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+A Framework for Active Haptic Guidance Using Robotic Haptic Proxies
+Niall L. Williams1, Jiasheng Li1, and Ming C. Lin1
+https://gamma.umd.edu/active haptic guidance/
+Abstract— Haptic feedback is an important component of
+creating an immersive virtual experience. Traditionally, haptic
+forces are rendered in response to the user’s interactions
+with the virtual environment. In this work, we explore the
+idea of rendering haptic forces in a proactive manner, with
+the explicit intention to influence the user’s behavior through
+compelling haptic forces. To this end, we present a framework
+for active haptic guidance in mixed reality, using one or more
+robotic haptic proxies to influence user behavior and deliver
+a safer and more immersive virtual experience. We provide
+details on common challenges that need to be overcome when
+implementing active haptic guidance, and discuss example
+applications that show how active haptic guidance can be used
+to influence the user’s behavior. Finally, we apply active haptic
+guidance to a virtual reality navigation problem, and conduct a
+user study that demonstrates how active haptic guidance creates
+a safer and more immersive experience for users.
+I. INTRODUCTION
+In mixed reality (MR), the user is at least partially im-
+mersed in a 3D, computer-generated environment. Included
+within the mixed reality spectrum are augmented reality
+and virtual reality (VR). A major factor that makes MR a
+unique medium is that it is interactive—the user is able to
+interact with the virtual environment (VE) through position-
+tracking sensors that update the VE according to the user’s
+movements in the physical environment (PE). For example,
+when the user moves their head in the real world, the position
+of the camera in the virtual world moves as well. Interactions
+like these help to make users feel like they are really in the
+VE that they see through the head-mounted display (HMD).
+One key component to increasing the user’s sense of presence
+in a VE is to improve the perceptual stimuli matching [8],
+wherein the user is provided with perceptual information that
+matches their actions (e.g. the viewing perspective updates
+as the user moves their head). In this work, we focus on
+the sense of touch provided by mechanical haptic feedback
+and how we can use robots to provide more realistic haptic
+sensations to improve the sense of immersion and safety in
+mixed reality.
+Robotic technology has in fact been used to provide haptic
+feedback in MR to improve the sense of virtual touch and
+virtual manipulation [10]. For example, MR can enhance
+robotics via telepresence, wherein humans can remotely
+operate robot to high precision using immersive controls
+afforded by VR.
+*This work is partially supported by National Science Foundation and
+Lin’s professorship.
+1Authors are with the Department of Computer Science, University of
+Maryland, College Park. {niallw, jsli, lin}@umd.edu
+Fig. 1.
+An image of a user in the physical environment (left) and virtual
+environment (right) in our implementation of active haptic guidance. The
+user is tethered to a robot in the physical environment and to a virtual dog
+companion in the virtual environment. The robot provides haptic feedback to
+the user according to the virtual companion’s movements, which improves
+the user’s sense of presence in the virtual world and encourages the user to
+avoid the boundaries of the virtual reality system’s tracked space.
+In this paper, we introduce the possibility of using robots
+to enhance the virtual experience through haptic feedback.
+Specifically, we use robots to guide the user as they navigate
+through a VE, and reconfigure and virtually expand the
+PE to align with the VE; we achieve this through manual
+haptic feedback that directs the user’s locomotion behavior
+in the VE, thereby making the virtual experience more
+immersive and safer. To this end, we introduce the concept
+of active haptic guidance, which describes the problem of
+reconfiguring one or multiple robots in the PE in real time
+such that they provide haptic feedback to guide the user
+and influence their actions and motion in the VE, with the
+ultimate goal of improving the user’s safety or level of
+immersion in MR. One major challenge with robots for active
+haptic feedback in MR is that the physical robots and their
+virtual counterparts must be co-located relative to the user,
+in order to provide the correct haptic feedback that aligns
+with the virtual counterpart. This problem can be exacerbated
+when the environments/interactions are dynamic (i.e. the
+physical and virtual haptic proxy must move synchronously)
+or when there is a decoupling between the user’s physical and
+virtual locations (as is common with some VR interaction
+techniques like redirected walking [24]).
+Main contributions: We introduce the concept of ac-
+tive haptic guidance for improved virtual locomotion, and
+conduct a user study to show an example of how active
+haptic guidance can be used to improve a user’s safety and
+arXiv:2301.05311v1  [cs.RO]  12 Jan 2023
+
+feelings of immersion in a virtual experience. Our framework
+is general, so it can be applied to use cases other than
+locomotion, and we provide examples of other possible use-
+cases for active haptic guidance. Our main contributions
+include:
+• A formal description of the active haptic guidance
+problem and details on common challenges that are
+faced when implementing active haptic guidance. Ac-
+tive haptic guidance involves using robots to provide
+realistic haptic feedback to users in mixed reality, with
+the goal of influencing users’ behaviors to improve
+their safety and/or sense of presence in the virtual
+environment.
+• An prototype realization and user study showing the
+benefits of active haptic guidance. In our study, par-
+ticipants completed a virtual navigation task using real
+walking, either with or without active haptic guidance.
+Our results show that active haptic guidance can signif-
+icantly improve the virtual experience by reducing the
+number of “breaks in presence” and keeping them a safe
+distance away from physical objects for longer.
+II. BACKGROUND AND RELATED WORK
+Haptic feedback can be utilized in any mixed reality
+setting, but in this work we mainly discuss applications of
+haptics to virtual reality (VR) settings, since our implementa-
+tion was done in VR. In VR, the user wears a head-mounted
+display (HMD) through which they view a 3D, computer-
+generated virtual environment (VE) [15]. The user’s position
+in the physical environment (PE) is tracked, so that whenever
+the user updates their position in the PE, the position
+of the virtual camera updates accordingly to provide an
+accurate viewing perspective of the VE. VR is an interactive
+experience, meaning that the user does not passively observe
+the virtual content, but instead the environment changes in
+response to the user’s actions and movements. When the
+virtual experience feels sufficiently real, the user experiences
+a sense of presence, which describes the subjective feeling of
+really being in the environment [31]. Factors that contribute
+to a user’s feelings of presence and immersion in a VE
+include the HMD refresh rate [3], the environment realism
+and visual quality [34], and perceptual stimuli matching
+[8], [33] (the process of providing users with perceptual
+information that matches their actions in the VE). In this
+paper, we focus on providing haptic stimuli for perceptual
+stimulus matching to improve the user’s experience in VR.
+Haptic feedback can be provided in a passive or an active
+manner. With passive haptics, objects are placed in the PE
+such that they align with the locations of objects in the VE,
+resulting in haptic feedback when the user tries to touch
+objects in the VE [11]. Conversely, active haptics involves
+a haptic proxy that dynamically alters its configuration in
+real time to provide the appropriate haptic force feedback,
+depending on the user’s interactions with the VE. It is
+common to use robotic systems to render haptic forces. For
+example, Zhang et al. [36] used a robotic arm to provide
+haptic feedback during object assembly by aligning the arm’s
+end effector with the handheld proxy. Siu et al. [28] used
+an array of actuated pins to match the contours of virtual
+objects. Similarly, Zhao et al. [37] used robotic assembly
+to construct tangible representations of virtual objects, made
+from magnetically attached blocks. To recreate the feelings of
+grasping virtual objects, Kovacs et al. [18] used a wrist-worn
+device to provide on-demand haptic feedback when users
+grip virtual objects, while Sinclair et al. [27] used a force-
+resisting, handheld controller to render haptic forces for rigid
+and compliant objects. Suzuki et al. [32] used mobile robots
+to rearrange physical furniture to align with virtual furniture
+as the user moved through a virtual world. Robotic systems
+have also been used to aid in navigation through VEs, via
+handheld canes that use vibrations to provide information
+about the VE [38], [19], [29], mechanical staircases [13] to
+simulate uneven virtual terrain, or mobile tiles that simulate
+infinite walking in any direction [12].
+The majority of prior work on active haptics for mixed
+reality requires the user to initiate interactions with the VE
+before the haptic forces are rendered. That is, the haptic
+forces are triggered by the user’s interactions with the VE,
+so it is the user’s actions that dictate when haptic forces
+are rendered. In this work, we make the distinction of using
+active haptics specifically to direct the user and influence
+their behavior in the VE (in addition to providing a more
+immersive experience, as all haptics aims to do). We define
+this use of haptics as active haptic guidance, since it is the
+haptic forces that direct the user’s behaviors, rather than
+the other way around. We note that there already exists
+research on “haptic guidance,” which Feygin et al. use to
+refer to haptic feedback that is used to help people learn
+motor skills [7]. The distinction between our work on active
+haptic guidance and Feygin et al.’s work is that we use haptic
+feedback to discreetly influence the user’s behavior in an
+effort to enhance their feelings of presence and level of safety
+in a mixed reality experience, while Feygin et al. use haptics
+to teach people motor skills.
+III. PROBLEM DESCRIPTION
+Here we describe the active haptic guidance problem, as
+well as constraints that need to be satisfied to effectively
+utilize haptics to guide users in MR.
+A. Definitions
+In virtual reality, the user is located in a physical envi-
+ronment (PE) and a virtual environment (VE) at the same
+time. Each environment consists of objects (either physical
+objects or virutal objects represented by textured meshes)
+and agents (the users and robots). Note that it is common
+to refer to virtual humans and animals as agents, but in this
+work we will consider all components of the VE as generic
+objects for simplicity, and we use “agents” to refer only to
+humans and robots in the PE.
+Let O = {o1, o2, ..., oi} be a set of polygonal objects,
+where each object o is a mesh with vertices in R3. Let
+U = {u1, u2, ..., uj} be the set of users in an environment.
+Here, u represents the user’s state in an environment, and
+
+usually describes their position and orientation in said envi-
+ronment. For example, we can define u = {p, θ}, where
+p ∈ R2 represents their position in the 2D plane and
+θ ∈ [0, 2π) represents their orientation in the environment.
