ground state, we can antici-  pate the possibility that Model-X and Model-Y can have  other novel phases in the high frustration regime J < Jc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content='  This makes both our models worthy of further investiga-  tions, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' the nature of these additional phases and the  phase transitions which might feature the exotic decon-  fined criticality [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content='  Further Possibilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content='  We have introduced Model-X,  defined on one quasicrystal, which admits an exact dimer  ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' However, the question remains: Are there  other quasicrystals with exact dimer ground states?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' The  answer is yes, and we are now going to outline how to  construct such systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' Since the prescription we present  here applies to graphs, it is suitable for all types of sys-  tems – crystalline, quasicrystalline, and amorphous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' We  begin with a graph where each vertex is connected to four  other vertices (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' In the next step, the connec-  tions are all decorated by adding a new vertex (the filled  circles in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' These new vertices will be our actual  sites carrying the spins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' After that, one applies a gen-  eralized version of star-triangle-type transformation (we  deem it as ×-□ transformation) to decimate the original  vertices and form connections between the new vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content='  In our spin model these new connections act as inter-  dimer bonds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' At this point, we have produced a graph  where each vertex is part of two generalized quadran-  gles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' In our final step, we connect one diagonal of each  quadrangle, which serves as our dimer bonds, while en-  suring that any two dimers do not share a site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' Thus, we  construct a system that can host an exact dimer ground  state, which is the product state of singlets on the dimer  bonds, and the proof is similar to SSM [15], MLM [16],  and Model-X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' In the simplest case, one assigns equal  strength to all inter-dimer couplings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' All the intra-dimer  couplings also have equal strength but are different from  the inter-dimer couplings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' However, more complex mod-  els can also be defined on such a graph that allow exact  dimer states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' Note that the last step can result in two  4  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content='  A general scheme for obtaining a model with an exact dimer ground state from a graph where each vertex has  four connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content='  First, we decorate the connections with new vertices (marked by filled circles), which is followed by a  generalized star-triangle-type transformation to decimate all the old vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' From there, one can generate two possible graphs  by connecting a subset of the diagonals of the newly generated quadrangles (keeping in mind that no site can be part of two  diagonals), both of which can host an exact dimer state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content='  independent graphs (see the last panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' 3), both  ideal for our purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content='  Taking the square and the kagome lattice as exam-  ples of four coordinated lattices, and then following our  procedure one obtains the SSM and the MLM, respec-  tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' To construct such quasicrystals and amorphous  systems, however, the initial difficulty is to acquire a sys-  tem with coordination number 4 on which our prescrip-  tion can be implemented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' This is simple for amorphous  systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' One can draw random straight lines on a plane  and, in general, this would result in a system that has a  coordination number of 4, when one considers the inter-  sections as vertices and the line segments between them  as edges [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' For quasicrystals, one can start with an  existing quasicrystal made up of quadrangles, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' the  Penrose rhomb tiling [26, 27], place a vertex in the mid-  dle of each tile, and then connect all pairs of vertices if  their corresponding tiles share an edge, and thus, one can  obtain a 4-coordinated quasicrystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content='  Conclusion and Outlook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' We have introduced Model-  X, which is defined on a quasicrystal made out of  hexagons, rhombi, and rectangles, and studied its exact  dimer ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' To the best of our knowledge, no  such model on quasicrystals has been reported so far to  exhibit such a property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' We also create a crystal using the  same tiles and do similar studies on that as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' Finally,  we lay out a general scheme for constructing crystals,  quasicrystals, and amorphous systems, that can admit  an exact dimer ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content='  The Model-X opens up several questions which require  further investigations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content='  First, one needs to understand  how the aperiodicity influences the stability of the dimer  state (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' Second would be the study of the nature of  the other phases, and the possible phase transitions in  Model-X, also with the bond anisotropy Js ̸= Jl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' The  third is the investigation of Model-X in a finite magnetic  field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content='  The lattice versions of exact dimer models, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content='  the SSM and the MLM, show a series of spin-density  wave and multi-triplet bound-state crystal-based magne-  tization plateaux [39–44], behind all of which the lattice  periodicity plays a pivotal role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' One can still speculate  the formation of two and three-triplet bound states in  Model-X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' However, how the aperiodicity of the model  would affect the magnetization process in this system is  an extremely tempting question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' Lastly, a material re-  alization of Model-X will be highly sought out for, in  general, similar to how the SSM and its experimental  realizations have played a central role in numerous the-  oretical and experimental developments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content='  Additionally,  our scheme for creating systems with exact dimer ground  states will significantly advance the study of amorphous  spin systems, a subject in which the exact solution has  just lately begun to emerge [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content='  Acknowledgments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content='  The author acknowledges Tobias  M¨uller and Ronny Thomale for useful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' The  work is supported by the Deutsche Forschungsgemein-  schaft (DFG, German Research Foundation) through  Project-ID 258499086-SFB 1170 and the W¨urzburg-  Dresden Cluster of Excellence on Complexity and Topol-  ogy in Quantum Matter – ct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content='qmat Project-ID 390858490-  EXC 2147.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
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+page_content='  Seufert,  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content='  M¨uller,  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content='  Mila,  and  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' Thomale, 2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content='08264v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
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+page_content=' D’Ornellas, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' Hodson, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' Natori,  and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' Knolle, 2208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content='08246v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content='  Supplemental Materials: Exact ground state of a two-dimensional quasicrystal  quantum antiferromagnet  Pratyay Ghosh1, ∗  1Institut f¨ur Theoretische Physik und Astrophysik and W¨urzburg-Dresden Cluster of Excellence ct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content='qmat,  Universit¨at W¨urzburg, Am Hubland Campus S¨ud, W¨urzburg 97074, Germany  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content='  DETAILS OF DMRG CALCULATIONS  The supplement contains further details about our DMRG calculations which are performed using iTENSOR  library [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' For both model-X and Y we choose a 144 site cluster (see model-X in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' S1 and model-Y in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' S2)  with open boundary and perform 24 sweeps with a maximum bond dimension of 1024.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' The results, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' the ground  state energy and the spin-spin correlations on a few selected bonds, are shown in Fig S3, which shows a clear first  order transition out of the exact dimer phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content='  Figure S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' The 144-site Model-X cluster used to perform the DMRG calculations mentioned in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
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+page_content=' The DMRG results for Model-X: (a) the ground state energy per site and (b) the spin-spin correlations on selected  bonds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
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+page_content=' S1  and S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' In both calculations we have used a open boundary condition for a fair comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content='  3  Figure S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' An example of a amorphous system with exact ground state constructed via the prescription put forward in the  main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' The dashed lines are placed randomly on the plane to create an amorphous system with coordination number 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content='  EXAMPLE OF A AMORPHOUS SYSTEM WITH EXACT GROUND STATE  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' S4 shows a amorphous system with a exact dimer ground state which a product of singlets on the thick green  bonds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' This is created by using the procedure explained in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content='  [1] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' Fishman, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' White, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
+page_content=' Stoudenmire, The itensor software library for tensor network calculations (2020),  arXiv:2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFIT4oBgHgl3EQfrSul/content/2301.11331v1.pdf'}
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+LARGE MUSIC RECOMMENDATION STUDIES FOR SMALL TEAMS
+Kyle Robinson
+Dan Brown
+David R. Cheriton School of Computer Science, University of Waterloo, Canada
+kyle.robinson@uwaterloo.ca, dan.brown@uwaterloo.ca
+ABSTRACT
+Running live music recommendation studies without di-
+rect industry partnerships can be a prohibitively daunting
+task, especially for small teams. In order to help future re-
+searchers interested in such evaluations, we present a num-
+ber of struggles we faced in the process of generating our
+own such evaluation system alongside potential solutions.
+These problems span the topics of users, data, computa-
+tion, and application architecture.
+1. RUNNING A LIVE RECOMMENDATION
+STUDY
+There are clearly benefits to evaluating music recom-
+mender systems with real users [1]. In our recent paper
+analysing user perceptions of diversity in music recom-
+mendation we found that mere accuracy evaluations are
+not necessarily good indicators of individual track ratings,
+and overall list satisfaction is not a function of individual
+track ratings alone [2]. Insights such as this are not new;
+a decade and a half ago McNee et al. informally argued
+that there must be more emphasis put on user-centric rec-
+ommender system evaluation [3], yet just two years ago
+Dacrema et al. highlighted a disturbing lack of attention to
+evaluation even in strictly offline analyses [4].
+User studies and online analyses require significantly
+more resources and time than strictly offline analyses. In
+the hope of assisting researchers completing live evalua-
+tions of their methods, we present some of the struggles
+faced in developing our recent study, and their resolutions.
+For further reading on live evaluation of recommender sys-
+tems we refer readers to the relevant chapters of the Rec-
+ommender Systems Handbook [1,5].
+1.1 General Architecture
+The goals of our system were twofold: to generate up-to-
+date music recommendations for previously unseen par-
+ticipants using the same models described in pre-existing
+research, and to generate these recommendations on de-
+mand.
+The final system consisted of an online appli-
+© K. Robinson, and D. Brown. Licensed under a Creative
+Commons Attribution 4.0 International License (CC BY 4.0). Attribu-
+tion: K. Robinson, and D. Brown, “Large Music Recommendation Stud-
+ies for Small Teams”, in Proc. of the 22nd Int. Society for Music Infor-
+mation Retrieval Conf., Online, 2021.
+cation which, after obtaining consent, collected partici-
+pants’ listening histories from the LastFM API, fed their
+data through several recommendation algorithms to obtain
+top-n lists, obtained music previews and metadata from
+the Spotify API, and displayed song previews alongside
+song-specific appropriateness questions and global sum-
+mary questions. We implemented the system using a Flask
+backend which served static HTML and Javascript 1 . The
+study was hosted on a single AWS EC2 server instance us-
+ing Elastic Beanstalk.
+1.2 Users and Data
+1.2.1 Up-to-date Training Data
+A first challenge is that for collaborative filtering recom-
+mendation (especially for music) training data must be col-
+lected as close to the study as possible in order ensure that
+participant data is known by the model. Real data is also
+difficult to locate and obtain.
+In the domain of music, LastFM continues to provide
+a useful API for user-song listening event (LE) data col-
+lection 2 , though collection is not always a straightfor-
+ward process. We collected a base data set from pseudo-
+randomly selected users to train our model by crawling the
+public social graph of friends lists. Contacting authors of
+prior research utilizing data sets which fit our needs proved
+to be vital in developing a method of data collection. Al-
+though their data was not up-to-date, we were able to de-
+velop our own collection method after corresponding. This
+data was topped-up before each subsequent study, and ap-
+propriately randomized.
+1.2.2 Live Participant Data
+An especially interesting challenge was that we needed to
+be able to collect data from participants as they connected
+to the system. This data also needed to be as current as
+possible.
+Our solution to this problem was to use a media-
+tracking application. LastFM and its associated API also
+worked well for this purpose. For smaller pools of par-
+ticipants, we helped them register an account and manu-
+ally monitored it over a collection period of a few weeks.
+For larger pools of participants we specified in recruit-
+ment and consent materials that they must have an exist-
+1 An un-maintained repository of our application can be found at
+https://github.com/Stack-Attack/music_rec_div_study
+2 The
+LastFM
+API
+documentation
+can
+be
+accessed
+at https://www.last.fm/api
+arXiv:2301.13388v1  [cs.HC]  31 Jan 2023
+
+ing account containing some minimum number of listening
+events/plays/scrobbles. Amazon Mechanical Turk specifi-
+cally does not allow researchers to ask participants to log
+into any accounts, but because LastFM accounts are pub-
+licly accessible by default, data can be accessed with only
+a username. It is worth noting that in our case, some par-
+ticipants appear to have created new accounts to complete
+the study without being prompted to do so.
+1.2.3 Showing Music Recommendations
+To evaluate a recommendation, participants need to be able
+to listen to it!
+Music previews can typically be accessed without hav-
+ing to authenticate with a music service. We used the Spo-
+tify API to obtain 30 second previews with album art in the
+form of HTML iframes embedded in the page 3 . The rel-
+evant track previews were retrieved by searching for tracks
+using their exact song and artist names as well as the re-
+gion a participant was connecting form. We discarded the
+small portion of tracks that did not return any results; this
+is likely unavoidable.
+1.3 Computation
+1.3.1 Model Training
+We trained two different ML models for our project, one
+more traditional model based on matrix factorization, and
+one more modern approach based on neural networks.
+Writing the necessary code to implement models effi-
+ciently is time-consuming and error prone. Additionally,
+training models on huge data sets seems infeasible due to
+size and dimensionality.
+Using open-source libraries can save time and help alle-
+viate the risk of errors impacting results, though they may
+mis-implement key algorithmic features, or be difficult to
+extend. Comparing multiple implementations online can
+help, but one must ensure to follow any licenses and refer-
+ence the source.
+1.3.2 Data Handling
+In order to reduce the size of data, we filtered out irrelevant
+items and users. By filtering out tracks with 10 or fewer
+LEs we reduced the number of unique tracks by 82% while
+only decreasing LE count by 6%. Even after filtering, our
+Variational Autoencoder (MultVAE) was too large to fit in
+GPU memory, and so we trained multiple model variations
+concurrently using CPU’s in order to make up for lost time.
+Some models may simply be infeasible without access to
+High Performance Computing (HPC) resources.
+Other
+models may simply not be suited to real-world implemen-
+tations without significant structural changes and/or pre-
+processing steps. This is simply the reality of evaluating
+models on real populations.
+3 Details on embedding Spotify music previews can be found
+at https://developer.spotify.com/documentation/widgets/guides/adding-a-
+spotify-embed/
+1.3.3 Complex Architecture and Resource Requirements
+The size of trained models is too large to fit in memory,
+especially if a new model is loaded for each server connec-
+tion.
+The size of trained models can be very large even af-
+ter removing unnecessary data (i.e., neural network opti-
+mizer information). Our trained MultVAE model was over
+6.5GB in size. Luckily, online cloud computing platforms
+often offer specific instances with large amounts of dedi-
+cated memory at the expense of processing power. These
+instances are a great fit for running user-studies which will
+inherently have a low number of concurrent users. As of
+now, simple hosting services such as Heroku will be infea-
+sible due to the memory requirements [6]. AWS Elastic
+Beanstalk, however, provides very cost effective solutions
+in the form of memory optimized EC2 instances [7].
+Even with the low number of concurrent users, there
+will still be some asynchronous computation required. The
+ideal, yet complex, solution to this problem is to decou-
+ple the longer tasks (recommendation and data collection)
+from the main server using a separate worker process or
+even server.
+Developing this architecture can be time-
+consuming, expensive, and unnecessary for such small
+temporary applications. We found success by limiting the
+server to one Python process, and running data collection
+and recommendation on separate threads using the built in
+concurrent.futures library 4 . As live user data collection
+was input/output bound it did not block the server from
+handling requests, and as most recommendation and pro-
+cessing tasks utilized multi-core optimized libraries these
+tasks were also handled relatively quickly. This kind of ar-
+chitecture would certainly not work for large-scale appli-
+cations, but was ideal for our small user-count study due to
+its simplicity and efficient use of only one compute node.
+1.4 Summary
+The benefits of evaluating music recommender systems on
+real users are as intuitive as they are founded in empirical
+evaluation [1]. Without industry collaboration, and at min-
+imal cost, we were able to develop a music recommenda-
+tion system which could generate and present recommen-
+dations to new users within a single un-moderated interac-
+tive session. To assist and encourage future researchers in
+developing similar systems, we have described some of the
+challenges and solutions to problems encountered along
+the way. Among the problems we addressed were train-
+ing data collection, live user data collection, and obtain-
+ing music previews. We also discussed our technical im-
+plementation; specifically dealing with issues of memory
+management and availability. In general, we hope that re-
+searchers embrace collaboration with others to better base
+our analysis of recommender systems in users themselves.
+Our key message is that independent user studies on music
+recommendation are both important and achievable.
+4 In practice, we used the Flask-Executor python library to manage our
+futures: https://pypi.org/project/Flask-Executor/
+
+2. REFERENCES
+[1] B. P. Knijnenburg and M. C. Willemsen, “Evalu-
+ating recommender systems with user experiments,”
+in Recommender Systems Handbook, Second Edition.
+Springer, 2015, pp. 309–352.
+[2] K. Robinson and D. Brown, “Quantitative User Per-
+ceptions of Music Recommendation List Diversity,” in
+Proceedings of the 22nd International Society for Mu-
+sic Information Retrieval Conference, 2021.
+[3] S. M. McNee, J. Riedl, and J. A. Konstan, “Being ac-
+curate is not enough: How accuracy metrics have hurt
+recommender systems,” in Conference on Human Fac-
+tors in Computing Systems - Proceedings, 2006, pp.
+1097–1101.
+[4] M. F. Dacrema, P. Cremonesi, and D. Jannach, “Are
+we really making much progress? A worrying anal-
+ysis of recent neural recommendation approaches,” in
+Proceedings of the 13th ACM Conference on Recom-
+mender Systems, 2019, pp. 101–109.
+[5] A. Gunawardana and G. Shani, “Evaluating recom-
+mender systems,” in Recommender Systems Hand-
+book, Second Edition.
+Springer, 2015, pp. 265–308.
+[6] Heroku, “Dyno Types,” https://devcenter.heroku.com/
+articles/dyno-types, 2021.
+[7] A. W. Services, “EC2 Instance Types,” https://aws.
+amazon.com/ec2/instance-types/, 2021.
+
diff --git a/p9FQT4oBgHgl3EQfsja5/content/tmp_files/load_file.txt b/p9FQT4oBgHgl3EQfsja5/content/tmp_files/load_file.txt
new file mode 100644
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@@ -0,0 +1,154 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf,len=153
+page_content='LARGE MUSIC RECOMMENDATION STUDIES FOR SMALL TEAMS  Kyle Robinson  Dan Brown  David R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content=' Cheriton School of Computer Science, University of Waterloo, Canada  kyle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content='robinson@uwaterloo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content='ca, dan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content='brown@uwaterloo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content='ca  ABSTRACT  Running live music recommendation studies without di-  rect industry partnerships can be a prohibitively daunting  task, especially for small teams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content=' In order to help future re-  searchers interested in such evaluations, we present a num-  ber of struggles we faced in the process of generating our  own such evaluation system alongside potential solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content='  These problems span the topics of users, data, computa-  tion, and application architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content=' RUNNING A LIVE RECOMMENDATION  STUDY  There are clearly benefits to evaluating music recom-  mender systems with real users [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content=' In our recent paper  analysing user perceptions of diversity in music recom-  mendation we found that mere accuracy evaluations are  not necessarily good indicators of individual track ratings,  and overall list satisfaction is not a function of individual  track ratings alone [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content=' Insights such as this are not new;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content='  a decade and a half ago McNee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content=' informally argued  that there must be more emphasis put on user-centric rec-  ommender system evaluation [3], yet just two years ago  Dacrema et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content=' highlighted a disturbing lack of attention to  evaluation even in strictly offline analyses [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content='  User studies and online analyses require significantly  more resources and time than strictly offline analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content=' In  the hope of assisting researchers completing live evalua-  tions of their methods, we present some of the struggles  faced in developing our recent study, and their resolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content='  For further reading on live evaluation of recommender sys-  tems we refer readers to the relevant chapters of the Rec-  ommender Systems Handbook [1,5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content='1 General Architecture  The goals of our system were twofold: to generate up-to-  date music recommendations for previously unseen par-  ticipants using the same models described in pre-existing  research, and to generate these recommendations on de-  mand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content='  The final system consisted of an online appli-  © K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content=' Robinson, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content=' Brown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content=' Licensed under a Creative  Commons Attribution 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content='0 International License (CC BY 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content='0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content=' Attribu-  tion: K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content=' Robinson, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content=' Brown, “Large Music Recommendation Stud-  ies for Small Teams”, in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content=' of the 22nd Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content=' Society for Music Infor-  mation Retrieval Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content=', Online, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content='  cation which, after obtaining consent, collected partici-  pants’ listening histories from the LastFM API, fed their  data through several recommendation algorithms to obtain  top-n lists, obtained music previews and metadata from  the Spotify API, and displayed song previews alongside  song-specific appropriateness questions and global sum-  mary questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content=' We implemented the system using a Flask  backend which served static HTML and Javascript 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content=' The  study was hosted on a single AWS EC2 server instance us-  ing Elastic Beanstalk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content='2 Users and Data  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content='1 Up-to-date Training Data  A first challenge is that for collaborative filtering recom-  mendation (especially for music) training data must be col-  lected as close to the study as possible in order ensure that  participant data is known by the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content=' Real data is also  difficult to locate and obtain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content='  In the domain of music, LastFM continues to provide  a useful API for user-song listening event (LE) data col-  lection 2 , though collection is not always a straightfor-  ward process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content=' We collected a base data set from pseudo-  randomly selected users to train our model by crawling the  public social graph of friends lists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content=' Contacting authors of  prior research utilizing data sets which fit our needs proved  to be vital in developing a method of data collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content=' Al-  though their data was not up-to-date, we were able to de-  velop our own collection method after corresponding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content=' This  data was topped-up before each subsequent study, and ap-  propriately randomized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content='2 Live Participant Data  An especially interesting challenge was that we needed to  be able to collect data from participants as they connected  to the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content=' This data also needed to be as current as  possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content='  Our solution to this problem was to use a media-  tracking application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content=' LastFM and its associated API also  worked well for this purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content=' For smaller pools of par-  ticipants, we helped them register an account and manu-  ally monitored it over a collection period of a few weeks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content='  For larger pools of participants we specified in recruit-  ment and consent materials that they must have an exist-  1 An un-maintained repository of our application can be found at  https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content='com/Stack-Attack/music_rec_div_study  2 The  LastFM  API  documentation  can  be  accessed  at https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content='last.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content='fm/api  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content='13388v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content='HC]  31 Jan 2023  ing account containing some minimum number of listening  events/plays/scrobbles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content=' Amazon Mechanical Turk specifi-  cally does not allow researchers to ask participants to log  into any accounts, but because LastFM accounts are pub-  licly accessible by default, data can be accessed with only  a username.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content=' It is worth noting that in our case, some par-  ticipants appear to have created new accounts to complete  the study without being prompted to do so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content='3 Showing Music Recommendations  To evaluate a recommendation, participants need to be able  to listen to it!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content='  Music previews can typically be accessed without hav-  ing to authenticate with a music service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content=' We used the Spo-  tify API to obtain 30 second previews with album art in the  form of HTML iframes embedded in the page 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content=' The rel-  evant track previews were retrieved by searching for tracks  using their exact song and artist names as well as the re-  gion a participant was connecting form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content=' We discarded the  small portion of tracks that did not return any results;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content=' this  is likely unavoidable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content='3 Computation  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content='1 Model Training  We trained two different ML models for our project, one  more traditional model based on matrix factorization, and  one more modern approach based on neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content='  Writing the necessary code to implement models effi-  ciently is time-consuming and error prone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content=' Additionally,  training models on huge data sets seems infeasible due to  size and dimensionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content='  Using open-source libraries can save time and help alle-  viate the risk of errors impacting results, though they may  mis-implement key algorithmic features, or be difficult to  extend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content=' Comparing multiple implementations online can  help, but one must ensure to follow any licenses and refer-  ence the source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content='2 Data Handling  In order to reduce the size of data, we filtered out irrelevant  items and users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content=' By filtering out tracks with 10 or fewer  LEs we reduced the number of unique tracks by 82% while  only decreasing LE count by 6%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content=' Even after filtering, our  Variational Autoencoder (MultVAE) was too large to fit in  GPU memory, and so we trained multiple model variations  concurrently using CPU’s in order to make up for lost time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content='  Some models may simply be infeasible without access to  High Performance Computing (HPC) resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content='  Other  models may simply not be suited to real-world implemen-  tations without significant structural changes and/or pre-  processing steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content=' This is simply the reality of evaluating  models on real populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content='  3 Details on embedding Spotify music previews can be found  at https://developer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content='spotify.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content='com/documentation/widgets/guides/adding-a-  spotify-embed/  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content='3 Complex Architecture and Resource Requirements  The size of trained models is too large to fit in memory,  especially if a new model is loaded for each server connec-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content='  The size of trained models can be very large even af-  ter removing unnecessary data (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content=', neural network opti-  mizer information).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content=' Our trained MultVAE model was over  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content='5GB in size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content=' Luckily, online cloud computing platforms  often offer specific instances with large amounts of dedi-  cated memory at the expense of processing power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content=' These  instances are a great fit for running user-studies which will  inherently have a low number of concurrent users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content=' As of  now, simple hosting services such as Heroku will be infea-  sible due to the memory requirements [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content=' AWS Elastic  Beanstalk, however, provides very cost effective solutions  in the form of memory optimized EC2 instances [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content='  Even with the low number of concurrent users, there  will still be some asynchronous computation required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content=' The  ideal, yet complex, solution to this problem is to decou-  ple the longer tasks (recommendation and data collection)  from the main server using a separate worker process or  even server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content='  Developing this architecture can be time-  consuming, expensive, and unnecessary for such small  temporary applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content=' We found success by limiting the  server to one Python process, and running data collection  and recommendation on separate threads using the built in  concurrent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content='futures library 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content=' As live user data collection  was input/output bound it did not block the server from  handling requests, and as most recommendation and pro-  cessing tasks utilized multi-core optimized libraries these  tasks were also handled relatively quickly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content=' This kind of ar-  chitecture would certainly not work for large-scale appli-  cations, but was ideal for our small user-count study due to  its simplicity and efficient use of only one compute node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content='4 Summary  The benefits of evaluating music recommender systems on  real users are as intuitive as they are founded in empirical  evaluation [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content=' Without industry collaboration, and at min-  imal cost, we were able to develop a music recommenda-  tion system which could generate and present recommen-  dations to new users within a single un-moderated interac-  tive session.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content=' To assist and encourage future researchers in  developing similar systems, we have described some of the  challenges and solutions to problems encountered along  the way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content=' Among the problems we addressed were train-  ing data collection, live user data collection, and obtain-  ing music previews.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content=' We also discussed our technical im-  plementation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content=' specifically dealing with issues of memory  management and availability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content=' In general, we hope that re-  searchers embrace collaboration with others to better base  our analysis of recommender systems in users themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content='  Our key message is that independent user studies on music  recommendation are both important and achievable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content='  4 In practice, we used the Flask-Executor python library to manage our  futures: https://pypi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content='org/project/Flask-Executor/  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content=' REFERENCES  [1] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content=' Knijnenburg and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content=' Willemsen, “Evalu-  ating recommender systems with user experiments,”  in Recommender Systems Handbook, Second Edition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content='  Springer, 2015, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content=' 309–352.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content='  [2] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content=' Robinson and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content=' Brown, “Quantitative User Per-  ceptions of Music Recommendation List Diversity,” in  Proceedings of the 22nd International Society for Mu-  sic Information Retrieval Conference, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content='  [3] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content=' McNee, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content=' Riedl, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content=' Konstan, “Being ac-  curate is not enough: How accuracy metrics have hurt  recommender systems,” in Conference on Human Fac-  tors in Computing Systems - Proceedings, 2006, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content='  1097–1101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content='  [4] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content=' Dacrema, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content=' Cremonesi, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content=' Jannach, “Are  we really making much progress?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content=' A worrying anal-  ysis of recent neural recommendation approaches,” in  Proceedings of the 13th ACM Conference on Recom-  mender Systems, 2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content=' 101–109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content='  [5] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content=' Gunawardana and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content=' Shani, “Evaluating recom-  mender systems,” in Recommender Systems Hand-  book, Second Edition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content='  Springer, 2015, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content=' 265–308.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content='  [6] Heroku, “Dyno Types,” https://devcenter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content='heroku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content='com/  articles/dyno-types, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content='  [7] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content=' Services, “EC2 Instance Types,” https://aws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content='  amazon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
+page_content='com/ec2/instance-types/, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FQT4oBgHgl3EQfsja5/content/2301.13388v1.pdf'}
diff --git a/pNAzT4oBgHgl3EQfqv3l/content/tmp_files/2301.01635v1.pdf.txt b/pNAzT4oBgHgl3EQfqv3l/content/tmp_files/2301.01635v1.pdf.txt
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+1
+SPTS v2: Single-Point Scene Text Spotting
+Yuliang Liu1, Jiaxin Zhang2, Dezhi Peng3, Mingxin Huang3, Xinyu Wang4, Jingqun Tang2, Can Huang2,
+Dahua Lin5, Chunhua Shen6, Xiang Bai1, Lianwen Jin∗3,
+Abstract—End-to-end scene text spotting has made significant progress due to its intrinsic synergy between text detection and
+recognition. Previous methods commonly regard manual annotations such as horizontal rectangles, rotated rectangles, quadrangles,
+and polygons as a prerequisite, which are much more expensive than using single-point. For the first time, we demonstrate that training
+scene text spotting models can be achieved with an extremely low-cost single-point annotation by the proposed framework, termed
+SPTS v2. SPTS v2 reserves the advantage of the auto-regressive Transformer with an Instance Assignment Decoder (IAD) through
+sequentially predicting the center points of all text instances inside the same predicting sequence, while with a Parallel Recognition
+Decoder (PRD) for text recognition in parallel. These two decoders share the same parameters and are interactively connected with a
+simple but effective information transmission process to pass the gradient and information. Comprehensive experiments on various
+existing benchmark datasets demonstrate the SPTS v2 can outperform previous state-of-the-art single-point text spotters with fewer
+parameters while achieving 14× faster inference speed. Most importantly, within the scope of our SPTS v2, extensive experiments
+further reveal an important phenomenon that single-point serves as the optimal setting for the scene text spotting compared to
+non-point, rectangular bounding box, and polygonal bounding box. Such an attempt provides a significant opportunity for scene text
+spotting applications beyond the realms of existing paradigms. Code is available at https://github.com/shannanyinxiang/SPTS.
+Index Terms—Scene text spotting, Transformer, Single-point annotation
+!
+1
+INTRODUCTION
+S
+Cene text reading techniques have made great strides in
+recent years. Given an image, text spotters can simulta-
+neously locate and recognize the textual content, enabling
+many real-world applications such as document digitaliza-
+tion, intelligent assistants, and autopilot. Basically, bound-
+ing boxes such as rectangles, quadrilaterals, and polygons
+are commonly employed to represent the text of different
+shapes. However, the fact that humans can intuitively read
+texts without such a defined region encourages the de-
+velopment of a bounding-box-free text spotter, lifting the
+limitations imposed by bounding-box annotations.
+As shown in Fig. 1, the previous methods use a bound-
+ing box consisting of a series of coordinates to define the
+instance-level text, where the enclosed region is considered
+a positive sample. With its simplicity and straightforward-
+ness, the bounding box has become the favored annotation
+format for many other vision tasks. However, unlike the
+target in object detection tasks that are usually presented
+in a defined appearance, text instances may appear in ar-
+bitrary shapes due to the different typographies and fonts.
+Therefore, it is required to use bounding boxes containing
+more coordinates, such as polygons, to label these arbi-
+trarily shaped texts. Otherwise, considerable noise might
+be involved, which may negatively impact the recognition
+performance. For example, Total-Text [1] uses up to 20 coor-
+dinates while SCUT-CTW1500 [2] uses up to 28 coordinates
+to annotate a single curved scene text instance. Although
+using polygons can, to some extent, alleviate the problem of
+noise in labeling arbitrarily shaped text, it greatly increases
+the annotation cost. To solve these issues, this paper pro-
+1Huazhong University of Science and Technology.
+2ByteDance.
+3South
+China University of Technology.
+4University of Adelaide.
+5The Chinese
+University of Hong Kong.
+6Zhejiang University.
+Part of this work was
+done when YL was with Chinese University of Hong Kong.
+NEXT
+42km
+Ways of Indication
+1. Pointing a direction → "NEXT" 
+2. Left top x1,y1; Right top. x2, y2,  
+    Left bot. x3,y3; Right bot. x4, y4 → "42km" 
+Fig. 1 – Existing OCR methods typically use bounding boxes
+to represent the text area. However, inspired by how humans
+can intuitively read texts without such a defined region, this
+paper demonstrates that a single attention point is sufficient for
+guiding the model to learn a strong scene text spotter.
+poses a brand-new manner of supervision for scene text
+spotters by using a single guidance point. As shown in
+Fig. 1, each of the texts is indicated by a single point within
+the instance. Such a streamlined representation breaks the
+limitation of bounding boxes, enabling the model to access
+the pixels in vicinity freely and further learn to discriminate
+the boundary between texts. Also, it considerably saves the
+cost of annotation compared to the polygons.
+In the past decade, the focus of research on scene text
+spotting has shifted from horizontal [3], [4] and multi-
+oriented text [2], [5], [6] to arbitrarily shaped text [7], [8],
+as reflected in the transition from rectangular and quadri-
+lateral annotations to more compact but more expensive
+polygons. As shown in Fig. 2, the rectangular bounding
+arXiv:2301.01635v1  [cs.CV]  4 Jan 2023
+
+NEXT
+42km2
+(a) Rectangle (55s)
+(b) Quadrilateral (96s)
+(c) Character (581s)
+(d) Polygon (172s)
+(e) Single-Point (11s)
+Fig. 2 – Different annotation styles and their time cost (for all the text instances in the sample image) measured by the
+LabelMe1tool. Green areas are positive samples, while red dashed boxes are noises that may be possibly included. Note the time
+is measured by the average of three annotators. For representations other than point, they normally need to zoom in for alignment
+of the exact location, which consumes non-trivial annotation efforts.
+boxes are prone to involve other text instances, which may
+confuse the subsequent scene text recognition. Furthermore,
+many efforts have been made to develop more sophisticated
+representations to fit arbitrarily shaped text instances [7],
+[9], [10], [11], [12], [13]. For example, as shown in Fig. 3,
+Mask TextSpotter [14] utilizes boundary polygons to localize
+the text region. Text Dragon [10] utilizes character-level
+bounding boxes to generate centerlines for enabling the
+prediction of local geometry attributes, ABCNet [7] con-
+verts polygon annotations to Bezier curves for representing
+curved text instances, and Text Snake [13] describes text
+instances by a series of ordered disks centered at symmetric
+axes. These heuristic representations are carefully designed
+by knowledgeable experts. Although they have been shown
+to be effective for aligning features between text detec-
+tion and recognition modules, the reliance on manually-
+designed rules undeniably undermines the generalizability.
+Specifically, specified network architectures and modules
+are required to process the features and annotations, such as
+variants of RoI modules and post-processing mechanisms.
+In addition, as shown in Fig. 2, the above representations
+that rely on annotations of polygonal or character bounding
+boxes are costly for labeling, while the proposed single point
+can halve the cost.