+Let R
+=
+{r1, r2, ..., rk} be the set of robots in an
+environment, and let A = {U ∪ R} be set of all agents.
+Each of these sets O, U, R, and A may be empty.
+We define an environment E as a set of obstacles and
+agents; that is, E = {O, A}. To differentiate between the
+PE and VE, we denote the PE as EP = {OP , AP } and the
+VE as EV = {OV , AV }. For each user in virtual reality,
+they will have a representation in both the PE and VE, so
+|UP | = |UV | = n, where n is the number of users in virtual
+reality. Since we only consider agents to be users and robots
+in this work, |AV | = n (i.e., the only agents in the VE are
+the users). In the VE, there are some objects that the user
+is likely to interact with, which will improve their sense of
+presence in the environment. We define this set of “objects
+of interest” O ⊂ OV as the set of virtual objects for which
+we render haptic forces when the user interacts with them.
+With these definitions of the PE and VE, we can now de-
+scribe the two main conditions that need to be met to provide
+active haptic guidance to users in MR. First, the robots in the
+physical environment need to provide the appropriate haptic
+feedback to influence the user’s configuration. Second, we
+need to ensure that the robots that provide haptic feedback
+are co-located (relative to the user) with the virtual objects
+of interest with which the haptic forces are associated.
+B. Influential Haptics Constraint
+The first condition that needs to be met in order to
+implement active haptic guidance is that the rendered haptic
+forces should influence the user’s behavior such that they
+update their physical and virtual configurations. We dub
+this constraint the influential haptics (IH) constraint. For
+simplicity, we formalize this constraint using one user, one
+robot, and one virtual object of interest, but this constraint
+applies to any group of agents and virtual objects for which
+we render haptic forces.
+Given the user’s physical and virtual configurations uP
+and uV , a virtual object of interest o, and a robot r that
+provides haptic feedback for o, we wish to render a haptic
+force F that compels the user to update uP and uV to
+some goal configurations u∗
+P and u∗
+V . Thus, fulfilling the
+IH constraint requires completing the following steps:
+1) Compute the goal configurations u∗
+P and u∗
+V .
+2) Detect or initiate an interaction I between o and uV .
+3) Update the configuration of r to render a haptic force
+F(I, uV , uP , u∗
+P , u∗
+V , r) that minimizes an objective
+function f(uV , uP , u∗
+P , u∗
+V ).
+In practice, computing F(I, uV , uP , u∗
+P , u∗
+V , r) depends
+heavily on the mechanics of the haptic proxy r and the ob-
+jective function f(uV , uP , u∗
+P , u∗
+V ). The objective function is
+usually a distance function that measures the error between
+uP and uV , and it depends on the user’s configuration
+space. By rendering F, the user hopefully updates their
+configuration such that they move closer to u∗
+P and u∗
+V .
+Computing u∗
+P and u∗
+V is a matter of determining how we
+want the user to behave. In mixed reality (MR), two main
+reasons to influence the user’s behavior are to ensure their
+safety and to deliver a more immersive experience. In MR
+systems, the user tries to navigate through the PE and the VE
+at the same time, but the PE is partially or fully occluded.
+Thus, in order to prevent the user from bumping into physical
+objects that they cannot see, locomotion interfaces for MR
+usually display a notification that prompts them to reposition
+themself to a safer position away from nearby objects. By
+using haptics to warn users (either overtly or subtly), we can
+decrease the likelihood that the user collides with unseen
+physical obstacles or exits the designated tracking area.
+In addition to ensuring user safety, influencing the user’s
+behavior can be useful for improving the user’s sense of
+presence in the VE. In MR, providing perceptual stimuli
+that align with the content rendered on the visual display
+enhances the user’s feeling that they are really in the VE that
+they are seeing. To this end, haptic feedback can significantly
+improve the user’s sense of presence in the VE [11]. In the
+case of active haptic guidance, the haptic feedback can be
+used as an additional narrative element that encourages users
+to explore a particular area or interact with particular objects
+in the VE (e.g. pairing visual distractors [21] with haptic
+feedback to direct the user’s attention).
+C. Relative Co-location Constraint
+The second main constraint that should be met when
+using active haptic guidance is that the physical robots
+that render the haptic forces and their associated virtual
+objects should be co-located relative to the user. That is,
+the position of the robot and the virtual object should be the
+same relative to the user’s configuration in the PE and VE.
+This is done to ensure that the user perceives a congruent VE
+that is augmented by haptic forces, rather than perceiving a
+VE along with misaligned haptic forces, which may break
+their sense of presence in the virtual experience. We call this
+the relative co-location (RC) constraint.
+Given the user’s physical and virtual configurations uP
+and uV , a virtual object of interest o, and a robot r that
+provides haptic feedback for o, we wish to update r such
+that we minimize the error in the relative positions between
+uV and o and uP and r. Fulfilling the RC constraint requires
+completing the following steps:
+1) Compute the configurations of o and r relative to uV
+and uP , respectively. Usually, these are just positions
+po and pr of o and r relative to the user in the
+respective environment.
+2) Compute a goal configuration r∗ for the haptic proxy
+that minimizes an objective function f(po, pr).
+3) Update the configuration of r to move it towards r∗.
+In practice, updating the robot’s configuration in step #3 is
+a motion planning problem where we aim to find a path
+through the configuration space that brings r close to r∗,
+and it depends on the mechanics of the haptic proxy.
+Since MR is an interactive technology, the relative posi-
+tions po and pr are constantly changing as the user explores
+
+and interacts with the VE. Thus, evaluating and fulfilling the
+RC constraint must be done constantly to ensure that any per-
+ceptual stimuli mismatch is minimized. Failure to adequately
+meet this constraint can degrade the user experience, since
+it increases the likelihood that the user notices a discrepancy
+between visual stimuli and the haptic stimuli [14], [20].
+Furthermore, knowing how much error between their relative
+positions the user will tolerate is a subjective measure [2],
+[17], so it is usually not the case that the robot must reach
+r∗ exactly. Note that this relative co-location constraint is
+not unique to the active haptic guidance problem (unlike
+subsection III-B); other work on active haptics for virtual
+reality also has to deal with the problem of ensuring the
+co-location of robotic agents and their virtual counterparts.
+IV. PROTOTYPE REALIZATION EXAMPLES
+In this section, we provide details on our prototype im-
+plementation of an application of active haptic guidance. In
+particular, we implement an active haptic-driven locomotion
+application to provide a safer and more immersive virtual
+navigation experience for users. We discuss other potential
+use-cases for active haptic guidance in the supplementary
+materials posted on our project page.
+A. Natural Walking in Virtual Reality
+In VR, it is common for the PE to be much smaller than
+the VE. To enable users to explore large VEs, many different
+locomotion interfaces such as teleportation, joystick naviga-
+tion, and walking-in-place have been developed [6]. Ideally,
+users explore the VE using natural, everyday walking since
+it improves their sense of presence [33] and performance
+in tasks [9], [22], [26]. One technique that enables natural
+walking in VR is redirected walking (RDW) [24].
+RDW works by slowly rotating the VE around the user’s
+virtual camera while they walk, which causes them to
+unconsciously adjust their physical trajectory to counteract
+the VE rotations and remain on their intended path in the
+VE. It works because the human perceptual system tends to
+believe the user’s visual stimuli over other stimuli (proprio-
+ceptive, vestibular, etc.) when they conflict, as is the case in
+RDW [23]. Using RDW, we can steer the user along paths in
+the PE that direct them away from objects and edges of the
+tracked space, resulting in a safer and more immersive virtual
+experience. To help mask the VE rotations, researchers make
+use of distractors which grab the user’s attention to decrease
+the likelihood that they attend to the rotations of the VE [4],
+[21], [35]. In our prototype implementation, we use a virtual
+dog as a distractor in conjunction with a RDW algorithm
+known as steer-to-center, which rotates the VE such that the
+user is steered towards the center of the PE at all times [23].
+B. Virtual Experience and Equipment
+For our implementation, a user u1 completed a navi-
+gation task in a virtual city and had a virtual dog as a
+companion (only a single user participated at a time, so
+|UP |
+=
+|UV |
+=
+1). Additionally, u1 held a position-
+tracked leash that was tethered to a differential wheeled robot
+r1. The PE was an empty rectangular room with four walls
+(represented by the boundaries of the VR tracking space).
+Thus, EP
+=
+{OP , AP }, where AP
+=
+{u1, r1}. The
+virtual dog served as a distractor and was the only object
+of interest in EV (|O| = 1), meaning that the robot only
+rendered haptic forces associated with the virtual dog.
+Our application was implemented using one HTC VIVE
+Cosmos VR HMD with two VIVE trackers, and one robot
+car (ELEGOO UNO Robot Car kit). We attached one VIVE
+tracker to the robot to track its location and orientation data,
+and the other was attached to the leash handle to calculate
+the distance between u1 and r1. The robot was equipped
+with an HC-06 Bluetooth LE adapter, which connected to
+the PC to transmit robot movement commands. The Unreal
+4.22 game engine was used to render the VE.
+C. Virtual Companion and Robot Behavior
+Here we describe the behavior of the virtual dog com-
+panion and how the robot matches the virtual companion’s
+movements and provides haptic feedback.
+1) Virtual Dog Companion Behavior: The virtual dog has
+two main behavior states: following and distracting. When
+the user walks around and is not at risk of leaving the
+tracking space, the dog is in follow mode. In this mode,
+the dog walks slightly ahead of the user as they walk, and
+remains in one spot when the user stands still.
+When the user reaches a boundary of the tracked space, the
+VR system initiates what is called a reset, wherein the user
+reorients themself such that they face towards the inside of
+the tracking space in the PE. To ensure that their orientation
+in the VE is not altered, the VR system applies redirection
+that effectively cancels out their physical rotation in the
+virtual space. When a reset is initiated, the virtual dog enters
+distract mode. In distract mode, we compute a goal position
+in the VE for the dog to move towards. The idea behind
+distract mode is that the user is likely to pay attention to the
+virtual dog as it runs to a goal position, which allows the
+system to apply stronger redirection (away from the obstacles
+in the PE) without interfering with the user’s experience [21].