+In the past few years, some researchers [15], [16], [17],
+[18] have explored training the OCR models with coarse
+annotations in a weakly-supervised manner. These meth-
+ods can mainly be separated into two categories, i.e., (1)
+bootstrapping labels to finer granularity [15], [16] and (2)
+training with partial annotations [17], [18]. The former
+usually derives character-level labels from word-level or
+line-level annotations; thus, the models could enjoy the
+well-understood advantage of character-level supervision
+without introducing overhead costs. The latter is committed
+to achieving competitive performance with fewer training
+samples. However, both methods still rely on the bounding
+box annotations. A recent study [19] demonstrates that
+point-only annotation for scene text can still achieve com-
+petitive performance in scene text spotting tasks.
+One of the underlying problems to replace the bounding
+box with a simpler annotation format, such as a single-point,
+is that most text spotters rely on RoI-like sampling strategies
+to extract the shared backbone features. For example, Li
+et al. [20] and Mask TextSpotter require box and mask
+prediction inside an RoI [21]; ABCNet [7] proposes Bezier-
+(a) Mask TextSpotter [14]
+(b) Text Dragon [10]
+(c) ABCNet [7]
+(d) Text Snake [13]
+Fig. 3 – Some recent representations of text instances. (a)
+Mask TextSpotter utilizes character-level and polygonal bound-
+ing boxes to predict the compact bounding box. (b) Text
+Dragon [10] employs character-level bounding box to generate
+text centerline. (c) ABCNet [7] converts polygon annotations to
+Bezier curves. (d) Text Snake [13] describes text instances by a
+series of ordered disks centered at symmetric axes.
+Align to wrap the curved representation into the horizontal
+format, while TextDragon [10] introduces RoISlide to unify
+the detection and recognition heads.
+In this paper, inspired by the recent success of a
+sequence-based object detector Pix2Seq [22], we show that
+the text spotter can be trained with a single point. Due
+to the concise form of annotation, annotation time can be
+significantly saved, e.g., it only takes less than one-fiftieth of
+the time to label single-points for the sample image shown
+in Fig. 2 than annotating character-level bounding boxes.
+Another motivating factor in selecting point annotation is
+that a clean and efficient OCR pipeline can be developed,
+discarding the complex post-processing module and roi-
+based sampling strategies; thus, the ambiguity introduced
+by RoIs (see red dashed regions in Fig. 2) can be alleviated.
+However, adopting single-point representation is still
+challenging. The previous state-of-the-art single-point text
+spotter (SPTS) in our conference version [19] uses the
+auto-regressive Transformer to generate the long sequence
+
+JEWI
+ELR
+MAI
+RKEYITALIAN
+EST2015
+g
+HAHNDORFEST2015
+HAHNDORE
+DINNERHAHNDORF
+NNERITALIAN
+EST2015
+g
+HAHNDORFEST2015
+HAHNDORE
+DINNERHAHNDORF
+NNERITALIAN
+EST2015
+g
+HAHNDORFEST2015
+HAHNDORE
+DINNERHAHNDORF
+NNERITALIAN
+EST2015
+g
+HAHNDORFEST2015
+HAHNDORE
+DINNERHAHNDORF
+NNERITALIAN
+CUISINE
+EST2015
+HAHNDORFC3
+for all text instances. Here, point-only prediction is very
+straightforward and can handle unordered text instances
+as a whole. Thus it can learn to avoid repeated predic-
+tions, specified label assign methods like bipartite matching
+in DETR [23], or complicated post-processing like non-
+maximum-suppression (NMS) [24]. Although our confer-
+ence version SPTS [19] is effective, the inference speed of it is
+extremely low, especially the images where many instances
+are simultaneously included.
+Therefore, to take advantage of auto-regressive predic-
+tion and preserve high efficiency at the same time, we design
+SPTS v2, which significantly improves the inference speed while
+achieving higher performance.
+Specifically, we observe that
+the long sequence of the result is mainly occupied by the
+text recognition, and thus we can only predict the location,
+e.g., x and y for each instance auto-regressively, termed as
+an indicator, in the first instance assignment decoder (IAD);
+while for the second parallel recognition decoder (PRD),
+each indicator is responsible for its subsequent text recogni-
+tion prediction, which can be implemented in parallel. The
+rationale behind this is that the IAD is to solve implicit
+label assignment while PRD is for parallel prediction of
+the responsible text recognition result given the features
+after the label assignment. To enable the gradient flow from
+recognition features to the IAD stage, we propose a simple
+yet effective information transmission method that inte-
+grates the embeddings of the text location and the features
+in IAD stage, which is demonstrated essential to the success
+of the SPTS v2.
+We summarize our contributions as follows:
+•
+For the first time, we demonstrate that the text
+spotters can be supervised by a simple yet effec-
+tive single-point representation and the text recogni-
+tion information. Such a straightforward annotation
+paradigm can considerably reduce the labeling costs,
+avoid the label inconsistent, and make it possible to
+access large-scale OCR data due to the low-cost an-
+notation format. More importantly, we demonstrate
+that under such pipeline, the point annotation can
+be served as the optimal setting for the text spotting
+compared to non-point, rectangular bounding box,
+and polygonal bounding box.
+•
+We form the text spotting as a language modeling
+task, which requires only the identical cross-entropy
+loss. Benefiting from such a concise pipeline, the
+complex post-processing and sampling strategies de-
+signed based on prior knowledge can be discarded,
+showing great potential in terms of flexibility and
+generality.
+•
+SPTS v2 introduces a new Instance Assignment De-
+coder (IAD) as well as the Parallel Recognition De-
+coder (PRD) by sharing the same parameters, which
+are incorporated by a simple but effective informa-
+tion transmission method. SPTS v2 outperforms the
+conference version SPTS, with fewer parameters and
+14x faster inference speed.
+•
+Extensive experiments and ablations conducted on
+five benchmarks, i.e., ICDAR 2013 [25], ICDAR
+2015 [26], Total-Text [1], SCUT-CTW1500 [27], and
+Inverse-Text [28] involving both horizontal and ar-
+bitrarily shaped texts demonstrate the competitive
+performance of our method against previous state-
+of-the-arts.
+2
+RELATED WORK
+In the past decades, a variety of annotation styles have
+been proposed focusing on various scenarios of scene text
+spotting, including letters represented by stroke-level or
+character-level bounding box [8], [14], [21], [29], [30], hor-
+izontal text [25], [31] represented by rectangles (Fig. 2(a)),
+multi-oriented text [26], [32] represented by quadrilaterals
+(Fig. 2(b)), arbitrary-shaped text [1], [27], [33] represented
+by polygons (Fig. 2(d)), and other novel representation such
+as single-point [19], [34] or non-point [19], [35].
+2.1
+Character-level Scene Text Spotter
+In the early stage, many classical methods require character-
+level annotations to train the model. Wang et al. [36] uses a
+character classifier based on the HOG [37] features to read
+text. Bissacco et al. [38] combine DNN with HOG features to
+build a character classifier system for text recognition. The
+follow-up work [39] further develops a convolutional neural
+network to be the character classifier. The above methods are
+adjusted to the horizontal text using character-level annota-
+tions. Some researchers attempt to extend the character-level
+scene text spotter to handle the arbitrarily-shaped text. Mask
+TextSpotter [8] designs a character segmentation module to
+locate and recognize the character; its improved version [14],
+[21] significantly reduce the cost of manual annotations.
+CharNet [29] proposes a one-stage framework to boost the
+text spotting performance by utilizing character-level anno-
+tations. CFRATS [40] locates the character regions and sends
+the information of the character regions to the attention-
+based recognizer. The MANGO [30] develops a position-
+aware mask attention module to generate a location mask
+for the character and use a sequence decoder to obtain the
+recognition results.
+2.2
+Rectangle-based Scene Text Spotter
+The rectangle-based scene text spotter plays an important
+role in the early stage of the task. Weinman et al. [41]
+propose a text spotting system that first generates the text
+proposal and then use an independent word recognition
+model to extract the text content. Li et al. [20] adapt a
+generic object detectors framework Faster R-CNN [42] to
+detect the rectangle-shape text and bridge the detector
+and CTC-based [43] recognizer through sharing backbone.
+Its enhanced version [44], equipped with a 2D attention
+recognition module, is designed to handle the irregular text.
+Gupta et al. [45] employ FCRN to detect the rectangle boxes
+and use a word classifier as the recognizer. Recently, Liao
+et al. [46] propose a text spotting systems based on the
+TextBoxes [4] and the CRNN [47], which are used to locate
+and recognize the words, respectively. Shi et al. [48] use the
+TextBoxes as the detector to obtain the detection results and
+propose a new recognizer ASTER [49] which develops a
+Thin-Plate Spline transformation as rectification network to
+rectify the recognition image.
+1. https://github.com/wkentaro/labelme
+
+4
+2.3
+Multi-oriented Scene Text Spotter
+Recent methods develop multi-oriented scene text spot-
+ters to handle text instances with complex shapes. Busta
+et al. [50] propose a Deep TextSpotter which uses the
+YOLOv2 [51] as the detector to detect the multi-oriented
+text and use a CTC-based recognizer to transform the recog-
+nition features into a sequence of characters. FOTS [52]
+proposes a new RoI operation, termed RoI Rotate, to trans-
+form oriented text recognition features to regular ones from
+quadrilateral detection results. He et al. [53] propose a simi-
+lar framework to locate the instance of the text. They further
+develop a Text-Alignment to sample rotated features into
+the horizontal feature and use an attention-based recognizer
+to improve the performance.
+2.4
+Arbitrarily-shaped Scene Text Spotter
+Arbitrarily-shaped scene text spotting is challenging due to
+the diversity of text, such as various shapes, colors, fonts,
+and languages, which has attracted increasing attention. In
+this aspect, the text is normally annotated with polygons
+of arbitrary shapes. Recently, Qin et al. [54] propose an
+RoI Masking to suppress the background noise for the
+recognition features and use a 2D attention-based recog-
+nizer to read the arbitrarily-shaped text from recognition
+features. Wang et al. [55] design a spotting system, termed
+PAN++, based on the fast detector PAN [56]. TextNet [57]
+predicts the quadrilateral text proposal to locate the text
+and develops a perspective RoI transformation process to
+rectify the quadrilateral features. Feng et al. [10] describe
+the text instances as a series of quadrangles and propose
+the RoISlide to connect the quadrangles for text recognition.
+Wang et al. [58] detect the oriented rectangular box and
+transform the oriented rectangular box into a boundary.
+The boundary is served as the fiducial points for the Thin-
+Plate-Spline transformation to rectify the irregular text into
+a regular one. Qiao et al. [59] use a similar way which
+develops a segmentation detector to generate the fiducial
+points. ABCNet [7] uses a one-stage detector [60], which
+incorporates the parameterized Bezier curves to represent
+the text instance as well as a new RoI operation (Beiz-
+erAlign) for sampling the arbitrarily-shaped text features
+into the horizontal format. Its improved version [61] adopts
+the BiFPN [62] as the backbone and uses an attention-
+based recognizer to further improve the performance. Swin-
+TextSpotter [63] further exploits a new way of synergy be-
+tween the detection and recognition, termed the Recognition
+Conversion module, to make detection differentiable with
+recognition loss. TESTR [64] designs a single encoder and
+dual decoder structure based on the Deformable-DETR [65]
+to remove the hand-designed components. TTS [66] adds an
+RNN recognition head in Deformable-DETR and proposes
+a weakly-supervised method using only text transcription
+annotations to train the model. ABINet++ [67] uses the
+framework in [61] and further use a recognizer with an au-
+tonomous, bidirectional, and iterative language model [68]
+to improve the performance.
+2.5
+Point-based Scene Text Spotter
+In this paper, we propose a new Single-Point Text Spotting
+method, which is, to the best of our knowledge, the first
+scene text spotter that does not rely on bounding box
+annotations. Specifically, each text instance is represented
+by a single point (see Fig. 2(e)) with a meager cost. The fact
+that this point does not need to be accurately marked in
+practice further demonstrates the possibility of text spotting
+learning in a weakly supervised manner, shedding a new
+scene text spotting paradigm with potential values toward
+applications of a much larger scale.
+3
+METHODOLOGY
+Most of the existing text spotting algorithms require cus-
+tomized modules to bridge the detection and recognition
+blocks, where backbone features are cropped and shared be-
+tween detection and recognition heads, e.g., BezierAlign [7],
+RoISlide [10], and RoIMasking [21].
+Inspired by the Pix2Seq [22], we cast the generic object
+detection problem as a language modeling task, based on
+an intuitive assumption that if a deep model knows what
+and where the target is, it can be taught to tell the results
+by the desired sequence. It forms a concise pipeline, where
+annotations with different attributes such as location coor-
+dinates and object categories can be integrated into a single
+sequence with only cross-entropy loss, enabling an end-to-
+end trainable framework without task-specific modules (e.g.,
+Region Proposal Networks and RoI pooling layers), which
+can thus be naturally adapted to the text spotting task. In
+this paper, we propose Single-Point Text Spotter (SPTS) v2.
+SPTS v2 tackles text detection and recognition as a sequence
+prediction task, solvable by a much more straightforward
+pipeline.
+Unlike Pix2Seq, we propose to form the predicting se-
+quence through the Instance Assignment Decoder (IAD)
+and the Parallel Recognition Decoder (PRD); the former
+IAD keeps the underlying essence of pix2seq by unifying
+all instances inside the identical sequence, while the PRD
+uses the isolated instance features generated from the IAD
+to predict the text contents in parallel, which significantly
+reduces the length of the sequence. More importantly, we
+propose to use the information transmission method to pass
+the gradient of the recognition results to different instances,
+which is essential to the final performance of the SPTS
+v2. Specifically, as shown in Fig. 4, each input image is
+first encoded by CNN and Transformer encoders to extract
+visual and contextual features. Then, the captured features
+are decoded by a Transformer decoder, where tokens are
+predicted in an auto-regressive manner. Unlike previous
+algorithms, we further simplify the bounding box to a
+corner point located at the upper left of the first character
+or the center of the text instance, described in Fig. 5, in the
+text instance. Benefiting from such a simple yet effective
+representation, the modules carefully designed based on
+prior knowledge, such as grouping strategies utilized in
+segmentation-based methods and feature sampling blocks
+equipped in box-based text spotters, can be eschewed.
+3.1
+Sequence Construction
+To express the target text instances by a sequence, it is
+required to convert the continuous descriptions (e.g., bound-
+ing boxes) to a discretized space. To this end, we simplify the
+
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+Fig. 4 – Overall framework of the proposed SPTS v2. The visual and contextual features are first extracted by a series of CNN and
+Transformer encoders. Then, the features are auto-regressively decoded into a sequence that contains localization and recognition
+information through IAD and PRD, respectively. For IAD, it predicts coordinates of all center points of text instances inside the
+same sequence, while for the PRD, the recognition results are predicted in parallel. Note that IAD shares identical parameters
+with PRD, and thus no additional parameters are introduced for the PRD stage.
+(a) Top-left
+(b) Central
+(c) Random
+Fig. 5 – Indicated points (red color) using different positions.
+<
+<SOS>
+x
+y
+x
+...
+x
+y
+x
+y
+...
+y
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+Input Seq.
+x
+y
+x
+...
+x
+y
+x
+...
+Output Seq.
+<EOS>
+y
+>
+...
+y
+...
+Fig. 6 – Input and output sequences of the Instance Assign-
+ment Decoder (IAD). Different colors represent different text
+instances.
+bounding box to a single point and use the variable-length
+transcription instead of the single-token object category.
+Although SPTS [19] has been proven effective, the main
+limitation is that the long sequence length will significantly
+slow down the inference speed. This is because the recogni-
+tion results normally are fixed to the maximum length equal
+to 25 and 100 for word-level and line-level text instances,
+respectively. To this end, in SPTS v2, we design the In-
+stance Assignment Decoder (IAD) and Parallel Recognition
+Decoder (PRD) to overcome such limitations.
+3.2
+Instance Assignment Decoder
+The auto-regressive decoder is known to be effective in
+literature [19], [22]; however, it is intuitive that this is a
+time-consuming solution given the long sequence of the
+Discretized Targets 
+⌊x1/w×nbins⌋, ⌊y1/h×nbins⌋ 
+⌊x2/w×nbins⌋, ⌊y2/h×nbins⌋ 
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+P,I,Z,Z,E,R,I,A,<PAD>,.. 
+P,A,R,A,D,I,S,O,<PAD>,..
+B,I,R,R,E,R,I,A,<PAD>,.. 
+Coordinates
+Transcription
+x1,y1
+x2,y2
+x3,y3
+Original Labels 
+PIZZERIA
+PARADISO
+BIRRERIA
+Coordinates
+Transcription
+Discretization
+ex3 ey3
+B
+I
+R
+R
+E
+R
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+A
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+Transcription
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+feats.
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+Target Sequence
+PIZZERIA
+PARADISO
+BIRRERIA
+Information Transmission
+∅
+∅
+∅
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+∅
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+∅
+ Parallel Recognition
+Decoder (PRD) 
+Fig. 7 – Input and output sequences of the Parallel Recogni-
+tion Decoder (PRD). Different colors represent different text
+instances. Given the features generated from information trans-
+mission, the recognition results are predicted in parallel until
+reaching the maximum length or the EOS symbols.
+text instances. To improve efficiency, the SPTS v2 divides
+detection and recognition into a two-stage workflow by
+sharing the same Transformer decoder. The first stage is
+called Instance Assignment Decoder (IAD). In the first stage,
+SPTS v2 only decodes the center point for every text instance
+until the end of the sequence comes. An intuitive pipeline is
+shown in Fig. 6.
+Specifically, the continuous coordinates of the central
+point of the text instance are uniformly discretized into
+integers between [1, nbins], where nbins controls the degree
+of discretization. For example, an image with a long side
+of 800 pixels requires only nbins = 800 to achieve zero
+quantization error. Note that the central point of the text
+instance is obtained by averaging the upper and lower
+midpoints as shown in Fig. 5 (b). As so far, a text instance
+can thereby be represented by a sequence of three parts, i.e.,
+[x, y, t], where (x, y) are the discretized coordinates and t is
+the transcription text that will be predicted in PRD. Notably,
+the transcriptions are inherently discrete, i.e., each of the
+
+SU
+AGUASU
+AGUAransformerSU
+AGUASU
+AGUAranstormelDREAMDREAMDREAMBIRRE6
+characters represents a category.
+<SOS> and <EOS> tokens are inserted into the head
+and tail of the sequence, indicating the start and the end
+of a sequence, respectively. Therefore, given an image that
+contains N text instances, the constructed sequence will in-
+clude 2N discrete tokens, where the text instances would be
+randomly ordered. In fact, as shown in previous works [22],
+the randomly ordered text instances can be effectively
+learned, and thus it achieves the label assignment for dif-
+ferent hidden features inconspicuously, which subtly avoids
+an explicit label assignment like using bipartite matching
+that plays a vital role for the DETR series [23], [65], [69],
+[70]. In fact, compared with other label assignments, the
+instance assignment is intuitively more efficient. The dense
+label assignment methods [7], [14] use the non-maximum
+suppression (NMS) to select the suitable detection results
+for recognition. The bipartite matching label assignment
+methods [64], [66] use a maximum number of instances
+to detect and recognize texts, which consumes additional
+computation for the empty text instances.
+3.3
+Parallel Recognition Decoder
+With the help of the IAD, we separate the different text
+instances. The content of different text instances will be ob-
+tained at the same time in the Parallel Recognition Decoder.
+Different from the generic object detection that categorizes
+objects into fixed categories, recognizing the text content
+is a sequence classification problem that has a variable
+length of the target sequence. This may cause misalignment
+issues and can consume more computational resources. To
+eliminate such problems, we first pad or truncate the texts
+to a fixed length K, where the <PAD> token is used
+to fill the vacancy for shorter text instances. In addition,
+supposing there are ncls categories of characters (e.g., 97 for
+English characters and symbols), the vocabulary size of the
+dictionary used to tokenize the sequence can be calculated
+as ncls + 3, where the extra three classes are for <PAD>,
+<SOS>, and <EOS> tokens. Empirically, we set the K and
+nbins to 25 (or 100 for SCUT-CTW1500) and 1,000, respec-
+tively, in our experiments. Moreover, the maximum value of
+nti is set to 60, which means the sequence containing more
+than 60 text instances will be truncated. An illustration of
+the PRD is shown in Fig. 7.
+We assume that one image includes N text instances,
+and every instance includes the maximum number of K
+characters. It takes Nv1 loops for SPTS to predict this image,
+where Nv1 is defined as:
+Nv1 = (2 + K) · N + 1.
+(1)
+While for SPTS v2, it only needs Nv2 for-loops, where Nv2
+is:
+Nv2 = 2 · N + K + 1,
+(2)
+with a K · (N − 1) reduction. In our implementation, N
+and K are set to 60 and 25, respectively. In this case, SPTS
+requires 1,621 auto-regressive loops while SPTS v2 requires
+only 146 loops, with 91.0% (1475/1621) reduction rate of
+the number of loops. Actually, inside the PRD, SPTS v2 can
+also take an early end if all instances have met the end of
+sequence symbol. Through PRD, the inference speed can be
+significantly improved.
+3.4
+Information Transmission
+The parameters of the above two decoders are shared
+and supervised by the detection and recognition gradients.
+However, there is information loss between different text
+instances. In the conference version [19], the information of
+the previously detected text can be sensed by the recognition
+token and the gradient of text recognition can be passed
+on to supervise the predictions of different text instances.
+Such interaction is also important for the Parallel Recog-
+nition Decoder in SPTS v2 to find the correct position of
+the text. To address this issue, we propose an information
+transmission method. Formally, we first extract the hidden
+text instance location features (short for feat. in Fig. 4) and
+the corresponding prediction results of the text location (e.g.,
+x1, y1). Then, we convert the text instance location results
+into an embeddings which is then added to the text instance
+location features. The process can be formulated as follow:
+embedxi = embedding(xi), i = 0, 1, 2, ...n.
+(3)
+embedyi = embedding(yi), i = 0, 1, 2, ...n.
+(4)
+exi = featxi + embedxi,
+(5)
+eyi = featyi + embedyi.
+(6)
+With the help of the information transmission, the gradient
+of later text recognition can be passed on to different text
+instances by the featxi or featyi, and the information of the
+previously detected text can be sensed by the recognition
+token in PRD stage through the features. PRD takes these
+prior information as the first two queries to instruct the
+decoder and thus recognize all the text instances in parallel,
+as shown in Fig. 7. Such a straightforward transmission is
+essential to the SPTS v2 in practice.
+3.5
+Model Training
+Since the SPTS v2 is trained to predict tokens, it only
+requires to maximize the likelihood loss at training time,
+which can be written as:
+L = max
+L
+�
+i=1
+wi log P(˜si|I, s1:i),
+(7)
+where I is the input image, ˜s is the output sequence, s is
+the input sequence, L is the length of the sequence, and wi
+is the weight of the likelihood of the i-th token, which is
+empirically set to 1. For both IAD and PRD, they share the
+same Transformer and require only the cross-entropy loss,
+maintaining a concise pipeline.
+3.6
+Inference
+At the inference stage, SPTS v2 first sequentially predicts the
+tokens of location in IAD until the end of the sequence token
+<EOS> occurs. Then, the information transmission will
+integrate the detection features to auto-regressively predict
+the text contents in parallel. The predicted sequence will
+subsequently be divided into multiple segments. Therefore,
+the tokens can be easily translated into point coordinates
+and transcriptions, yielding the text spotting results. In
+addition, the likelihood of all tokens in the corresponding
+segment is averaged and assigned as a confidence score to
+filter the original outputs, effectively removing redundant
+and false-positive predictions.
+
+7
+4
+EXPERIMENTS
+We report the experimental results on five benchmarks, in-
+cluding horizontal dataset ICDAR 2013 [25], multi-oriented
+dataset ICDAR 2015 [26], and arbitrarily shaped datasets
+Total-Text
+[1]
+and
+SCUT-CTW1500
+[27],
+and
+Inverse-
+Text [28] dataset.
+4.1
+Datasets
+Curved Synthetic Dataset 150k. It is admitted that the per-
+formance of text spotters can be improved by pre-training
+on synthesized samples. Following previous work [7], we
+use the 150k synthetic images generated by the Synth-
+Text [45] toolbox, which contains around one-third of curved
+texts and two-thirds of horizontal instances.
+ICDAR 2013 [25] contains 229 training and 233 testing
+samples, while the images are primarily captured in a
+controlled environment, where most of the texts are hori-
+zontally presented and explicitly focused.
+ICDAR 2015 [26] consists of 1,000 training and 500
+testing images that were incidentally captured, containing
+multi-oriented text instances presented in complicated back-
+grounds with strong variations in blurring, distortions, etc.
+Total-Text [1] includes 1,255 training and 300 testing
+images, where at least one curved sample is presented in
+each image and annotated with polygonal bounding boxes
+at the word level.
+SCUT-CTW1500 [27] is another widely used benchmark
+designed for spotting arbitrarily shaped scene text, which
+involves 1,000 and 500 images for training and testing. The
+text instances are labeled by polygons at the text-line level.
+Inverse-Text [28] is a recently proposed dataset focused
+on arbitrary-shape scene text with about 40% inverse-like
+instances, containing 500 testing images. Following the pre-
+vious work [28], we test this dataset with the model trained
+on Total-Text.
+4.2
+Evaluation Protocol
+The existing evaluation protocol of text spotting tasks con-
+sists of two steps. Firstly, the intersection over union (IoU)
+scores between ground-truth (GT) and detected boxes are
+calculated; and only if the IoU score is larger than a desig-
+nated threshold (usually set to 0.5), the boxes are matched.
+Then, the recognized content inside each matched bounding
+box is compared with the GT transcription; only if the
+predicted text is the same as the GT will it contribute to
+the end-to-end accuracy. However, in the proposed method,
+each text instance is represented by a single point; thus,
+the evaluation metric based on the IoU is not available to
+measure the performance. Meanwhile, comparing the local-
+ization performance between bounding-box-based methods
+and the proposed point-based methods might be unfair,
+e.g., directly treating points inside a bounding box as true
+positives may overestimate the detection performance. To
+this end, we propose a new evaluation metric to ensure
+a relatively fair comparison to existing approaches, which
+mainly considers the end-to-end accuracy as it reflects both
+detection and recognition performance (failure detections
+usually lead to incorrect recognition results). Specifically, as
+shown in Fig. 8, we modified the text instance matching rule
+262
+232
+-
+-
+243
+58
+-
+-
+45
+255
+-
+-
+208
+333
+-
+-
+Distance Matrix
+Fig. 8 – Illustration of the point-based evaluation metric. Dia-
+monds are predicted points, and circles represent ground truth.
+TABLE 1 – Comparison of the end-to-end recognition perfor-
+mance evaluated by the proposed point-based metric and box-
+based metric. Results are reproduced using official codes.
+Method
+Total-Text
+SCUT-CTW1500
+Box
+Point
+Box
+Point
+ABCNetv1 [7]
+67.2
+67.4
+53.5
+53.0
+ABCNetv2 [61]
+71.7
+71.9
+57.6
+57.1
+by replacing the IoU metric with a distance metric, i.e., the
+predicted point that has the nearest distance to the central
+point of the GT box would be selected, and the recognition
+results will be measured by the same full-matching rules
+used in existing benchmarks. Only one predicted point with
+the highest confidence would be matched to the ground
+truth; others are then marked as false positives.
+To explore whether the proposed evaluation protocol can
+genuinely represent the model accuracy, Table 1 compares
+the end-to-end recognition accuracy of ABCNetv1 [7] and
+ABCNetv2 [61] on Total-Text [1] and SCUT-CTW1500 [27]
+under two metrics, i.e., the commonly used bounding box
+metric that is based on IoU, and the proposed point-based
+metric. The results demonstrate that the point-based evalu-
+ation protocol can well reflect the performance, where the
+difference between the values evaluated by box-based and
+point-based metrics is no more than 0.5%. For example, the
+ABCNetv1 model achieves 53.5% and 53.0% scores on the
+SCUT-CTW1500 dataset under the two metrics, respectively.
+Therefore, we use the point-based metric to evaluate the
+proposed SPTS v2 in the following experiments.
+4.3
+Implemented Details
+The model is first pretrained on a combination dataset that
+includes Curved Synthetic Dataset 150k [7], MLT-2017 [71],
+ICDAR 2013 [25], ICDAR 2015 [26], and Total-Text [1] for
+150 epochs, which is optimized by the AdamW [72] with
+an initial learning rate of 5 × 10−4, while the learning
+rate is linearly decayed to 1 × 10−5. After pretraining, the
+model is then fine-tuned on the training split of each target
+dataset for another 200 epochs, with a fixed learning rate of
+1 × 10−5. The entire model is distributively trained on 16
+NVIDIA A100 GPUs with a batch size of 8 per GPU. Note
+that the effective batch size is 64 because two independent
+augmentations are performed on each image in a mini-
+batch, following [22], [73]. In addition, we utilize ResNet-
+50 as the backbone network, while both the Transformer
+encoder and decoder consist of 6 layers with eight heads.
+Regarding the architecture of the Transformer, we adopt the
+Pre-LN Transformer [74]. During training, the short size of
+
+GHT
+H
+URY
+TAGE
+Vintage8
+TABLE 2 – Ablation studies on Total-Text w.r.t. the designs.
+“None” represents lexicon-free. “Full” represents that we use
+all the words that appeared in the test set. The Feat and token
+represent the items on the right of Equation 5, respectively.
+Shared represents sharing the parameters of the IAD and PRD.
+Method
+Token
+Feat
+Shared
+Total-Text
+None
+Full
+Baseline
+✓
+✓
+66.4
+78.4
+Baseline
+✓
+✓
+66.1
+75.9
+Baseline
+✓
+✓
+68.1
+80.0
+Baseline
+✓
+✓
+✓
+68.5
+81.4
+TABLE 3 – Ablation study of the position of the indicated point.
+Position
+E2E Total-Text
+E2E SCUT-CTW1500
+None
+Full
+None
+Full
+Central
+74.2
+82.4
+63.6
+83.8
+Top-left
+71.6
+79.7
+61.4
+82.0
+Random
+73.2
+80.8
+62.3
+81.1
+the input image is randomly resized to a range from 640 to
+896 (intervals of 32) while keeping the longer side shorter
+than 1,600 pixels, following to previous methods. Random
+cropping and rotating are employed for data augmentation.
+At the inference stage, we resize the short edge to 1,000
+while keeping the longer side shorter than 1824 pixels,
+following the previous works [7], [61], [64].
+4.4
+Ablation Study
+4.4.1
+Ablation Study of Designs
+We first conduct ablation studies to evaluate different de-
+signs of SPTS v2. Because the PRD requires started tokens
+to predict the recognition results in parallel, at least the
+hidden features (termed Feat) or the embeddings of the
+locations (termed Token) are required in the baseline setting.
+The results are shown in Table 2. We can see that without
+sharing the parameters of IAD and PRD, the performance
+encounters a slight drop, with 0.4% reduction in terms of the
+None metric of the Total-Text dataset. In addition, according
+to lines 1, 2, and 4 of the table, integrating the Token and
+Feat can further improve the performance, e.g., 3% and 5.5%
+higher than independently using the Token and Feat, respec-
+tively, in terms of the Full metric. The results demonstrate
+the importance of the information transmission. We use the
+pre-trained model to test the results.
+4.4.2
+Ablation Study of the Position of The Indicated Point
+Intuitively, all points in the region enclosed by the bounding
+box should be able to represent the target text instance. To
+explore the differences, we conduct ablation studies that
+use three different strategies to get the indicated points
+(see Fig. 5), i.e., the central point obtained by averaging
+the upper and lower midpoints, the top-left corner, and the
+random point inside the box. It should be noted that we
+use the corresponding ground-truth here to calculate the
+distance matrix for evaluating the performance to ensure
+the fair comparison, i.e., the distance to the ground-truth
+top-left point is used for top-left, the distance to the ground-
+truth central point for central, and the closest distance to the
+ground-truth polygon for random.
+TABLE 4 – Comparison with different shapes of bounding
+box. Np is the number of parameters required to describe the
+location of text instances by different representations.
+Variants
+Total-Text
+SCUT-CTW1500
+Np
+None
+Full
+None
+Full
+SPTS-Bezier
+60.6
+71.6
+52.6
+73.9
+16
+SPTS-Rect
+71.6
+80.4
+62.2
+82.0
+4
+SPTS-Non-Point
+64.7
+71.9
+55.4
+74.3
+0
+SPTS-Point
+74.2
+82.4
+63.6
+83.8
+2
+SPTS v2-Bezier
+63.2
+73.6
+52.3
+70.2
+16
+SPTS v2-Rect
+72.6
+79.5
+55.0
+71.5
+4
+SPTS v2-Point
+75.0
+82.6
+64.4
+84.0
+2
+The results are shown in Table 3, where the result of
+left-top is the worst. The result of random is close to central
+with approximately 1% worse in terms of the None metric.
+Although the central point shows the best performance
+against other formats, it suggests that the performance is
+not very sensitive to the positions of the point annotation.
+4.4.3
+Comparison Between Different Representations
+The proposed method can be easily extended to produce
+bounding boxes by modifying the point coordinates to
+bounding box locations during sequence construction. Here,
+we conduct ablation studies to explore the influence by only
+changing representations of the text instances. Specifically,
+four variants are explored, including 1) the Bezier curve
+bounding box; 2) the rectangular bounding box; 3) the
+indicated point; and 4) non-point. Note for the non-point
+representation, we only implement the results using SPTS,
+because it is hard to implement using SPTS v2, which
+requires the prediction of the location for the PRD stage.
+Since we only focus on end-to-end performance here, to
+minimize the impact of the detection results, each method
+uses corresponding representations to match the GT box
+in the evaluation. That is, the single-point model uses the
+evaluation metrics introduced in Sec. 4.2, i.e., distance be-
+tween points; the predictions of SPTS/v2-Rect are matched
+to the circumscribed rectangle of the polygonal annotations;
+the SPTS/v2-Bezier adopts the original metric that matches
+polygon boxes; and the evaluation metric for non-point can
+be referred to Sec. 5.1. As shown in Table 4, the SPTS/v2-
+point achieves the best performance on both the Total-
+Text and SCUT-CTW1500 datasets, outperforming the other
+representations by a large margin. Such experimental results
+suggest that a low-cost annotation, i.e., the indicated point, is
+capable of providing supervision for the text spotting task.
+Here, to safely ground such findings, we further provide
+analysis as follows:
+•
+The results of SPTS-Rect and SPTS-Bezier are ob-
+tained using the same training schedule as SPTS-
+Point. To further explore if the former may require
+a longer training schedule, we compare the SPTS-
+Bezier trained for 2× epochs with SPTS-Point in
+Table 5. It can be seen that the SPTS-Bezier with
+2× epochs does not significantly outperform the
+counterpart with 1× epochs and is still inferior to
+the SPTS-Point with 1× epochs. In addition, using a
+longer schedule even results in lower performance
+on SCUT-CTW1500 for SPTS-Bezier in terms of the
+
+9
+TABLE 5 – Comparison of different representations of text
+instances using a longer schedule.