+During a reset, the goal position is selected by first
+computing the vector from the user towards the center of the
+physical space. The goal position is then determined to be
+either the endpoint of this vector in the VE, or a virtual object
+near the vector’s endpoint that was labeled as a potential
+goal position during development. Potential goal positions
+are virtual objects that a dog would be likely to interact with,
+such as a fire hydrant or a lamp post. If the vector intersects
+with a virtual object (e.g. a virtual building) and there are no
+potential goal objects nearby, the goal position is simply the
+point furthest along the vector that does not intersect with
+any objects. See Figure 2 for a visualization of this process.
+2) Robot Haptic Proxy Behavior: The physical robot’s
+main purpose is to provide haptic feedback to make the user’s
+virtual experience feel more immersive and to encourage the
+user to walk away from nearby objects or tracking space
+boundaries in the PE. In both follow and distract mode, the
+physical robot needs to update its position such that it is
+
+Fig. 2.
+Our method of automatically choosing a suitable virtual goal position for the virtual companion. When the user gets close to a boundary of
+the physical space, they need to be reoriented away from the boundary before they continue walking. In order to pick a goal destination for the virtual
+companion and robotic haptic proxy, we cast a ray from the physical user to the center of the tracked space and then superimpose this vector onto the
+user’s virtual position. If the endpoint of this vector is near a pre-defined potential goal position, that is chosen as the current goal position. Otherwise, we
+choose the furthest point along the vector that does not intersect with any objects in the virtual environment.
+aligned with the position of the virtual dog, relative to the
+user in either environment. Checking if a position update is
+necessary is easily achieved by computing the vector from
+the virtual user to the virtual dog and comparing it to the
+vector from the user’s HMD and the robot.
+To compute the trajectory that the robot will follow, we
+compute a circular arc path based on the robot’s position,
+forward direction, and destination position (determined by
+the relative position of the virtual dog and user). The ideal
+path for a differential drive robot is a circular arc since it only
+requires one set of wheel velocities [5]. The wheel velocities
+are computed with the ratio
+2rd
+2r−d, where r is the arc radius
+and d is the distance between the robot wheels. Note that we
+do not use typical PID-based drift correction due to possible
+unexpected complications that may arise from the tethering
+to the user [1], [25], [30].
+D. Maintaining Active Haptic Guidance Constraints
+This section describes how our active haptic-drive loco-
+motion application satisfies the IH and RC constraints.
+1) Directing Users With Haptic Feedback:
+Since the
+virtual object of interest is a dog, the user is attached to the
+robot by an elastic tether that resembles a leash. When the
+robot moves away from the user in the PE, it simulates the
+sensation of a dog tugging on its leash, thereby improving the
+realism of the virtual experience. Additionally, this tugging
+encourages the user to follow the robot rather than “fight” it,
+allowing us to further influence the user’s movement patterns
+in the PE and VE. By triggering the robot to move away from
+the user and towards the center of the PE when they get too
+close to the tracking space boundaries, the tugging force on
+the leash encourages the user to turn and walk towards the
+robot and away from the tracked space boundaries.
+2) Maintaining Co-location: Normally, maintaining rela-
+tive co-location between a haptic proxy and a virtual object
+is a matter of updating the position of the haptic proxy
+whenever the virtual object’s position changes. We also do
+this in our implementation by updating the position of the
+robot to match the movements of the virtual dog. However,
+our implementation requires additional work to maintain co-
+location due to a new problem which we refer to as the haptic
+proxy distortion (HPD) problem.
+Virtual environment 
+before rotation.
+Virtual environment after 
+rotation.
+Physical environment 
+and superimposed
+virtual relative positions.
+Fig. 3. A visualization of the haptic proxy distortion problem. Left: Initially,
+the virtual user and virtual companion have a particular relative position.
+Middle: After rotating the virtual environment around the virtual user, the
+relative position of the companion changes since the companion is rotated
+along with the rest of the environment. Right: In the physical space, the
+haptic proxy has not been updated, so its position coincides with the virtual
+companion’s relative position before rotation (opaque robot and vector). The
+new relative position of the virtual companion, which the haptic proxy needs
+to match, is shown as the translucent robot and dashed-line vector.
+In our implementation, we make use of a locomotion
+interface called redirected walking (RDW) that enables nat-
+ural walking in VR. RDW works by rotating the entire VE
+around the virtual camera that represents the user’s viewpoint
+in the VE. Consequently, the virtual dog companion may
+change its position relative to the virtual user without the dog
+actually moving to a new destination in the VE (see Figure 3).
+Thus, as we apply redirection, the relative position of the
+virtual dog changes constantly, while the relative position
+of the physical robot does not. To resolve this discrepancy
+in relative position, we check the relative positions of the
+virtual dog and physical robot on each frame, and update
+the robot’s destination in the PE to minimize the difference
+in relative position. The user will perceive this as the haptic
+proxy “sliding” across the floor around them, which might
+result in unsmooth motion that may detract from the user
+experience. In practice, this did not seem to be a major
+problem for users, but we acknowledge that there may be
+better solutions to the HPD problem, and leave that for future
+work. This HPD problem adds onto the errors in relative co-
+location between the haptic proxy and the virtual companion,
+which makes it harder to satisfy the RC constraint. Note
+that the HPD problem is not specific to our implementation;
+this problem is present in any application that uses haptic
+proxies and creates a mismatch between the user’s positions
+in the physical and virtual environments, as is common for
+
+Physical Environment
+Virtual Environment
+Virtual Environment
+Virtual Environment
+Virtual Environment
+UTU
+User reached the tracked space boundary, so a
+Superimpose the physical user-to-center
+If there is a potential pre-defined goal
+If there is no pre-defined goal position near
+If the superimposed user-to-center vector
+reorientation is required. Compute the vector
+vector onto the virtual user to determine the
+position near the endpoint of the user-to-
+the endpoint of the user-to-center vector,
+intersects with a virtual object, use the
+from the user to the center of the physical
+goal position of the virtual companion.
+center vector (e.g., a fire hydrant), set that as
+use the vector endpoint as the goal position.
+furthest non-intersecting point along the
+" vector).
+the goal position.
+vector as the goal position.locomotion interfaces for mixed reality.
+V. EXPERIMENTS & RESULTS
+A. Experiment Design and Procedure
+To evaluate the effectiveness of our implementation of
+active haptic-driven locomotion prototype, we conducted a
+user study where participants completed a navigation task.
+The study design was approved by our university’s Insti-
+tutional Review Board. The goal of our user study was to
+evaluate how effective the haptic guidance was at improving
+users’ sense of presence in the VE and keeping users away
+from the boundaries of the VR system’s tracked space. We
+used a between participants design, where one group of
+participants completed a navigation task with active haptic
+guidance enabled, and the other group completed the same
+task without any haptic guidance. The navigation task had
+a time limit of 5 minutes and 30 seconds, after which the
+experiment ended regardless of if the participant reached the
+goal destination. Participants were unaware of this time limit
+so that they did not rush to complete the task. We recruited 20
+participants (13 male, 5 female, 2 participants did not report)
+through online advertising and oral recruitment. Participants’
+ages ranged from 18 to 28 (µ = 24.59, σ = 2.37). All
+participants were able to walk without any assistance.
+The study consisted of three sections, and lasted about 15
+minutes for each participant. First, we debriefed participants
+on the experiment procedures and had them complete a pre-
+study Simulator Sickness Questionnaire (SSQ) [16]. Next,
+the user put on the HMD and completed the task in the
+VE. The VE was a city environment with several streets and
+blocks, and was populated with common objects such as bus
+stops, stores, park squares, and virtual humans that roamed
+around the environment (see Figure 1 and the supplementary
+video). To mask any potentially distracting noises from the
+robot as it moves, participants wore headphones and back-
+ground music was played for the duration of the experiment
+task. Participants started the task at one intersection in the
+city, and their task was to reach a green question mark in
+the environment that indicated their destination, which was
+one block away from the their starting position. During the
+experiment, we recorded how many times users reached the
+bounds of the PE and the time taken to complete the task.
+Once participants finished the task, they completed another
+SSQ survey and a questionnaire with questions on a 7-point
+Likert scale that measured their sense of presence in the VE
+(7 = high presence, 1 = low presence). Finally, the experiment
+was ended with open-ended questions where participants
+could provide additional comments.
+B. Results
+The metrics we used to measure the effectiveness of our
+active haptic-driven locomotion interface were the number of
+breaks in presence (BiPs), the completion rate and time taken
+to complete the task, and participants’ subjective feelings of
+presence in the VE. A BiP is incurred when the user reaches
+the boundaries of the tracking space and they are forced to
+reorient away from the boundary before continuing to walk.
+BiPs
+Time (s)
+Presence
+Completed
+Haptics
+µ
+σ
+µ
+σ
+µ
+σ
+Total #
+With
+0.90
+0.74
+195.20
+22.25
+4.63
+1.77
+10
+Without
+18.90
+5.17
+309.40
+65.14
+3.57
+1.64
+1
+TABLE I
+Performance results from our user study. THE “WITH HAPTICS”
+GROUP OF PARTICIPANTS INCURRED SIGNIFICANTLY fewer BREAKS IN
+PRESENCE (“BIPS” COLUMN), COMPLETED THE EXPERIMENT MUCH
+more quickly (“TIME” COLUMN) AND WITH MUCH higher SUCCESS
+RATES (“COMPLETED” COLUMN), AND REPORTED A higher SENSE OF
+PRESENCE IN THE VIRTUAL EXPERIENCE (“PRESENCE” COLUMN).
+THESE RESULTS SHOW THAT HAPTIC GUIDANCE CAN BE EFFECTIVE FOR
+IMPROVING USERS’ VIRTUAL EXPERIENCE.
+Based on the results in Table I, the presence of our active
+haptic guidance companion resulted in significantly fewer
+BiPs, notably lower completion times and higher completion
+rates, and slightly higher (and above-average) presence lev-
+els. Meanwhile, participants who completed the navigation
+task without any haptic guidance incurred a large number of
+BiPs, did not finish the task in time, and reported below-
+average levels of presence. These results support the notion
+that active haptic guidance can be used to help keep users
+safe and feel more immersed in mixed reality experiences.
+VI. CONCLUSIONS & FUTURE WORK
+In this work, we presented the active haptic guidance
+problem for mixed reality (MR), which describes the use
+of one or more robots to provide haptic feedback to users
+in order to create a richer virtual experience for the user,
+while also influencing the user’s behavior to improve their
+safety and immersion in the virtual world. As a prototype
+realization, we implemented active haptic guidance in a VR
+locomotion application that enables the user to explore a
+large VE while located in a much smaller PE. By combining
+active haptic guidance and redirected walking, we increased
+the effective area of the PE while also decreasing the
+likelihood that the user exits the VR system’s tracked area.
+The concept of active haptic guidance is general and can be
+applied MR applications other than locomotion; we discuss
+other potential use cases for active haptic guidance in the
+supplementary materials on our project page.
+Limitations and Future Work: One limitation of our
+work is the haptic proxy distortion problem, in which the
+haptic proxy and the associated virtual object can become
+desynchronized due to mismatches between the user’s phys-
+ical and virtual configurations. Solving this problem requires
+continuously updating the position of the haptic proxy, and
+our proposed solution in this work is likely not the most
+optimized solution. Additionally, our system uses only a
+rough estimation of drift to readjust the haptic proxy position,
+instead of a more accurate method like PID-based drift
+correction. Future work in this area should investigate the
+use of more realistic companions and behavior models, and
+should explore how active haptic guidance can be applied
+to other types of VR experiences with different applications,
+such as social mixed reality settings with other users.
+
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+APPENDIX
+Additional Applications of Active Haptic Guidance: Here
+we discuss other potential applications of active haptic guid-
+ance for immersive applications:
+• Wood Carving Application: In wood carving, the grain
+of the wood will impact the direction in which the artist
+carves the wood. That is, sometimes the artist will carve
+“with the grain” and sometimes will carve “against the
+grain.” Using active haptics, one could accurately render
+the different resistance forces that arise from carving
+with or against the grain of a virtual wooden block,
+which will in turn influence the way in which the user
+carves their virtual wooden sculpture. In addition to
+providing a more realistic experience, this could be
+used to guide the user to create a more appealing final
+sculpture (e.g. by altering the direction of the grain to
+subtly change their hand movements, which will change
+the shape of the final carved surface).
+• Immersive Cooperative Application: A major appeals
+of mixed reality experiences is the ability to connect
+with other users in shared virtual experiences. Important
+to these shared experiences is the ability to touch the
+other person, which can provide a greater sense of
+companionship and connection between users. Haptic
+forces can be used to encourage users to interact with
+or follow other users who are also present in their virtual
+experience, which may improve the users’ sense of
+presence in the experience due to the enhanced realism.
+• Virtual Cooking Training Application: Given a seated
+VR experience where the user is practicing their cook-
+ing skills in a virtual environment, a mobile, tabletop
+robot can provide haptic feedback that represents feed-
+back provided by cooking utensils. For example, when
+spreading brownie batter in a baking pan, the user will
+feel haptic forces when the virtual spreading utensil gets
+too close to the edges of the virtual baking pan. These
+forces could be rendered using a mobile robot with a
+flat surface that serves as a wall that the user’s physical
+hand will bump into.
+

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+arXiv:2301.11383v1  [math.RT]  26 Jan 2023
+TENSOR PRODUCTS AND INTERTWINING OPERATORS
+BETWEEN TWO UNISERIAL REPRESENTATIONS OF
+THE GALILEAN LIE ALGEBRA sl(2) ⋉ hn
+LEANDRO CAGLIERO AND IV´AN G´OMEZ RIVERA
+Abstract. Let sl(2) ⋉ hn, n ≥ 1, be the Galilean Lie algebra over a
+field of characteristic zero, where hn is the Heisenberg Lie algebra of
+dimension 2n+ 1, and sl(2) acts on hn so that hn ≃ V (2n− 1)⊕ V (0) as
+sl(2)-modules (here V (k) denotes the irreducible sl(2)-module of highest
+weight k). The isomorphism classes of uniserial
+�
+sl(2)⋉hn
+�
+-modules are
+known.
+In this paper we study the tensor product of two uniserial represen-
+tations of sl(2) ⋉ hn. Among other things, we obtain the sl(2)-module
+structure of the socle of V ⊗W and we describe the space of intertwining
+operators Homsl(2)⋉hn(V, W ), where V and W are uniserial representa-
+tions of sl(2) ⋉ hn. This article extends a previous work in which we
+obtained analogous results for the Lie algebra sl(2) ⋉ am where am is
+the abelian Lie algebra and sl(2) acts so that am ≃ V (m − 1) as sl(2)-
+modules.
+1. Introduction
+This article is part of a project whose general goal is to understand to
+what extent there is a class of finite-dimensional representations of (non-
+semisimple) Lie algebras that is small enough, so that it members can be
+described in a reasonably efficient way in terms of uniserial representations,
+and large enough to include many representations that appear in problems
+of interest. This naturally leads to consider the tensor products of unise-
+rial representations. A typical example of a representation we would like
+to describe thoroughly is a cohomology space associated to an algebra (as-
+sociative algebra or non-semisimple Lie algebra), viewed as a module over
+its whole Lie algebra of derivations (which is, in general, non-semisimple).
+Understanding this action becomes specially important if the cohomology
+space has a Gerstenhaber or a Poisson structure.
+Why uniserial representations as building blocks?
+We recall that, for
+associative algebras, the class of uniserial modules is very relevant, a foun-
+dational result here is due to T. Nakayama [24] (see also [1] or[2]) and it
+states that every finitely generated module over a serial ring is a direct sum
+of uniserial modules. For more information in the associative case we refer
+the reader mainly to [1, 2, 27], and also [3, 21, 25]. In Lie algebra case, even
+2010 Mathematics Subject Classification. 17B10, 18M20, 22E27.
+Key words and phrases. non-semisimple Lie algebras, uniserial representations, socle,
+tensor product, intertwining operators.
+This research was partially supported by an NSERC grant, CONICET PIP 112-2013-
+01-00511, PIP 112-2012-01-00501, MinCyT C´ordoba, FONCYT Pict2013 1391, SeCyT-
+UNC 33620180100983CB.
+1
+
+2
+LEANDRO CAGLIERO AND IV´AN G´OMEZ RIVERA
+though very little is known about uniserial representations, in the articles
+[8, 9, 10, 11, 5, 4, 6, 15] we and other authors have classified all finite dimen-
+sional uniserial representations for some different families of Lie algebras g.
+These classifications show that the class of uniserial representations of g is
+rather small and treatable in the universe of all the indecomposable modules.
+Also, in [20], it is shown how the infinite dimensional uniserial representa-
+tions of certain special linear groups obtained in [28] appear naturally in
+cohomology spaces.
+We point out that many other authors work on the idea of describing
+or classifying a special class of indecomposable representations of (non-
+semisimple) Lie algebras whose members might be used as building blocks
+for describing more general representations. For instance, A. Piard [26] ana-
+lyzed thoroughly the indecomposable modules U, of the complex Lie algebra
+sl(2)⋉C2, such that U/rad(U) is irreducible. More recently, various families
+of indecomposable modules over various types of non-semisimple Lie alge-
+bras have been constructed and/or classified, see for instance [12, 13, 14, 16,
+19, 17, 18, 22].
+This article and [7] are motivated by the challenge of describing, in a
+standard way, the tensor product of two uniserial g-modules in terms of
+uniserial g-modules. In contrast to the Nakayama case, these tensor products
+are not at all a direct sum of uniserials, there are many cases where the tensor
+product of two uniserial g-modules is an indecomposable g-module but not
+uniserial.
+1.1. Main results. In this paper, all Lie algebras and representations con-
+sidered are assumed to be finite dimensional over a field F of characteristic
+zero.
+For n ≥ 0, we denote by an the abelian Lie algebra of dimension
+n, by hn the Heisenberg Lie algebra of dimension 2n + 1, and by V (n)
+the irreducible sl(2)-module with highest weight n (dim V (n) = n + 1).
+In addition, for n ≥ 1, sl(2) ⋉ an denotes the Lie algebra obtained by
+letting sl(2) act so that an ≃ V (n − 1), and sl(2) ⋉ hn denotes the Lie
+algebra where hn ≃ V (2n − 1) ⊕ V (0) as sl(2)-modules. We notice that
+sl(2) ⋉ a2n−1 is isomorphic to the quotient sl(2) ⋉ hn mod its 1-dimensional
+center z
+�
+sl(2) ⋉ hn
+�
+≃ V (0).
+In this work we study the structure of the tensor product of two uniserial
+representations of sl(2)⋉hn. The classification of all the isomorphism classes
+of uniserial
+�
+sl(2) ⋉ hn
+�
+-modules was obtained in [4]. This classification is
+reviewed with details in §3 and a rough description of it is the following:
+• Non-faithful
+�
+sl(2) ⋉ hn
+�
+-modules. Since z
+�
+sl(2) ⋉ hn
+�
+acts trivially on
+them, they are in correspondence with the uniserial
+�
+sl(2)⋉a2n−1
+�
+-modules
+In turn, these where classified in [8]:
+– A general family E(a, b), where a, b are non-negative integers with cer-
+tain restrictions (depending on n). The composition length of E(a, b)
+is 2.
+– A general family Z(a, ℓ) and its duals. Here a and ℓ are a non-negative
+integers. The composition length of Z(a, ℓ) is ℓ + 1.
+– Some exceptional modules with composition lengths 3 and 4.