+Variants
+Epochs
+Total-Text
+CTW1500
+Np
+None
+Full
+None
+Full
+SPTS-Bezier
+1×
+60.6
+71.6
+52.6
+73.9
+16
+SPTS-Bezier
+2×
+62.9
+74.4
+51.1
+74.3
+16
+SPTS-Point
+1×
+74.2
+82.4
+63.6
+83.8
+2
+MAHMUTPASA
+MAHMUTPASA
+Fig. 9 – The receptive field can be beneficial for the final
+recognition. Upper: the result of ABCNet v2. Lower: rough
+receptive field of our method.
+None metric, which suggests the training schedule
+may not be the case.
+•
+To further eliminate the influence of the different
+metrics, we also directly adopt the center point inside
+the rectangular or Bezier-curved bounding box to test
+the same point metric as our method. The results are
+shown in Table 7, which show that the variance is
+still consistent with the conclusion of Table 1, i.e., the
+result of the point metric is close to that of the box or
+polygonal based metrics in terms of the None metric.
+•
+As we can observe from previous scene text spot-
+ting method [61], sometimes the recognition results
+can still be accurate even if the detection result is
+inaccurate, like missing some of the regions of the
+characters, as shown in the top of Fig. 9. This is
+because the alignment for text recognition is based
+on the feature space, in which the cropped features
+have enough receptive fields for the text contents.
+Such phenomenon can also support our finding: as
+shown in the bottom of Fig. 9, because the image
+is globally encoded in our method, an approximate
+location could be enough for the model to capture
+the desired features in vicinity, which may further
+release the power of the Transformer.
+4.4.4
+Order of Text Instances
+As described in Sec. 3, the text instances are randomly
+ordered in the constructed sequence. Here, we further in-
+vestigate the impact of the order of text instances. The
+performances on Total-Text and SCUT-CTW1500 of different
+ordering strategies are presented in Table 6. The “Area” and
+“Dist2ori” mean that text instances are sorted by the area
+and the distance to the top-left origin in descending order,
+TABLE 6 – Ablation study of different ordering strategies of
+text instances in the sequence construction.
+Order
+Total-Text
+SCUT-CTW1500
+None
+Full
+None
+Full
+Area
+70.7
+79.2
+59.0
+75.3
+Topdown
+73.2
+81.3
+62.7
+79.7
+Dist2ori
+72.1
+81.8
+61.1
+79.6
+Random
+74.2
+82.4
+63.6
+83.8
+TABLE 7 – Comparison of the end-to-end recognition perfor-
+mance evaluated by the proposed point-based metric and box-
+based metric.
+Order
+Total-Text
+SCUT-CTW1500
+None
+Full
+None
+Full
+boxes
+72.6
+79.5
+55.0
+71.5
+boxes-point
+72.9
+81.1
+56.5
+77.7
+polygon
+63.2
+73.6
+52.3
+70.2
+polygon-point
+64.9
+76.0
+52.6
+79.5
+TABLE 8 – End-to-end recognition results and detection results
+on Total-Text. “None” represents lexicon-free. “Full” represents
+that we use all the words that appeared in the test set. Decoder
+1 represents using one layer for the decoder instead of using six
+layers.
+Method
+Total-Text End-to-End
+None
+Full
+R18 decoder 1
+11.5
+26.0
+R18
+60.6
+74.3
+R34 decoder 1
+22.7
+49.0
+R34
+64.7
+76.7
+R50 decoder 1
+53.7
+65.5
+R50
+68.5
+81.4
+respectively. The “Topdown” indicates that text instances
+are arranged from top to bottom. It can be seen that the ran-
+dom order for our method achieves the best performance,
+which may be explained by the improved robustness due
+to the different sequences constructed for the same image at
+different iterations.
+4.4.5
+Ablation Study of Different Settings
+We further conduct ablation studies w.r.t. depth of the
+sharing decoder layers of both IAD and PRD and various
+backbones for our framework on Total-Text, as shown in
+Table 8. We observe that using ResNet-34 as backbone
+surpasses ResNet-18 by 4.1% in terms of the None metric.
+Using ResNet-50 as backbone can outperform ResNet-34 by
+a further 3.8%. In addition, we find that the number of
+decoder layers may greatly influence the performance for
+different backbones. For example, with ResNet-18, ResNet-
+34, and ResNet-50 as backbones, decreasing the number of
+decoder layers from 6 to 1 leads to consistent 49.1%, 42%,
+and 15.2% performance declining in terms of the None
+metric for the Total-Text dataset. We use the pretrained
+model to test the results.
+
+AHMUTPASAPAS10
+TABLE 9 – Comparison between the end-to-end recognition results of the SPTS and NPTS models.
+Method
+Total-Text
+SCUT-CTW1500
+ICDAR 2013
+ICDAR 2015
+None
+Full
+None
+Full
+S
+W
+G
+S
+W
+G
+SPTS
+74.2
+82.4
+63.6
+83.8
+93.3
+91.7
+88.5
+77.5
+70.2
+65.8
+SPTS v2
+75.0
+82.6
+64.4
+84.0
+93.9
+91.8
+88.6
+81.2
+74.3
+68.0
+NPTS
+64.7
+71.9
+55.4
+74.3
+90.4
+84.9
+80.2
+69.4
+60.3
+55.6
+Ridge
+Toymakers
+Rootin
+org
+MUSEUM
+THEKEEN
+EXIT
+CREATION
+ANKYLOSAURUS
+11
+TEN YEAR CLUB
+SKY'S THE LIMIT
+THE
+BLOOMING GROWE
+OF WASHINGTON ILLE YOUTH
+FOOTBALL E CHERIEBLEADING
+In Memory of St Marthew K
+POLICE
+KL
+HOME DEPARTME
+Welcome to Kelly Field
+university
+Know
+Student
+anyone
+Accounts
+this
+starting
+you
+year
+Do
+OF
+JOSEPH
+HEART
+CONRAD
+DARKNESS
+Passages
+Theatre
+Point
+Meeting
+L2
+LORS
+CORNER
+PLAZA
+DS
+SED
+HALL
+1746
+VITAE
+LITITA
+LINDEN
+FOUNDED
+NON
+SCHOLOTALE
+PENIUM
+ISMIND
+SINOYB
+XNVWINDOSE
+MOTS
+ECCHYMOSE
+ESUJIVONS
+LES
+Fig. 10 – Qualitative results on the scene text benchmarks.
+Images are selected from Total-Text (first row), SCUT-CTW1500
+(second row), ICDAR 2013 (third row), ICDAR 2015 (fourth
+row), and Inverse-Text (fifth row). Best viewed on screen.
+TABLE 10 – End-to-end recognition results on ICDAR 2013.
+“S”, “W”, and “G” represent recognition with “Strong”,
+“Weak”, and “Generic” lexicon, respectively.
+Method
+IC13 End-to-End
+Para.
+FPS
+S
+W
+G
+Bounding Box-based methods
+Jaderberg et al. [75]
+86.4
+–
+–
+–
+–
+Textboxes [4]
+91.6
+89.7
+83.9
+–
+–
+Deep Text Spotter [50]
+89.0
+86.0
+77.0
+–
+–
+Li et al. [20]
+91.1
+89.8
+84.6
+–
+–
+MaskTextSpotter [8]
+92.2
+91.1
+86.5
+45.5M
+4.8
+Point-based methods
+SPTS
+93.3
+91.7
+88.5
+36.5M
+0.4
+SPTS v2
+93.9
+91.8
+88.6
+36.0M
+5.6
+4.5
+Comparison with Existing Methods on Scene Text
+Benchmarks
+4.5.1
+Horizontal-Text Dataset
+Table 10 compares the proposed method with existing meth-
+ods on the widely used ICDAR 2013 [25] benchmark. Our
+method achieves the best performance under all three lexi-
+cons while achieving 14x faster than the previous state-of-
+the-art single-point-based method with fewer parameters.
+4.5.2
+Multi-Oriented Dataset
+The quantitative results of the ICDAR 2015 [26] dataset are
+shown in Table 12. A performance gap between the pro-
+posed method and state-of-the-art methods can be found.
+The proposed method can not accurately recognize tiny
+texts because it directly predicts the sequence based on the
+low-resolution high-level features without RoI operations.
+Quantitatively, if the texts with an area smaller than 3000
+(after resizing) are ignored during evaluation, the F-measure
+with generic lexicons on ICDAR 2015 will be improved to
+77.5. Furthermore, current state-of-the-art methods on IC-
+DAR 2015 usually adopt larger image sizes during training
+and testing. For example, the short sides of the testing
+images are resized to 1440 pixels, while the long sides
+are shorter than 4000 pixels. As shown in Table 11, the
+performance of SPTSv2 on ICDAR 2015 with a larger testing
+size is much better than that with a smaller testing size.
+4.5.3
+Arbitrarily Shaped Dataset
+We further compare our method with existing approaches
+on the benchmarks containing arbitrarily shaped texts, in-
+cluding Total-Text [1] and SCUT-CTW1500 [27]. As shown
+in Table 12, for single-point-based methods, our method
+achieves state-of-the-art performance. Additionally, Table 12
+shows that our method achieves superior results on the
+long text-line-based SCUT-CTW1500 dataset, which further
+demonstrates that the single-point could be strong enough
+1. https://github.com/aim-uofa/AdelaiDet
+
+Ridge
+RootinHEKEEN
+ANKYLOSAURUS
+CREATION
+MUSEUM
+EXIT11
+OrRS
+H
+C
+E
+ARPOLICE
+In MemoryofSgt.MatthewKellyStudentAccounts
+Do you Know anyone
+starting university
+this year?HEART OF DARKNESS
+JOSEPHCONRADMeeting PointPLAZASCHOL
+NONLES
+MOTS
+DES
+ES11
+TABLE 11 – End-to-end recognition results on ICDAR 2015.
+“S”, “W”, and “G” represent recognition with “Strong”,
+“Weak”, and “Generic” lexicon, respectively. Bold indicates the
+state of the art, and underline indicates the second best.
+Method
+IC15 End-to-End
+S
+W
+G
+Bounding Box-based methods
+FOTS [52]
+81.1
+75.9
+60.8
+Mask TextSpotter [14]
+83.0
+77.7
+73.5
+CharNet [29]
+83.1
+79.2
+69.1
+TextDragon [10]
+82.5
+78.3
+65.2
+Mask TextSpotter v3 [21]
+83.3
+78.1
+74.2
+MANGO [30]
+81.8
+78.9
+67.3
+ABCNetV2 [61]
+82.7
+78.5
+73.0
+PAN++ [55]
+82.7
+78.2
+69.2
+Point-based methods
+SPTS (1000)
+77.5
+70.2
+65.8
+SPTS (1440)
+79.5
+74.1
+70.2
+SPTS v2 (720)
+73.2
+65.0
+57.3
+SPTS v2 (1000)
+81.2
+74.3
+68.0
+SPTS v2 (1440)
+81.7
+75.6
+70.3
+IRE DEPT
+Rycbus
+F
+Y
+N
+STATES
+1790
+COAS
+BININ
+CITANZA
+SNOUTS
+DOMAINS
+BRISHIBUI
+Fig. 11 – Error analysis of our method.
+to guide the text spotting. For the challenging Inverse-Text,
+our method further achieves the competitive performance
+without using specific rotation augmentation, demonstrat-
+ing its robustness to deal with rotated arbitrarily-shaped
+text. Note that, compared to SwinTextSpotter, we do not use
+a stronger rotation augmentation for fair comparison with
+other methods, which is crucial to the final performance of
+this dataset [28].
+4.5.4
+Summary
+In summary, the proposed method can achieve competi-
+tive performance compared with previous text spotters on
+several benchmarks. Especially on the two curved datasets,
+i.e., SCUT-CTW1500 [27], the proposed method outperforms
+some recently proposed methods by a large margin. The
+reason why our methods can achieve better accuracy on
+arbitrary-shaped texts might be: (1) The proposed method
+discards the task-specific modules (e.g., RoI modules) de-
+signed based on prior knowledge; therefore, the recognition
+accuracy is decoupled with the detection results, i.e., our
+COMPANY
+WALKER
+BREWING
+FIRESTONE
+FE
+TRESTONE
+PERSON
+PRECIOUS
+IS
+EVERY
+CARMINES
+CARMINES
+CARMINES
+Turkish
+Welcome
+Delight
+TICKET!
+KNOW
+YOUR
+First
+BAR
+food
+UENUC
+CAFE
+DUnn
+Raffles
+City
+giordano
+SPING
+ladies
+Fig. 12 – Qualitative results of the NPTS model on several scene
+text benchmarks. Images are selected from Total-Text (first row),
+SCUT-CTW1500 (second row), ICDAR 2013 (third row), and
+ICDAR 2015 (fourth row). Best view on screen.
+method can achieve acceptable recognition results even the
+detection position is shifted. On the other hand, the features
+fed to the recognition module are sampled based on the
+ground-truth position during training but from detection
+results during testing, which leads to feature misalignment.
+However, by tackling the spotting task in a sequence mod-
+eling manner, the proposed method eliminates such issues,
+thus showing more robustness on especially long text-line
+based arbitrarily shaped datasets.
+Some of the visualization results of five datasets are
+shown in Fig. 10. From the figure, we can see that the
+method shows robustness in curved, dense, highly-rotated,
+and long text. Especially in the rightmost image of the
+second row, the multi-oriented dense long text may interfere
+with each other and thus result in miss recall of some
+instances for some bounding-box based method, while for
+our method, as there is only a single point for location
+indication, such interference is intuitively less occurred, and
+thus all instances are correctly spotted.
+5
+DISCUSSION
+We further conduct experiments to comprehensively evalu-
+ate the limitations and other property of our method.
+5.1
+No-Point Text Spotting
+The experiments suggest that the detection and recognition
+may have been decoupled. Based on the results, we further
+show that our method can be converged even without the
+supervision of the single point annotations. The No-Point
+Text Spotting (NPTS) model is obtained by removing the
+coordinates of the indicated points from the constructed
+sequence. Fig. 12 shows the qualitative results of NPTS,
+which indicates the model may have learned the ability to
+implicitly find out the locations of the text merely based
+
+IS PRECIOUSCARMINE'S
+CARMINESTurkish
+DelightKNOW YOUR
+First
+TICKET!MOnDo
+CAFC
+BAR
+Ucnue
+foodRafflesCitygiordanoladies
+SPINCFIRESTONE WALKER
+★BREWING COMPANY★12
+TABLE 12 – End-to-end text spotting results on Total-Text, SCUT-CTW1500, ICDAR2015 and Inverse-Text. ‘None’ means lexicon-
+free. ‘Full’ indicates that we use all the words appeared in the test set. ‘S’, ‘W’, and ‘G’ represent recognition with ‘Strong’, ‘Weak’,
+and ‘Generic’ lexicon, respectively.
+Methods
+Total-Text
+SCUT-CTW1500
+ICDAR 2015 End-to-End
+Inverse-Text
+None
+Full
+None
+Full
+S
+W
+G
+None
+Full
+Bounding Box-based methods
+Mask TextSpotter [14]
+65.3
+77.4
+–
+–
+83.0
+77.7
+73.5
+39.0
+43.5
+Unconstrained [54]
+67.8
+–
+–
+–
+–
+–
+–
+–
+–
+CharNet [29]
+66.2
+–
+–
+–
+80.1
+74.5
+62.2
+–
+–
+FOTS [52]
+–
+–
+21.1
+39.7
+83.6
+79.1
+65.3
+–
+–
+TextDragon [10]
+48.8
+74.8
+39.7
+72.4
+82.5
+78.3
+65.2
+–
+–
+Text Perceptron [10]
+69.7
+78.3
+57.0
+–
+80.5
+76.6
+65.1
+–
+–
+ABCNet [7]
+64.2
+75.7
+45.2
+74.1
+–
+–
+–
+22.2
+34.3
+Boundary TextSpotter [58]
+65.0
+76.1
+–
+–
+79.7
+75.2
+64.1
+–
+–
+Mask TextSpotter v3 [21]
+71.2
+78.4
+–
+–
+83.3
+78.1
+74.2
+–
+–
+PGNet [76]
+63.1
+–
+–
+–
+83.3
+78.3
+63.5
+–
+–
+MANGO [30]
+72.9
+83.6
+58.9
+78.7
+81.8
+78.9
+67.3
+–
+–
+ABCNet v2 [61]
+70.4
+78.1
+57.5
+77.2
+82.7
+78.5
+73.0
+34.5
+47.4
+PAN++ [61]
+68.6
+78.6
+–
+–
+82.7
+78.2
+69.2
+–
+–
+TESTR [64]
+73.3
+83.9
+56.0
+81.5
+85.2
+79.4
+73.6
+34.2
+41.6
+SwinTextSpotter [63]
+74.3
+84.1
+51.8
+77.0
+83.9
+77.3
+70.5
+55.4
+67.9
+TTS [66]
+78.2
+86.3
+–
+–
+85.2
+81.7
+77.4
+–
+–
+GLASS [77]
+79.9
+86.2
+–
+–
+84.7
+80.1
+76.3
+–
+–
+Boundary TextSpotter’22 [78]
+66.2
+78.4
+46.1
+73.0
+82.5
+77.4
+71.7
+–
+–
+SRSTS [79]
+78.8
+86.3
+–
+–
+85.6
+81.7
+74.5
+-
+-
+Point-based methods
+TOSS [35]
+65.1
+74.8
+54.2
+65.3
+65.9
+59.6
+52.4
+-
+-
+SPTS [19]
+74.2
+82.4
+63.6
+83.8
+77.5
+70.2
+65.8
+38.3
+46.2
+SPTS v2
+75.0
+82.6
+64.4
+84.0
+81.7
+75.6
+70.3
+40.0
+46.7
+on the transcriptions. The comparison between the end-to-
+end recognition results of the SPTS/v2 and NPTS models is
+presented in Table 9. For the evaluation metric, the distance
+matrix between the predicted and GT points is replaced
+with an edit distance matrix between the predicted and
+GT transcriptions. Other parts are the same as described in
+Sec. 4.2. Despite the obvious gap between our method and
+NPTS, the preliminary results achieved by NPTS are still
+very encouraging. On the other hand, it suggests that the
+location indication is necessary for the text spotting task.
+5.2
+Failure Cases
+We further conduct error analysis on the incorrectly pre-
+dicted results. We visualize four typical errors in Fig. 11. For
+the left-top image, the error occurs in severe perspective dis-
+tortion text and the interference of the illumination. For the
+bottom-left image, error occurs in some rotated characters,
+e.g., the “U” is mistakenly recognized to “I”in the rightmost
+text. For the bottom-right image, the inverse text at the top
+is detected but no any recognition result is produced by our
+method, which indicates the method falls short in perceiving
+such inverse case. For the right-top image, the errors occur
+because the full stop symbols separate the characters. With
+incorrect distinction between different text instances, even
+though recognition is not limited by text boundaries, it still
+fails to recognize the text instance. We can find that the
+detection plays an important role in text spotting.
+6
+CONCLUSION
+We have proposed SPTS v2, a new scene text spotting
+paradigm that shows an extremely low-cost single-point
+annotation can be successfully used to train a powerful text
+spotter. SPTS v2 is based on a very concise Transformer-
+based framework, in which the detection and recognition
+of the text are simply formulated as language sequences,
+requiring only the cross-entropy loss without feature align-
+ment nor additional post-processing strategies. It includes
+an instance assignment decoder (IAD) which reserves the
+advantage of unifying all text instances inside the identical
+sequence, and a parallel recognition decoder (PRD) as well
+as the simple but effective information transmission method
+for significantly reducing the length of the sequence. Note
+both the IAD and PRD share exact the same parameters.
+With less parameters, SPTS v2 outperforms previous state-
+of-the-art single-point text spotter (SPTS) meanwhile with
+14x faster for the inference speed. Extensive experiments
+demonstrate that such point-based method can still achieve
+competitive results.
+We believe this is a brand-new attempt that for the first
+time, completely avoiding using any location supervision
+other than single-point can provide such inspiring results.
+Most importantly, under the scope of our auto-regressive
+based framework, we validate that the single-point might
+be served as the optimal setting comparing to polygon,
+rectangle, and non-point representations. This suggests that
+a rough position cue may better release the power of the
+Transformer instead of explicit constrains such as RoI pool-
+ing that may introduce the quantified error. We hope such
+findings may shed some new possibility toward more robust
+point-based methods for a wide range of applications.
+Although this work may question the necessity of the
+box annotations in text spotting, we claim that the box an-
+notations are still very valuable in many cases. For example,
+for layout analysis or text digitization, the precise location of
+the text instances are of great important. In addition, some
+
+13
+downstream text-related tasks such as text erasing or editing
+also require the accurate location. In fact, we have tried to
+visualize the attention maps to see if the bounding box can
+be inferred from point and the recognition supervision, but
+the results are very unsatisfactory. How to generate bound-
+ing box based on single-point and recognition information
+is still a very challenging task, which is worthy of further
+exploration in the future.
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diff --git a/pNAzT4oBgHgl3EQfqv3l/content/tmp_files/load_file.txt b/pNAzT4oBgHgl3EQfqv3l/content/tmp_files/load_file.txt
new file mode 100644
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@@ -0,0 +1,1994 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf,len=1993
+page_content='1  SPTS v2: Single-Point Scene Text Spotting  Yuliang Liu1, Jiaxin Zhang2, Dezhi Peng3, Mingxin Huang3, Xinyu Wang4, Jingqun Tang2, Can Huang2,  Dahua Lin5, Chunhua Shen6, Xiang Bai1, Lianwen Jin∗3,  Abstract—End-to-end scene text spotting has made significant progress due to its intrinsic synergy between text detection and  recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' Previous methods commonly regard manual annotations such as horizontal rectangles, rotated rectangles, quadrangles,  and polygons as a prerequisite, which are much more expensive than using single-point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' For the first time, we demonstrate that training  scene text spotting models can be achieved with an extremely low-cost single-point annotation by the proposed framework, termed  SPTS v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' SPTS v2 reserves the advantage of the auto-regressive Transformer with an Instance Assignment Decoder (IAD) through  sequentially predicting the center points of all text instances inside the same predicting sequence, while with a Parallel Recognition  Decoder (PRD) for text recognition in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' These two decoders share the same parameters and are interactively connected with a  simple but effective information transmission process to pass the gradient and information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' Comprehensive experiments on various  existing benchmark datasets demonstrate the SPTS v2 can outperform previous state-of-the-art single-point text spotters with fewer  parameters while achieving 14× faster inference speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' Most importantly, within the scope of our SPTS v2, extensive experiments  further reveal an important phenomenon that single-point serves as the optimal setting for the scene text spotting compared to  non-point, rectangular bounding box, and polygonal bounding box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' Such an attempt provides a significant opportunity for scene text  spotting applications beyond the realms of existing paradigms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' Code is available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='com/shannanyinxiang/SPTS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  Index Terms—Scene text spotting, Transformer, Single-point annotation  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  1  INTRODUCTION  S  Cene text reading techniques have made great strides in  recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' Given an image, text spotters can simulta-  neously locate and recognize the textual content, enabling  many real-world applications such as document digitaliza-  tion, intelligent assistants, and autopilot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' Basically, bound-  ing boxes such as rectangles, quadrilaterals, and polygons  are commonly employed to represent the text of different  shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' However, the fact that humans can intuitively read  texts without such a defined region encourages the de-  velopment of a bounding-box-free text spotter, lifting the  limitations imposed by bounding-box annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' 1, the previous methods use a bound-  ing box consisting of a series of coordinates to define the  instance-level text, where the enclosed region is considered  a positive sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' With its simplicity and straightforward-  ness, the bounding box has become the favored annotation  format for many other vision tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' However, unlike the  target in object detection tasks that are usually presented  in a defined appearance, text instances may appear in ar-  bitrary shapes due to the different typographies and fonts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  Therefore, it is required to use bounding boxes containing  more coordinates, such as polygons, to label these arbi-  trarily shaped texts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' Otherwise, considerable noise might  be involved, which may negatively impact the recognition  performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' For example, Total-Text [1] uses up to 20 coor-  dinates while SCUT-CTW1500 [2] uses up to 28 coordinates  to annotate a single curved scene text instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' Although  using polygons can, to some extent, alleviate the problem of  noise in labeling arbitrarily shaped text, it greatly increases  the annotation cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' To solve these issues, this paper pro-  1Huazhong University of Science and Technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  2ByteDance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  3South  China University of Technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  4University of Adelaide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  5The Chinese  University of Hong Kong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  6Zhejiang University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  Part of this work was  done when YL was with Chinese University of Hong Kong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  NEXT  42km  Ways of Indication  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' Pointing a direction → "NEXT"  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' Left top x1,y1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' Right top.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' x2, y2,  Left bot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' x3,y3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' Right bot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' x4, y4 → "42km"  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' 1 – Existing OCR methods typically use bounding boxes  to represent the text area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' However, inspired by how humans  can intuitively read texts without such a defined region, this  paper demonstrates that a single attention point is sufficient for  guiding the model to learn a strong scene text spotter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  poses a brand-new manner of supervision for scene text  spotters by using a single guidance point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' As shown in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' 1, each of the texts is indicated by a single point within  the instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' Such a streamlined representation breaks the  limitation of bounding boxes, enabling the model to access  the pixels in vicinity freely and further learn to discriminate  the boundary between texts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' Also, it considerably saves the  cost of annotation compared to the polygons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  In the past decade, the focus of research on scene text  spotting has shifted from horizontal [3], [4] and multi-  oriented text [2], [5], [6] to arbitrarily shaped text [7], [8],  as reflected in the transition from rectangular and quadri-  lateral annotations to more compact but more expensive  polygons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' 2, the rectangular bounding  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='01635v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='CV]  4 Jan 2023  NEXT  42km2  (a) Rectangle (55s)  (b) Quadrilateral (96s)  (c) Character (581s)  (d) Polygon (172s)  (e) Single-Point (11s)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' 2 – Different annotation styles and their time cost (for all the text instances in the sample image) measured by the  LabelMe1tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' Green areas are positive samples, while red dashed boxes are noises that may be possibly included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' Note the time  is measured by the average of three annotators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' For representations other than point, they normally need to zoom in for alignment  of the exact location, which consumes non-trivial annotation efforts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  boxes are prone to involve other text instances, which may  confuse the subsequent scene text recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' Furthermore,  many efforts have been made to develop more sophisticated  representations to fit arbitrarily shaped text instances [7],  [9], [10], [11], [12], [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' For example, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' 3,  Mask TextSpotter [14] utilizes boundary polygons to localize  the text region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' Text Dragon [10] utilizes character-level  bounding boxes to generate centerlines for enabling the  prediction of local geometry attributes, ABCNet [7] con-  verts polygon annotations to Bezier curves for representing  curved text instances, and Text Snake [13] describes text  instances by a series of ordered disks centered at symmetric  axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' These heuristic representations are carefully designed  by knowledgeable experts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' Although they have been shown  to be effective for aligning features between text detec-  tion and recognition modules, the reliance on manually-  designed rules undeniably undermines the generalizability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  Specifically, specified network architectures and modules  are required to process the features and annotations, such as  variants of RoI modules and post-processing mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  In addition, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' 2, the above representations  that rely on annotations of polygonal or character bounding  boxes are costly for labeling, while the proposed single point  can halve the cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  In the past few years, some researchers [15], [16], [17],  [18] have explored training the OCR models with coarse  annotations in a weakly-supervised manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' These meth-  ods can mainly be separated into two categories, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=', (1)  bootstrapping labels to finer granularity [15], [16] and (2)  training with partial annotations [17], [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' The former  usually derives character-level labels from word-level or  line-level annotations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' thus, the models could enjoy the  well-understood advantage of character-level supervision  without introducing overhead costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' The latter is committed  to achieving competitive performance with fewer training  samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' However, both methods still rely on the bounding  box annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' A recent study [19] demonstrates that  point-only annotation for scene text can still achieve com-  petitive performance in scene text spotting tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  One of the underlying problems to replace the bounding  box with a simpler annotation format, such as a single-point,  is that most text spotters rely on RoI-like sampling strategies  to extract the shared backbone features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' For example, Li  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' [20] and Mask TextSpotter require box and mask  prediction inside an RoI [21];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' ABCNet [7] proposes Bezier-  (a) Mask TextSpotter [14]  (b) Text Dragon [10]  (c) ABCNet [7]  (d) Text Snake [13]  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' 3 – Some recent representations of text instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' (a)  Mask TextSpotter utilizes character-level and polygonal bound-  ing boxes to predict the compact bounding box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' (b) Text  Dragon [10] employs character-level bounding box to generate  text centerline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' (c) ABCNet [7] converts polygon annotations to  Bezier curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' (d) Text Snake [13] describes text instances by a  series of ordered disks centered at symmetric axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  Align to wrap the curved representation into the horizontal  format, while TextDragon [10] introduces RoISlide to unify  the detection and recognition heads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  In this paper, inspired by the recent success of a  sequence-based object detector Pix2Seq [22], we show that  the text spotter can be trained with a single point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' Due  to the concise form of annotation, annotation time can be  significantly saved, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=', it only takes less than one-fiftieth of  the time to label single-points for the sample image shown  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' 2 than annotating character-level bounding boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  Another motivating factor in selecting point annotation is  that a clean and efficient OCR pipeline can be developed,  discarding the complex post-processing module and roi-  based sampling strategies;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' thus, the ambiguity introduced  by RoIs (see red dashed regions in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' 2) can be alleviated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  However, adopting single-point representation is still  challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' The previous state-of-the-art single-point text  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='spotter (SPTS) in our conference version [19] uses the  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
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+page_content='CUISINE  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
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+page_content='HAHNDORFC3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='for all text instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' Here, point-only prediction is very  straightforward and can handle unordered text instances  as a whole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' Thus it can learn to avoid repeated predic-  tions, specified label assign methods like bipartite matching  in DETR [23], or complicated post-processing like non-  maximum-suppression (NMS) [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' Although our confer-  ence version SPTS [19] is effective, the inference speed of it is  extremely low, especially the images where many instances  are simultaneously included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  Therefore, to take advantage of auto-regressive predic-  tion and preserve high efficiency at the same time, we design  SPTS v2, which significantly improves the inference speed while  achieving higher performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  Specifically, we observe that  the long sequence of the result is mainly occupied by the  text recognition, and thus we can only predict the location,  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=', x and y for each instance auto-regressively, termed as  an indicator, in the first instance assignment decoder (IAD);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  while for the second parallel recognition decoder (PRD),  each indicator is responsible for its subsequent text recogni-  tion prediction, which can be implemented in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' The  rationale behind this is that the IAD is to solve implicit  label assignment while PRD is for parallel prediction of  the responsible text recognition result given the features  after the label assignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' To enable the gradient flow from  recognition features to the IAD stage, we propose a simple  yet effective information transmission method that inte-  grates the embeddings of the text location and the features  in IAD stage, which is demonstrated essential to the success  of the SPTS v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  We summarize our contributions as follows: For the first time, we demonstrate that the text  spotters can be supervised by a simple yet effec-  tive single-point representation and the text recogni-  tion information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' Such a straightforward annotation  paradigm can considerably reduce the labeling costs,  avoid the label inconsistent, and make it possible to  access large-scale OCR data due to the low-cost an-  notation format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' More importantly, we demonstrate  that under such pipeline, the point annotation can  be served as the optimal setting for the text spotting  compared to non-point, rectangular bounding box,  and polygonal bounding box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' We form the text spotting as a language modeling  task, which requires only the identical cross-entropy  loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' Benefiting from such a concise pipeline, the  complex post-processing and sampling strategies de-  signed based on prior knowledge can be discarded,  showing great potential in terms of flexibility and  generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' SPTS v2 introduces a new Instance Assignment De-  coder (IAD) as well as the Parallel Recognition De-  coder (PRD) by sharing the same parameters, which  are incorporated by a simple but effective informa-  tion transmission method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' SPTS v2 outperforms the  conference version SPTS, with fewer parameters and  14x faster inference speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' Extensive experiments and ablations conducted on  five benchmarks, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=', ICDAR 2013 [25], ICDAR  2015 [26], Total-Text [1], SCUT-CTW1500 [27], and  Inverse-Text [28] involving both horizontal and ar-  bitrarily shaped texts demonstrate the competitive  performance of our method against previous state-  of-the-arts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  2  RELATED WORK  In the past decades, a variety of annotation styles have  been proposed focusing on various scenarios of scene text  spotting, including letters represented by stroke-level or  character-level bounding box [8], [14], [21], [29], [30], hor-  izontal text [25], [31] represented by rectangles (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' 2(a)),  multi-oriented text [26], [32] represented by quadrilaterals  (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' 2(b)), arbitrary-shaped text [1], [27], [33] represented  by polygons (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' 2(d)), and other novel representation such  as single-point [19], [34] or non-point [19], [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='1  Character-level Scene Text Spotter  In the early stage, many classical methods require character-  level annotations to train the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' [36] uses a  character classifier based on the HOG [37] features to read  text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' Bissacco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' [38] combine DNN with HOG features to  build a character classifier system for text recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' The  follow-up work [39] further develops a convolutional neural  network to be the character classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' The above methods are  adjusted to the horizontal text using character-level annota-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' Some researchers attempt to extend the character-level  scene text spotter to handle the arbitrarily-shaped text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' Mask  TextSpotter [8] designs a character segmentation module to  locate and recognize the character;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' its improved version [14],  [21] significantly reduce the cost of manual annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  CharNet [29] proposes a one-stage framework to boost the  text spotting performance by utilizing character-level anno-  tations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' CFRATS [40] locates the character regions and sends  the information of the character regions to the attention-  based recognizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' The MANGO [30] develops a position-  aware mask attention module to generate a location mask  for the character and use a sequence decoder to obtain the  recognition results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='2  Rectangle-based Scene Text Spotter  The rectangle-based scene text spotter plays an important  role in the early stage of the task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' Weinman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' [41]  propose a text spotting system that first generates the text  proposal and then use an independent word recognition  model to extract the text content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' [20] adapt a  generic object detectors framework Faster R-CNN [42] to  detect the rectangle-shape text and bridge the detector  and CTC-based [43] recognizer through sharing backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  Its enhanced version [44], equipped with a 2D attention  recognition module, is designed to handle the irregular text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  Gupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' [45] employ FCRN to detect the rectangle boxes  and use a word classifier as the recognizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' Recently, Liao  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' [46] propose a text spotting systems based on the  TextBoxes [4] and the CRNN [47], which are used to locate  and recognize the words, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' Shi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' [48] use the  TextBoxes as the detector to obtain the detection results and  propose a new recognizer ASTER [49] which develops a  Thin-Plate Spline transformation as rectification network to  rectify the recognition image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='com/wkentaro/labelme  4  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='3  Multi-oriented Scene Text Spotter  Recent methods develop multi-oriented scene text spot-  ters to handle text instances with complex shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' Busta  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' [50] propose a Deep TextSpotter which uses the  YOLOv2 [51] as the detector to detect the multi-oriented  text and use a CTC-based recognizer to transform the recog-  nition features into a sequence of characters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' FOTS [52]  proposes a new RoI operation, termed RoI Rotate, to trans-  form oriented text recognition features to regular ones from  quadrilateral detection results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' [53] propose a simi-  lar framework to locate the instance of the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' They further  develop a Text-Alignment to sample rotated features into  the horizontal feature and use an attention-based recognizer  to improve the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='4  Arbitrarily-shaped Scene Text Spotter  Arbitrarily-shaped scene text spotting is challenging due to  the diversity of text, such as various shapes, colors, fonts,  and languages, which has attracted increasing attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' In  this aspect, the text is normally annotated with polygons  of arbitrary shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' Recently, Qin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' [54] propose an  RoI Masking to suppress the background noise for the  recognition features and use a 2D attention-based recog-  nizer to read the arbitrarily-shaped text from recognition  features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' [55] design a spotting system, termed  PAN++, based on the fast detector PAN [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' TextNet [57]  predicts the quadrilateral text proposal to locate the text  and develops a perspective RoI transformation process to  rectify the quadrilateral features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' Feng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' [10] describe  the text instances as a series of quadrangles and propose  the RoISlide to connect the quadrangles for text recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' [58] detect the oriented rectangular box and  transform the oriented rectangular box into a boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  The boundary is served as the fiducial points for the Thin-  Plate-Spline transformation to rectify the irregular text into  a regular one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' Qiao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' [59] use a similar way which  develops a segmentation detector to generate the fiducial  points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' ABCNet [7] uses a one-stage detector [60], which  incorporates the parameterized Bezier curves to represent  the text instance as well as a new RoI operation (Beiz-  erAlign) for sampling the arbitrarily-shaped text features  into the horizontal format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' Its improved version [61] adopts  the BiFPN [62] as the backbone and uses an attention-  based recognizer to further improve the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' Swin-  TextSpotter [63] further exploits a new way of synergy be-  tween the detection and recognition, termed the Recognition  Conversion module, to make detection differentiable with  recognition loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' TESTR [64] designs a single encoder and  dual decoder structure based on the Deformable-DETR [65]  to remove the hand-designed components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' TTS [66] adds an  RNN recognition head in Deformable-DETR and proposes  a weakly-supervised method using only text transcription  annotations to train the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' ABINet++ [67] uses the  framework in [61] and further use a recognizer with an au-  tonomous, bidirectional, and iterative language model [68]  to improve the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='5  Point-based Scene Text Spotter  In this paper, we propose a new Single-Point Text Spotting  method, which is, to the best of our knowledge, the first  scene text spotter that does not rely on bounding box  annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' Specifically, each text instance is represented  by a single point (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' 2(e)) with a meager cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' The fact  that this point does not need to be accurately marked in  practice further demonstrates the possibility of text spotting  learning in a weakly supervised manner, shedding a new  scene text spotting paradigm with potential values toward  applications of a much larger scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  3  METHODOLOGY  Most of the existing text spotting algorithms require cus-  tomized modules to bridge the detection and recognition  blocks, where backbone features are cropped and shared be-  tween detection and recognition heads, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=', BezierAlign [7],  RoISlide [10], and RoIMasking [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  Inspired by the Pix2Seq [22], we cast the generic object  detection problem as a language modeling task, based on  an intuitive assumption that if a deep model knows what  and where the target is, it can be taught to tell the results  by the desired sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' It forms a concise pipeline, where  annotations with different attributes such as location coor-  dinates and object categories can be integrated into a single  sequence with only cross-entropy loss, enabling an end-to-  end trainable framework without task-specific modules (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=',  Region Proposal Networks and RoI pooling layers), which  can thus be naturally adapted to the text spotting task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' In  this paper, we propose Single-Point Text Spotter (SPTS) v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  SPTS v2 tackles text detection and recognition as a sequence  prediction task, solvable by a much more straightforward  pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  Unlike Pix2Seq, we propose to form the predicting se-  quence through the Instance Assignment Decoder (IAD)  and the Parallel Recognition Decoder (PRD);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' the former  IAD keeps the underlying essence of pix2seq by unifying  all instances inside the identical sequence, while the PRD  uses the isolated instance features generated from the IAD  to predict the text contents in parallel, which significantly  reduces the length of the sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' More importantly, we  propose to use the information transmission method to pass  the gradient of the recognition results to different instances,  which is essential to the final performance of the SPTS  v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' Specifically, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' 4, each input image is  first encoded by CNN and Transformer encoders to extract  visual and contextual features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' Then, the captured features  are decoded by a Transformer decoder, where tokens are  predicted in an auto-regressive manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' Unlike previous  algorithms, we further simplify the bounding box to a  corner point located at the upper left of the first character  or the center of the text instance, described in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' 5, in the  text instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' Benefiting from such a simple yet effective  representation, the modules carefully designed based on  prior knowledge, such as grouping strategies utilized in  segmentation-based methods and feature sampling blocks  equipped in box-based text spotters, can be eschewed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='1  Sequence Construction  To express the target text instances by a sequence, it is  required to convert the continuous descriptions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=', bound-  ing boxes) to a discretized space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' To this end, we simplify the  5  ������������1  ������������1  ������������2  ������������3  ������������������������  ������������������������������������  Deformable  Transformer  CNN  …  …  feat1  ������������������������������������  feat2  feat3  featk  feat1  feat2  featk  featk-1  ……' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  …  ……' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='…  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='Instance Classification  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='featlast  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='������������1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='������������1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='������������2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='������������������������  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='eos  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='������������1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
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+page_content=' Then, the features are auto-regressively decoded into a sequence that contains localization and recognition  information through IAD and PRD, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' For IAD, it predicts coordinates of all center points of text instances inside the  same sequence, while for the PRD, the recognition results are predicted in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' Note that IAD shares identical parameters  with PRD, and thus no additional parameters are introduced for the PRD stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  (a) Top-left  (b) Central  (c) Random  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' 5 – Indicated points (red color) using different positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  <  <SOS>  x  y  x  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  x  y  x  y  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  y  Randomly Ordered Instances  Input Seq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  x  y  x  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  x  y  x  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  Output Seq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  <EOS>  y  >  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  y  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' 6 – Input and output sequences of the Instance Assign-  ment Decoder (IAD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' Different colors represent different text  instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  bounding box to a single point and use the variable-length  transcription instead of the single-token object category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  Although SPTS [19] has been proven effective, the main  limitation is that the long sequence length will significantly  slow down the inference speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' This is because the recogni-  tion results normally are fixed to the maximum length equal  to 25 and 100 for word-level and line-level text instances,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' To this end, in SPTS v2, we design the In-  stance Assignment Decoder (IAD) and Parallel Recognition  Decoder (PRD) to overcome such limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='2  Instance Assignment Decoder  The auto-regressive decoder is known to be effective in  literature [19], [22];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' however, it is intuitive that this is a  time-consuming solution given the long sequence of the  Discretized Targets  ⌊x1/w×nbins⌋, ⌊y1/h×nbins⌋  ⌊x2/w×nbins⌋, ⌊y2/h×nbins⌋  ⌊x3/w×nbins⌋, ⌊y3/h×nbins⌋  P,I,Z,Z,E,R,I,A,<PAD>,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='.  P,A,R,A,D,I,S,O,<PAD>,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='.  B,I,R,R,E,R,I,A,<PAD>,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='.  Coordinates  Transcription  x1,y1  x2,y2  x3,y3  Original Labels  PIZZERIA  PARADISO  BIRRERIA  Coordinates  Transcription  Discretization  ex3 ey3  B  I  R  R  E  R  I  A  ∅  Transcription  <PAD>  feats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  <EOS>  Target Sequence  PIZZERIA  PARADISO  BIRRERIA  Information Transmission  ∅  ∅  ∅  ∅  ∅  >  ∅  >  >  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  ex2 ey2  P  A  R  A  D  I  S  O  ∅  ∅  ∅  ∅  ∅  ∅  ∅  ex1 ey1  P  I  Z  Z  E  R  I  A  ∅  ∅  ∅  ∅  ∅  ∅  ∅  Parallel Recognition  Decoder (PRD)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' 7 – Input and output sequences of the Parallel Recogni-  tion Decoder (PRD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' Different colors represent different text  instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' Given the features generated from information trans-  mission, the recognition results are predicted in parallel until  reaching the maximum length or the EOS symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  text instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' To improve efficiency, the SPTS v2 divides  detection and recognition into a two-stage workflow by  sharing the same Transformer decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' The first stage is  called Instance Assignment Decoder (IAD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' In the first stage,  SPTS v2 only decodes the center point for every text instance  until the end of the sequence comes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' An intuitive pipeline is  shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  Specifically, the continuous coordinates of the central  point of the text instance are uniformly discretized into  integers between [1, nbins], where nbins controls the degree  of discretization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' For example, an image with a long side  of 800 pixels requires only nbins = 800 to achieve zero  quantization error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' Note that the central point of the text  instance is obtained by averaging the upper and lower  midpoints as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' 5 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' As so far, a text instance  can thereby be represented by a sequence of three parts, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=',  [x, y, t], where (x, y) are the discretized coordinates and t is  the transcription text that will be predicted in PRD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' Notably,  the transcriptions are inherently discrete, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=', each of the  SU  AGUASU  AGUAransformerSU  AGUASU  AGUAranstormelDREAMDREAMDREAMBIRRE6  characters represents a category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  <SOS> and <EOS> tokens are inserted into the head  and tail of the sequence, indicating the start and the end  of a sequence, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' Therefore, given an image that  contains N text instances, the constructed sequence will in-  clude 2N discrete tokens, where the text instances would be  randomly ordered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' In fact, as shown in previous works [22],  the randomly ordered text instances can be effectively  learned, and thus it achieves the label assignment for dif-  ferent hidden features inconspicuously, which subtly avoids  an explicit label assignment like using bipartite matching  that plays a vital role for the DETR series [23], [65], [69],  [70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' In fact, compared with other label assignments, the  instance assignment is intuitively more efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' The dense  label assignment methods [7], [14] use the non-maximum  suppression (NMS) to select the suitable detection results  for recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' The bipartite matching label assignment  methods [64], [66] use a maximum number of instances  to detect and recognize texts, which consumes additional  computation for the empty text instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='3  Parallel Recognition Decoder  With the help of the IAD, we separate the different text  instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' The content of different text instances will be ob-  tained at the same time in the Parallel Recognition Decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  Different from the generic object detection that categorizes  objects into fixed categories, recognizing the text content  is a sequence classification problem that has a variable  length of the target sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' This may cause misalignment  issues and can consume more computational resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' To  eliminate such problems, we first pad or truncate the texts  to a fixed length K, where the <PAD> token is used  to fill the vacancy for shorter text instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' In addition,  supposing there are ncls categories of characters (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=', 97 for  English characters and symbols), the vocabulary size of the  dictionary used to tokenize the sequence can be calculated  as ncls + 3, where the extra three classes are for <PAD>,  <SOS>, and <EOS> tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' Empirically, we set the K and  nbins to 25 (or 100 for SCUT-CTW1500) and 1,000, respec-  tively, in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' Moreover, the maximum value of  nti is set to 60, which means the sequence containing more  than 60 text instances will be truncated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' An illustration of  the PRD is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  We assume that one image includes N text instances,  and every instance includes the maximum number of K  characters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' It takes Nv1 loops for SPTS to predict this image,  where Nv1 is defined as:  Nv1 = (2 + K) · N + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  (1)  While for SPTS v2, it only needs Nv2 for-loops, where Nv2  is:  Nv2 = 2 · N + K + 1,  (2)  with a K · (N − 1) reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' In our implementation, N  and K are set to 60 and 25, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' In this case, SPTS  requires 1,621 auto-regressive loops while SPTS v2 requires  only 146 loops, with 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='0% (1475/1621) reduction rate of  the number of loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' Actually, inside the PRD, SPTS v2 can  also take an early end if all instances have met the end of  sequence symbol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' Through PRD, the inference speed can be  significantly improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='4  Information Transmission  The parameters of the above two decoders are shared  and supervised by the detection and recognition gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  However, there is information loss between different text  instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' In the conference version [19], the information of  the previously detected text can be sensed by the recognition  token and the gradient of text recognition can be passed  on to supervise the predictions of different text instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  Such interaction is also important for the Parallel Recog-  nition Decoder in SPTS v2 to find the correct position of  the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' To address this issue, we propose an information  transmission method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' Formally, we first extract the hidden  text instance location features (short for feat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' 4) and  the corresponding prediction results of the text location (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=',  x1, y1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' Then, we convert the text instance location results  into an embeddings which is then added to the text instance  location features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' The process can be formulated as follow:  embedxi = embedding(xi), i = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  (3)  embedyi = embedding(yi), i = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  (4)  exi = featxi + embedxi,  (5)  eyi = featyi + embedyi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  (6)  With the help of the information transmission, the gradient  of later text recognition can be passed on to different text  instances by the featxi or featyi, and the information of the  previously detected text can be sensed by the recognition  token in PRD stage through the features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' PRD takes these  prior information as the first two queries to instruct the  decoder and thus recognize all the text instances in parallel,  as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' Such a straightforward transmission is  essential to the SPTS v2 in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='5  Model Training  Since the SPTS v2 is trained to predict tokens, it only  requires to maximize the likelihood loss at training time,  which can be written as:  L = max  L  �  i=1  wi log P(˜si|I, s1:i),  (7)  where I is the input image, ˜s is the output sequence, s is  the input sequence, L is the length of the sequence, and wi  is the weight of the likelihood of the i-th token, which is  empirically set to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' For both IAD and PRD, they share the  same Transformer and require only the cross-entropy loss,  maintaining a concise pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='6  Inference  At the inference stage, SPTS v2 first sequentially predicts the  tokens of location in IAD until the end of the sequence token  <EOS> occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' Then, the information transmission will  integrate the detection features to auto-regressively predict  the text contents in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' The predicted sequence will  subsequently be divided into multiple segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' Therefore,  the tokens can be easily translated into point coordinates  and transcriptions, yielding the text spotting results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' In  addition, the likelihood of all tokens in the corresponding  segment is averaged and assigned as a confidence score to  filter the original outputs, effectively removing redundant  and false-positive predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  7  4  EXPERIMENTS  We report the experimental results on five benchmarks, in-  cluding horizontal dataset ICDAR 2013 [25], multi-oriented  dataset ICDAR 2015 [26], and arbitrarily shaped datasets  Total-Text  [1]  and  SCUT-CTW1500  [27],  and  Inverse-  Text [28] dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='1  Datasets  Curved Synthetic Dataset 150k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' It is admitted that the per-  formance of text spotters can be improved by pre-training  on synthesized samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' Following previous work [7], we  use the 150k synthetic images generated by the Synth-  Text [45] toolbox, which contains around one-third of curved  texts and two-thirds of horizontal instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  ICDAR 2013 [25] contains 229 training and 233 testing  samples, while the images are primarily captured in a  controlled environment, where most of the texts are hori-  zontally presented and explicitly focused.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  ICDAR 2015 [26] consists of 1,000 training and 500  testing images that were incidentally captured, containing  multi-oriented text instances presented in complicated back-  grounds with strong variations in blurring, distortions, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  Total-Text [1] includes 1,255 training and 300 testing  images, where at least one curved sample is presented in  each image and annotated with polygonal bounding boxes  at the word level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  SCUT-CTW1500 [27] is another widely used benchmark  designed for spotting arbitrarily shaped scene text, which  involves 1,000 and 500 images for training and testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' The  text instances are labeled by polygons at the text-line level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  Inverse-Text [28] is a recently proposed dataset focused  on arbitrary-shape scene text with about 40% inverse-like  instances, containing 500 testing images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' Following the pre-  vious work [28], we test this dataset with the model trained  on Total-Text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='2  Evaluation Protocol  The existing evaluation protocol of text spotting tasks con-  sists of two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' Firstly, the intersection over union (IoU)  scores between ground-truth (GT) and detected boxes are  calculated;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' and only if the IoU score is larger than a desig-  nated threshold (usually set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='5), the boxes are matched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  Then, the recognized content inside each matched bounding  box is compared with the GT transcription;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' only if the  predicted text is the same as the GT will it contribute to  the end-to-end accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' However, in the proposed method,  each text instance is represented by a single point;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' thus,  the evaluation metric based on the IoU is not available to  measure the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' Meanwhile, comparing the local-  ization performance between bounding-box-based methods  and the proposed point-based methods might be unfair,  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=', directly treating points inside a bounding box as true  positives may overestimate the detection performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' To  this end, we propose a new evaluation metric to ensure  a relatively fair comparison to existing approaches, which  mainly considers the end-to-end accuracy as it reflects both  detection and recognition performance (failure detections  usually lead to incorrect recognition results).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' Specifically, as  shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' 8, we modified the text instance matching rule  262  232 243  58 45  255 208  333 Distance Matrix  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' 8 – Illustration of the point-based evaluation metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' Dia-  monds are predicted points, and circles represent ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  TABLE 1 – Comparison of the end-to-end recognition perfor-  mance evaluated by the proposed point-based metric and box-  based metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' Results are reproduced using official codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  Method  Total-Text  SCUT-CTW1500  Box  Point  Box  Point  ABCNetv1 [7]  67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='2  67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='4  53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='5  53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='0  ABCNetv2 [61]  71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='7  71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='9  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='6  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='1  by replacing the IoU metric with a distance metric, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=', the  predicted point that has the nearest distance to the central  point of the GT box would be selected, and the recognition  results will be measured by the same full-matching rules  used in existing benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' Only one predicted point with  the highest confidence would be matched to the ground  truth;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' others are then marked as false positives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  To explore whether the proposed evaluation protocol can  genuinely represent the model accuracy, Table 1 compares  the end-to-end recognition accuracy of ABCNetv1 [7] and  ABCNetv2 [61] on Total-Text [1] and SCUT-CTW1500 [27]  under two metrics, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=', the commonly used bounding box  metric that is based on IoU, and the proposed point-based  metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' The results demonstrate that the point-based evalu-  ation protocol can well reflect the performance, where the  difference between the values evaluated by box-based and  point-based metrics is no more than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' For example, the  ABCNetv1 model achieves 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='5% and 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='0% scores on the  SCUT-CTW1500 dataset under the two metrics, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  Therefore, we use the point-based metric to evaluate the  proposed SPTS v2 in the following experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='3  Implemented Details  The model is first pretrained on a combination dataset that  includes Curved Synthetic Dataset 150k [7], MLT-2017 [71],  ICDAR 2013 [25], ICDAR 2015 [26], and Total-Text [1] for  150 epochs, which is optimized by the AdamW [72] with  an initial learning rate of 5 × 10−4, while the learning  rate is linearly decayed to 1 × 10−5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' After pretraining, the  model is then fine-tuned on the training split of each target  dataset for another 200 epochs, with a fixed learning rate of  1 × 10−5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' The entire model is distributively trained on 16  NVIDIA A100 GPUs with a batch size of 8 per GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' Note  that the effective batch size is 64 because two independent  augmentations are performed on each image in a mini-  batch, following [22], [73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' In addition, we utilize ResNet-  50 as the backbone network, while both the Transformer  encoder and decoder consist of 6 layers with eight heads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  Regarding the architecture of the Transformer, we adopt the  Pre-LN Transformer [74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' During training, the short size of  GHT  H  URY  TAGE  Vintage8  TABLE 2 – Ablation studies on Total-Text w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' the designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  “None” represents lexicon-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' “Full” represents that we use  all the words that appeared in the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' The Feat and token  represent the items on the right of Equation 5, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  Shared represents sharing the parameters of the IAD and PRD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  Method  Token  Feat  Shared  Total-Text  None  Full  Baseline  ✓  ✓  66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='4  78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='4  Baseline  ✓  ✓  66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='1  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='9  Baseline  ✓  ✓  68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='1  80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='0  Baseline  ✓  ✓  ✓  68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='5  81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='4  TABLE 3 – Ablation study of the position of the indicated point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  Position  E2E Total-Text  E2E SCUT-CTW1500  None  Full  None  Full  Central  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='2  82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='4  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='6  83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='8  Top-left  71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='6  79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='7  61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='4  82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='0  Random  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='2  80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='8  62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='3  81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='1  the input image is randomly resized to a range from 640 to  896 (intervals of 32) while keeping the longer side shorter  than 1,600 pixels, following to previous methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' Random  cropping and rotating are employed for data augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  At the inference stage, we resize the short edge to 1,000  while keeping the longer side shorter than 1824 pixels,  following the previous works [7], [61], [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='4  Ablation Study  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='1  Ablation Study of Designs  We first conduct ablation studies to evaluate different de-  signs of SPTS v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' Because the PRD requires started tokens  to predict the recognition results in parallel, at least the  hidden features (termed Feat) or the embeddings of the  locations (termed Token) are required in the baseline setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  The results are shown in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' We can see that without  sharing the parameters of IAD and PRD, the performance  encounters a slight drop, with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='4% reduction in terms of the  None metric of the Total-Text dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' In addition, according  to lines 1, 2, and 4 of the table, integrating the Token and  Feat can further improve the performance, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=', 3% and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='5%  higher than independently using the Token and Feat, respec-  tively, in terms of the Full metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' The results demonstrate  the importance of the information transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' We use the  pre-trained model to test the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='2  Ablation Study of the Position of The Indicated Point  Intuitively, all points in the region enclosed by the bounding  box should be able to represent the target text instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' To  explore the differences, we conduct ablation studies that  use three different strategies to get the indicated points  (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' 5), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=', the central point obtained by averaging  the upper and lower midpoints, the top-left corner, and the  random point inside the box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' It should be noted that we  use the corresponding ground-truth here to calculate the  distance matrix for evaluating the performance to ensure  the fair comparison, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=', the distance to the ground-truth  top-left point is used for top-left, the distance to the ground-  truth central point for central, and the closest distance to the  ground-truth polygon for random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  TABLE 4 – Comparison with different shapes of bounding  box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' Np is the number of parameters required to describe the  location of text instances by different representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  Variants  Total-Text  SCUT-CTW1500  Np  None  Full  None  Full  SPTS-Bezier  60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='6  71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='6  52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='6  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='9  16  SPTS-Rect  71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='6  80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='4  62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='2  82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='0  4  SPTS-Non-Point  64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='7  71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='9  55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='4  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='3  0  SPTS-Point  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='2  82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='4  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='6  83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='8  2  SPTS v2-Bezier  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='2  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='6  52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='3  70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='2  16  SPTS v2-Rect  72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='6  79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='5  55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='0  71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='5  4  SPTS v2-Point  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='0  82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='6  64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='4  84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='0  2  The results are shown in Table 3, where the result of  left-top is the worst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' The result of random is close to central  with approximately 1% worse in terms of the None metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  Although the central point shows the best performance  against other formats, it suggests that the performance is  not very sensitive to the positions of the point annotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='3  Comparison Between Different Representations  The proposed method can be easily extended to produce  bounding boxes by modifying the point coordinates to  bounding box locations during sequence construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' Here,  we conduct ablation studies to explore the influence by only  changing representations of the text instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' Specifically,  four variants are explored, including 1) the Bezier curve  bounding box;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' 2) the rectangular bounding box;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' 3) the  indicated point;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' and 4) non-point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' Note for the non-point  representation, we only implement the results using SPTS,  because it is hard to implement using SPTS v2, which  requires the prediction of the location for the PRD stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  Since we only focus on end-to-end performance here, to  minimize the impact of the detection results, each method  uses corresponding representations to match the GT box  in the evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' That is, the single-point model uses the  evaluation metrics introduced in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=', distance be-  tween points;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' the predictions of SPTS/v2-Rect are matched  to the circumscribed rectangle of the polygonal annotations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  the SPTS/v2-Bezier adopts the original metric that matches  polygon boxes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' and the evaluation metric for non-point can  be referred to Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' As shown in Table 4, the SPTS/v2-  point achieves the best performance on both the Total-  Text and SCUT-CTW1500 datasets, outperforming the other  representations by a large margin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' Such experimental results  suggest that a low-cost annotation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=', the indicated point, is  capable of providing supervision for the text spotting task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  Here, to safely ground such findings, we further provide  analysis as follows: The results of SPTS-Rect and SPTS-Bezier are ob-  tained using the same training schedule as SPTS-  Point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' To further explore if the former may require  a longer training schedule, we compare the SPTS-  Bezier trained for 2× epochs with SPTS-Point in  Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' It can be seen that the SPTS-Bezier with  2× epochs does not significantly outperform the  counterpart with 1× epochs and is still inferior to  the SPTS-Point with 1× epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' In addition, using a  longer schedule even results in lower performance  on SCUT-CTW1500 for SPTS-Bezier in terms of the  9  TABLE 5 – Comparison of different representations of text  instances using a longer schedule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  Variants  Epochs  Total-Text  CTW1500  Np  None  Full  None  Full  SPTS-Bezier  1×  60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='6  71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='6  52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='6  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='9  16  SPTS-Bezier  2×  62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='9  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='4  51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='1  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='3  16  SPTS-Point  1×  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='2  82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='4  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='6  83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='8  2  MAHMUTPASA  MAHMUTPASA  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' 9 – The receptive field can be beneficial for the final  recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' Upper: the result of ABCNet v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' Lower: rough  receptive field of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  None metric, which suggests the training schedule  may not be the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' To further eliminate the influence of the different  metrics, we also directly adopt the center point inside  the rectangular or Bezier-curved bounding box to test  the same point metric as our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' The results are  shown in Table 7, which show that the variance is  still consistent with the conclusion of Table 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=', the  result of the point metric is close to that of the box or  polygonal based metrics in terms of the None metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' As we can observe from previous scene text spot-  ting method [61], sometimes the recognition results  can still be accurate even if the detection result is  inaccurate, like missing some of the regions of the  characters, as shown in the top of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' This is  because the alignment for text recognition is based  on the feature space, in which the cropped features  have enough receptive fields for the text contents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  Such phenomenon can also support our finding: as  shown in the bottom of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' 9, because the image  is globally encoded in our method, an approximate  location could be enough for the model to capture  the desired features in vicinity, which may further  release the power of the Transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='4  Order of Text Instances  As described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' 3, the text instances are randomly  ordered in the constructed sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' Here, we further in-  vestigate the impact of the order of text instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' The  performances on Total-Text and SCUT-CTW1500 of different  ordering strategies are presented in Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' The “Area” and  “Dist2ori” mean that text instances are sorted by the area  and the distance to the top-left origin in descending order,  TABLE 6 – Ablation study of different ordering strategies of  text instances in the sequence construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  Order  Total-Text  SCUT-CTW1500  None  Full  None  Full  Area  70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='7  79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='2  59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='0  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='3  Topdown  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='2  81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='3  62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='7  79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='7  Dist2ori  72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='1  81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='8  61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='1  79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='6  Random  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='2  82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='4  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='6  83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='8  TABLE 7 – Comparison of the end-to-end recognition perfor-  mance evaluated by the proposed point-based metric and box-  based metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  Order  Total-Text  SCUT-CTW1500  None  Full  None  Full  boxes  72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='6  79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='5  55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='0  71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='5  boxes-point  72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='9  81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='1  56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='5  77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='7  polygon  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='2  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='6  52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='3  70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='2  polygon-point  64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='9  76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='0  52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='6  79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='5  TABLE 8 – End-to-end recognition results and detection results  on Total-Text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' “None” represents lexicon-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' “Full” represents  that we use all the words that appeared in the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' Decoder  1 represents using one layer for the decoder instead of using six  layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  Method  Total-Text End-to-End  None  Full  R18 decoder 1  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='5  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='0  R18  60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='6  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='3  R34 decoder 1  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='7  49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='0  R34  64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='7  76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='7  R50 decoder 1  53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='7  65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='5  R50  68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='5  81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='4  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' The “Topdown” indicates that text instances  are arranged from top to bottom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' It can be seen that the ran-  dom order for our method achieves the best performance,  which may be explained by the improved robustness due  to the different sequences constructed for the same image at  different iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='5  Ablation Study of Different Settings  We further conduct ablation studies w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' depth of the  sharing decoder layers of both IAD and PRD and various  backbones for our framework on Total-Text, as shown in  Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' We observe that using ResNet-34 as backbone  surpasses ResNet-18 by 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='1% in terms of the None metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  Using ResNet-50 as backbone can outperform ResNet-34 by  a further 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='8%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' In addition, we find that the number of  decoder layers may greatly influence the performance for  different backbones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' For example, with ResNet-18, ResNet-  34, and ResNet-50 as backbones, decreasing the number of  decoder layers from 6 to 1 leads to consistent 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='1%, 42%,  and 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='2% performance declining in terms of the None  metric for the Total-Text dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' We use the pretrained  model to test the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  AHMUTPASAPAS10  TABLE 9 – Comparison between the end-to-end recognition results of the SPTS and NPTS models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  Method  Total-Text  SCUT-CTW1500  ICDAR 2013  ICDAR 2015  None  Full  None  Full  S  W  G  S  W  G  SPTS  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='2  82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='4  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='6  83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='8  93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='3  91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='7  88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='5  77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='5  70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='2  65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='8  SPTS v2  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='0  82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
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+page_content=' Our  method achieves the best performance under all three lexi-  cons while achieving 14x faster than the previous state-of-  the-art single-point-based method with fewer parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='2  Multi-Oriented Dataset  The quantitative results of the ICDAR 2015 [26] dataset are  shown in Table 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' A performance gap between the pro-  posed method and state-of-the-art methods can be found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  The proposed method can not accurately recognize tiny  texts because it directly predicts the sequence based on the  low-resolution high-level features without RoI operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  Quantitatively, if the texts with an area smaller than 3000  (after resizing) are ignored during evaluation, the F-measure  with generic lexicons on ICDAR 2015 will be improved to  77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' Furthermore, current state-of-the-art methods on IC-  DAR 2015 usually adopt larger image sizes during training  and testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' For example, the short sides of the testing  images are resized to 1440 pixels, while the long sides  are shorter than 4000 pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' As shown in Table 11, the  performance of SPTSv2 on ICDAR 2015 with a larger testing  size is much better than that with a smaller testing size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='3  Arbitrarily Shaped Dataset  We further compare our method with existing approaches  on the benchmarks containing arbitrarily shaped texts, in-  cluding Total-Text [1] and SCUT-CTW1500 [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' As shown  in Table 12, for single-point-based methods, our method  achieves state-of-the-art performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' Additionally, Table 12  shows that our method achieves superior results on the  long text-line-based SCUT-CTW1500 dataset, which further  demonstrates that the single-point could be strong enough  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='com/aim-uofa/AdelaiDet  Ridge  RootinHEKEEN  ANKYLOSAURUS  CREATION  MUSEUM  EXIT11  OrRS  H  C  E  ARPOLICE  In MemoryofSgt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='MatthewKellyStudentAccounts  Do you Know anyone  starting university  this year?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='HEART OF DARKNESS  JOSEPHCONRADMeeting PointPLAZASCHOL  NONLES  MOTS  DES  ES11  TABLE 11 – End-to-end recognition results on ICDAR 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  “S”, “W”, and “G” represent recognition with “Strong”,  “Weak”, and “Generic” lexicon, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' Bold indicates the  state of the art, and underline indicates the second best.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  Method  IC15 End-to-End  S  W  G  Bounding Box-based methods  FOTS [52]  81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='1  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='9  60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='8  Mask TextSpotter [14]  83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='0  77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='7  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='5  CharNet [29]  83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='1  79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='2  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='1  TextDragon [10]  82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='5  78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='3  65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='2  Mask TextSpotter v3 [21]  83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='3  78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='1  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='2  MANGO [30]  81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='8  78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='9  67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='3  ABCNetV2 [61]  82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='7  78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='5  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='0  PAN++ [55]  82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='7  78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='2  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='2  Point-based methods  SPTS (1000)  77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='5  70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='2  65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='8  SPTS (1440)  79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='5  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='1  70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='2  SPTS v2 (720)  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='2  65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='0  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='3  SPTS v2 (1000)  81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='2  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='3  68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='0  SPTS v2 (1440)  81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='7  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='6  70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='3  IRE DEPT  Rycbus  F  Y  N  STATES  1790  COAS  BININ  CITANZA  SNOUTS  DOMAINS  BRISHIBUI  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' 11 – Error analysis of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  to guide the text spotting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' For the challenging Inverse-Text,  our method further achieves the competitive performance  without using specific rotation augmentation, demonstrat-  ing its robustness to deal with rotated arbitrarily-shaped  text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' Note that, compared to SwinTextSpotter, we do not use  a stronger rotation augmentation for fair comparison with  other methods, which is crucial to the final performance of  this dataset [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='4  Summary  In summary, the proposed method can achieve competi-  tive performance compared with previous text spotters on  several benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' Especially on the two curved datasets,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=', SCUT-CTW1500 [27], the proposed method outperforms  some recently proposed methods by a large margin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' The  reason why our methods can achieve better accuracy on  arbitrary-shaped texts might be: (1) The proposed method  discards the task-specific modules (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=', RoI modules) de-  signed based on prior knowledge;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' therefore, the recognition  accuracy is decoupled with the detection results, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=', our  COMPANY  WALKER  BREWING  FIRESTONE  FE  TRESTONE  PERSON  PRECIOUS  IS  EVERY  CARMINES  CARMINES  CARMINES  Turkish  Welcome  Delight  TICKET!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  KNOW  YOUR  First  BAR  food  UENUC  CAFE  DUnn  Raffles  City  giordano  SPING  ladies  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' 12 – Qualitative results of the NPTS model on several scene  text benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' Images are selected from Total-Text (first row),  SCUT-CTW1500 (second row), ICDAR 2013 (third row), and  ICDAR 2015 (fourth row).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' Best view on screen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  method can achieve acceptable recognition results even the  detection position is shifted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' On the other hand, the features  fed to the recognition module are sampled based on the  ground-truth position during training but from detection  results during testing, which leads to feature misalignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  However, by tackling the spotting task in a sequence mod-  eling manner, the proposed method eliminates such issues,  thus showing more robustness on especially long text-line  based arbitrarily shaped datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  Some of the visualization results of five datasets are  shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' From the figure, we can see that the  method shows robustness in curved, dense, highly-rotated,  and long text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' Especially in the rightmost image of the  second row, the multi-oriented dense long text may interfere  with each other and thus result in miss recall of some  instances for some bounding-box based method, while for  our method, as there is only a single point for location  indication, such interference is intuitively less occurred, and  thus all instances are correctly spotted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  5  DISCUSSION  We further conduct experiments to comprehensively evalu-  ate the limitations and other property of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='1  No-Point Text Spotting  The experiments suggest that the detection and recognition  may have been decoupled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' Based on the results, we further  show that our method can be converged even without the  supervision of the single point annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' The No-Point  Text Spotting (NPTS) model is obtained by removing the  coordinates of the indicated points from the constructed  sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=" 12 shows the qualitative results of NPTS,  which indicates the model may have learned the ability to  implicitly find out the locations of the text merely based  IS PRECIOUSCARMINE'S  CARMINESTurkish  DelightKNOW YOUR  First  TICKET!" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='MOnDo  CAFC  BAR  Ucnue  foodRafflesCitygiordanoladies  SPINCFIRESTONE WALKER  ★BREWING COMPANY★12  TABLE 12 – End-to-end text spotting results on Total-Text, SCUT-CTW1500, ICDAR2015 and Inverse-Text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' ‘None’ means lexicon-  free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' ‘Full’ indicates that we use all the words appeared in the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' ‘S’, ‘W’, and ‘G’ represent recognition with ‘Strong’, ‘Weak’,  and ‘Generic’ lexicon, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  Methods  Total-Text  SCUT-CTW1500  ICDAR 2015 End-to-End  Inverse-Text  None  Full  None  Full  S  W  G  None  Full  Bounding Box-based methods  Mask TextSpotter [14]  65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='3  77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
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+page_content='2  82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='4  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='6  83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='8  77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='5  70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='2  65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='8  38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='3  46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='2  SPTS v2  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='0  82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='6  64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='4  84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='0  81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='7  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='6  70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='3  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='0  46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='7  on the transcriptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' The comparison between the end-to-  end recognition results of the SPTS/v2 and NPTS models is  presented in Table 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' For the evaluation metric, the distance  matrix between the predicted and GT points is replaced  with an edit distance matrix between the predicted and  GT transcriptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' Other parts are the same as described in  Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' Despite the obvious gap between our method and  NPTS, the preliminary results achieved by NPTS are still  very encouraging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' On the other hand, it suggests that the  location indication is necessary for the text spotting task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='2  Failure Cases  We further conduct error analysis on the incorrectly pre-  dicted results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' We visualize four typical errors in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' For  the left-top image, the error occurs in severe perspective dis-  tortion text and the interference of the illumination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' For the  bottom-left image, error occurs in some rotated characters,  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=', the “U” is mistakenly recognized to “I”in the rightmost  text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' For the bottom-right image, the inverse text at the top  is detected but no any recognition result is produced by our  method, which indicates the method falls short in perceiving  such inverse case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' For the right-top image, the errors occur  because the full stop symbols separate the characters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' With  incorrect distinction between different text instances, even  though recognition is not limited by text boundaries, it still  fails to recognize the text instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' We can find that the  detection plays an important role in text spotting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  6  CONCLUSION  We have proposed SPTS v2, a new scene text spotting  paradigm that shows an extremely low-cost single-point  annotation can be successfully used to train a powerful text  spotter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' SPTS v2 is based on a very concise Transformer-  based framework, in which the detection and recognition  of the text are simply formulated as language sequences,  requiring only the cross-entropy loss without feature align-  ment nor additional post-processing strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' It includes  an instance assignment decoder (IAD) which reserves the  advantage of unifying all text instances inside the identical  sequence, and a parallel recognition decoder (PRD) as well  as the simple but effective information transmission method  for significantly reducing the length of the sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' Note  both the IAD and PRD share exact the same parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  With less parameters, SPTS v2 outperforms previous state-  of-the-art single-point text spotter (SPTS) meanwhile with  14x faster for the inference speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' Extensive experiments  demonstrate that such point-based method can still achieve  competitive results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  We believe this is a brand-new attempt that for the first  time, completely avoiding using any location supervision  other than single-point can provide such inspiring results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  Most importantly, under the scope of our auto-regressive  based framework, we validate that the single-point might  be served as the optimal setting comparing to polygon,  rectangle, and non-point representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' This suggests that  a rough position cue may better release the power of the  Transformer instead of explicit constrains such as RoI pool-  ing that may introduce the quantified error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' We hope such  findings may shed some new possibility toward more robust  point-based methods for a wide range of applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content='  Although this work may question the necessity of the  box annotations in text spotting, we claim that the box an-  notations are still very valuable in many cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' For example,  for layout analysis or text digitization, the precise location of  the text instances are of great important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' In addition, some  13  downstream text-related tasks such as text erasing or editing  also require the accurate location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' In fact, we have tried to  visualize the attention maps to see if the bounding box can  be inferred from point and the recognition supervision, but  the results are very unsatisfactory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
+page_content=' How to generate bound-  ing box based on single-point and recognition information  is still a very challenging task, which is worthy of further  exploration in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNAzT4oBgHgl3EQfqv3l/content/2301.01635v1.pdf'}
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diff --git a/pNFST4oBgHgl3EQfNTg4/content/tmp_files/2301.13747v1.pdf.txt b/pNFST4oBgHgl3EQfNTg4/content/tmp_files/2301.13747v1.pdf.txt
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+arXiv:2301.13747v1  [math.CO]  31 Jan 2023
+ALGEBRAIC IDENTITIES ON Q-HARMONIC NUMBERS AND
+Q-BINOMIAL COEFFICIENTS
+SAID ZRIAA AND MOHAMMED MOUC¸OUF
+Abstract. The aim of this paper is to present a general algebraic identity. Applying this
+identity, we provide several formulas involving the q-binomial coefficients and the q-harmonic
+numbers. We also recover some known identities including an algebraic identity of D. Y.