+The modules Z(a, ℓ) and their duals are referred to as modules of type Z.
+
+UNISERIAL REPRESENTATIONS OF THE LIE ALGEBRA sl(2) ⋉ hn
+3
+• Faithful
+�
+sl(2) ⋉ hn
+�
+-modules. All of them have composition length 3.
+– For n = 1: Two families denoted by FU +
+a and FU −
+a , with a an integer
+that satisfies a ≥ 0 and a ≥ 1, respectively.
+– For n = 2: Only four equivalence classes, they are denoted by FU(0,3,0),
+FU(1,4,1), FU(1,2,1) and FU(4,3,4).
+– For any n ≥ 3: Only three equivalence classes, they are denoted by
+FU(0,m,0), FU(1,m+1,1) and FU(1,m−1,1), here m = 2n − 1.
+All faithful uniserial modules (except FU(4,3,4), n = 2) are, in some sense,
+of a similar type and are referred to as standard faithful modules. The
+�
+sl(2) ⋉ h2
+�
+-module FU(4,3,4) is quite exceptional.
+The modules E(a, b) constitute the building blocks of all the other unise-
+rial modules: all uniserials can be obtained by combining the modules E(a, b)
+in a subtle way governed by the zeros of the 6j-symbols (see [5, 8]). As a
+consequence, the
+�
+sl(2) ⋉ hn
+�
+-module structure of the tensor product of two
+uniserial representations of sl(2) ⋉ hn depends strongly on the
+�
+sl(2) ⋉ hn
+�
+-
+module structure of E(a, b) ⊗ E(c, d) which is already quite involved.
+In [7] we stated a conjecture that provides the description of the socle of
+E(a, b)⊗E(c, d) for any a, b, c, d (see Conjecture 4.2 below) and we proved the
+part of it that was necessary to obtain the socle of V ⊗W and the intertwining
+operators Homsl(2)⋉hn(V, W) where V and W are uniserial representations
+of sl(2) ⋉ hn of type Z (in fact, we dealt in [7] with uniserial representations
+of sl(2) ⋉ am, instead of sl(2) ⋉ hn, recall that, for modules of type Z, the
+action of z
+�
+sl(2) ⋉ hn
+�
+is trivial). In particular we proved that the socle
+of V ⊗ W is multiplicity free as sl(2)-modules. As an application of these
+results, we proved in [7] that if V and W are
+�
+sl(2) ⋉ hn
+�
+-modules of type
+Z, then V and W are determined from V ⊗ W. Moreover, we provided a
+procedure to identify the corresponding parameters a and ℓ of V and W
+from V ⊗ W. This is a rare property, even if the factors are irreducible, it
+is not frequent that the factors V and W are determined from V ⊗ W (see
+[23] and references within).
+In this paper we extend the results of [7] obtaining the socle of V ⊗ W
+and the intertwining operators Homsl(2)⋉hn(V, W) when both V and W are
+standard faithful uniserial
+�
+sl(2) ⋉ hn
+�
+-modules, or when one of them is
+standard faithful and the other one is uniserial of type Z. In contrast to the
+non-faithful case, in the standard faithful case it may happen that the socle
+of V ⊗W is not multiplicity free as sl(2)-modules (this occurs when V = W).
+As a consequence, if V and W are isomorphic standard faithful uniserials,
+then the space of intertwining operators Homsl(2)⋉hn(V, W) is 2-dimensional.
+The main step toward these results requires to make a considerable advance
+in the proof of Conjecture 4.2. See the comments after it to know what is
+still open about this conjecture.
+The paper is organized as follows.
+In §2 we review some basic facts
+about uniserial representations of Lie algebras and recall all the necessary
+definitions and formulas involving the Clebsch-Gordan coefficients. In §3 we
+review the classification of all uniserial representations of the Lie algebras
+sl(2)⋉an (obtained in [8]) and sl(2)⋉hn (obtained in [4]). The main section
+
+4
+LEANDRO CAGLIERO AND IV´AN G´OMEZ RIVERA
+of the paper is §4, and we obtain in it the sl(2)-module structure of the
+socle of the tensor product of two (non-exceptional) uniserial
+�
+sl(2) ⋉ hn
+�
+-
+modules V and W: Theorem 4.1 recalls the case when V and W are of
+type Z (obtained in [7]), Conjecture 4.2 deals with all the possible cases of
+composition length 2, Theorem 4.3 confirms part of Conjecture 4.2 needed to
+prove Theorems 4.6 and 4.7, Theorem 4.6 gives the socle when V is of type
+Z and W is standard faithful, and Theorem 4.7 describes the socle when
+both V and W are standard faithful. Finally, in §5 we obtain the space of
+intertwining operators from the results in §4. Our proof of Theorem 4.3 is
+technical and long, it requires to consider some linear systems with entries
+given by the Clebsch-Gordan coefficients, and thus we decided to devote §6
+to it.
+2. Preliminaries
+2.1. The Clebsch-Gordan coefficients. Recall that F is a field of char-
+acteristic zero and that all Lie algebras and representations are assumed to
+be finite dimensional over F. Let
+(2.1)
+e =
+�
+0
+1
+0
+0
+�
+,
+h =
+�
+1
+0
+0
+−1
+�
+,
+f =
+�
+0
+0
+1
+0
+�
+be the standard basis of sl(2). Let V (a) be the irreducible sl(2)-module with
+highest weight a ≥ 0. We fix a basis {va
+0, . . . , va
+a} of V (a) relative to which
+the basis {e, h, f} acts as follows:
+e va
+k =
+�
+a
+2
+�a
+2 + 1
+�
+−
+�a
+2 − k + 1
+� �a
+2 − k
+�
+va
+k−1,
+h va
+k =(a − 2k)va
+k,
+f va
+k =
+�
+a
+2
+�a
+2 + 1
+�
+−
+�a
+2 − k − 1
+� �a
+2 − k
+�
+va
+k+1,
+where 0 ≤ k ≤ a and va
+−1 = va
+a+1 = 0. The basis {va
+0, . . . , va
+a} has been chosen
+in a convenient way to introduce below the Clebsch-Gordan coefficients.
+Note that if we denote by (x)a the matrix of x ∈ sl(2) relative to the basis
+{va
+0, . . . , va
+a}, then {(e)1, (h)1, (f)1} are as in (2.1), and
+(e)2 =
+
+
+0
+√
+2
+0
+0
+0
+√
+2
+0
+0
+0
+
+ ,
+(h)2 =
+
+
+2
+0
+0
+0
+0
+0
+0
+0
+−2
+
+ ,
+(f)2 =
+
+
+0
+0
+0
+√
+2
+0
+0
+0
+√
+2
+0
+
+ .
+This means that we may assume that {v2
+0, v2
+1, v2
+2} = {−e,
+√
+2
+2 h, f}.
+We know that V (a) ≃ V (a)∗ as sl(2)-modules. More precisely, if {(va
+0)∗, . . . , (va
+a)∗}
+is the dual basis of {va
+0, . . . , va
+a} then the map
+V (a) → V (a)∗
+va
+k �→ (−1)a−k(va
+a−k)∗
+(2.2)
+gives an explicit sl(2)-isomorphism.
+It is well known that the decomposition of the tensor product of two
+irreducible sl(2)-modules V (a) and V (b) is
+(2.3)
+V (a) ⊗ V (b) ≃ V (a + b) ⊕ V (a + b − 2) ⊕ · · · ⊕ V (|a − b|).
+
+UNISERIAL REPRESENTATIONS OF THE LIE ALGEBRA sl(2) ⋉ hn
+5
+This is the well known Clebsch-Gordan formula.
+The Clebsch-Gordan coefficients
+CG(j1, m1; j2, m2 | j3, m3)
+are defined below and they provide an explicit sl(2)-embedding V (c) →
+V (a) ⊗ V (b) which is the following
+V (c) → V (a) ⊗ V (b)
+vc
+k �→ va,b,c
+k
+where, by definition,
+(2.4)
+va,b,c
+k
+=
+�
+i,j
+CG(a
+2, a
+2 − i; b
+2, b
+2 − j | c
+2, c
+2 − k) va
+i ⊗ vb
+j,
+where the sum runs over all i, j such that a
+2 − i + b
+2 − j = c
+2 − k (in fact, we
+could let i, j run freely since the Clebsch-Gordan coefficient involved is zero
+if a
+2 − i + b
+2 − j ̸= c
+2 − k). Since
+(2.5)
+Hom(V (b), V (a)) ≃ V (b)∗ ⊗ V (a) ≃ V (a) ⊗ V (b)
+it follows from (2.2) and (2.4) that the map V (c) → Hom(V (b), V (a)) given
+by
+vc
+k �→
+�
+i,j
+CG(a
+2, a
+2 − i; b
+2, b
+2 − j | c
+2, c
+2 − k) va
+i ⊗ vb
+j,
+�→
+�
+i,j
+(−1)b−jCG(a
+2, a
+2 − i; b
+2, b
+2 − j | c
+2, c
+2 − k) va
+i ⊗ (vb
+b−j)∗,
+�→
+�
+i,j
+(−1)jCG(a
+2, a
+2 − i; b
+2, − b
+2 + j | c
+2, c
+2 − k) (vb
+j)∗ ⊗ va
+i
+(2.6)
+is an sl(2)-module homomorphism.
+We now recall briefly the basic definitions and facts about the Clebsch-
+Gordan coefficients. We will mainly follow [29].
+Given three non-negative integers or half-integers j1, j2, j3, we say that
+they satisfy the triangle condition if j1 + j2 + j3 is an integer and they can
+be the side lengths of a (possibly degenerate) triangle (that is |j1 − j2| ≤
+j3 ≤ j1 + j2). We now define (see [29, §8.2, eq.(1)])
+∆(j1, j2, j3) =
+�
+(j1 + j2 − j3)!(j1 − j2 + j3)!(−j1 + j2 + j3)!
+(j1 + j2 + j3 + 1)!