+Zheng on q-Ap´ery numbers and we establish the q-analog of Euler’s formula. The proposed
+results may have important applications in the theory of q-supercongruences.
+1. Introduction
+For an indeterminate x the q-shifted factorial is usually defined by
+(x; q)0 = 1 and (x; q)n = (1 − x)(1 − qx) · · · (1 − qn−1x) for n = 1, 2, . . .
+The Gaussian q-binomial coefficient is correspondingly given by
+�n
+j
+�
+=
+(q; q)n
+(q; q)j(q; q)n−j
+for j = 0, 1, 2, . . ., n,
+The q-harmonic numbers are defined by
+Hn(q) =
+n
+�
+k=1
+1
+[k]q
+for n = 1, 2, . . .,
+where the q-numbers are given by
+[k] or [k]q := 1 − qk
+1 − q = 1 + q + · · · + qk−1
+Following Comtet [7], the complete Bell polynomials can be explicitly expressed as
+Bn(x1, x2, · · · , xn) =
+�
+m1+2m2+···+nmn=n
+n!
+m1!m2! · · · mn!
+�x1
+1!
+�m1�x2
+2!
+�m2
+· · ·
+�xn
+n!
+�mn
+During the last two decades, there has been an increasing interest in studying binomial
+sums and their q-analogues, one can consult recent papers, for example [6, 18, 19].
+For a
+comprehensive account of the q-series and its applications to numbers theory, combinatorics
+and special functions, we refer the reader to the excellent monograph by G. Gasper and M.
+Rahman [8].
+Recently, the study of the q-harmonic congruences turned into a very active
+research area (see e.g., [17] and the references therein).
+There exist numerous combinatorial identities involving q-binomial coefficients in the math-
+ematical literature (see e.g., [6, 12, 16, 18, 19]).
+Nowadays, there has been growing inter-
+est in deriving q-analogues of several combinatorial identities.
+This includes, for example,
+q-generalizations of some well known identities involving harmonic numbers.
+The Ap´ery numbers are defined by the following binomial sum
+A(n) =
+n
+�
+k=0
+�n
+k
+�2�n + k
+k
+�2
+Key words and phrases. Algebraic identities, q-binomial coefficients, q-harmonic numbers, complete Bell poly-
+nomials.
+1
+
+2
+SAID ZRIAA AND MOHAMMED MOUC¸OUF
+These numbers have many interesting properties that make them extremely useful in the proof
+of the irrationality of ζ(3) (see [3] for further details). Also the Ap´ery numbers have remarkable
+arithmetic properties [9]. F. Beukers conjectured [4] that
+A
+�p − 1
+2
+�
+≡ a(p) (modp2)
+where p is an odd prime and a(n) is determined by
+∞
+�
+k=1
+a(k)qk := q
+∞
+�
+k=1
+(1 − q2k)4(1 − q4k)4 = q − 4q3 − 2q5 + 24q7 + · · · .
+Beukers’ conjecture was later showed by S. Ahlgren, K. Ono [1], who reduce this statement in
+terms of the harmonic numbers Hn to the identity
+n
+�
+k=1
+�n
+k
+�2�n + k
+k
+�2�
+1 + 2kHn+k + 2kHn−k − 4kHk
+�
+= 0
+In order to give a classical proof of the last identity, Chu [5] presented the following algebraic
+identity
+n
+�
+j=0
+�n
+j
+�2�n + j
+j
+�2�
+−j
+(x + j)2 + 1 + 2j(Hn+j − Hj) + 2j(Hn−j − Hj)
+x + j
+�
+= x(1 − x)2
+n
+(x)2
+n+1
+which gives the desired formula in the limit.
+In order to prove irrationality results on the q-analog of ζ(3):
+ζq(3) =
++∞
+�
+k=1
+qk(1 + qk)
+(1 − qk)3
+The authors of [13] introduced implicitly a q-analog of the Ap´ery numbers AKRZ
+q
+(n) and they
+showed that
+AKRZ
+q
+(n) =
+n
+�
+k=0
+aq(n, k)
+qk
+where aq(n, k) can be defined via the following q-partial fraction decomposition
+(xq−n; q)2
+n
+(x; q)2
+n+1
+=
+n
+�
+k=0
+� aq(n, k)
+(1 − qkx)2 + bq(n, k)
+1 − qkx
+�
+(1.1)
+A. Straub [17] showed via the partial fraction decomposition technique that AKRZ
+q
+(n) have the
+following explicit q-binomial representation
+qn(2n+1)AKRZ
+q
+(n) =
+n
+�
+k=0
+q(n−k)2�n
+k
+�2�n + k
+k
+�2
+which reduces to Ap´ery numbers when q → 1.
+D. Y. Zheng [20] has recently introduced the q-Ap´ery numbers
+qn(n+1)AKRZ
+q
+(n) =
+n
+�
+k=0
+qk(k−2n)
+�n
+k
+�2�n + k
+k
+�2
+then he established an interesting algebraic identity
+x2n(q/x; q)2
+n
+(1 − x)(xq; q)2n
+=
+1
+1 − x+
+n
+�
+j=1
+�
+n
+j
+�2
+qj(j−2n)
+�
+qj − 1
+(1 − xqj)2 +1 − 4[j]Hj(q) + 2[j]Hn+j(q) + 2q[j]Hn−j(q−1)
+1 − xqj
+�
+(1.2)
+which is a q-extension of the Chu’s identity, and obtained the identity
+n
+�
+k=0
+qk(k−2n)
+�n
+k
+�2�n + k
+k
+�2�
+2Hk(q) − Hn+k(q) − qHn−k(q−1)
+�
+= 0
+
+3
+The purpose of this paper is to establish and develop important algebraic identities involving
+q-harmonic numbers and q-binomial coefficients, which may have important applications in the
+theory of q-supercongruences.
+2. Some identities of q-binomial coefficients
+We now state and prove one of the main results of this paper.
+Theorem 2.1. Let α1, α2, . . . , αs be distinct elements of C. For a positive integer m let P(x) =
+(x − α1)m(x − α2)m · · · (x − αs)m. For any polynomial Q(x) such that deg(Q) < deg(P), we
+have
+Q(x)
+P(x) =
+s
+�
+j=1
+m−1
+�
+i=0
+m−1−i
+�
+k=0
+(−1)kgj(αj)Bk(x1, · · · , xk)Q(i)(αj)
+i!k!(x − αj)m−i−k
+.
+(2.1)
+where
+xl = m(l − 1)!
+s
+�
+i=1,i̸=j
+1
+(αj − αi)l
+and gj(x) =
+s
+�
+i=1,i̸=j
+(x − αi)−m.
+Proof. Following [14, Eq.4], we can write
+Q(x) =
+s
+�
+j=1
+m−1
+�
+i=0
+1
+i!Q(i)(αj)Lji(x)[P].
+By virtue of Equation (2) of [14], it is clear that
+Lji(x)[P] = P(x)
+m−1−i
+�
+k=0
+g(k)
+j
+(αj)
+k!(x − αj)m−i−k
+If we combine the two previous identities, we get
+Q(x)
+P(x) =
+s
+�
+j=1
+m−1
+�
+i=0
+m−1−i
+�
+k=0
+g(k)
+j
+(αj)Q(i)(αj)
+i!k!(x − αj)m−i−k .
+Since
+gj(x) = φ(x) ◦ fj(x)
+where φ(x) = exp(mx) and fj(x) = ln(�s
+i=1,i̸=j(x − αi)−1). Then φ(k)(x) = mk exp(mx) and
+f (k)
+j
+(x) = (−1)k(k − 1)!Hk,αs[j](x), where Hl,αs[j](x) = �s
+i=1,i̸=j
+1
+(x−αi)l . By using the Fa`a di
+Bruno formula, we can easily prove that
+g(k)
+j
+(x) = (−1)kgj(x)
+�
+m1+2m2+···+kmk=k
+k!
+m1!m2! · · · mk!
+k
+�
+l=1
+�m(l − 1)!Hl,αs[j](x)
+l!
+�ml
+In particular
+g(k)
+j
+(αj) = (−1)kgj(αj)Bk(x1, · · · , xk)
+This gives the required result.
+□
+Taking αi = q−i in Theorem (2.1), we obtain after some minor manipulations the following
+theorem.
+Theorem 2.2. Let m and n positive integers and let Q(x) be a polynomial such that deg(Q) <
+(n + 1)m, we have
+(q; q)m
+n Q(x)
+(x; q)m
+n+1
+=
+n
+�
+j=0
+�n
+j
+�m
+qm(j+1
+2 )
+m−1
+�
+i=0
+m−1−i
+�
+k=0
+(−1)mj+iBk(x1, x2, · · · , xk)Q(i)(q−j)
+i!k!qj(i+k)(1 − xqj)m−i−k
+.
+where
+x1 = qjm
+1 − q
+�
+Hj(q) − qHn−j(q−1)
+�
+
+4
+SAID ZRIAA AND MOHAMMED MOUC¸OUF
+and
+xl = m(l − 1)!
+n
+�
+i=0,i̸=j
+qjl
+(1 − qj−i)l
+for l = 1, 2, . . ., m − 1,
+In view of Theorem (2.2), we establish interesting corollaries
+Corollary 2.3. Let n be a positive integer and let Q(x) be a polynomial such that deg(Q) <
+2(n + 1). We have
+(q; q)2
+nQ(x)
+(x; q)2
+n+1
+=
+n
+�
+j=0
+�n
+j
+�2
+qj(j+1)
+� Q(q−j)
+(1 − xqj)2 − q−jQ
+′(q−j)
+(1 − xqj) + 2Q(q−j)
+(1 − xqj)
+�Hj(q) − qHn−j(q−1)
+1 − q
+��
+Identity 2.4. Setting Q(x) = 1 in the last corollary, we obtain
+(q; q)2
+n
+(x; q)2
+n+1
+=
+n
+�
+j=0
+�n
+j
+�2
+qj(j+1)
+�
+1
+(1 − xqj)2 +
+2
+1 − xqj
+�Hj(q) − qHn−j(q−1)
+1 − q
+��
+In particular
+(q; q)2
+n =
+n
+�
+j=0
+�n
+j
+�2
+qj(j+1)
+�
+1 + 2
+�Hj(q) − qHn−j(q−1)
+1 − q
+��
+Identity 2.5.
+x2n(q/x; q)2
+n
+(1 − x)(xq; q)2n
+=
+1
+1 − x+
+n
+�
+j=1
+�n
+j
+�2
+qj(j−2n)
+�
+qj − 1
+(1 − xqj)2 +1 − 4[j]Hj(q) + 2[j]Hn+j(q) + 2q[j]Hn−j(q−1)
+1 − xqj
+�
+This identity recovers Zheng identity (1.2).
+Proof. Let
+Q(x) = (1 − x)(x − q)2 · · · (x − qn)2 = (1 − x)x2n(q/x; q)2
+n
+It is not difficult to verify that
+Q(x)
+(x; q)2
+n+1
+=
+x2n(q/x; q)2
+n
+(1 − x)(xq; q)2n
+,
+Q(q−j) = q−j(2n+1)(qj − 1)(q; q)2
+n
+�n + j
+j
+�2
+and
+Q
+′(q−j) = q−2nj(q; q)2
+n
+�n + j
+j
+�2�
+− 1 − 2[j]
+�
+Hn+j(q) − Hj(q)
+��
+Applying Corollary (2.3), we get after some simplifications the desired identity.
+□
+Identity 2.6.
+(xq−n; q)2
+n
+(x; q)2
+n+1
+=
+n
+�
+j=0
+�n
+j
+�2�n + j
+j
+�2
+qj(j+1)−n(n+2j+1)
+�
+1
+(1 − xqj)2 +4Hj(q) − 2qHn−j(q−1) − 2Hn+j(q)
+(1 − q)(1 − xqj)
+�
+This identity gives the explicit representation of (1.1).
+Proof. Let Q(x) = (xq−n; q)2
+n. We have Q(q−j) = q−n(n+2j+1)�n+j
+j
+�2 and
+Q
+′(q−j) = 2q−n(n+2j+1)+j
+�n + j
+j
+�2�Hn+j(q) − Hj(q)
+1 − q
+�
+Using Corollary (2.3), the identity follows.
+□
+Identity 2.7. Letting x = 0 in the last identity, we obtain
+n
+�
+j=0
+�n
+j
+�2�n + j
+j
+�2
+qj(j+1)−2nj
+�
+1 − q + 4Hj(q) − 2qHn−j(q−1) − 2Hn+j(q)
+�
+= qn(n+1)(1 − q)
+
+5
+Corollary 2.8. Let m, n, l be tree positive integers such that 0 ≤ l < (n + 1)m. Then we have
+(q; q)m
+n xl
+(x; q)m
+n+1
+=
+n
+�
+j=0
+�n
+j
+�m
+qm(j+1
+2 )
+m−1
+�
+i=0
+m−1−i
+�
+k=0
+�l
+i
+�(−1)mj+iBk(x1, x2, · · · , xk)
+k!qj(k+l)(1 − xqj)m−i−k
+.
+where
+xl = m(l − 1)!
+n
+�
+i=0,i̸=j
+qjl
+(1 − qj−i)l
+Corollary 2.9. Let m and n be positive integers. We have
+(q; q)m
+n
+(x; q)m
+n+1
+=
+n
+�
+j=0
+�n
+j
+�m
+qm(j+1
+2 )
+m−1
+�
+k=0
+(−1)mjBk(x1, x2, · · · , xk)
+k!qjk(1 − xqj)m−k
+.
+where
+xl = m(l − 1)!
+n
+�
+i=0,i̸=j
+qjl
+(1 − qj−i)l
+Corollary 2.10. Let Q(x) be a polynomial such that deg(Q) < n + 1. Then we have
+(q; q)nQ(x)
+(x; q)n+1
+=
+n
+�
+j=0
+(−1)j
+�n
+j
+�
+q(j+1
+2 ) Q(q−j)
+(1 − xqj).
+The limiting case of Theorem (2.2), is the following
+Theorem 2.11. Let m and n be two positive integers. Let Q(x) be a polynomial of degree l
+with leading coefficient al. Then we have the following curious identity:
+n
+�
+j=0
+�n
+j
+�m
+qm(j
+2)
+m−1
+�
+i=0
+(−1)mj+i+1Bm−1−i(x1, x2, · · · , xm−1−i)Q(i)(q−j)
+i!(m − 1 − i)!
+=
+�
+0
+if
+0 ≤ l < m(n + 1) − 1,
+(−1)m(n+1)(q; q)m
+n q−m(n+1
+2 )al
+if
+l = m(n + 1) − 1.
+where
+xl = m(l − 1)!
+n
+�
+i=0,i̸=j
+qjl
+(1 − qj−i)l
+When m = 1, the formula of Theorem (2.11) reads explicitly as
+Corollary 2.12. Let n be a positive integer and Q(x) be a polynomial of degree l with leading
+coefficient al. Then we have the following identity:
+n
+�
+j=0
+�n
+j
+�
+q(j
+2)(−1)jQ(q−j) =
+�
+0
+if
+0 ≤ l < n,
+(−1)n(q; q)nq−(n+1
+2 )al
+if
+l = n.
+Identity 2.13. Letting Q(x) = (x − 1)l in the last corollary, we obtain, after some simple
+calculations, that
+n
+�
+j=0
+�n
+j
+�
+q(
+j
+2)−jl(−1)j[j]l =
+
+
+
+0
+if
+0 ≤ l < n,
+(−1)n (q; q)n
+(1 − q)n q−(n+1
+2 )
+if
+l = n.
+We remark that in the limiting case q → 1 of this identity, we obtain the famous Euler’s
+formula[2, 10, 11, 15]. Therefore, this identity is the q-analog of Euler’s formula:
+n
+�
+j=0
+�n
+j
+�
+(−1)jjl =
+�
+0
+if
+0 ≤ l < n,
+(−1)nn!
+if
+l = n.
+Opting m = 2 in Theorem (2.11), we gain the following important result
+
+6
+SAID ZRIAA AND MOHAMMED MOUC¸OUF
+Corollary 2.14. Let n be a positive integer and Q(x) be a polynomial of degree l with leading
+coefficient al. Then the following identity holds:
+n
+�
+j=0
+�n
+j
+�2
+qj(j−1)
+�
+Q
+′(q−j) − 2qjQ(q−j)
+1 − q
+�
+Hj(q) − qHn−j(q−1)
+��
+=
+�
+0
+if
+0 ≤ l < 2n + 1,
+(q; q)2
+nq−n(n+1)al
+if
+l = 2n + 1.
+Identity 2.15. If Q(x) = 1, then the formula of Corollary (2.14) gives
+n
+�
+j=0
+�n
+j
+�2
+qj2�
+Hj(q) − qHn−j(q−1)
+�
+= 0
+Identity 2.16. Choosing Q(x) = (1 − x)(x − q)2 · · · (x − qn)2 in Corollary (2.14), we obtain
+n
+�
+j=0
+�n
+j
+�2�n + j
+j
+�2
+qj(j−1)−2nj
+�
+1 + 2[j]Hn+j(q) − 4[j]Hj(q) + 2qHn−j(q−1)
+��
+= q−n(n+1)
+It would be interesting to establish some identities by means of Corollary (2.10).
+Theorem 2.17. Let n be a positive integer and y any complex number. Then the following
+identity holds true:
+n
+�
+j=0
+(−1)j
+�n
+j
+�
+q(
+j+1
+2 )−jn
+(y; q)n+j
+(y; q)j(1 − xqj) = (q; q)n(y/x; q)nxn
+(x; q)n+1
+.
+Proof. Let Q(x) be the polynomial
+Q(x) = (x − y)(x − qy) · · · (x − yqn−1) = (y/x; q)nxn
+Since
+Q(q−j) = (yqj; q)nq−jn and (yqj; q)n = (y; q)n+j
+(y; q)j
+we get form Corollary (2.10) the desired formula.
+□
+Letting y = q in the last theorem, we obtain the following result
+Theorem 2.18. Let n be a positive integer, the following identities hold true
+(1)
+n
+�
+j=0
+(−1)j
+�n
+j
+��n + j
+j
+�q(
+j+1
+2 )−jn
+(1 − xqj) = xn(q/x; q)n
+(x; q)n+1
+.
+(2)
+n
+�
+j=0
+(−1)j
+�n
+j
+��n + j
+j
+� q(j+1
+2 )−jn
+(1 − qj+l) = 0.
+for l = 1, 2, . . . , n.
+(3)
+n−1
+�
+j=0
+(−1)n+1+j
+�n
+j
+��n + j
+j
+�(1 − qn+l)
+(1 − qj+l) q(
+j+1
+2 )+n2−jn =
+�2n
+n
+�
+q(
+n+1
+2 ).
+for l = 1, 2, . . . , n.
+(4)
+q(n+1
+2 ) =
+n
+�
+j=0
+(−1)n−j
+�n
+j
+��n + j
+j
+�
+q(j+1
+2 )−jn.
+Setting x = a in the first statement of Theorem (2.18), we recover Theorem 2 of [12].
+The follwoing result is an extension of Theorem 5 of [19].
+
+7
+Theorem 2.19. Let n be a positive integer. For m = 1, 2, . . . , n, we have
+n
+�
+j=0
+(−1)j
+�n
+j
+��n + j
+j
+�
+q(j
+2)−jn(Hm+j,q(x) − Hj,q(x)) = (−1)nq(n+1
+2 )
+m
+�
+i=1
+qi (xqi−n; q)n
+(xqi; q)n+1
+.
+where
+Hn,q(x) =
+n
+�
+i=1
+qi
+1 − xqi
+Proof. By the first identity of Theorem (2.18), it is easy to check
+n
+�
+j=0
+(−1)j
+�n
+j
+��n + j
+j
+�
+q(
+j
+2)−jn(Hm+j,q(x) − Hj,q(x)) =
+m
+�
+i=1
+qi (xqi)n(q/xqi; q)n
+(xqi; q)n+1
+.
+By means of
+(q/xqi; q)n = (−1)nqn−inx−nq(n
+2)(xqi−n; q)n
+we deduce the result.
+□
+Declarations
+Ethical Approval. Not applicable.
+Competing interests. No potential conflict of interest was reported by the authors.
+Authors’ contributions. The authors contributed equally.
+Funding. This research received no funding.
+Availability of data and materials. Not applicable.
+References
+[1] S. Ahlgren, K. Ono. A Gaussian hypergeometric series evaluation and Ap´ery number congruences, J. Reine
+Angew. Math. 518, 187-212 (2000).
+[2] H. Alzer and R. Chapman. On Boole’s formula for factorials. Australas. J. Combinatorics, 99, 333-336,
+(2014).
+[3] R. Ap´ery. Irrationalit´e de ζ(2) et ζ(3) (French), Ast´erisque 61 (1979), 11–13. Luminy Conference on Arith-
+metic.
+[4] F. Beukers. Another congruence for Ap´ery numbers, J. Number Theory 25, 201-210 (1987).
+[5] W. Chu. A binomial coefficient identity associated with Beukers’ conjecture on Ap´ery numbers, Electron. J.
+Comb. 11 (2004).
+[6] W. Chu and Y. You. Binomial symmetries inspired by Bruckman’s problem. Filomat 24, 41–46 (2010).
+[7] L. Comtet, Advanced combinatorics: the art of finite and infinite expansions. Reidel, Dordrecht 1974.
+[8] G. Gasper, M. Rahman. Basic hypergeometric series, Cambridge University Press, Cambridge, 1990.
+[9] I. Gessel. Some congruences for Ap´ery numbers, J. Number Theory 14, 3, 362–368 (1982).
+[10] H. W. Gould. Euler’s formula for the nth differences of powers, Am. Math. Mon. 85, 450-467 (1978).
+[11] E. A. Karatsuba. On an identity with binomial coefficients. Mathematical Notes, 105, 145-147, (2019).
+[12] E. Kili¸c, H. Prodinger. Evaluation of sums involving products of Gaussian q-binomial coefficients with
+applications to Fibonomial sums. Turj. J. Math. 41, 707-716 (2017).
+[13] C. Krattenthaler, T. Rivoal, and W. Zudilin. S´eries hyperg´eom´etriques basiques, q-analogues des valeurs de
+la fonction zˆeta et s´eries d’Eisenstein (French, with English and French summaries), J. Inst. Math. Jussieu
+5 (2006), 1, 53–79.
+[14] M. Mou¸couf, S. Zriaa, A new approach for computing the inverse of confluent Vandermonde matrices via
+Taylor’s expansion, Linear Multilinear Algebra. (2021), DOI:10.1080/03081087.2021.1940807.
+[15] C. Phoata. Boole’s formula as a consequence of Lagrange’s interpolating polynomial theorem. Integers, 8,
+Article A23, (2008).
+[16] J. X. Sharon, J. Zeng. A q-analog of dual sequences with applications. European Journal of Combinatorics,
+28, 214-227, (2007).
+[17] A. Straub. Supercongruences for polynomial analogs of the Ap´ery numbers. Proc. Am. Math. Soc. 147,
+1023–1036 (2019).
+[18] X. Wang, W. Chu. Harmonic number sums and q-analogues. International Journal of Computer Mathe-
+matics: Computer Systems Theory, 4(1), 48-56 (2019).
+[19] Q. Yan, C. Wei, X. Fan. q-generalizations of Mortenson’s identities and further identities. Ramanujan J.
+35, 131-139, (2014).
+
+8
+SAID ZRIAA AND MOHAMMED MOUC¸OUF
+[20] D. Y. Zheng. An algebraic identity on q-Ap´ery numbers. Discrete. Math, 311, 23-24 (2011).
+Said Zriaa and Mohammed Mouc¸ouf, University Chouaib Doukkali. Department of Mathematics,
+Faculty of science Eljadida, Morocco
+Email address: saidzriaa1992@gmail.com
+Email address: moucou@hotmail.com
+
diff --git a/pNFST4oBgHgl3EQfNTg4/content/tmp_files/load_file.txt b/pNFST4oBgHgl3EQfNTg4/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..56730598d712c54cfdce7befec75b61f75c88354
--- /dev/null
+++ b/pNFST4oBgHgl3EQfNTg4/content/tmp_files/load_file.txt
@@ -0,0 +1,390 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf,len=389
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='13747v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='CO]  31 Jan 2023  ALGEBRAIC IDENTITIES ON Q-HARMONIC NUMBERS AND  Q-BINOMIAL COEFFICIENTS  SAID ZRIAA AND MOHAMMED MOUC¸OUF  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' The aim of this paper is to present a general algebraic identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' Applying this  identity, we provide several formulas involving the q-binomial coefficients and the q-harmonic  numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' We also recover some known identities including an algebraic identity of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='  Zheng on q-Ap´ery numbers and we establish the q-analog of Euler’s formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' The proposed  results may have important applications in the theory of q-supercongruences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' Introduction  For an indeterminate x the q-shifted factorial is usually defined by  (x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' q)0 = 1 and (x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' q)n = (1 − x)(1 − qx) · · · (1 − qn−1x) for n = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='  The Gaussian q-binomial coefficient is correspondingly given by  �n  j  �  =  (q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' q)n  (q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' q)j(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' q)n−j  for j = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=', n,  The q-harmonic numbers are defined by  Hn(q) =  n  �  k=1  1  [k]q  for n = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=',  where the q-numbers are given by  [k] or [k]q := 1 − qk  1 − q = 1 + q + · · · + qk−1  Following Comtet [7], the complete Bell polynomials can be explicitly expressed as  Bn(x1, x2, · · · , xn) =  �  m1+2m2+···+nmn=n  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='  m1!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='m2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' · · · mn!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='  �x1  1!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='  �m1�x2  2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='  �m2  · ·  �xn  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='  �mn  During the last two decades, there has been an increasing interest in studying binomial  sums and their q-analogues, one can consult recent papers, for example [6, 18, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='  For a  comprehensive account of the q-series and its applications to numbers theory, combinatorics  and special functions, we refer the reader to the excellent monograph by G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' Gasper and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='  Rahman [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='  Recently, the study of the q-harmonic congruences turned into a very active  research area (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=', [17] and the references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='  There exist numerous combinatorial identities involving q-binomial coefficients in the math-  ematical literature (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=', [6, 12, 16, 18, 19]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='  Nowadays, there has been growing inter-  est in deriving q-analogues of several combinatorial identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='  This includes, for example,  q-generalizations of some well known identities involving harmonic numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='  The Ap´ery numbers are defined by the following binomial sum  A(n) =  n  �  k=0  �n  k  �2�n + k  k  �2  Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' Algebraic identities, q-binomial coefficients, q-harmonic numbers, complete Bell poly-  nomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='  1  2  SAID ZRIAA AND MOHAMMED MOUC¸OUF  These numbers have many interesting properties that make them extremely useful in the proof  of the irrationality of ζ(3) (see [3] for further details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' Also the Ap´ery numbers have remarkable  arithmetic properties [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' Beukers conjectured [4] that  A  �p − 1  2  �  ≡ a(p) (modp2)  where p is an odd prime and a(n) is determined by  ∞  �  k=1  a(k)qk := q  ∞  �  k=1  (1 − q2k)4(1 − q4k)4 = q − 4q3 − 2q5 + 24q7 + · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='  Beukers’ conjecture was later showed by S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' Ahlgren, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' Ono [1], who reduce this statement in  terms of the harmonic numbers Hn to the identity  n  �  k=1  �n  k  �2�n + k  k  �2�  1 + 2kHn+k + 2kHn−k − 4kHk  �  = 0  In order to give a classical proof of the last identity, Chu [5] presented the following algebraic  identity  n  �  j=0  �n  j  �2�n + j  j  �2�  −j  (x + j)2 + 1 + 2j(Hn+j − Hj) + 2j(Hn−j − Hj)  x + j  �  = x(1 − x)2  n  (x)2  n+1  which gives the desired formula in the limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='  In order to prove irrationality results on the q-analog of ζ(3):  ζq(3) =  +∞  �  k=1  qk(1 + qk)  (1 − qk)3  The authors of [13] introduced implicitly a q-analog of the Ap´ery numbers AKRZ  q  (n) and they  showed that  AKRZ  q  (n) =  n  �  k=0  aq(n, k)  qk  where aq(n, k) can be defined via the following q-partial fraction decomposition  (xq−n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' q)2  n  (x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' q)2  n+1  =  n  �  k=0  � aq(n, k)  (1 − qkx)2 + bq(n, k)  1 − qkx  �  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='1)  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' Straub [17] showed via the partial fraction decomposition technique that AKRZ  q  (n) have the  following explicit q-binomial representation  qn(2n+1)AKRZ  q  (n) =  n  �  k=0  q(n−k)2�n  k  �2�n + k  k  �2  which reduces to Ap´ery numbers when q → 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' Zheng [20] has recently introduced the q-Ap´ery numbers  qn(n+1)AKRZ  q  (n) =  n  �  k=0  qk(k−2n)  �n  k  �2�n + k  k  �2  then he established an interesting algebraic identity  x2n(q/x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' q)2  n  (1 − x)(xq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' q)2n  =  1  1 − x+  n  �  j=1  �  n  j  �2  qj(j−2n)  �  qj − 1  (1 − xqj)2 +1 − 4[j]Hj(q) + 2[j]Hn+j(q) + 2q[j]Hn−j(q−1)  1 − xqj  �  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='2)  which is a q-extension of the Chu’s identity, and obtained the identity  n  �  k=0  qk(k−2n)  �n  k  �2�n + k  k  �2�  2Hk(q) − Hn+k(q) − qHn−k(q−1)  �  = 0  3  The purpose of this paper is to establish and develop important algebraic identities involving  q-harmonic numbers and q-binomial coefficients, which may have important applications in the  theory of q-supercongruences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' Some identities of q-binomial coefficients  We now state and prove one of the main results of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' Let α1, α2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' , αs be distinct elements of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' For a positive integer m let P(x) =  (x − α1)m(x − α2)m · · · (x − αs)m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' For any polynomial Q(x) such that deg(Q) < deg(P), we  have  Q(x)  P(x) =  s  �  j=1  m−1  �  i=0  m−1−i  �  k=0  (−1)kgj(αj)Bk(x1, · · · , xk)Q(i)(αj)  i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' (x − αj)m−i−k  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='1)  where  xl = m(l − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='  s  �  i=1,i̸=j  1  (αj − αi)l  and gj(x) =  s  �  i=1,i̸=j  (x − αi)−m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' Following [14, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='4], we can write  Q(x) =  s  �  j=1  m−1  �  i=0  1  i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='Q(i)(αj)Lji(x)[P].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='  By virtue of Equation (2) of [14], it is clear that  Lji(x)[P] = P(x)  m−1−i  �  k=0  g(k)  j  (αj)  k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' (x − αj)m−i−k  If we combine the two previous identities, we get  Q(x)  P(x) =  s  �  j=1  m−1  �  i=0  m−1−i  �  k=0  g(k)  j  (αj)Q(i)(αj)  i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' (x − αj)m−i−k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='  Since  gj(x) = φ(x) ◦ fj(x)  where φ(x) = exp(mx) and fj(x) = ln(�s  i=1,i̸=j(x − αi)−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' Then φ(k)(x) = mk exp(mx) and  f (k)  j  (x) = (−1)k(k − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='Hk,αs[j](x), where Hl,αs[j](x) = �s  i=1,i̸=j  1  (x−αi)l .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' By using the Fa`a di  Bruno formula, we can easily prove that  g(k)  j  (x) = (−1)kgj(x)  �  m1+2m2+···+kmk=k  k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='  m1!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='m2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' · · · mk!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='  k  �  l=1  �m(l − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='Hl,αs[j](x)  l!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='  �ml  In particular  g(k)  j  (αj) = (−1)kgj(αj)Bk(x1, · · · , xk)  This gives the required result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='  □  Taking αi = q−i in Theorem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='1), we obtain after some minor manipulations the following  theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' Let m and n positive integers and let Q(x) be a polynomial such that deg(Q) <  (n + 1)m, we have  (q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' q)m  n Q(x)  (x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' q)m  n+1  =  n  �  j=0  �n  j  �m  qm(j+1  2 )  m−1  �  i=0  m−1−i  �  k=0  (−1)mj+iBk(x1, x2, · · · , xk)Q(i)(q−j)  i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='qj(i+k)(1 − xqj)m−i−k  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='  where  x1 = qjm  1 − q  �  Hj(q) − qHn−j(q−1)  �  4  SAID ZRIAA AND MOHAMMED MOUC¸OUF  and  xl = m(l − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='  n  �  i=0,i̸=j  qjl  (1 − qj−i)l  for l = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=', m − 1,  In view of Theorem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='2), we establish interesting corollaries  Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' Let n be a positive integer and let Q(x) be a polynomial such that deg(Q) <  2(n + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' We have  (q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' q)2  nQ(x)  (x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' q)2  n+1  =  n  �  j=0  �n  j  �2  qj(j+1)  � Q(q−j)  (1 − xqj)2 − q−jQ  ′(q−j)  (1 − xqj) + 2Q(q−j)  (1 − xqj)  �Hj(q) − qHn−j(q−1)  1 − q  ��  Identity 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' Setting Q(x) = 1 in the last corollary, we obtain  (q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' q)2  n  (x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' q)2  n+1  =  n  �  j=0  �n  j  �2  qj(j+1)  �  1  (1 − xqj)2 +  2  1 − xqj  �Hj(q) − qHn−j(q−1)  1 − q  ��  In particular  (q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' q)2  n =  n  �  j=0  �n  j  �2  qj(j+1)  �  1 + 2  �Hj(q) − qHn−j(q−1)  1 − q  ��  Identity 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='  x2n(q/x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' q)2  n  (1 − x)(xq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' q)2n  =  1  1 − x+  n  �  j=1  �n  j  �2  qj(j−2n)  �  qj − 1  (1 − xqj)2 +1 − 4[j]Hj(q) + 2[j]Hn+j(q) + 2q[j]Hn−j(q−1)  1 − xqj  �  This identity recovers Zheng identity (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' Let  Q(x) = (1 − x)(x − q)2 · · · (x − qn)2 = (1 − x)x2n(q/x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' q)2  n  It is not difficult to verify that  Q(x)  (x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' q)2  n+1  =  x2n(q/x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' q)2  n  (1 − x)(xq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' q)2n  ,  Q(q−j) = q−j(2n+1)(qj − 1)(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' q)2  n  �n + j  j  �2  and  Q  ′(q−j) = q−2nj(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' q)2  n  �n + j  j  �2�  − 1 − 2[j]  �  Hn+j(q) − Hj(q)  ��  Applying Corollary (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='3), we get after some simplifications the desired identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='  □  Identity 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='  (xq−n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' q)2  n  (x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' q)2  n+1  =  n  �  j=0  �n  j  �2�n + j  j  �2  qj(j+1)−n(n+2j+1)  �  1  (1 − xqj)2 +4Hj(q) − 2qHn−j(q−1) − 2Hn+j(q)  (1 − q)(1 − xqj)  �  This identity gives the explicit representation of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' Let Q(x) = (xq−n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' q)2  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' We have Q(q−j) = q−n(n+2j+1)�n+j  j  �2 and  Q  ′(q−j) = 2q−n(n+2j+1)+j  �n + j  j  �2�Hn+j(q) − Hj(q)  1 − q  �  Using Corollary (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='3), the identity follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='  □  Identity 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' Letting x = 0 in the last identity, we obtain  n  �  j=0  �n  j  �2�n + j  j  �2  qj(j+1)−2nj  �  1 − q + 4Hj(q) − 2qHn−j(q−1) − 2Hn+j(q)  �  = qn(n+1)(1 − q)  5  Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' Let m, n, l be tree positive integers such that 0 ≤ l < (n + 1)m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' Then we have  (q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' q)m  n xl  (x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' q)m  n+1  =  n  �  j=0  �n  j  �m  qm(j+1  2 )  m−1  �  i=0  m−1−i  �  k=0  �l  i  �(−1)mj+iBk(x1, x2, · · · , xk)  k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='qj(k+l)(1 − xqj)m−i−k  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='  where  xl = m(l − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='  n  �  i=0,i̸=j  qjl  (1 − qj−i)l  Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' Let m and n be positive integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' We have  (q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' q)m  n  (x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' q)m  n+1  =  n  �  j=0  �n  j  �m  qm(j+1  2 )  m−1  �  k=0  (−1)mjBk(x1, x2, · · · , xk)  k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='qjk(1 − xqj)m−k  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='  where  xl = m(l − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='  n  �  i=0,i̸=j  qjl  (1 − qj−i)l  Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' Let Q(x) be a polynomial such that deg(Q) < n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' Then we have  (q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' q)nQ(x)  (x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' q)n+1  =  n  �  j=0  (−1)j  �n  j  �  q(j+1  2 ) Q(q−j)  (1 − xqj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='  The limiting case of Theorem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='2), is the following  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' Let m and n be two positive integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' Let Q(x) be a polynomial of degree l  with leading coefficient al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' Then we have the following curious identity:  n  �  j=0  �n  j  �m  qm(j  2)  m−1  �  i=0  (−1)mj+i+1Bm−1−i(x1, x2, · · · , xm−1−i)Q(i)(q−j)  i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' (m − 1 − i)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='  =  �  0  if  0 ≤ l < m(n + 1) − 1,  (−1)m(n+1)(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' q)m  n q−m(n+1  2 )al  if  l = m(n + 1) − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='  where  xl = m(l − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='  n  �  i=0,i̸=j  qjl  (1 − qj−i)l  When m = 1, the formula of Theorem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='11) reads explicitly as  Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' Let n be a positive integer and Q(x) be a polynomial of degree l with leading  coefficient al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' Then we have the following identity:  n  �  j=0  �n  j  �  q(j  2)(−1)jQ(q−j) =  �  0  if  0 ≤ l < n,  (−1)n(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' q)nq−(n+1  2 )al  if  l = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='  Identity 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' Letting Q(x) = (x − 1)l in the last corollary, we obtain, after some simple  calculations, that  n  �  j=0  �n  j  �  q(  j  2)−jl(−1)j[j]l =  \uf8f1  \uf8f2  \uf8f3  0  if  0 ≤ l < n,  (−1)n (q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' q)n  (1 − q)n q−(n+1  2 )  if  l = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='  We remark that in the limiting case q → 1 of this identity, we obtain the famous Euler’s  formula[2, 10, 11, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' Therefore, this identity is the q-analog of Euler’s formula:  n  �  j=0  �n  j  �  (−1)jjl =  �  0  if  0 ≤ l < n,  (−1)nn!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='  if  l = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='  Opting m = 2 in Theorem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='11), we gain the following important result  6  SAID ZRIAA AND MOHAMMED MOUC¸OUF  Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' Let n be a positive integer and Q(x) be a polynomial of degree l with leading  coefficient al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' Then the following identity holds:  n  �  j=0  �n  j  �2  qj(j−1)  �  Q  ′(q−j) − 2qjQ(q−j)  1 − q  �  Hj(q) − qHn−j(q−1)  ��  =  �  0  if  0 ≤ l < 2n + 1,  (q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' q)2  nq−n(n+1)al  if  l = 2n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='  Identity 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' If Q(x) = 1, then the formula of Corollary (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='14) gives  n  �  j=0  �n  j  �2  qj2�  Hj(q) − qHn−j(q−1)  �  = 0  Identity 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' Choosing Q(x) = (1 − x)(x − q)2 · · · (x − qn)2 in Corollary (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='14), we obtain  n  �  j=0  �n  j  �2�n + j  j  �2  qj(j−1)−2nj  �  1 + 2[j]Hn+j(q) − 4[j]Hj(q) + 2qHn−j(q−1)  ��  = q−n(n+1)  It would be interesting to establish some identities by means of Corollary (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' Let n be a positive integer and y any complex number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' Then the following  identity holds true:  n  �  j=0  (−1)j  �n  j  �  q(  j+1  2 )−jn  (y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' q)n+j  (y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' q)j(1 − xqj) = (q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' q)n(y/x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' q)nxn  (x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' q)n+1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' Let Q(x) be the polynomial  Q(x) = (x − y)(x − qy) · · · (x − yqn−1) = (y/x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' q)nxn  Since  Q(q−j) = (yqj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' q)nq−jn and (yqj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' q)n = (y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' q)n+j  (y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' q)j  we get form Corollary (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='10) the desired formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='  □  Letting y = q in the last theorem, we obtain the following result  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' Let n be a positive integer, the following identities hold true  (1)  n  �  j=0  (−1)j  �n  j  ��n + j  j  �q(  j+1  2 )−jn  (1 − xqj) = xn(q/x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' q)n  (x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' q)n+1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='  (2)  n  �  j=0  (−1)j  �n  j  ��n + j  j  � q(j+1  2 )−jn  (1 − qj+l) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='  for l = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='  (3)  n−1  �  j=0  (−1)n+1+j  �n  j  ��n + j  j  �(1 − qn+l)  (1 − qj+l) q(  j+1  2 )+n2−jn =  �2n  n  �  q(  n+1  2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='  for l = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='  (4)  q(n+1  2 ) =  n  �  j=0  (−1)n−j  �n  j  ��n + j  j  �  q(j+1  2 )−jn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='  Setting x = a in the first statement of Theorem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='18), we recover Theorem 2 of [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='  The follwoing result is an extension of Theorem 5 of [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='  7  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' Let n be a positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' For m = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' , n, we have  n  �  j=0  (−1)j  �n  j  ��n + j  j  �  q(j  2)−jn(Hm+j,q(x) − Hj,q(x)) = (−1)nq(n+1  2 )  m  �  i=1  qi (xqi−n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' q)n  (xqi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' q)n+1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='  where  Hn,q(x) =  n  �  i=1  qi  1 − xqi  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' By the first identity of Theorem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='18), it is easy to check  n  �  j=0  (−1)j  �n  j  ��n + j  j  �  q(  j  2)−jn(Hm+j,q(x) − Hj,q(x)) =  m  �  i=1  qi (xqi)n(q/xqi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' q)n  (xqi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' q)n+1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='  By means of  (q/xqi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' q)n = (−1)nqn−inx−nq(n  2)(xqi−n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' q)n  we deduce the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='  □  Declarations  Ethical Approval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' Not applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='  Competing interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' No potential conflict of interest was reported by the authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='  Authors’ contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' The authors contributed equally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='  Funding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' This research received no funding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='  Availability of data and materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' Not applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='  References  [1] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' Ahlgren, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' Ono.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' A Gaussian hypergeometric series evaluation and Ap´ery number congruences, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' Reine  Angew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' 518, 187-212 (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='  [2] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' Alzer and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' Chapman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' On Boole’s formula for factorials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' Australas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' Combinatorics, 99, 333-336,  (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='  [3] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' Ap´ery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' Irrationalit´e de ζ(2) et ζ(3) (French), Ast´erisque 61 (1979), 11–13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' Luminy Conference on Arith-  metic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='  [4] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' Beukers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' Another congruence for Ap´ery numbers, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' Number Theory 25, 201-210 (1987).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='  [5] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' Chu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' A binomial coefficient identity associated with Beukers’ conjecture on Ap´ery numbers, Electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='  Comb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' 11 (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='  [6] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' Chu and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' You.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' Binomial symmetries inspired by Bruckman’s problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' Filomat 24, 41–46 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='  [7] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' Comtet, Advanced combinatorics: the art of finite and infinite expansions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' Reidel, Dordrecht 1974.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='  [8] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' Gasper, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' Rahman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' Basic hypergeometric series, Cambridge University Press, Cambridge, 1990.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='  [9] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' Gessel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' Some congruences for Ap´ery numbers, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' Number Theory 14, 3, 362–368 (1982).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='  [10] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' Gould.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' Euler’s formula for the nth differences of powers, Am.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' 85, 450-467 (1978).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='  [11] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' Karatsuba.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' On an identity with binomial coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' Mathematical Notes, 105, 145-147, (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='  [12] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' Kili¸c, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
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+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
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+page_content='  8  SAID ZRIAA AND MOHAMMED MOUC¸OUF  [20] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
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+page_content=' Zheng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
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+page_content=' Discrete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' Math, 311, 23-24 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='  Said Zriaa and Mohammed Mouc¸ouf, University Chouaib Doukkali.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content=' Department of Mathematics,  Faculty of science Eljadida, Morocco  Email address: saidzriaa1992@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='com  Email address: moucou@hotmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
+page_content='com' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFST4oBgHgl3EQfNTg4/content/2301.13747v1.pdf'}
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+arXiv:2301.05458v1  [math.OC]  13 Jan 2023
+ON THE MONOTONICITY OF THE STOPPING BOUNDARY
+FOR TIME-INHOMOGENEOUS OPTIMAL STOPPING PROBLEMS
+ALESSANDRO MILAZZO
+Abstract. We consider a class of time-inhomogeneous optimal stopping problems and we
+provide sufficient conditions on the data of the problem that guarantee monotonicity of the
+optimal stopping boundary.
+In our setting, time-inhomogeneity stems not only from the
+reward function but, in particular, from the time dependence of the drift coefficient of the
+one-dimensional stochastic differential equation (SDE) which drives the stopping problem. In
+order to obtain our results, we mostly employ probabilistic arguments: we use a comparison
+principle between solutions of the SDE computed at different starting times, and martingale
+methods of optimal stopping theory. We also show a variant of the main theorem, which
+weakens one of the assumptions and additionally relies on the renowned connection between
+optimal stopping and free-boundary problems.
+1. introdution
+In this paper we consider a general class of time-inhomogeneous optimal stopping problems
+and we provide simple sufficient conditions on the data of the problem that guarantee mono-
+tonicity of the optimal stopping boundary. The novelty of our work is to prove this result
+when the underlying process is time-inhomogeneous. In our setting, the underlying process is
+the unique strong solution of a one-dimensional stochastic differential equation (SDE) whose
+drift coefficient may be time-dependent. We first show how to obtain monotonicity of the op-
+timal stopping boundary when the reward function is time-homogeneous and then we extend
+the result to the case of a time-dependent reward function, when it is sufficiently regular to
+apply Ito’s formula. We focus our attention on finite-horizon optimal stopping problems but
+our methods clearly apply also to infinite-horizon optimal stopping problems, as the latter do
+not carry an additional time-dependence in the domain of the admissible stopping times.
+The behaviour of the optimal stopping boundary t �→ b(t) is crucial in order to fully
+characterise an optimal stopping problem. In particular, continuity and monotonicity of the
+map t �→ b(t) are two desirable properties. However, this regularity is usually studied on a
+case-by-case basis and the number of works that provide sufficient conditions to obtain these
+results in a general framework is limited. Classical tricks to show continuity of the stopping
+boundary are presented in [27] in various examples, whereas results in a general setting can
+be found in [4] (for one-dimensional diffusions) and [26] (for two-dimensional diffusions).
+Determining monotonicity of t �→ b(t) can be even a more relevant turning point. First, it is
+a helpful result in order to obtain its continuity (as shown, e.g., in [4]). Furthermore, when
+the underlying process is strong Markov, it implies that the optimal stopping time τ ∗
+t,x is a
+continuous function of the starting point (t, x) across the boundary1 or, equivalently, that the
+2020 Mathematics Subject Classification. 60G07, 60G40, 60J60, 49N30, 35R35.
+Key words and phrases. optimal stopping, monotone stopping boundary, time-inhomogeneous diffusions,
+partial information.
+1Here, we mean that if (t, x) = (t, b(t)) and (tn, xn) → (t, x) as n → ∞, then τ ∗
+tn,xn → τ ∗
+t,x as n → ∞, P-a.s.
+1
+
+2
+A. MILAZZO
+boundary is regular for the interior of the stopping set in the sense of diffusions (a concept
+extensively illustrated in [7]). This yields global C1-regularity of the value function, which is
+also a helpful result to characterise the stopping boundary (when continuous) as the unique
+continuous solution of a family of integral equations.
+An extensive probabilistic analysis
+of the geometry of a general class of optimal stopping problems, including continuity and
+monotonicity of the stopping boundary, is presented in [5] when the underlying diffusion and
+reward function are time-homogeneous. The shape of the continuation region is also studied
+under a general framework in [21]. However, their result on the monotonicity of t �→ b(t) (see
+Proposition 4.4 therein) holds only for time-homogeneous diffusions. One contribution of this
+paper is to extend this result to a class of time-inhomogeneous diffusions. Regularity and
+characterisation of the value function are obtained for time-inhomogeneous Markov processes
+in [25] and in [32], and for time-inhomogeneous Poisson processes in [19]. To the best of
+our knowledge, no study of the properties of the stopping boundary has been developed
+in a general setting for time-inhomogeneous diffusions. It is also worth mentioning several
+theoretical works on the behaviour of the stopping boundary and of the value function in the
+context American options. We cite, among others, [2], [9], [20], [22], [24] and [31].
+In order to obtain our results, we rely on probabilistic arguments.
+We first present a
+comparison principle between solutions of the underlying SDE computed at different starting
+times (see Lemma 3.1).
+Specifically, we show that if the drift coefficient t �→ µ(t, x) is
+monotone then the solutions of the SDE computed at different starting times are ordered.
+By means of this result and martingale methods of optimal stopping theory, we prove that
+if in addition a time-homogeneous reward function x �→ g(x) is non-decreasing then t �→
+v(t, x) is also monotone for every x ∈ R (see Theorem 4.1).
+In a variant of the theorem
+we show that if monotonicity of t �→ µ(t, x) does not hold for every x in the state space
+of the underlying process, we are able to weaken this condition and obtain the same result
+under a further assumption which involves the derivatives of the value function and it is
+implied by convexity of x �→ v(t, x) (see Theorem 4.2). This proof additionally relies on the
+renowned connection between optimal stopping and free-boundary problems. An example of
+time-inhomogeneous diffusions which perfectly fits the weaker monotonicity assumption (of
+Theorem 4.2) on t �→ µ(t, x) is given by Brownian bridges. Several works have investigated
+optimal stopping problems involving Brownian bridges and we cite, among others, [29], [15],
+[14], [12], [6], [17] and [3]. Both Theorem 4.1 and Theorem 4.2 lead to the monotonicity of the
+optimal stopping boundary t �→ b(t) (see Corollary 4.6). Then, we prove that monotonicity
+of t �→ b(t) can be obtained even when the reward function g depends on time (see Theorem
+5.4). This extension holds when g is sufficiently regular to apply Ito’s formula and under
+the additional assumption of monotonicity of t �→ Lg(t, x), where L denotes the infinitesimal
+generator of the underlying diffusion.
+Our methods are particularly suited to study optimal stopping problems under incomplete
+information. The common feature of these problems is a random variable whose outcome is
+unknown to the optimiser and which affects the drift of the underlying process and/or the
+reward function. The literature is vast and diverse in this field and we cite, among others,
+[30], [8], [10], [11], [12], [13], [16], [17] [18]. Our results apply, in particular, to models as in [12]
+and [17] where a random variable affects the drift of the underlying process and, in a Bayesian
+formulation of the problem, only the prior distribution of the random variable is known to the
+optimiser. As time evolves, the information obtained from observing the underlying process is
+used to update the initial beliefs about the unknown random variable. By filtering theory, the
+
+A RESULT FOR TIME-INHOMOGENEOUS OPTIMAL STOPPING PROBLEMS
+3
+underlying process can be expressed as a time-inhomogeneous diffusion whose time-dependent
+drift is the conditional expectation of the unknown random variable given the observations
+of the process, which can be obtained through the prior distribution. This, thus, fits into our
+framework, as we illustrate in Section 6.
+The rest of the paper is organised as follows. In Section 2 we formulate the starting problem
+and we recall some standard results on optimal stopping theory. In Section 3 we provide a
+comparison principle between solutions of the underlying SDE starting at different times,
+which will be later used in Section 4 to determine the monotonicity of the optimal stopping
+boundary. In Section 5 we extend the range of applicability for the results of Section 4 by
+considering stopping problems where also the reward functions may depend on time. Our
+methods are particularly suited to study a class of optimal stopping problems under partial
+information, which we describe in Section 6. We conclude by illustrating, in Section 7, some
+simple examples of optimal stopping problems where our results apply.
+2. Starting problem and background results
+Let (Ω, F, P) be a complete probability space with a filtration F := (Ft)t≥0 satisfying the
+usual conditions and let W := (Wt)t≥0 be a standard Brownian motion which is F-adapted.
+Let T ∈ (0, ∞) be a finite time horizon. In this paper we treat finite-horizon optimal stopping
+problems, but it will be clear that our methods apply also to the infinite-horizon analogues,
+where the time-dependence of the value function stems only from the drift coefficient of the
+underlying SDE and not from the domain of the admissible stopping times.
+Given an initial condition Xt = x ∈ R for t ∈ [0, T), let X = (Xs)s≥t be the time-
+inhomogeneous stochastic process described by
+(2.1)
+Xt+s = x +
+� s
+0
+µ(t + r, Xt+r)dr +
+� s
+0
+σ(Xt+r)dWr,
+s ∈ [0, T − t],
+where µ : [0, T] × R → R and σ : R → R are, respectively, the drift and diffusion coefficients.
+We assume that x �→ µ(t, x) is Lipschitz-continuous for every t ∈ [0, T] and that x �→ σ(x)
+satisfies the standard Yamada-Watanabe condition which guarantees the strong existence and
+uniqueness of the solution for the SDE (2.1) (see, e.g., [28, Theorem 40.1]). Namely, we assume
+that there exists an increasing function h : [0, ∞) → [0, ∞) such that
+� ε
+0
+h−1(s)ds = ∞,
+∀ ε > 0
+and
+(2.2)
+�
+σ(x) − σ(y)
+�2 ≤ h(|x − y|),
+∀ x, y ∈ R.
+In order to keep track of the initial condition Xt = x, we will sometimes denote the solution
+X of the SDE (2.1) by Xt,x.
+Given a (terminal) reward function g : R → R, we define the optimal stopping problem
+(2.3)
+v(t, x) := sup
+τ∈Tt
+E
+�
+g(Xt,x
+t+τ )
+�
+,
+(t, x) ∈ [0, T] × R,
+where Tt is the class of F-stopping times τ such that τ ∈ [0, T − t], P-a.s. To simplify the
+exposition, we start by considering stopping problems of the form (2.3). We then extend
+our results to stopping problems that include both a running reward function and a terminal
+reward function which may also depend on time (see Section 5).
+
+4
+A. MILAZZO
+Let C be the continuation region and its complement D := Cc be the stopping region,
+respectively, defined by
+C := {(t, x) ∈ [0, T] × R : v(t, x) > g(x)}
+and
+D := {(t, x) ∈ [0, T] × R : v(t, x) = g(x)}.
+We now state some mild assumptions for the optimal stopping problem (2.3).
+Assumption 2.1. The reward function g : R → R is upper semi-continuous, the value
+function v : [0, T] × R → R is continuous and we have that, for every (t, x) ∈ [0, T] × R,
+(2.4)
+E
+�
+sup
+s∈[t,T]
+��g(Xt,x
+s )
+��
+�
+< ∞.
+Under Assumption 2.1, we obtain the next three propositions, which are standard results
+in optimal stopping.
+Proposition 2.2. Let (t, x) ∈ [0, T] × R, then the stopping time
+(2.5)
+τ ∗ = τ ∗
+t,x := inf{s ∈ [0, T − t] : (t + s, Xt,x
+t+s) /∈ C}
+is optimal for the stopping problem (2.3).
+Proof. See, e.g., [27, Corollary 2.9].
+□
+Proposition 2.3. Let (t, x) ∈ [0, T] × R, then the process V := (Vs)s∈[0,T−t], defined by
+Vs = V t,x
+s
+:= v(t + s, Xt,x
+t+s),
+is a right-continuous supermartingale and the stopped process V ∗ := (Vs∧τ ∗)s∈[0,T−t] is a right-
+continuous martingale.
+Proof. See, e.g., [27, Theorem 2.4].
+□
+Let ∂t, ∂x and ∂xx denote the time derivative, the spatial derivative and the second spatial
+derivative, respectively, and let ∂C denote the boundary of C.
+Proposition 2.4. We have that v ∈ C1,2(C) and it solves the free-boundary problem
+�
+∂t + µ(t, x)∂x + 1
+2(σ(x))2∂xx
+�
+v(t, x) = 0,
+(t, x) ∈ C,
+(2.6)
+v(t, x) = g(x),
+(t, x) ∈ ∂C,
+Proof. By Assumption 2.1, C is an open set. Then the free-boundary problem (2.6) follows,
+e.g., by the same arguments as in the proof of [20, Proposition 2.6].
+□
+Remark 2.5. Continuity of v is not necessary to obtain Proposition 2.2 and Proposition 2.3
+but lower semi-continuity would be sufficient. Moreover, these two propositions may hold with
+no continuity assumption on v: they still hold if, e.g., g is continuous and non-negative and
+the integral condition (2.4) is satisfied (see, e.g., [23, Appendix D]). For the sake of simplicity,
+we assume continuity of v, which is necessary for Proposition 2.4.
+To avoid further initial conditions on the data of the problem, we also introduce the fol-
+lowing assumption.
+
+A RESULT FOR TIME-INHOMOGENEOUS OPTIMAL STOPPING PROBLEMS
+5
+Assumption 2.6. There exists a (lower) optimal stopping boundary for the problem (2.3),
+i.e., a function b : [0, T] → R that separates C from D. That is, we have
+C = {(t, x) ∈ [0, T) × R : x > b(t)}
+and
+D = {(t, x) ∈ [0, T) × R : x ≤ b(t)} ∪ {T} × R.
+Assumption 2.6 is usually proved by probabilistic arguments on a case-by-case basis (see,
+e.g., [20, Proposition 2.1]). It is easy to see that it holds if, e.g., x �→ v(t, x) − g(x) is non-
+decreasing.
+More general sufficient conditions that guarantee the existence of an optimal
+stopping boundary are shown in, e.g., [21, Theorem 4.3] and will be used later in Section 5.
+In this paper, we prove our results when a lower stopping boundary exists but it is clear that
+analogous arguments would follow when an upper stopping boundary exists instead.
+3. A comparison principle
+In this section we provide a comparison principle between solutions of the SDE (2.1) starting
+at different times, which will be applied in Section 4 to obtain monotonicity of the optimal
+stopping boundary.
+We denote by S ⊆ R the state space of the process X defined in (2.1). For every (t, x) ∈
+[0, T] × R, and for a non-empty set O ⊆ [0, T] × S, we define
+τO = τ t,x
+O := inf{s ≥ 0 : (t + s, Xt,x
+t+s) /∈ O} ∧ (T − t).
+Lemma 3.1. Let (t, x) ∈ [0, T] × R and let O ⊆ [0, T] × S be non-empty. Assume that
+(3.1)
+µ(s, y) ≤ µ(u, y),
+∀ (s, y) ∈ O,
+∀ u ∈ [0, s].
+Then, for every u ∈ [0, t], we have that
+P
+�
+Xt,x
+t+s∧τO ≤ Xu,x
+u+s∧τO,
+∀ s ∈ [0, T − t]
+�
+= 1,
+where τO = τ t,x
+O .
+Proof. Let X1
+s := Xt,x
+t+s, X2
+s := Xu,x
+u+s, µ1(s, y) := µ(t + s, y) and µ2(s, y) := µ(u + s, y) with
+u ∈ [0, t]. Thus, for i = 1, 2, we have
+Xi
+s = x +
+� s
+0
+µi(r, Xi
+r)dr +
+� s
+0
+σ(Xi
+r)dWr.
+Then, for Y := X1 − X2 by assumption (2.2), we obtain
+� s
+0
+h(Yr)−1
+1{Yr>0}d[Y ]r =
+� s
+0
+h(|X1
+r − X2
+r |)−1�
+σ(X1
+r ) − σ(X2
+r )
+�2
+1{Yr>0}dr ≤ s.
+Therefore, we have (see, e.g., [28, Ch. V, Prop. 39.3]) that L0
+s(Y ) = 0 for every s ∈ [0, T], where
+L0(Y ) denotes the local time of Y at 0. Thus, by Tanaka’s formula, for every s ∈ [0, T − t]
+we obtain
+�
+X1
+s∧τO − X2
+s∧τO
+�+ =
+� s∧τO
+0
+�
+µ1(r, X1
+r ) − µ2(r, X2
+r )
+�
+1{X1r −X2r >0}dr
++
+� s∧τO
+0
+�
+σ(X1
+r ) − σ(X2
+r )
+�
+1{X1r −X2r >0}dWr,
+
+6
+A. MILAZZO
+where τO = τ t,x
+O
+and (x)+ := max{x, 0}. Hence,
+0 ≤ E
+��
+X1
+s∧τO − X2
+s∧τO
+�+�
+= E
+� � s∧τO
+0
+�
+µ(t + r, X1
+r ) − µ(u + r, X2
+r )
+�
+1{X1r −X2r >0}dr
+�
+≤ E
+� � s∧τO
+0
+�
+µ(u + r, X1
+r ) − µ(u + r, X2
+r )
+�
+1{X1r −X2r >0}dr
+�
+≤ E
+� � s∧τO
+0
+K(u + r)
+�
+X1
+r − X2
+r
+�+dr
+�
+,
+where K(t) > 0 is the Lipschitz constant for x �→ µ(t, x) and the second to last inequality
+follows from assumption (3.1). Then, by Gronwall’s lemma, we obtain that
+E
+��
+X1
+s∧τO − X2
+s∧τO
+�+�
+= 0,
+∀ s ∈ [0, T − t],
+and by continuity of Y = X1 − X2 we reach the desired result.
+□
+Remark 3.2. Let (t, x) ∈ [0, T] × R. Notice that if O = [0, T] × S, then τ t,x
+O = T − t and so
+the result of Lemma 3.1 reads
+P
+�
+Xt,x
+t+s ≤ Xu,x
+u+s,
+∀ s ∈ [0, T − t]
+�
+= 1.
+4. Main results
+In this section we illustrate our main result for the optimal stopping problem (2.3), which
+provides monotonicity of t �→ v(t, x) and in turn implies monotonicity of the stopping bound-
+ary. This is obtained by means of Lemma 3.1 and will be presented in two versions (Theorem
+4.1 and Theorem 4.2) under different assumptions.
+Theorem 4.1. Let Assumption 2.1 hold. Moreover, assume that
+(i) x �→ g(x) is non-decreasing.
+(ii) t �→ µ(t, x) is non-increasing for every x ∈ S.
+Then, t �→ v(t, x) is non-increasing for every x ∈ R.
+Proof. Let (t, x) ∈ [0, T] × R and u ∈ [0, t]. By assumption (ii), we can apply Lemma 3.1 with
+O = [0, T] × S and obtain that
+(4.1)
+P
+�
+Xt,x
+t+s ≤ Xu,x
+u+s,
+∀ s ∈ [0, T − t]
+�
+= 1.
+By the (super)martingale property of the value function (recall Proposition 2.3) and since
+τ ∗ = τ ∗
+t,x is optimal for v(t, x) and sub-optimal for v(u, x), we have that
+v(t, x) − v(u, x) = V t,x
+0
+− V u,x
+0
+≤ E
+�
+V t,x
+τ ∗ − V u,x
+τ ∗
+�
+= E
+�
+v(t + τ ∗, Xt,x
+t+τ ∗) − v(u + τ ∗, Xu,x
+u+τ ∗)
+�
+≤ E
+�
+g(Xt,x
+t+τ ∗) − g(Xu,x
+u+τ ∗)
+�
+≤ 0,
+where to obtain the last inequality we have used assumption (i) and result (4.1). Hence,
+t �→ v(t, x) is non-increasing for every x ∈ R.
+□
+
+A RESULT FOR TIME-INHOMOGENEOUS OPTIMAL STOPPING PROBLEMS
+7
+We now show that we can weaken the monotonicity assumption on t �→ µ(t, x) but still
+obtain, under an additional assumption, the same result as in Theorem 4.1. This alternative
+version partially relies on the free-boundary problem (2.6) and turns out to be useful in some
+optimal stopping problems, as we will illustrate in Section 7.
+Let M := {(t, x) ∈ [0, T] × S : µ(t, x) < 0} and let us denote by Mc its complement, i.e.,
+(4.2)
+Mc := ([0, T] × S) \ M = {(t, x) ∈ [0, T] × S : µ(t, x) ≥ 0}.
+Throughout this paper we also assume that M is an open set, so that (t + τM, Xt+τM) ∈ Mc
+on {τM < T − t}, where recall that
+(4.3)
+τM = τ t,x
+M := inf{s ≥ 0 : (t + s, Xt,x
+t+s) /∈ M} ∧ (T − t).
+This holds if, e.g., µ is upper semi-continuous.
+Theorem 4.2. Let Assumption 2.1 hold. Moreover, assume that
+(i) x �→ g(x) is non-decreasing.