+if j1, j2, j3 satisfies the triangle condition; otherwise, we set ∆(j1, j2, j3) = 0.
+If in addition m1, m2 and m3 are three integers or half-integers then the
+corresponding Clebsch-Gordan coefficient
+CG(j1, m1; j2, m2|j3, m3)
+
+6
+LEANDRO CAGLIERO AND IV´AN G´OMEZ RIVERA
+is zero unless m1 + m2 = m3 and |mi| ≤ ji for i = 1, 2, 3. In this case, the
+following formula is valid for m3 ≥ 0 and j1 ≥ j2 (see [29, §8.2, eq.(3)])
+CG(j1, m1; j2, m2 | j3, m3) = ∆(j1, j2, j3)
+�
+(2j3 + 1)
+×
+�
+(j1 + m1)!(j1 − m1)!(j2 + m2)!(j2 − m2)!(j3 + m3)!(j3 − m3)!
+×
+�
+r
+(−1)r
+r!(j1+j2−j3−r)!(j1−m1−r)!(j2+m2−r)!(j3−j2+m1+r)!(j3−j1−m2+r)!,
+where the sum runs through all integers r for which the argument of every
+factorial is non-negative. If either m3 < 0 or j1 < j2 we have
+CG(j1, m1; j2, m2 | j3, m3) = (−1)j1+j2−j3 CG(j1, −m1; j2, −m2 | j3, −m3)
+= (−1)j1+j2−j3 CG(j2, m2; j1, m1 | j3, m3).
+(2.7)
+In addition, it also holds
+(2.8)
+CG(j1, m1; j2, m2 | j3, m3) = (−1)j1−m1
+�
+2j3 + 1
+2j2 + 1 CG(j1, m1; j3, −m3 | j2, −m2).
+In the following sections, we will need the following particular values of
+the Clebsch-Gordan coefficients. Here, a, b are integers and i = 0, . . . , a,
+j = 0, . . . , b.
+(2.9) CG(a
+2, a
+2 −i; b
+2, b
+2 −j | a+b
+2 , a+b
+2 −i−j) =
+�
+a!b!(a + b − i − j)!(i + j)!
+i!j!(a + b)!(a − i)!(b − j)! ,
+(2.10)
+CG(a
+2, a
+2 − i; b
+2, j − b
+2 | a−b
+2 , a−b
+2
+− i + j)
+= (−1)j
+�
+(a − i)! i! b! (a − b + 1)!
+(a + 1)! j! (b − j)! (a − b − i + j)! (i − j)!,
+(2.11)
+CG(a
+2, i − a
+2; b
+2, b
+2 − j | b−a
+2 , b−a
+2
++ i − j)
+= (−1)aCG( b
+2, b
+2 − j; a
+2, i − a
+2 | b−a
+2 , b−a
+2
++ i − j)
+= (−1)j
+�
+(b − j)! j! a! (b − a + 1)!
+(b + 1)! i! (a − i)! (b − a − j + i)! (j − i)!,
+(2.12)
+CG(a
+2, a
+2 − i; b
+2, b
+2 − j | a+b
+2
+− i − j, a+b
+2
+− i − j)
+= (−1)i
+�
+(a + b − 2i − 2j + 1)! (i + j)! (a − i)! (b − j)!
+(a + b − i − j + 1)! (a − i − j)! (b − i − j)! i! j!.
+2.2. Uniserial representations. Given a Lie algebra g, a g-module V is
+uniserial if it admits a unique composition series.
+In other words, V is
+uniserial if the socle series
+0 = soc0(V ) ⊂ soc1(V ) ⊂ · · · ⊂ socn(V ) = V
+
+UNISERIAL REPRESENTATIONS OF THE LIE ALGEBRA sl(2) ⋉ hn
+7
+is a composition series of V , that is, the socle factors soci(V )/soci−1(V ) are
+irreducible for all 1 ≤ i ≤ n. Recall that soc1(V ) = soc(V ) is the sum of all
+irreducible g-submodules of V and soci(V )/soci−1(V ) = soc(V/soci−1(V )).
+Note that for uniserial modules, the composition length of V coincides with
+its socle length.
+If the Levi decomposition of g is g = s⋉r, (with r the solvable radical and
+s semisimple) we may choose irreducible s-submodules Vi ⊂ V , 1 ≤ i ≤ n,
+such that
+(2.13)
+V = V1 ⊕ · · · ⊕ Vn
+with Vi ≃ soci(V )/soci−1(V ) as s-modules and
+rVi ⊂ V1 ⊕ · · · ⊕ Vi.
+In fact, if [s, r] = r, then rVi ⊂ V1 ⊕ · · · ⊕ Vi−1, see Lemma 2.2 below.
+Definition 2.1. We say that (2.13) is the socle decomposition of V . We
+point out that, in the socle decomposition of a g-module, the order of the
+summands is relevant.
+The proof of the following lemma can be found in [7, Lemmas 2.1 and
+2.2].
+Lemma 2.2. Assume that r = [s, r] and let V be a g-module. Then
+(1) soc(V ) = V r.
+(2) If V = V1⊕· · ·⊕Vn is a vector space decomposition such that soc(V ) = V1
+and rVk ⊂ Vk−1 for all k = 2, . . . , n, then sock(V ) = V1 ⊕ · · · ⊕ Vk for
+all k = 1, . . . , n.
+3. Uniserial representations of sl(2) ⋉ hn
+3.1. The Lie algebra sl(2) ⋉ hn. Let us fix n ≥ 1. We recall that the
+Heisenberg Lie algebra hn is the (2n + 1)-dimensional vector space with
+basis
+{x1, . . . , xn, x′
+1, . . . , x′
+n, z}
+with non-zero brackets [xi, x′
+i] = z for all i = 1, . . . , n. It is clear that the
+center of hn is generated by z. We know that sl(2) acts by derivations on
+hn in such a way that
+hn ≃ V (m) ⊕ V (0),
+m = 2n − 1,
+as sl(2)-modules, where V (0) corresponds to the center of hn and we may
+assume that V (m) corresponds to the subspace generated by {xi, x′
+i : i =
+1, . . . , n}.
+Notation 3.1. From now on, m will always be 2n−1 and thus am = hn/Fz
+as Lie algebras. In addition, we will denote by hn(m) the subspace of hn
+isomorphic to V (m) as sl(2)-modules.
+Hence, as sl(2)-modules, we have
+am ≃ hn(m) ≃ V (m).
+We may assume that z corresponds to the basis element v0
+0 of V (0). Sim-
+ilarly, let us denote by {e0, . . . , em} the basis of hn(m) corresponding to
+the basis {vm
+0 , . . . , vm
+m} of V (m). We may assume that z and {e0, . . . , em}
+have been chosen such that the bracket in hn is given by the projection
+V (m) ⊗ V (m) → V (0) (dual to the embedding (2.4)), that is
+
+8
+LEANDRO CAGLIERO AND IV´AN G´OMEZ RIVERA
+[ei, em−i] = CG(m
+2 , m
+2 − i; m
+2 , − m
+2 + i | 0, 0) z
+= (−1)i �
+1
+m+1 z.
+(3.1)
+It is clear that Fz is also the center of sl(2) ⋉ hn and
+�
+sl(2) ⋉ hn
+�
+/Fz ≃ sl(2) ⋉ am
+as Lie algebras.
+In [4], it is obtained the classification, up to isomorphism, of all uniserial
+representations of the Lie algebra sl(2) ⋉ hn. It is straightforward to see
+that a uniserial representation of sl(2) ⋉ hn is faithful if and only if z acts
+non-trivially. Therefore, the classification is given in two stages: the non-
+faithful and the faithful ones. The non-faithful ones are the same as those
+of the Lie algebra sl(2) ⋉ am ≃ sl(2) ⋉ V (m) which were classified earlier in
+[8, Theorem 10.1].
+We now recall this classification.
+3.2. The non-faithful
+�
+sl(2) ⋉ hn
+�
+-modules E(a, b). If a and b are non-
+negative integers such that m
+2 , a
+2, b
+2 satisfy the triangle condition, it follows
+from (2.3) and (2.5) that, up to scalar, there is a unique sl(2)-module ho-
+momorphism
+r = V (m) → Hom(V (b), V (a)).
+This produces an action of r on V (a)⊕V (b) such that r maps V (a) to 0 and
+V (b) to V (a) as follows
+(3.2)
+es vb
+j =
+a
+�
+i=0
+(−1)j CG(a
+2, a
+2 − i; b
+2, − b
+2 + j | m
+2 , m
+2 − s) va
+i ,
+s = 0, . . . , m.
+Note that this is, except for a sign, the same as (2.6). Note also that the
+above sum has, in fact, at most one summand, that is
+(3.3)
+es vb
+j =
+
+
+
+0,
+if i ̸= j + s + a−b−m
+2
+;
+(−1)jCG(a
+2, a
+2 − i; b
+2, − b
+2 + j | m
+2 , m
+2 − s) va
+i ,
+if i = j + s + a−b−m
+2
+.
+This action, combined with the action of sl(2) defines a uniserial
+�
+sl(2) ⋉
+am
+�
+-module structure with composition length 2 on
+E(a, b) = V (a) ⊕ V (b).
+It is straightforward to see that E(a, b)∗ ≃ E(b, a). The action given in
+(3.2) is the main building block for all other uniserial
+�
+sl(2) ⋉ am
+�
+-modules
+as follows.
+3.3. Non-faithful
+�
+sl(2) ⋉ hn
+�
+-modules of type Z. The above construc-
+tion can be extended to arbitrary composition length
+V (a0) ⊕ V (a1) ⊕ · · · ⊕ V (aℓ)
+only when the sequence {ai} is monotonic (increasing or decreasing) and
+|ai − ai−1| = m, for all i = 1, . . . , ℓ. More precisely, for the “increasing case”
+
+UNISERIAL REPRESENTATIONS OF THE LIE ALGEBRA sl(2) ⋉ hn
+9
+let α and ℓ be non-negative integers and let Z(α, ℓ) be the
+�
+sl(2) ⋉ am
+�
+-
+module defined by
+(3.4)
+Z(α, ℓ) = V (α) ⊕ V (α + m) ⊕ · · · ⊕ V (α + ℓm)
+as sl(2)-module with action of r sending
+0 ←− V (α) ←− V (α + 2m) ←− · · · ←− V (α + ℓm)
+as indicated in (3.2) (with a = α + (i − 1)m, b = α + im, for i = 1, . . . , ℓ).
+We point out that the above sequence serves as an indication of the action
+of r, there is no chain complex involved.
+We notice that Z(α, 0) = V (α) (r acts trivially) and Z(α, 1) = E(α, α +
+m).
+The “decreasing case” corresponds to the dual modules Z(α, ℓ)∗. The
+modules Z(α, ℓ) and Z(α, ℓ)∗ are called of type Z and they are the unique
+isomorphism classes of uniserial
+�
+sl(2) ⋉ am
+�
+-modules of composition length
+ℓ + 1 for ℓ ≥ 4.
+3.4. Non-faithful
+�
+sl(2) ⋉ hn
+�
+-modules of exceptional type (compo-
+sition lengths 2, 3 and 4). The modules E(a, b) with |a − b| ̸= m are not
+of type Z and we consider them of exceptional type (of composition lengths
+2). For composition lengths 3 and 4 there are very few possible ways to
+“combine” the modules E(a, b) so that we do not fall in type Z.
+For composition length equal to 3, given 0 ≤ c < 2m and c ≡ 2m mod 4,
+let
+E3(c) = V (0) ⊕ V (m) ⊕ V (c)
+as sl(2)-modules with action of r sending
+0
+V (0)
+V (m)
+V (c)
+with the maps V (c) → V (m) and V (m) → V (0) given by (3.2).
+For composition length equal to 4, if m ≡ 0 mod 4, there is a family of
+�
+sl(2)⋉am
+�
+-modules, parameterized by a non-zero scalar t ∈ F, with a fixed
+socle decomposition. This is defined by
+E4(t) = V (0) ⊕ V (m) ⊕ V (m) ⊕ V (0)
+as sl(2)-modules with action of r, sending each irreducible component as
+shown by the arrows
+0
+V (0)
+V (m)
+V (m)
+V (0)
+where the horizontal arrows are given by (3.2) and the bent arrow is t times
+(3.2).
+3.5. Classification of all non-faithful
+�
+sl(2)⋉hn
+�
+-modules. As we said
+at the beginning of the section, a uniserial representation of sl(2) ⋉ hn is
+faithful if and only if z acts non-trivially. Thus, the non-faithful uniserial
+�
+sl(2) ⋉ hn
+�
+-modules are in correspondence, via the projection sl(2) ⋉ hn →
+sl(2) ⋉ am, with the uniserial representations of sl(2) ⋉ am.
+These were
+classified in [8, Theorem 10.1], we summarize that result in the following
+theorem.
+
+10
+LEANDRO CAGLIERO AND IV´AN G´OMEZ RIVERA
+Theorem 3.2. The following list describes all the isomorphism classes of
+non-faithful uniserial representations of sl(2) ⋉ hn.
+Length 1.
+Z(a, 0) = V (a), a ≥ 0 (here r acts trivially).
+Length 2.
+E(a, b), with a + b ≡ m mod 2 and 0 ≤ |a − b| ≤ m ≤ a + b.
+Length 3.
+Z(a, 2), Z(a, 2)∗, a ≥ 0; and
+E3(c) with c ≡ 2m mod 4 and 0 ≤ c < 2m.
+Length 4.
+Z(a, 3), Z(a, 3)∗, a ≥ 0; and
+E4(t), with t ∈ F (this exists only if m ≡ 0 mod 4).
+Length ℓ ≥ 5.
+Z(a, ℓ − 1), Z(a, ℓ − 1)∗, a ≥ 0.
+3.6. Faithful
+�
+sl(2) ⋉ hn
+�
+-modules. The faithful uniserial
+�
+sl(2) ⋉ hn
+�
+-
+modules were classified, up to isomorphism, in [4, Theorems 3.5 and 5.2].
+It turns out that there are no faithful uniserial
+�
+sl(2) ⋉ hn
+�
+-modules of
+composition length different from 3. Moreover, if
+V = V (a0) ⊕ V (a1) ⊕ V (a2)
+is socle decomposition (see Definition 2.1) of a faithful uniserial
+�
+sl(2)⋉hn
+�
+-
+module, then a0 = a2 and an explicit representative of each class can be
+obtained by conveniently combining the modules E(a, b) for some specific
+values of a and b as we explain below.
+Let us start with the sl(2)-module V = V (a0) ⊕ V (a1) ⊕ V (a2) with
+a2 = a0 such that m
+2 , a0
+2 , a1
+2 satisfy the triangle condition. We now indicate
+how to obtain an action of hn = hn(m) ⊕ Fz on V so that V becomes a
+faithful uniserial
+�
+sl(2) ⋉ hn
+�
+-module. Although we know that a2 = a0 we
+keep the notation a2 because we need to indicate that V (a0) is the socle of
+V and V (a2) corresponds to the third socle factor of V .
+First, let hn(m) act on V as follows
+0 ←− V (a0) ←− V (a1) ←− V (a2)
+where the actions V (a1) → V (a0) and V (a2) → V (a1) are given by (3.2)
+(with a = a0, b = a1 and a = a1, b = a2 respectively). This action of hn(m)
+on V can be extended to hn only in the following cases. In all of them,
+a0 = a2 and z acts as an sl(2)-isomorphism V (a2) → V (a0).
+(i) For n = 1 (that is m = 1), (a0, a1, a2) must be
+(a0, a0 + 1, a0),
+a0 ≥ 0;
+(a0, a0 − 1, a0),
+a0 ≥ 1.
+Let us call, respectively, FU +
+a0 and FU −
+a0 the first and second
+�
+sl(2)⋉
+hn
+�
+-modules above.
+(ii) For n = 2 (that is m = 3), (a0, a1, a2) must be
+(0, 3, 0), (1, 4, 1), (1, 2, 1), (4, 3, 4).
+We call these modules FU(0,3,0), FU(1,4,1), FU(1,2,1) and FU(4,3,4)
+respectively.
+
+UNISERIAL REPRESENTATIONS OF THE LIE ALGEBRA sl(2) ⋉ hn
+11
+(iii) If n ≥ 3 (that is m ≥ 5), (a0, a1, a2) must be
+(0, m, 0), (1, m + 1, 1), (1, m − 1, 1).
+We call these modules FU(0,m,0), FU(1,m+1,1) and FU(1,m−1,1) re-
+spectively.
+In these modules, the action of the center Fz is given by
+z va2
+j
+=
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+−2√m + 1
+a + 1
+va0
+j ,
+if V =
+
+
+
+
+
+FU +
+a and m = 1,
+FU(0,3,0), FU(1,4,1) and m = 3,
+FU(0,m,0), FU(1,m+1,1) and m ≥ 5.
+2√m + 1
+a + 1
+va0
+j ,
+if V =
+
+
+
+
+
+FU −
+a and m = 1,
+FU(1,2,1) and m = 3,
+FU(1,m−1,1) and m ≥ 5.
+−4
+5va0
+j ,
+if V = FU(4,3,4) and m = 3.
+(3.5)
+The next theorem summarizes this information and was proved in [4].
+Theorem 3.3. The following list describes all the isomorphism classes of
+faithful uniserial representations of sl(2) ⋉ hn.
+(i) For n = 1: FU +
+a , a ≥ 0, and FU −
+a , a ≥ 1.
+(ii) For n = 2: FU(0,3,0), FU(1,4,1), FU(1,2,1) and FU(4,3,4).
+(iii) For n ≥ 3: FU(0,m,0), FU(1,m+1,1) and FU(1,m−1,1) (m = 2n − 1).
+Each of these modules is isomorphic to its own dual.
+We close this section pointing out that all the faithful uniserial
+�
+sl(2)⋉hn
+�
+-
+modules, except FU(4,3,4), belong, in some sense, to the same kind of modules
+(we will say something more about this after Conjecture 4.2). Therefore, we
+introduce the following definition.
+Definition 3.4. All faithful uniserial
+�
+sl(2) ⋉ hn
+�
+-modules that are not
+isomorphic to FU(4,3,4) will be referred to as standard faithful uniserials.
+4. The tensor product of two uniserial
+�
+sl(2) ⋉ hn
+�
+-modules
+In [7, Theorem 3.5] we obtained the sl(2)-module structure of the socle
+of the tensor product of two uniserial
+�
+sl(2) ⋉ a(m)
+�
+-modules of type Z.
+Therefore (see §3), we already know the sl(2)-module structure of the socle
+of the tensor product V1 ⊗ V2 of two uniserial
+�
+sl(2) ⋉ hn
+�
+-modules in the
+cases where V1 and V2 are non-faithful of type Z. We summarize this in
+Theorem 4.1 below.
+In this section we obtain the sl(2)-module structure of the socle of the
+tensor product V1 ⊗ V2 when both V1 and V2 are standard faithful uniserial
+modules, or when one of them is standard faithful and the other one is
+uniserial of type Z. From this, we derive a complete description of the space
+of intertwining operators Homsl(2)⋉hn(V1, V2) in all these cases.
+
+12
+LEANDRO CAGLIERO AND IV´AN G´OMEZ RIVERA
+4.1. The non-faithful case and the crucial conjecture. Let us recall
+some of the results obtained in [7], mainly Conjecture 3.4 and Theorem 3.5
+that describes the sl(2)-module structure of the socle of the tensor product
+of two uniserial
+�
+sl(2) ⋉ a
+�
+-modules of type Z.
+If U is a
+�
+sl(2)⋉hn
+�
+-module, since hn = [sl(2), hn], it follows from Lemma
+2.2 that
+(4.1)
+soc(U) = U hn.