+(ii) µ(t, x) ≤ µ(t − ε, x) for every (t, x) ∈ M, ε ∈ (0, t).
+(iii) σ2(x)∂xxv(t, x) ≥ −2µ(t, x)∂xv(t, x) for every (t, x) ∈ C ∩ Mc.
+Then, t �→ v(t, x) is non-increasing for every x ∈ R.
+Remark 4.3. Since x �→ Xt,x is non-decreasing (see, e.g., [28, Ch. V, Th. 43.1]) and, under
+the assumptions of Theorem 4.2, x �→ g(x) is non-decreasing, we also have that x �→ v(t, x)
+is non-decreasing. Thus, notice that assumption (iii) holds, in particular, if x �→ v(t, x) is
+convex.
+This is in turn implied by convexity of x �→ Xt,x and of x �→ g(x).
+Therefore,
+assumption (iii) of Theorem 4.2 can be substituted by convexity of x �→ Xt,x and of x �→ g(x).
+However, if σ(x) is sufficiently small or if µ(t, x)∂xv(t, x) is sufficiently large on Mc, then we
+may not need x �→ v(t, x) to be convex in order to satisfy assumption (iii).
+Proof. We prove the result of the theorem in two steps. We first show that ∂tv(t, x) ≤ 0 for
+every (t, x) /∈ ∂C and we then prove that this implies that t �→ v(t, x) is non-increasing for
+every x ∈ R.
+Step 1. If (t, x) ∈ D \ ∂C then v(t, x) = g(x) and so ∂tv(t, x) = 0. If (t, x) ∈ C ∩ Mc (we
+can skip this step if Mc = ∅), then by (2.6)
+∂tv(t, x) + µ(t, x)∂xv(t, x) + 1
+2(σ(x))2∂xxv(t, x) = 0,
+and, by assumption (iii), we obtain
+(4.4)
+∂tv(t, x) ≤ 0,
+∀ (t, x) ∈ C ∩ Mc.
+To conclude the proof we consider (t, x) ∈ C ∩ M (we can skip this step if M = ∅). By
+assumption (ii) we can apply Lemma 3.1 with O = M and, for every ε ∈ (0, t), we obtain
+(4.5)
+P
+�
+Xt,x
+t+s∧τM ≤ Xt−ε,x
+t−ε+s∧τM,
+∀ s ∈ [0, T − t]
+�
+= 1,
+where τM = τ t,x
+M is defined in (4.3). Let ε ∈ (0, t), τ ∗ = τ ∗
+t,x (recall (2.5)) and ρ := τ ∗ ∧ τM.
+By the (super)martingale property of the value function (recall Proposition 2.3) and since τ ∗
+
+8
+A. MILAZZO
+is optimal for v(t, x) and ρ is sub-optimal for v(t − ε, x), we have that
+v(t, x) − v(t − ε, x) ≤ E
+�
+v(t + ρ, Xt,x
+t+ρ) − v(t − ε + ρ, Xt−ε,x
+t−ε+ρ)
+�
+≤ E
+�
+1{τ ∗≤τM}
+�
+g(Xt,x
+t+τ ∗) − g(Xt−ε,x
+t−ε+τ ∗)
+��
++ E
+�
+1{τM<τ ∗}
+�
+v(t + τM, Xt,x
+t+τM) − v(t − ε + τM, Xt−ε,x
+t−ε+τM)
+��
+= E
+�
+1{τ ∗≤τM}
+�
+g(Xt,x
+t+τ ∗) − g(Xt−ε,x
+t−ε+τ ∗)
+��
++ E
+�
+1{τM<τ ∗}
+�
+v(t + τM, Xt,x
+t+τM) − v(t − ε + τM, Xt,x
+t+τM)
+��
++ E
+�
+1{τM<τ ∗}
+�
+v(t − ε + τM, Xt,x
+t+τM) − v(t − ε + τM, Xt−ε,x
+t−ε+τM)
+��
+≤ E
+�
+1{τM<τ ∗}
+�
+v(t + τM, Xt,x
+t+τM) − v(t − ε + τM, Xt,x
+t+τM)
+��
+,
+where to obtain the last inequality we have used assumption (i) and result (4.5) for the first
+term; result (4.5) and the fact that x �→ v(t, x) is non-decreasing (recall Remark 4.3) for the
+third term. Dividing by ε, letting ε → 0 and applying dominated convergence theorem (by
+assumption (2.4)), we obtain
+∂tv(t, x) ≤ E
+�
+1{τM<τ ∗}∂tv(t + τM, Xt,x
+t+τM)
+�
+≤ 0,
+where the last inequality follows from (4.4). Hence, ∂tv(t, x) ≤ 0 also for (t, x) ∈ C ∩ M and
+the proof of Step 1 is completed.
+Step 2. If (t, x) ∈ D, then v(t, x) = g(x) and, since v(s, x) ≥ g(x) for every (s, x) ∈ [0, T]×R,
+then v(s, x) ≥ v(t, x) for every s ∈ [0, t].
+Now let (t, x) ∈ C. We want to show that also (s, x) ∈ C for every s ∈ [0, t], which by Step 1
+would imply that v(s, x) ≥ v(t, x) for every s ∈ [0, t] and would conclude the proof. Assume,
+by contradiction, that
+Dt,x := {s ∈ [0, t] : (s, x) ∈ D} ̸= ∅
+and let t0 := sup Dt,x.
+Recall that, since (t, x) ∈ C, we have v(t, x) > g(x).
+Since v is
+continuous (by Assumption 2.1), then t0 < t and t0 ∈ Dt,x, i.e., (t0, x) ∈ D and so v(t0, x) =
+g(x). Moreover, by definition of t0, we have (s, x) ∈ C for every s ∈ (t0, t] and so
+v(t, x) − v(s, x) =
+� t
+t0
+∂tv(s, x)ds ≤ 0,
+∀ s ∈ (t0, t],
+where the last inequality follows from Step 1. Hence, by continuity of v, we have that v(t0, x) ≥
+v(t, x). This leads to a contradiction, as we would obtain
+g(x) = v(t0, x) ≥ v(t, x) > g(x).
+□
+Remark 4.4. If assumption (ii) in Theorem 4.1 (and similarly for assumptions (ii) and (iii)
+in Theorem 4.2) is substituted by a symmetric assumption (i.e., if t �→ µ(t, x) is increasing)
+then, in infinite-horizon problems, we would obtain a symmetric result, i.e., t �→ v(t, x) would
+be increasing. However, this is, in general, not the case for finite-horizon problems. In that
+context we would have two opposite driving effects as time increases: the drift µ that increases
+and the stopping time domain Tt that shrinks. The former leads to an increase of the value
+function with respect to time, whereas the latter leads to a decrease of the value function
+
+A RESULT FOR TIME-INHOMOGENEOUS OPTIMAL STOPPING PROBLEMS
+9
+with respect to time. In order to study the monotonicity of t �→ v(t, x) in such problems, it
+would be necessary (and, perhaps, not sufficient) to have a quantitative information on the
+monotonicity of t �→ µ(t, x).
+Remark 4.5. In some cases it is possible to apply a pure PDE approach, as in (4.4), and to
+derive monotonicity of t �→ v(t, x) also when the diffusion coefficient may be time-dependent.
+Consider the same SDE as in (2.1) but when also σ may be a function of time, i.e., σ :
+[0, T] × R → R. If µ(t, x) ≥ 0 for every (t, x) ∈ [0, T] × S, then Mc = [0, T] × S (recall (4.2)).
+Thus, under assumptions (i) and (iii) of Theorem 4.2 and in the same way as in (4.4), we
+would obtain
+∂tv(t, x) ≤ 0,
+∀ (t, x) /∈ ∂C.
+Monotonicity of t �→ v(t, x) then follows as in Step 2 of the proof of Theorem 4.2.
+Monotonicity of t �→ v(t, x), which follows from either Theorem 4.1 or Theorem 4.2, then
+yields monotonicity of the optimal stopping boundary.
+Corollary 4.6. If t �→ v(t, x) is non-increasing for every x ∈ R and Assumption 2.6 holds,
+then the optimal stopping boundary t �→ b(t) is non-decreasing.
+Proof. Let (t, x) ∈ C. Then, v(t, x) > g(x) and, since t �→ v(t, x) is non-increasing, we obtain
+that v(s, x) ≥ v(t, x) > g(x) for every s ∈ [0, t]. Hence, also (s, x) ∈ C and thus t �→ b(t) is
+non-decreasing.
+□
+Remark 4.7. Monotonicity of t �→ v(t, x) is also a helpful result to obtain continuity of the
+stopping boundary (see, e.g., arguments as in [4, Section 3] and [6, Lemma 4]).
+5. Extension to time-dependent reward functions
+In this section we show how to obtain monotonicity of the stopping boundary for more gen-
+eral time-inhomogeneous optimal stopping problems, which include a running reward function
+and a terminal reward function that may also depend on time. We consider the same under-
+lying framework of Section 2 but we study the optimal stopping problem
+(5.1)
+v(t, x) := sup
+τ∈Tt
+E
+� � τ
+0
+f(t + s, Xt,x
+t+s)ds + g(t + τ, Xt,x
+t+τ)
+�
+,
+(t, x) ∈ [0, T] × R,
+where X = Xt,x is defined in (2.1), f : [0, T] × R → R is a running reward function and
+g : [0, T] × R → R is a terminal reward function. For this problem the continuation region C
+and the stopping region D are defined, respectively, by
+C := {(t, x) ∈ [0, T] × R : v(t, x) > g(t, x)}
+and
+D := {(t, x) ∈ [0, T] × R : v(t, x) = g(t, x)}.
+We then introduce the following assumption.
+Assumption 5.1. We have that g ∈ C1,2([0, T] × R), the value function v : [0, T] × R → R
+is continuous and, for every (t, x) ∈ [0, T] × R,
+E
+�
+sup
+s∈[t,T]
+���
+� s
+0
+f(r, Xt,x
+r )dr + g(s, Xt,x
+s )
+���
+�
+< ∞.
+
+10
+A. MILAZZO
+Notice that Assumption 5.1 is the analogous of Assumption 2.1 except for the stronger
+regularity of g. Under this regularity, we can apply Ito’s formula and obtain that
+g(t + s, Xt+s) = g(t, x) +
+� s
+0
+Lg(t + r, Xt+r)dr,
+∀ s ∈ [0, T − t],
+where L is defined by
+Lg(t, x) :=
+�
+∂t + µ(t, x)∂x + 1
+2(σ(x))2∂xx
+�
+g(t, x).
+The function w(t, x) := v(t, x) − g(t, x) is, thus, the value function for the optimal stopping
+problem
+(5.2)
+w(t, x) = sup
+τ∈Tt
+E
+� � τ
+0
+h(t + s, Xt,x
+t+s)ds
+�
+,
+(t, x) ∈ [0, T] × R,
+where h(t, x) := f(t, x) + Lg(t, x). Then, notice that
+C = {(t, x) ∈ [0, T] × R : w(t, x) > 0}
+and
+D = {(t, x) ∈ [0, T] × R : w(t, x) = 0}.
+Analogously to Section 2, under Assumption 5.1, we have that the stopping time
+τ ∗ = τ ∗
+t,x := inf{s ∈ [0, T − t] : (t + s, Xt,x
+t+s) /∈ C}
+is optimal for the problem (5.1) (and thus also for the problem (5.2)). Moreover, we obtain
+that the process V := (Vs)s∈[0,T−t], defined by
+(5.3)
+Vs = V t,x
+s
+:=
+� s
+0
+h(t + r, Xt,x
+t+r)dr + w(t + s, Xt,x
+t+s),
+is a right-continuous supermartingale and the process V ∗ := (Vs∧τ ∗)s∈[0,T−t] is a right-
+continuous martingale.
+Remark 5.2. For the sake of simplicity, we have assumed g ∈ C1,2([0, T] × R) but one may
+consider different (weaker) conditions in order to apply Ito’s formula and reformulate the
+stopping problem (5.1) into the stopping problem (5.2).
+We then study the optimal stopping problem (5.1) by means of the equivalent problem
+(5.2). We have the following result on the existence of an optimal stopping boundary.
+Proposition 5.3. Assume that x �→ h(t, x) is non-decreasing for every t ∈ [0, T], then there
+exists a lower optimal stopping boundary for the problem (5.1), i.e., a function b : [0, T) →
+R ∪ {±∞} such that
+C = {(t, x) ∈ [0, T) × R : x > b(t)}
+and
+D = {(t, x) ∈ [0, T) × R : x ≤ b(t)} ∪ {T} × R.
+Proof. Since x �→ h(t, x) is non-decreasing for every t ∈ [0, T], then x �→ w(t, x) is non-
+decreasing for every t ∈ [0, T]. Hence, if (t, x1) ∈ D, then (t, x2) ∈ D for every x2 ∈ (−∞, x1].
+Therefore, for t ∈ [0, T), the function
+b(t) := sup{x ∈ R : w(t, x) = 0}
+is a lower optimal stopping boundary for the stopping problem (5.2) and, thus, also for
+(5.1).
+□
+
+A RESULT FOR TIME-INHOMOGENEOUS OPTIMAL STOPPING PROBLEMS
+11
+We can now obtain monotonicity of the optimal stopping boundary also for the more general
+class of time-inhomogeneous optimal stopping problems in (5.1). Recall that S denotes the
+state space of X, and that we define h(t, x) := f(t, x) + Lg(t, x) and
+Lg(t, x) :=
+�
+∂t + µ(t, x)∂x + 1
+2(σ(x))2∂xx
+�
+g(t, x).
+Theorem 5.4. Let Assumption 5.1 hold. Moreover, assume that
+(i) x �→ h(t, x) is non-decreasing for every t ∈ [0, T] and t �→ h(t, x) is non-increasing for
+every x ∈ S.
+(ii) t �→ µ(t, x) is non-increasing for every x ∈ S.
+Then, t �→ w(t, x) is non-increasing for every x ∈ R and so the optimal stopping boundary
+t �→ b(t) is non-decreasing.
+Proof. Let (t, x) ∈ [0, T] × R and u ∈ [0, t]. By assumption (ii), we can apply Lemma 3.1 with
+O = [0, T] × S and obtain that
+(5.4)
+P
+�
+Xt,x
+t+s ≤ Xu,x
+u+s,
+∀ s ∈ [0, T − t]
+�
+= 1.
+By the (super)martingale property (5.3) of V and since τ ∗ = τ ∗
+t,x is optimal for w(t, x) and
+sub-optimal for w(u, x), we have that
+w(t, x) − w(u, x) = V t,x
+0
+− V u,x
+0
+≤ E
+�
+V t,x
+τ ∗ − V u,x
+τ ∗
+�
+= E
+� � τ ∗
+0
+�
+h(t + s, Xt,x
+t+s) − h(u + s, Xu,x
+u+s)
+�
+ds
+�
++ E
+�
+w(t + τ ∗, Xt,x
+t+τ ∗) − w(u + τ ∗, Xu,x
+u+τ ∗)
+�
+≤ E
+� � τ ∗
+0
+�
+h(t + s, Xt,x
+t+s) − h(u + s, Xt,x
+t+s)
+�
+ds
+�
++ E
+� � τ ∗
+0
+�
+h(u + s, Xt,x
+t+s) − h(u + s, Xu,x
+u+s)
+�
+ds
+�
+≤ 0,
+where the last inequality follows from assumption (i) and result (5.4). Hence, t �→ w(t, x) is
+non-increasing for every x ∈ R. The monotonicity of t �→ b(t) (whose existence is guaranteed
+by Proposition 5.3) is, thus, obtained by the same arguments as in the proof of Corollary
+4.6.
+□
+Remark 5.5. Notice that the proof of [21, Proposition 4.4], which provides monotonicity of
+the optimal stopping boundary, holds only if the underlying process is time-homogeneous. Our
+Theorem 5.4 extends that result to time-inhomogeneous optimal stopping problems, under the
+additional assumption that t �→ µ(t, x) is non-increasing for every x ∈ R.
+6. Optimal stopping under incomplete information
+Our methods are particularly suited to study optimal stopping problems under incomplete
+information. To this purpose, in this section, we provide some background material on this
+class of problems and in Section 7 we will look into a specific example.
+The common feature of these stopping problems is a random variable whose outcome is
+unknown to the optimiser and which affects the drift of the underlying process and/or the
+payoff function. The literature is vast and diverse in this field and we cite, among others,
+
+12
+A. MILAZZO
+[30], [8], [10], [11], [12], [13], [16], [17] [18]. We focus, in particular, on models as in [12] and
+[17] where a random variable affects the drift of the underlying process and, in a Bayesian
+formulation of the problem, only the prior distribution of the random variable is known to the
+optimiser. As time evolves, the information obtained from observing the underlying process
+is used to update the initial beliefs about the unknown random variable.
+Let (Ω, F, P) be a complete probability space on which it is defined a standard Brownian
+motion W := (Wt)t≥0 and a real-valued random variable Y with (prior) probability distribu-
+tion ν. Let T ∈ (0, ∞) be a finite time horizon. The underlying process X evolves according
+to
+(6.1)
+dXt = h(Y )dt + dWt,
+X0 = x ∈ R,
+where h is a measurable function such that
+�
+R |h(y)|ν(dy) < ∞. Given a reward function
+g : [0, T] × R → R, we can then define the stopping problem
+(6.2)
+V := sup
+τ∈[0,T]
+E
+�
+g(τ, Xτ )
+�
+,
+where τ is a stopping time with respect to (FX
+t )t∈[0,T], the augmented filtration generated by
+X.
+It is well-known from filtering theory (see, e.g., [1, Proposition 3.16]) that E[h(Y )|FX
+t ] =
+E[h(Y )|Xt] =: f(t, Xt) where
+(6.3)
+f(t, x) :=
+�
+R h(y)exy−y2t/2ν(dy)
+�
+R exy−y2t/2ν(dy)
+.
+Moreover,
+(6.4)
+dXs = f(s, Xs)ds + dBs,
+where Bt :=
+� t
+0
+�
+h(Y ) − E[h(Y )|FX
+s ]
+�
+ds + Wt is an FX-Brownian motion (see, e.g., [1, Propo-
+sition 2.30]) known as the “innovation process”. By means of (6.4), we can embed the original
+stopping problem (6.2) into a Markovian framework and define the value function
+(6.5)
+v(t, x) :=
+sup
+τ∈[0,T−t]
+E
+�
+g(t + τ, Xt,x
+t+τ)
+�
+,
+where τ is a stopping time with respect to the augmented filtration generated by B, and Xt,x
+follows the dynamics in (6.4) with Xt,x
+t
+= x ∈ R. The optimal stopping problem (6.5) is
+now in the form of (5.1) and we may, thus, apply Theorem 5.4 to obtain monotonicity of its
+stopping boundary. The form of the drift coefficient f(t, x) defined in (6.3) highly depends
+on the prior distribution ν and in Section 7 we will look at a simple example where the
+monotonicity required in assumption (ii) of Theorem 5.4 holds.
+7. Examples
+In this section we show some simple examples of time-inhomogeneous optimal stopping
+problems where our results apply. We consider different underlying time-inhomogeneous dif-
+fusions of the form (2.1) that give rise to corresponding optimal stopping problems of the
+form (2.3). We then determine under which conditions on the data of the problems we can
+apply our theorems and obtain monotonicity of the stopping boundary. We fix a complete
+probability space (Ω, F, P) with a filtration F := (Ft)t≥0 satisfying the usual conditions. We
+let W := (Wt)t≥0 be a standard Brownian motion which is F-adapted and T ∈ (0, ∞) be a
+
+A RESULT FOR TIME-INHOMOGENEOUS OPTIMAL STOPPING PROBLEMS
+13
+finite time horizon. For simplicity, in the following examples we also let Assumption 2.1 and
+Assumption 2.6 hold.
+7.1. Applications of Theorem 4.1. We begin by considering two simple time-inhomogeneous
+diffusions and the corresponding optimal stopping problems of the form (2.3). For the reward
+function of these two examples we only assume that x �→ g(x) is non-decreasing, as required
+in Theorem 4.1.
+First, for (t, x) ∈ [0, T]×R, let X = Xt,x be a Brownian motion with time-dependent drift,
+described by
+Xt+s = x +
+� s
+0
+µ(t + r)dr + σWs,
+s ∈ [0, T − t].
+If t �→ µ(t) is non-increasing, then we can apply Theorem 4.1 and obtain that t �→ v(t, x)
+is non-increasing for every x ∈ R. Thus, Corollary 4.6 guarantees that the corresponding
+optimal stopping boundary t �→ b(t) is non-decreasing.
+Now, for (t, x) ∈ [0, T] × (0, ∞), let X = Xt,x be a geometric Brownian motion with
+time-dependent drift, described by
+Xt+s = x +
+� s
+0
+γ(t + r)Xt+rdr +
+� s
+0
+σXt+rdWr,
+s ∈ [0, T − t].
+Its state space is S = (0, ∞) and its drift is µ(t, x) := xγ(t). If t �→ γ(t) is non-increasing,
+we can apply Theorem 4.1 and obtain that t �→ v(t, x) is non-increasing for every x ∈ R.
+Thus, Corollary 4.6 guarantees that the corresponding optimal stopping boundary t �→ b(t)
+is non-decreasing.
+Notice that the assumptions on the monotonicity of the drifts for the previous two examples
+could be weakened by considering assumption (ii) of Theorem 4.2 instead.
+However, this
+comes with the cost of adding assumption (iii) of Theorem 4.2 (which is implied by convexity
+of x �→ Xt,x and of x �→ g(x), recall Remark 4.3).
+7.2. Applications of Theorem 4.2. For some underlying time-inhomogeneous diffusion it
+happens that assumption (ii) of Theorem 4.1 does not hold. In these cases it is useful to
+consider Theorem 4.2, which weakens this assumption. We now show an example of this
+situation. This application is even more suited to our techniques as no specific assumption
+on the drift of the underlying process is required in order to apply Theorem 4.2. This is the
+case when the underlying time-inhomogeneous diffusion is a Brownian bridge and an example
+of this optimal stopping problem, where g(x) = ex, is studied in [6]. For our example we
+assume that x �→ g(x) is non-decreasing (as required by assumption (i) of Theorem 4.2) and,
+for simplicity, that it is convex (to satisfy assumption (iii) of Theorem 4.2, but convexity is
+not strictly necessary, as explained in Remark 4.3).
+For (t, x) ∈ [0, T] × R, let X = Xt,x be a Brownian bridge pinned at 0 at time T ∈ (t, ∞),
+whose dynamics are described by
+(7.1)
+Xt+s = x −
+� s
+0
+Xt+r
+T − t − rdr + σWs,
+s ∈ [0, T − t),
+with XT = 0. It is easy to check that the unique strong solution to this SDE is given by
+Xt+s = (1 − t − s)
+�
+x
+1 − t +
+� s
+0
+1
+1 − t − rdWr
+�
+,
+s ∈ [0, T − t).
+
+14
+A. MILAZZO
+Hence, x �→ Xt,x is linear (and thus convex) and together with convexity of x �→ g(x) we
+have that assumption (iii) of Theorem 4.2 is satisfied (recall Remark 4.3). In order to apply
+Theorem 4.2, we are only left to check that assumption (ii) is satisfied. The drift coefficient
+of the SDE (7.1) is µ(t, x) := −x/(T − t). Thus, we have that
+M := {(t, x) ∈ [0, T) × R : µ(t, x) < 0} = {(t, x) ∈ [0, T) × R : x > 0}
+and
+µ(t, x) = −
+x
+T − t ≤ −
+x
+T − t + ε = µ(t − ε, x),
+(t, x) ∈ M,
+ε ∈ (0, t).
+Therefore, also assumption (ii) of Theorem 4.2 holds. Hence, we can apply Theorem 4.2 and
+obtain that t �→ v(t, x) is non-increasing for every x ∈ R. Thus, Corollary 4.6 guarantees that
+the corresponding optimal stopping boundary t �→ b(t) is non-decreasing.
+Further examples can be shown to satisfy the assumptions of Theorem 4.2. For instance,
+optimal stopping problems where the underlying diffusion follows the Vasicek model (i.e., an
+Ornstein-Ulhenbeck process) or the Cox-Ingersoll-Ross model can be studied in their time-
+inhomogeneous version, i.e., when the long-term mean is allowed to be time-dependent.
+7.3. An application to optimal stopping under incomplete information. To conclude,
+we consider an example of an optimal stopping problem under incomplete information as in
+Section 6, for which we want to apply Theorem 4.1. Assume that X follows the dynamics as
+in (6.1) with h(y) = y, i.e.,
+dXt = Y dt + dWt,
+where Y is a real-valued random variable with prior distribution ν which has finite first
+moment. Then, by filtering theory, we have that
+dXt = f(t, Xt)dt + dBt,
+where
+f(t, x) :=
+�
+R yexy−y2t/2ν(dy)
+�
+R exy−y2t/2ν(dy) ,
+and B is a Brownian motion with respect to the filtration generated by X. We can then
+consider corresponding optimal stopping problems as in (6.5), where the reward function
+x �→ g(x) is non-decreasing (so that it satisfies assumption (i) of Theorem 4.1). We now want
+to look at an example where also assumption (ii) of Theorem 4.1 is satisfied, i.e., where the
+drift t �→ f(t, x) is non-increasing, so that we can obtain monotonicity of the optimal stopping
+boundary. Let us consider a simple example of a two-point prior distribution ν = pδl+(1−p)δr,
+where δx is the Dirac delta centred in x ∈ R, −∞ < l < r < ∞ and p ∈ (0, 1). Then,
+f(t, x) = plelx−l2t/2 + (1 − p)rerx−r2t/2
+pelx−l2t/2 + (1 − p)erx−r2t/2 .
+For assumption (ii) of Theorem to hold, we would like to obtain ∂tf ≤ 0. After some algebra,
+we have that
+∂tf(t, x) = −1
+2
+p(1 − p)e(l+r)x−(l2+r2)t/2(r − l)2(r + l)
+�
+pelx−l2t/2 + (1 − p)erx−r2t/2�2
+.
+If r ≥ −l, then ∂tf(t, x) ≤ 0 for every (t, x) ∈ [0, T] × R. Therefore, we can apply Theorem
+4.1, and obtain that t �→ v(t, x) is non-increasing for every x ∈ R.
+Thus, Corollary 4.6
+guarantees monotonicity of the corresponding stopping boundary also for this example of
+optimal stopping problem under incomplete information. Notice that this is only one example
+
+A RESULT FOR TIME-INHOMOGENEOUS OPTIMAL STOPPING PROBLEMS
+15
+of prior distribution and different priors may be investigated. For instance, Theorem 4.1 can
+also be applied to the optimal stopping of a Brownian bridge with unknown pinning point
+whose prior is normal µ ∼ N(m, γ2) (see, [12, Section 5.1]).
+Acknowledgments
+I wish to thank S. Villeneuve for encouraging me to pursue this research idea, and T. De
+Angelis and E. Ekstr¨om for the enlightening discussions.
+References
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+DEN.
+Email address: alessandro.milazzo@math.uu.se
+
diff --git a/qtE5T4oBgHgl3EQfJg5m/content/tmp_files/load_file.txt b/qtE5T4oBgHgl3EQfJg5m/content/tmp_files/load_file.txt
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@@ -0,0 +1,785 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf,len=784
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='05458v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='OC]  13 Jan 2023  ON THE MONOTONICITY OF THE STOPPING BOUNDARY  FOR TIME-INHOMOGENEOUS OPTIMAL STOPPING PROBLEMS  ALESSANDRO MILAZZO  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' We consider a class of time-inhomogeneous optimal stopping problems and we  provide sufficient conditions on the data of the problem that guarantee monotonicity of the  optimal stopping boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  In our setting, time-inhomogeneity stems not only from the  reward function but, in particular, from the time dependence of the drift coefficient of the  one-dimensional stochastic differential equation (SDE) which drives the stopping problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' In  order to obtain our results, we mostly employ probabilistic arguments: we use a comparison  principle between solutions of the SDE computed at different starting times, and martingale  methods of optimal stopping theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' We also show a variant of the main theorem, which  weakens one of the assumptions and additionally relies on the renowned connection between  optimal stopping and free-boundary problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' introdution  In this paper we consider a general class of time-inhomogeneous optimal stopping problems  and we provide simple sufficient conditions on the data of the problem that guarantee mono-  tonicity of the optimal stopping boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' The novelty of our work is to prove this result  when the underlying process is time-inhomogeneous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' In our setting, the underlying process is  the unique strong solution of a one-dimensional stochastic differential equation (SDE) whose  drift coefficient may be time-dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' We first show how to obtain monotonicity of the op-  timal stopping boundary when the reward function is time-homogeneous and then we extend  the result to the case of a time-dependent reward function, when it is sufficiently regular to  apply Ito’s formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' We focus our attention on finite-horizon optimal stopping problems but  our methods clearly apply also to infinite-horizon optimal stopping problems, as the latter do  not carry an additional time-dependence in the domain of the admissible stopping times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  The behaviour of the optimal stopping boundary t �→ b(t) is crucial in order to fully  characterise an optimal stopping problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' In particular, continuity and monotonicity of the  map t �→ b(t) are two desirable properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' However, this regularity is usually studied on a  case-by-case basis and the number of works that provide sufficient conditions to obtain these  results in a general framework is limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' Classical tricks to show continuity of the stopping  boundary are presented in [27] in various examples, whereas results in a general setting can  be found in [4] (for one-dimensional diffusions) and [26] (for two-dimensional diffusions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  Determining monotonicity of t �→ b(t) can be even a more relevant turning point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' First, it is  a helpful result in order to obtain its continuity (as shown, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=', in [4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' Furthermore, when  the underlying process is strong Markov, it implies that the optimal stopping time τ ∗  t,x is a  continuous function of the starting point (t, x) across the boundary1 or, equivalently, that the  2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' 60G07, 60G40, 60J60, 49N30, 35R35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' optimal stopping, monotone stopping boundary, time-inhomogeneous diffusions,  partial information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  1Here, we mean that if (t, x) = (t, b(t)) and (tn, xn) → (t, x) as n → ∞, then τ ∗  tn,xn → τ ∗  t,x as n → ∞, P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  1  2  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' MILAZZO  boundary is regular for the interior of the stopping set in the sense of diffusions (a concept  extensively illustrated in [7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' This yields global C1-regularity of the value function, which is  also a helpful result to characterise the stopping boundary (when continuous) as the unique  continuous solution of a family of integral equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  An extensive probabilistic analysis  of the geometry of a general class of optimal stopping problems, including continuity and  monotonicity of the stopping boundary, is presented in [5] when the underlying diffusion and  reward function are time-homogeneous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' The shape of the continuation region is also studied  under a general framework in [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' However, their result on the monotonicity of t �→ b(t) (see  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='4 therein) holds only for time-homogeneous diffusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' One contribution of this  paper is to extend this result to a class of time-inhomogeneous diffusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' Regularity and  characterisation of the value function are obtained for time-inhomogeneous Markov processes  in [25] and in [32], and for time-inhomogeneous Poisson processes in [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' To the best of  our knowledge, no study of the properties of the stopping boundary has been developed  in a general setting for time-inhomogeneous diffusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' It is also worth mentioning several  theoretical works on the behaviour of the stopping boundary and of the value function in the  context American options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' We cite, among others, [2], [9], [20], [22], [24] and [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  In order to obtain our results, we rely on probabilistic arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  We first present a  comparison principle between solutions of the underlying SDE computed at different starting  times (see Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  Specifically, we show that if the drift coefficient t �→ µ(t, x) is  monotone then the solutions of the SDE computed at different starting times are ordered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  By means of this result and martingale methods of optimal stopping theory, we prove that  if in addition a time-homogeneous reward function x �→ g(x) is non-decreasing then t �→  v(t, x) is also monotone for every x ∈ R (see Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  In a variant of the theorem  we show that if monotonicity of t �→ µ(t, x) does not hold for every x in the state space  of the underlying process, we are able to weaken this condition and obtain the same result  under a further assumption which involves the derivatives of the value function and it is  implied by convexity of x �→ v(t, x) (see Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' This proof additionally relies on the  renowned connection between optimal stopping and free-boundary problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' An example of  time-inhomogeneous diffusions which perfectly fits the weaker monotonicity assumption (of  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='2) on t �→ µ(t, x) is given by Brownian bridges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' Several works have investigated  optimal stopping problems involving Brownian bridges and we cite, among others, [29], [15],  [14], [12], [6], [17] and [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' Both Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='1 and Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='2 lead to the monotonicity of the  optimal stopping boundary t �→ b(t) (see Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' Then, we prove that monotonicity  of t �→ b(t) can be obtained even when the reward function g depends on time (see Theorem  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' This extension holds when g is sufficiently regular to apply Ito’s formula and under  the additional assumption of monotonicity of t �→ Lg(t, x), where L denotes the infinitesimal  generator of the underlying diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  Our methods are particularly suited to study optimal stopping problems under incomplete  information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' The common feature of these problems is a random variable whose outcome is  unknown to the optimiser and which affects the drift of the underlying process and/or the  reward function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' The literature is vast and diverse in this field and we cite, among others,  [30], [8], [10], [11], [12], [13], [16], [17] [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' Our results apply, in particular, to models as in [12]  and [17] where a random variable affects the drift of the underlying process and, in a Bayesian  formulation of the problem, only the prior distribution of the random variable is known to the  optimiser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' As time evolves, the information obtained from observing the underlying process is  used to update the initial beliefs about the unknown random variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' By filtering theory, the  A RESULT FOR TIME-INHOMOGENEOUS OPTIMAL STOPPING PROBLEMS  3  underlying process can be expressed as a time-inhomogeneous diffusion whose time-dependent  drift is the conditional expectation of the unknown random variable given the observations  of the process, which can be obtained through the prior distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' This, thus, fits into our  framework, as we illustrate in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  The rest of the paper is organised as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' In Section 2 we formulate the starting problem  and we recall some standard results on optimal stopping theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' In Section 3 we provide a  comparison principle between solutions of the underlying SDE starting at different times,  which will be later used in Section 4 to determine the monotonicity of the optimal stopping  boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' In Section 5 we extend the range of applicability for the results of Section 4 by  considering stopping problems where also the reward functions may depend on time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' Our  methods are particularly suited to study a class of optimal stopping problems under partial  information, which we describe in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' We conclude by illustrating, in Section 7, some  simple examples of optimal stopping problems where our results apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' Starting problem and background results  Let (Ω, F, P) be a complete probability space with a filtration F := (Ft)t≥0 satisfying the  usual conditions and let W := (Wt)t≥0 be a standard Brownian motion which is F-adapted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  Let T ∈ (0, ∞) be a finite time horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' In this paper we treat finite-horizon optimal stopping  problems, but it will be clear that our methods apply also to the infinite-horizon analogues,  where the time-dependence of the value function stems only from the drift coefficient of the  underlying SDE and not from the domain of the admissible stopping times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  Given an initial condition Xt = x ∈ R for t ∈ [0, T), let X = (Xs)s≥t be the time-  inhomogeneous stochastic process described by  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='1)  Xt+s = x +  � s  0  µ(t + r, Xt+r)dr +  � s  0  σ(Xt+r)dWr,  s ∈ [0, T − t],  where µ : [0, T] × R → R and σ : R → R are, respectively, the drift and diffusion coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  We assume that x �→ µ(t, x) is Lipschitz-continuous for every t ∈ [0, T] and that x �→ σ(x)  satisfies the standard Yamada-Watanabe condition which guarantees the strong existence and  uniqueness of the solution for the SDE (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='1) (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=', [28, Theorem 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' Namely, we assume  that there exists an increasing function h : [0, ∞) → [0, ∞) such that  � ε  0  h−1(s)ds = ∞,  ∀ ε > 0  and  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='2)  �  σ(x) − σ(y)  �2 ≤ h(|x − y|),  ∀ x, y ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  In order to keep track of the initial condition Xt = x, we will sometimes denote the solution  X of the SDE (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='1) by Xt,x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  Given a (terminal) reward function g : R → R, we define the optimal stopping problem  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='3)  v(t, x) := sup  τ∈Tt  E  �  g(Xt,x  t+τ )  �  ,  (t, x) ∈ [0, T] × R,  where Tt is the class of F-stopping times τ such that τ ∈ [0, T − t], P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' To simplify the  exposition, we start by considering stopping problems of the form (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' We then extend  our results to stopping problems that include both a running reward function and a terminal  reward function which may also depend on time (see Section 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  4  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' MILAZZO  Let C be the continuation region and its complement D := Cc be the stopping region,  respectively, defined by  C := {(t, x) ∈ [0, T] × R : v(t, x) > g(x)}  and  D := {(t, x) ∈ [0, T] × R : v(t, x) = g(x)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  We now state some mild assumptions for the optimal stopping problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' The reward function g : R → R is upper semi-continuous, the value  function v : [0, T] × R → R is continuous and we have that, for every (t, x) ∈ [0, T] × R,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='4)  E  �  sup  s∈[t,T]  ��g(Xt,x  s )  ��  �  < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  Under Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='1, we obtain the next three propositions, which are standard results  in optimal stopping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' Let (t, x) ∈ [0, T] × R, then the stopping time  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='5)  τ ∗ = τ ∗  t,x := inf{s ∈ [0, T − t] : (t + s, Xt,x  t+s) /∈ C}  is optimal for the stopping problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' See, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=', [27, Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  □  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' Let (t, x) ∈ [0, T] × R, then the process V := (Vs)s∈[0,T−t], defined by  Vs = V t,x  s  := v(t + s, Xt,x  t+s),  is a right-continuous supermartingale and the stopped process V ∗ := (Vs∧τ ∗)s∈[0,T−t] is a right-  continuous martingale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' See, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=', [27, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  □  Let ∂t, ∂x and ∂xx denote the time derivative, the spatial derivative and the second spatial  derivative, respectively, and let ∂C denote the boundary of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' We have that v ∈ C1,2(C) and it solves the free-boundary problem  �  ∂t + µ(t, x)∂x + 1  2(σ(x))2∂xx  �  v(t, x) = 0,  (t, x) ∈ C,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='6)  v(t, x) = g(x),  (t, x) ∈ ∂C,  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' By Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='1, C is an open set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' Then the free-boundary problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='6) follows,  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=', by the same arguments as in the proof of [20, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  □  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' Continuity of v is not necessary to obtain Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='2 and Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='3  but lower semi-continuity would be sufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' Moreover, these two propositions may hold with  no continuity assumption on v: they still hold if, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=', g is continuous and non-negative and  the integral condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='4) is satisfied (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=', [23, Appendix D]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' For the sake of simplicity,  we assume continuity of v, which is necessary for Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  To avoid further initial conditions on the data of the problem, we also introduce the fol-  lowing assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  A RESULT FOR TIME-INHOMOGENEOUS OPTIMAL STOPPING PROBLEMS  5  Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' There exists a (lower) optimal stopping boundary for the problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='3),  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=', a function b : [0, T] → R that separates C from D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' That is, we have  C = {(t, x) ∈ [0, T) × R : x > b(t)}  and  D = {(t, x) ∈ [0, T) × R : x ≤ b(t)} ∪ {T} × R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='6 is usually proved by probabilistic arguments on a case-by-case basis (see,  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=', [20, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' It is easy to see that it holds if, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=', x �→ v(t, x) − g(x) is non-  decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  More general sufficient conditions that guarantee the existence of an optimal  stopping boundary are shown in, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=', [21, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='3] and will be used later in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  In this paper, we prove our results when a lower stopping boundary exists but it is clear that  analogous arguments would follow when an upper stopping boundary exists instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' A comparison principle  In this section we provide a comparison principle between solutions of the SDE (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='1) starting  at different times, which will be applied in Section 4 to obtain monotonicity of the optimal  stopping boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  We denote by S ⊆ R the state space of the process X defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' For every (t, x) ∈  [0, T] × R, and for a non-empty set O ⊆ [0, T] × S, we define  τO = τ t,x  O := inf{s ≥ 0 : (t + s, Xt,x  t+s) /∈ O} ∧ (T − t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' Let (t, x) ∈ [0, T] × R and let O ⊆ [0, T] × S be non-empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' Assume that  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='1)  µ(s, y) ≤ µ(u, y),  ∀ (s, y) ∈ O,  ∀ u ∈ [0, s].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  Then, for every u ∈ [0, t], we have that  P  �  Xt,x  t+s∧τO ≤ Xu,x  u+s∧τO,  ∀ s ∈ [0, T − t]  �  = 1,  where τO = τ t,x  O .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' Let X1  s := Xt,x  t+s, X2  s := Xu,x  u+s, µ1(s, y) := µ(t + s, y) and µ2(s, y) := µ(u + s, y) with  u ∈ [0, t].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' Thus, for i = 1, 2, we have  Xi  s = x +  � s  0  µi(r, Xi  r)dr +  � s  0  σ(Xi  r)dWr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  Then, for Y := X1 − X2 by assumption (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='2), we obtain  � s  0  h(Yr)−1  1{Yr>0}d[Y ]r =  � s  0  h(|X1  r − X2  r |)−1�  σ(X1  r ) − σ(X2  r )  �2  1{Yr>0}dr ≤ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  Therefore, we have (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=', [28, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' V, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='3]) that L0  s(Y ) = 0 for every s ∈ [0, T], where  L0(Y ) denotes the local time of Y at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' Thus, by Tanaka’s formula, for every s ∈ [0, T − t]  we obtain  �  X1  s∧τO − X2  s∧τO  �+ =  � s∧τO  0  �  µ1(r, X1  r ) − µ2(r, X2  r )  �  1{X1r −X2r >0}dr  +  � s∧τO  0  �  σ(X1  r ) − σ(X2  r )  �  1{X1r −X2r >0}dWr,  6  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' MILAZZO  where τO = τ t,x  O  and (x)+ := max{x, 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' Hence,  0 ≤ E  ��  X1  s∧τO − X2  s∧τO  �+�  = E  � � s∧τO  0  �  µ(t + r, X1  r ) − µ(u + r, X2  r )  �  1{X1r −X2r >0}dr  �  ≤ E  � � s∧τO  0  �  µ(u + r, X1  r ) − µ(u + r, X2  r )  �  1{X1r −X2r >0}dr  �  ≤ E  � � s∧τO  0  K(u + r)  �  X1  r − X2  r  �+dr  �  ,  where K(t) > 0 is the Lipschitz constant for x �→ µ(t, x) and the second to last inequality  follows from assumption (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' Then, by Gronwall’s lemma, we obtain that  E  ��  X1  s∧τO − X2  s∧τO  �+�  = 0,  ∀ s ∈ [0, T − t],  and by continuity of Y = X1 − X2 we reach the desired result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  □  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' Let (t, x) ∈ [0, T] × R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' Notice that if O = [0, T] × S, then τ t,x  O = T − t and so  the result of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='1 reads  P  �  Xt,x  t+s ≤ Xu,x  u+s,  ∀ s ∈ [0, T − t]  �  = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' Main results  In this section we illustrate our main result for the optimal stopping problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='3), which  provides monotonicity of t �→ v(t, x) and in turn implies monotonicity of the stopping bound-  ary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' This is obtained by means of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='1 and will be presented in two versions (Theorem  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='1 and Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='2) under different assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' Let Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='1 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' Moreover, assume that  (i) x �→ g(x) is non-decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  (ii) t �→ µ(t, x) is non-increasing for every x ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  Then, t �→ v(t, x) is non-increasing for every x ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' Let (t, x) ∈ [0, T] × R and u ∈ [0, t].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' By assumption (ii), we can apply Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='1 with  O = [0, T] × S and obtain that  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='1)  P  �  Xt,x  t+s ≤ Xu,x  u+s,  ∀ s ∈ [0, T − t]  �  = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  By the (super)martingale property of the value function (recall Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='3) and since  τ ∗ = τ ∗  t,x is optimal for v(t, x) and sub-optimal for v(u, x), we have that  v(t, x) − v(u, x) = V t,x  0  − V u,x  0  ≤ E  �  V t,x  τ ∗ − V u,x  τ ∗  �  = E  �  v(t + τ ∗, Xt,x  t+τ ∗) − v(u + τ ∗, Xu,x  u+τ ∗)  �  ≤ E  �  g(Xt,x  t+τ ∗) − g(Xu,x  u+τ ∗)  �  ≤ 0,  where to obtain the last inequality we have used assumption (i) and result (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' Hence,  t �→ v(t, x) is non-increasing for every x ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  □  A RESULT FOR TIME-INHOMOGENEOUS OPTIMAL STOPPING PROBLEMS  7  We now show that we can weaken the monotonicity assumption on t �→ µ(t, x) but still  obtain, under an additional assumption, the same result as in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' This alternative  version partially relies on the free-boundary problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='6) and turns out to be useful in some  optimal stopping problems, as we will illustrate in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  Let M := {(t, x) ∈ [0, T] × S : µ(t, x) < 0} and let us denote by Mc its complement, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=',  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='2)  Mc := ([0, T] × S) \\ M = {(t, x) ∈ [0, T] × S : µ(t, x) ≥ 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  Throughout this paper we also assume that M is an open set, so that (t + τM, Xt+τM) ∈ Mc  on {τM < T − t}, where recall that  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='3)  τM = τ t,x  M := inf{s ≥ 0 : (t + s, Xt,x  t+s) /∈ M} ∧ (T − t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  This holds if, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=', µ is upper semi-continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' Let Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='1 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' Moreover, assume that  (i) x �→ g(x) is non-decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  (ii) µ(t, x) ≤ µ(t − ε, x) for every (t, x) ∈ M, ε ∈ (0, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  (iii) σ2(x)∂xxv(t, x) ≥ −2µ(t, x)∂xv(t, x) for every (t, x) ∈ C ∩ Mc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  Then, t �→ v(t, x) is non-increasing for every x ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' Since x �→ Xt,x is non-decreasing (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=', [28, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' V, Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='1]) and, under  the assumptions of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='2, x �→ g(x) is non-decreasing, we also have that x �→ v(t, x)  is non-decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' Thus, notice that assumption (iii) holds, in particular, if x �→ v(t, x) is  convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  This is in turn implied by convexity of x �→ Xt,x and of x �→ g(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  Therefore,  assumption (iii) of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='2 can be substituted by convexity of x �→ Xt,x and of x �→ g(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  However, if σ(x) is sufficiently small or if µ(t, x)∂xv(t, x) is sufficiently large on Mc, then we  may not need x �→ v(t, x) to be convex in order to satisfy assumption (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' We prove the result of the theorem in two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' We first show that ∂tv(t, x) ≤ 0 for  every (t, x) /∈ ∂C and we then prove that this implies that t �→ v(t, x) is non-increasing for  every x ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' If (t, x) ∈ D \\ ∂C then v(t, x) = g(x) and so ∂tv(t, x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' If (t, x) ∈ C ∩ Mc (we  can skip this step if Mc = ∅), then by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='6)  ∂tv(t, x) + µ(t, x)∂xv(t, x) + 1  2(σ(x))2∂xxv(t, x) = 0,  and, by assumption (iii), we obtain  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='4)  ∂tv(t, x) ≤ 0,  ∀ (t, x) ∈ C ∩ Mc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  To conclude the proof we consider (t, x) ∈ C ∩ M (we can skip this step if M = ∅).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' By  assumption (ii) we can apply Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='1 with O = M and, for every ε ∈ (0, t), we obtain  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='5)  P  �  Xt,x  t+s∧τM ≤ Xt−ε,x  t−ε+s∧τM,  ∀ s ∈ [0, T − t]  �  = 1,  where τM = τ t,x  M is defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' Let ε ∈ (0, t), τ ∗ = τ ∗  t,x (recall (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='5)) and ρ := τ ∗ ∧ τM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  By the (super)martingale property of the value function (recall Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='3) and since τ ∗  8  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' MILAZZO  is optimal for v(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' x) and ρ is sub-optimal for v(t − ε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' we have that  v(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' x) − v(t − ε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' x) ≤ E  �  v(t + ρ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' Xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='x  t+ρ) − v(t − ε + ρ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' Xt−ε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='x  t−ε+ρ)  �  ≤ E  �  1{τ ∗≤τM}  �  g(Xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='x  t+τ ∗) − g(Xt−ε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='x  t−ε+τ ∗)  ��  + E  �  1{τM<τ ∗}  �  v(t + τM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' Xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='x  t+τM) − v(t − ε + τM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' Xt−ε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='x  t−ε+τM)  ��  = E  �  1{τ ∗≤τM}  �  g(Xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='x  t+τ ∗) − g(Xt−ε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='x  t−ε+τ ∗)  ��  + E  �  1{τM<τ ∗}  �  v(t + τM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' Xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='x  t+τM) − v(t − ε + τM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' Xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='x  t+τM)  ��  + E  �  1{τM<τ ∗}  �  v(t − ε + τM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' Xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='x  t+τM) − v(t − ε + τM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' Xt−ε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='x  t−ε+τM)  ��  ≤ E  �  1{τM<τ ∗}  �  v(t + τM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' Xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='x  t+τM) − v(t − ε + τM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' Xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='x  t+τM)  ��  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  where to obtain the last inequality we have used assumption (i) and result (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='5) for the first  term;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' result (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='5) and the fact that x �→ v(t, x) is non-decreasing (recall Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='3) for the  third term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' Dividing by ε, letting ε → 0 and applying dominated convergence theorem (by  assumption (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='4)), we obtain  ∂tv(t, x) ≤ E  �  1{τM<τ ∗}∂tv(t + τM, Xt,x  t+τM)  �  ≤ 0,  where the last inequality follows from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' Hence, ∂tv(t, x) ≤ 0 also for (t, x) ∈ C ∩ M and  the proof of Step 1 is completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' If (t, x) ∈ D, then v(t, x) = g(x) and, since v(s, x) ≥ g(x) for every (s, x) ∈ [0, T]×R,  then v(s, x) ≥ v(t, x) for every s ∈ [0, t].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  Now let (t, x) ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' We want to show that also (s, x) ∈ C for every s ∈ [0, t], which by Step 1  would imply that v(s, x) ≥ v(t, x) for every s ∈ [0, t] and would conclude the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' Assume,  by contradiction, that  Dt,x := {s ∈ [0, t] : (s, x) ∈ D} ̸= ∅  and let t0 := sup Dt,x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  Recall that, since (t, x) ∈ C, we have v(t, x) > g(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  Since v is  continuous (by Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='1), then t0 < t and t0 ∈ Dt,x, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=', (t0, x) ∈ D and so v(t0, x) =  g(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' Moreover, by definition of t0, we have (s, x) ∈ C for every s ∈ (t0, t] and so  v(t, x) − v(s, x) =  � t  t0  ∂tv(s, x)ds ≤ 0,  ∀ s ∈ (t0, t],  where the last inequality follows from Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' Hence, by continuity of v, we have that v(t0, x) ≥  v(t, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' This leads to a contradiction, as we would obtain  g(x) = v(t0, x) ≥ v(t, x) > g(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  □  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' If assumption (ii) in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='1 (and similarly for assumptions (ii) and (iii)  in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='2) is substituted by a symmetric assumption (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=', if t �→ µ(t, x) is increasing)  then, in infinite-horizon problems, we would obtain a symmetric result, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=', t �→ v(t, x) would  be increasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' However, this is, in general, not the case for finite-horizon problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' In that  context we would have two opposite driving effects as time increases: the drift µ that increases  and the stopping time domain Tt that shrinks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' The former leads to an increase of the value  function with respect to time, whereas the latter leads to a decrease of the value function  A RESULT FOR TIME-INHOMOGENEOUS OPTIMAL STOPPING PROBLEMS  9  with respect to time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' In order to study the monotonicity of t �→ v(t, x) in such problems, it  would be necessary (and, perhaps, not sufficient) to have a quantitative information on the  monotonicity of t �→ µ(t, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' In some cases it is possible to apply a pure PDE approach, as in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='4), and to  derive monotonicity of t �→ v(t, x) also when the diffusion coefficient may be time-dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  Consider the same SDE as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='1) but when also σ may be a function of time, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=', σ :  [0, T] × R → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' If µ(t, x) ≥ 0 for every (t, x) ∈ [0, T] × S, then Mc = [0, T] × S (recall (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  Thus, under assumptions (i) and (iii) of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='2 and in the same way as in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='4), we  would obtain  ∂tv(t, x) ≤ 0,  ∀ (t, x) /∈ ∂C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  Monotonicity of t �→ v(t, x) then follows as in Step 2 of the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  Monotonicity of t �→ v(t, x), which follows from either Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='1 or Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='2, then  yields monotonicity of the optimal stopping boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' If t �→ v(t, x) is non-increasing for every x ∈ R and Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='6 holds,  then the optimal stopping boundary t �→ b(t) is non-decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' Let (t, x) ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' Then, v(t, x) > g(x) and, since t �→ v(t, x) is non-increasing, we obtain  that v(s, x) ≥ v(t, x) > g(x) for every s ∈ [0, t].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' Hence, also (s, x) ∈ C and thus t �→ b(t) is  non-decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  □  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' Monotonicity of t �→ v(t, x) is also a helpful result to obtain continuity of the  stopping boundary (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=', arguments as in [4, Section 3] and [6, Lemma 4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' Extension to time-dependent reward functions  In this section we show how to obtain monotonicity of the stopping boundary for more gen-  eral time-inhomogeneous optimal stopping problems, which include a running reward function  and a terminal reward function that may also depend on time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' We consider the same under-  lying framework of Section 2 but we study the optimal stopping problem  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='1)  v(t, x) := sup  τ∈Tt  E  � � τ  0  f(t + s, Xt,x  t+s)ds + g(t + τ, Xt,x  t+τ)  �  ,  (t, x) ∈ [0, T] × R,  where X = Xt,x is defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='1), f : [0, T] × R → R is a running reward function and  g : [0, T] × R → R is a terminal reward function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' For this problem the continuation region C  and the stopping region D are defined, respectively, by  C := {(t, x) ∈ [0, T] × R : v(t, x) > g(t, x)}  and  D := {(t, x) ∈ [0, T] × R : v(t, x) = g(t, x)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  We then introduce the following assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  Assumption 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' We have that g ∈ C1,2([0, T] × R), the value function v : [0, T] × R → R  is continuous and, for every (t, x) ∈ [0, T] × R,  E  �  sup  s∈[t,T]  ���  � s  0  f(r, Xt,x  r )dr + g(s, Xt,x  s )  ���  �  < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  10  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' MILAZZO  Notice that Assumption 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='1 is the analogous of Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='1 except for the stronger  regularity of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' Under this regularity, we can apply Ito’s formula and obtain that  g(t + s, Xt+s) = g(t, x) +  � s  0  Lg(t + r, Xt+r)dr,  ∀ s ∈ [0, T − t],  where L is defined by  Lg(t, x) :=  �  ∂t + µ(t, x)∂x + 1  2(σ(x))2∂xx  �  g(t, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  The function w(t, x) := v(t, x) − g(t, x) is, thus, the value function for the optimal stopping  problem  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='2)  w(t, x) = sup  τ∈Tt  E  � � τ  0  h(t + s, Xt,x  t+s)ds  �  ,  (t, x) ∈ [0, T] × R,  where h(t, x) := f(t, x) + Lg(t, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' Then, notice that  C = {(t, x) ∈ [0, T] × R : w(t, x) > 0}  and  D = {(t, x) ∈ [0, T] × R : w(t, x) = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  Analogously to Section 2, under Assumption 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='1, we have that the stopping time  τ ∗ = τ ∗  t,x := inf{s ∈ [0, T − t] : (t + s, Xt,x  t+s) /∈ C}  is optimal for the problem (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='1) (and thus also for the problem (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' Moreover, we obtain  that the process V := (Vs)s∈[0,T−t], defined by  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='3)  Vs = V t,x  s  :=  � s  0  h(t + r, Xt,x  t+r)dr + w(t + s, Xt,x  t+s),  is a right-continuous supermartingale and the process V ∗ := (Vs∧τ ∗)s∈[0,T−t] is a right-  continuous martingale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' For the sake of simplicity, we have assumed g ∈ C1,2([0, T] × R) but one may  consider different (weaker) conditions in order to apply Ito’s formula and reformulate the  stopping problem (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='1) into the stopping problem (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  We then study the optimal stopping problem (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='1) by means of the equivalent problem  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' We have the following result on the existence of an optimal stopping boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' Assume that x �→ h(t, x) is non-decreasing for every t ∈ [0, T], then there  exists a lower optimal stopping boundary for the problem (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='1), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=', a function b : [0, T) →  R ∪ {±∞} such that  C = {(t, x) ∈ [0, T) × R : x > b(t)}  and  D = {(t, x) ∈ [0, T) × R : x ≤ b(t)} ∪ {T} × R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' Since x �→ h(t, x) is non-decreasing for every t ∈ [0, T], then x �→ w(t, x) is non-  decreasing for every t ∈ [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' Hence, if (t, x1) ∈ D, then (t, x2) ∈ D for every x2 ∈ (−∞, x1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  Therefore, for t ∈ [0, T), the function  b(t) := sup{x ∈ R : w(t, x) = 0}  is a lower optimal stopping boundary for the stopping problem (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='2) and, thus, also for  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  □  A RESULT FOR TIME-INHOMOGENEOUS OPTIMAL STOPPING PROBLEMS  11  We can now obtain monotonicity of the optimal stopping boundary also for the more general  class of time-inhomogeneous optimal stopping problems in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' Recall that S denotes the  state space of X, and that we define h(t, x) := f(t, x) + Lg(t, x) and  Lg(t, x) :=  �  ∂t + µ(t, x)∂x + 1  2(σ(x))2∂xx  �  g(t, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' Let Assumption 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='1 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' Moreover, assume that  (i) x �→ h(t, x) is non-decreasing for every t ∈ [0, T] and t �→ h(t, x) is non-increasing for  every x ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  (ii) t �→ µ(t, x) is non-increasing for every x ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  Then, t �→ w(t, x) is non-increasing for every x ∈ R and so the optimal stopping boundary  t �→ b(t) is non-decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' Let (t, x) ∈ [0, T] × R and u ∈ [0, t].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' By assumption (ii), we can apply Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='1 with  O = [0, T] × S and obtain that  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='4)  P  �  Xt,x  t+s ≤ Xu,x  u+s,  ∀ s ∈ [0, T − t]  �  = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  By the (super)martingale property (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='3) of V and since τ ∗ = τ ∗  t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='x is optimal for w(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' x) and  sub-optimal for w(u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' we have that  w(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' x) − w(u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' x) = V t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='x  0  − V u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='x  0  ≤ E  �  V t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='x  τ ∗ − V u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='x  τ ∗  �  = E  � � τ ∗  0  �  h(t + s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' Xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='x  t+s) − h(u + s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' Xu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='x  u+s)  �  ds  �  + E  �  w(t + τ ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' Xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='x  t+τ ∗) − w(u + τ ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' Xu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='x  u+τ ∗)  �  ≤ E  � � τ ∗  0  �  h(t + s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' Xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='x  t+s) − h(u + s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' Xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='x  t+s)  �  ds  �  + E  � � τ ∗  0  �  h(u + s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' Xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='x  t+s) − h(u + s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' Xu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='x  u+s)  �  ds  �  ≤ 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  where the last inequality follows from assumption (i) and result (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' Hence, t �→ w(t, x) is  non-increasing for every x ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' The monotonicity of t �→ b(t) (whose existence is guaranteed  by Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='3) is, thus, obtained by the same arguments as in the proof of Corollary  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  □  Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' Notice that the proof of [21, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='4], which provides monotonicity of  the optimal stopping boundary, holds only if the underlying process is time-homogeneous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' Our  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='4 extends that result to time-inhomogeneous optimal stopping problems, under the  additional assumption that t �→ µ(t, x) is non-increasing for every x ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' Optimal stopping under incomplete information  Our methods are particularly suited to study optimal stopping problems under incomplete  information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' To this purpose, in this section, we provide some background material on this  class of problems and in Section 7 we will look into a specific example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  The common feature of these stopping problems is a random variable whose outcome is  unknown to the optimiser and which affects the drift of the underlying process and/or the  payoff function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' The literature is vast and diverse in this field and we cite, among others,  12  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' MILAZZO  [30], [8], [10], [11], [12], [13], [16], [17] [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' We focus, in particular, on models as in [12] and  [17] where a random variable affects the drift of the underlying process and, in a Bayesian  formulation of the problem, only the prior distribution of the random variable is known to the  optimiser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' As time evolves, the information obtained from observing the underlying process  is used to update the initial beliefs about the unknown random variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  Let (Ω, F, P) be a complete probability space on which it is defined a standard Brownian  motion W := (Wt)t≥0 and a real-valued random variable Y with (prior) probability distribu-  tion ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' Let T ∈ (0, ∞) be a finite time horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' The underlying process X evolves according  to  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='1)  dXt = h(Y )dt + dWt,  X0 = x ∈ R,  where h is a measurable function such that  �  R |h(y)|ν(dy) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' Given a reward function  g : [0, T] × R → R, we can then define the stopping problem  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='2)  V := sup  τ∈[0,T]  E  �  g(τ, Xτ )  �  ,  where τ is a stopping time with respect to (FX  t )t∈[0,T], the augmented filtration generated by  X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  It is well-known from filtering theory (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=', [1, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='16]) that E[h(Y )|FX  t ] =  E[h(Y )|Xt] =: f(t, Xt) where  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='3)  f(t, x) :=  �  R h(y)exy−y2t/2ν(dy)  �  R exy−y2t/2ν(dy)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  Moreover,  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='4)  dXs = f(s, Xs)ds + dBs,  where Bt :=  � t  0  �  h(Y ) − E[h(Y )|FX  s ]  �  ds + Wt is an FX-Brownian motion (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=', [1, Propo-  sition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='30]) known as the “innovation process”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' By means of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='4), we can embed the original  stopping problem (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='2) into a Markovian framework and define the value function  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='5)  v(t, x) :=  sup  τ∈[0,T−t]  E  �  g(t + τ, Xt,x  t+τ)  �  ,  where τ is a stopping time with respect to the augmented filtration generated by B, and Xt,x  follows the dynamics in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='4) with Xt,x  t  = x ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' The optimal stopping problem (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='5) is  now in the form of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='1) and we may, thus, apply Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='4 to obtain monotonicity of its  stopping boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' The form of the drift coefficient f(t, x) defined in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='3) highly depends  on the prior distribution ν and in Section 7 we will look at a simple example where the  monotonicity required in assumption (ii) of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='4 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' Examples  In this section we show some simple examples of time-inhomogeneous optimal stopping  problems where our results apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' We consider different underlying time-inhomogeneous dif-  fusions of the form (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='1) that give rise to corresponding optimal stopping problems of the  form (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' We then determine under which conditions on the data of the problems we can  apply our theorems and obtain monotonicity of the stopping boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' We fix a complete  probability space (Ω, F, P) with a filtration F := (Ft)t≥0 satisfying the usual conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' We  let W := (Wt)t≥0 be a standard Brownian motion which is F-adapted and T ∈ (0, ∞) be a  A RESULT FOR TIME-INHOMOGENEOUS OPTIMAL STOPPING PROBLEMS  13  finite time horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' For simplicity, in the following examples we also let Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='1 and  Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='6 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' Applications of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' We begin by considering two simple time-inhomogeneous  diffusions and the corresponding optimal stopping problems of the form (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' For the reward  function of these two examples we only assume that x �→ g(x) is non-decreasing, as required  in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  First, for (t, x) ∈ [0, T]×R, let X = Xt,x be a Brownian motion with time-dependent drift,  described by  Xt+s = x +  � s  0  µ(t + r)dr + σWs,  s ∈ [0, T − t].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  If t �→ µ(t) is non-increasing, then we can apply Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='1 and obtain that t �→ v(t, x)  is non-increasing for every x ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' Thus, Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='6 guarantees that the corresponding  optimal stopping boundary t �→ b(t) is non-decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  Now, for (t, x) ∈ [0, T] × (0, ∞), let X = Xt,x be a geometric Brownian motion with  time-dependent drift, described by  Xt+s = x +  � s  0  γ(t + r)Xt+rdr +  � s  0  σXt+rdWr,  s ∈ [0, T − t].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  Its state space is S = (0, ∞) and its drift is µ(t, x) := xγ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' If t �→ γ(t) is non-increasing,  we can apply Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='1 and obtain that t �→ v(t, x) is non-increasing for every x ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  Thus, Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='6 guarantees that the corresponding optimal stopping boundary t �→ b(t)  is non-decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  Notice that the assumptions on the monotonicity of the drifts for the previous two examples  could be weakened by considering assumption (ii) of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='2 instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  However, this  comes with the cost of adding assumption (iii) of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='2 (which is implied by convexity  of x �→ Xt,x and of x �→ g(x), recall Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' Applications of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' For some underlying time-inhomogeneous diffusion it  happens that assumption (ii) of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='1 does not hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' In these cases it is useful to  consider Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='2, which weakens this assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' We now show an example of this  situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' This application is even more suited to our techniques as no specific assumption  on the drift of the underlying process is required in order to apply Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' This is the  case when the underlying time-inhomogeneous diffusion is a Brownian bridge and an example  of this optimal stopping problem, where g(x) = ex, is studied in [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' For our example we  assume that x �→ g(x) is non-decreasing (as required by assumption (i) of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='2) and,  for simplicity, that it is convex (to satisfy assumption (iii) of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='2, but convexity is  not strictly necessary, as explained in Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  For (t, x) ∈ [0, T] × R, let X = Xt,x be a Brownian bridge pinned at 0 at time T ∈ (t, ∞),  whose dynamics are described by  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='1)  Xt+s = x −  � s  0  Xt+r  T − t − rdr + σWs,  s ∈ [0, T − t),  with XT = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' It is easy to check that the unique strong solution to this SDE is given by  Xt+s = (1 − t − s)  �  x  1 − t +  � s  0  1  1 − t − rdWr  �  ,  s ∈ [0, T − t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  14  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' MILAZZO  Hence, x �→ Xt,x is linear (and thus convex) and together with convexity of x �→ g(x) we  have that assumption (iii) of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='2 is satisfied (recall Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' In order to apply  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='2, we are only left to check that assumption (ii) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' The drift coefficient  of the SDE (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='1) is µ(t, x) := −x/(T − t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' Thus, we have that  M := {(t, x) ∈ [0, T) × R : µ(t, x) < 0} = {(t, x) ∈ [0, T) × R : x > 0}  and  µ(t, x) = −  x  T − t ≤ −  x  T − t + ε = µ(t − ε, x),  (t, x) ∈ M,  ε ∈ (0, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  Therefore, also assumption (ii) of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='2 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' Hence, we can apply Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='2 and  obtain that t �→ v(t, x) is non-increasing for every x ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' Thus, Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='6 guarantees that  the corresponding optimal stopping boundary t �→ b(t) is non-decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  Further examples can be shown to satisfy the assumptions of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' For instance,  optimal stopping problems where the underlying diffusion follows the Vasicek model (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=', an  Ornstein-Ulhenbeck process) or the Cox-Ingersoll-Ross model can be studied in their time-  inhomogeneous version, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=', when the long-term mean is allowed to be time-dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' An application to optimal stopping under incomplete information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' To conclude,  we consider an example of an optimal stopping problem under incomplete information as in  Section 6, for which we want to apply Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' Assume that X follows the dynamics as  in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='1) with h(y) = y, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=',  dXt = Y dt + dWt,  where Y is a real-valued random variable with prior distribution ν which has finite first  moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' Then, by filtering theory, we have that  dXt = f(t, Xt)dt + dBt,  where  f(t, x) :=  �  R yexy−y2t/2ν(dy)  �  R exy−y2t/2ν(dy) ,  and B is a Brownian motion with respect to the filtration generated by X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' We can then  consider corresponding optimal stopping problems as in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='5), where the reward function  x �→ g(x) is non-decreasing (so that it satisfies assumption (i) of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' We now want  to look at an example where also assumption (ii) of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='1 is satisfied, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=', where the  drift t �→ f(t, x) is non-increasing, so that we can obtain monotonicity of the optimal stopping  boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' Let us consider a simple example of a two-point prior distribution ν = pδl+(1−p)δr,  where δx is the Dirac delta centred in x ∈ R, −∞ < l < r < ∞ and p ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' Then,  f(t, x) = plelx−l2t/2 + (1 − p)rerx−r2t/2  pelx−l2t/2 + (1 − p)erx−r2t/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  For assumption (ii) of Theorem to hold, we would like to obtain ∂tf ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' After some algebra,  we have that  ∂tf(t, x) = −1  2  p(1 − p)e(l+r)x−(l2+r2)t/2(r − l)2(r + l)  �  pelx−l2t/2 + (1 − p)erx−r2t/2�2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  If r ≥ −l, then ∂tf(t, x) ≤ 0 for every (t, x) ∈ [0, T] × R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' Therefore, we can apply Theorem  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='1, and obtain that t �→ v(t, x) is non-increasing for every x ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  Thus, Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='6  guarantees monotonicity of the corresponding stopping boundary also for this example of  optimal stopping problem under incomplete information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' Notice that this is only one example  A RESULT FOR TIME-INHOMOGENEOUS OPTIMAL STOPPING PROBLEMS  15  of prior distribution and different priors may be investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' For instance, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='1 can  also be applied to the optimal stopping of a Brownian bridge with unknown pinning point  whose prior is normal µ ∼ N(m, γ2) (see, [12, Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  Acknowledgments  I wish to thank S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' Villeneuve for encouraging me to pursue this research idea, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' De  Angelis and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content=' Ekstr¨om for the enlightening discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
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+page_content=' Milazzo: Department of Mathematics, Uppsala University, Box 480, 75106, Uppsala, SWE-  DEN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='  Email address: alessandro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
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+page_content='uu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
+page_content='se' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQfJg5m/content/2301.05458v1.pdf'}
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