+Therefore, if V = V (a0) ⊕ . . . ⊕ V (aℓ) and W = V (b0) ⊕ . . . ⊕ V (bℓ′) are the
+socle decomposition of two
+�
+sl(2) ⋉ hn
+�
+-modules, then
+soc(V ⊗ W) =
+ℓ+ℓ′
+�
+t=0
+
+soc(V ⊗ W) ∩
+�
+i+j=t
+V (ai) ⊗ V (bj)
+
+
+=
+ℓ+ℓ′
+�
+t=0
+
+ �
+i+j=t
+V (ai) ⊗ V (bj)
+
+
+hn
+.
+(4.2)
+For t = 0, . . . , ℓ + ℓ′, we define
+St = St(V, W) =
+
+�
+i+j
+V (ai) ⊗ V (bj)
+
+
+hn
+.
+Hence, as sl(2)-modules,
+soc(V ⊗ W) =
+ℓ+ℓ′
+�
+t=0
+St
+and
+S0 = soc(V ) ⊗ soc(W) = V (a0) ⊗ V (b0).
+The following theorem is the same as [7, Theorem 3.5] but stated in terms
+of non-faithful uniserial
+�
+sl(2) ⋉ hn
+�
+-modules.
+Theorem 4.1. Let V1 = V (a0) ⊕ . . . ⊕ V (aℓ) and V2 = V (b0) ⊕ . . . ⊕ V (bℓ′)
+be socle decomposition of two non-faithful uniserial
+�
+sl(2) ⋉ hn
+�
+-modules of
+type Z. Then, St = 0 for all t > min{ℓ, ℓ′},
+S0 ≃
+min{a0,b0}
+�
+k=0
+V (a0 + b0 − 2k),
+and, for t = 1, . . . , min{ℓ, ℓ′}, we have
+(i) If V1 = Z(a0, ℓ) and V2 = Z(b0, ℓ′), then
+St ≃ V (a0 + b0 + tm).
+(ii) If V1 = Z(a0, ℓ) and V2 = Z(bℓ′, ℓ′)∗, then
+St ≃
+�
+0,
+if tm > b0 − a0;
+V (b0 − a0 − tm),
+if tm ≤ b0 − a0.
+(iii) If V1 = Z(aℓ, ℓ)∗ and V2 = Z(bℓ′, ℓ′)∗, then St = 0.
+
+UNISERIAL REPRESENTATIONS OF THE LIE ALGEBRA sl(2) ⋉ hn
+13
+One of the main steps towards proving the above theorem was to prove
+certain instances of the following conjecture (see [7, Conjecture 3.4]).
+Conjecture 4.2. Let V1 = E(a, b) and V2 = E(c, d) (two uniserial sl(2) ⋉
+a(m)-modules of length 2) and assume that a < c, or a = c and b ≤ d. Then
+S2 = 0 in all cases and S1 = 0 except in the following cases.
+• Case 1: [a, b] = [0, m]. Here S1 ≃ V (d).
+• Cases 2: Here a > 0.
+– Case 2.1: a+b = c+d = m with d−a = b−c ≥ 0. Here S1 ≃ V (d−a).
+– Case 2.2: b − a = d − c = m. Here S1 ≃ V (d + a).
+– Case 2.3: b−a = c−d = m with d−a = c−b ≥ 0. Here S1 ≃ V (d−a).
+• Case 3: [c, d] = [b, a]. Here S1 ≃ V (0).
+In order to prove Theorem 4.1, we proved in [7, Theorem 3.3] the cases
+2.2 and 2.3 (and certain converse statement). Now, in this paper, we need
+to prove part of case 1 and case 2.1 of the conjecture (together with certain
+converse statement) in order to prove Theorems 4.6 and 4.7. This is estab-
+lished in Theorem 4.3 below. We point out that this theorem leaves out
+the uniserial
+�
+sl(2) ⋉ a(3)
+�
+-modules E(3, 4) and E(4, 3), and a consequence
+of this is that Theorems 4.6 and 4.7 are restricted to the standard faith-
+ful modules, leaving the exceptional faithful uniserial
+�
+sl(2) ⋉ h2
+�
+-module
+FU(4,3,4) out of our results.
+Theorem 4.3. Let V1 = E(a, b) with a + b = m and a, b ̸= 0. Let V2 =
+V (c)⊕V (d) be the socle decomposition of a uniserial
+�
+sl(2)⋉V (m)
+�
+-module.
+Then
+(i) If V2 ≃ E(c, d) with c + d = m and 0 < a ≤ c < m, then, as sl(2)-
+modules,
+S1(V1, V2) ≃
+�
+V (d − a),
+if d − a = b − c ≥ 0
+0,
+otherwise.
+(ii) If V2 ≃ Z(c, 1) ≃ E(c, c + m), then, as sl(2)-modules,
+S1(V1, V2) ≃
+�
+V (b),
+if c = 0
+0,
+if c ̸= 0 .
+(iii) If V2 ≃ Z(d, 1)∗ ≃ E(d + m, d), then S1(V1, V2) = 0.
+The proof of this result is very technical and it will be given in §6.
+4.2. The faithful case. We will now focus on the tensor product of two
+uniserial
+�
+sl(2)⋉hn
+�
+-modules where one of the factors is a standard faithful
+uniserial.
+Let V = V (a0) ⊕ V (a1) ⊕ V (a2) be the socle decomposition of a faithful
+uniserial
+�
+sl(2)⋉hn
+�
+-module and set W = V (b0)⊕. . .⊕V (bℓ), with ℓ ≥ 1, be
+the socle decomposition of a (not necessarily faithful) uniserial
+�
+sl(2) ⋉ hn
+�
+-
+module. By Theorems 3.2 and 3.3 we have that
+hn(m) · V (ai) ⊂ V (ai−1) and hn(m) · V (bj) ⊂ V (bj−1) ⊕ V (bj−2)
+z · V (ai) ⊂ V (ai−2) and z · V (bj) ⊂ V (bj−2)
+
+14
+LEANDRO CAGLIERO AND IV´AN G´OMEZ RIVERA
+for all i = 0, 1, 2 and 0 ≤ j ≤ ℓ (for convenience we assume V (ai) = V (bj) =
+0 if i, j < 0).
+Given v ∈ V (ai) ⊗ V (bj), let
+(4.3)
+esv = (esv)1 + (esv)2 + (esv)3 and zv = (zv)1 + (zv)2
+where
+(esv)1 ∈ V (ai−1) ⊗ V (bj),
+(zv)1 ∈ V (ai−2) ⊗ V (bj),
+(esv)2 ∈ V (ai) ⊗ V (bj−1),
+(zv)2 ∈ V (ai) ⊗ V (bj−2),
+(esv)3 ∈ V (ai) ⊗ V (bj−2).
+Note that (esv)3 = 0 if W is not isomorphic to E4 and that (zv)2 = 0 if W
+is not a faithful uniserial module.
+Lemma 4.4. Let V1 = V (a0)⊕V (a1)⊕V (a2) and V2 = V (b0)⊕. . . ⊕V (bℓ),
+with ℓ ≥ 1, be the socle decomposition of two uniserial
+�
+sl(2) ⋉ hn
+�
+-modules,
+where V1 is faithful (not necessarily standard). If v0 ∈ V (ai0) ⊗ V (bj0) is a
+highest weight vector then:
+(i) (esv0)1 = 0 for all s = 0, . . . , m if and only if i0 = 0.
+(ii) (esv0)2 = 0 for all s = 0, . . . , m if and only if j0 = 0.
+(iii) (zv0)1 = 0 if and only if i0 ̸= 2.
+(iv) If V2 is faithful, then (zv0)2 = 0 if and only if j0 ̸= 2.
+Proof. Since the action of hn(m) on any uniserial
+�
+sl(2) ⋉ hn
+�
+is the same
+as the action of am in the corresponding
+�
+sl(2) ⋉ am
+�
+-module, cases (i) and
+(ii) are immediate consequences of [7, Lemma 3.1].
+By symmetry, it sufficient to prove (iii) to obtain (iv), so let us prove (iii).
+If c is the weight of v0, we can assume that v0 = v
+ai0,bj0,c
+0
+. It follows from
+(2.4) and (4.3) that
+(zv0)1 = (zv
+ai0,bj0,c
+0
+)1
+=
+�
+i+j= ai0+bj0−c
+2
+CG(ai0
+2 , ai0
+2 − i; bj0
+2 , bj0
+2 − j | c
+2, c
+2) zv
+ai0
+i
+⊗ v
+bj0
+j
+.
+(4.4)
+From the definition of the modules FU ±
+a
+for n = 1; FU(0,3,0),FU(1,4,1),
+FU(1,2,1) and FU(4,3,4) for n = 2; and the modules FU(0,m,0), FU(1,m+1,1)
+and FU1,m−1,1 for n ≥ 3 (here m = 2n − 1), we know that zv
+ai0
+i
+= 0 if
+i0 ̸= 2. Therefore, if i0 ̸= 2 then
+(zv0)1 = 0.
+On the other hand, if i0 = 2, we know from (3.5) that zva2
+i
+= λva0
+i , where λ
+is a non-zero scalar independent of i, 0 ≤ i ≤ a2. Thus, the equation (4.4)
+becomes
+(zv0)1 = λ
+�
+i+j= a2+bj0−c
+2
+CG(a2
+2 , a2
+2 − i;
+bj0
+2 ,
+bj0
+2 − j | c
+2, c
+2) va0
+i
+⊗ v
+bj0
+j
+.
+
+UNISERIAL REPRESENTATIONS OF THE LIE ALGEBRA sl(2) ⋉ hn
+15
+In this sum, the term corresponding to i = 0, has a non-zero Clebsch-Gordan
+coefficient, indeed
+CG(a2
+2 , a2
+2 ; bj0
+2 , c−a2
+2
+| c
+2, c
+2) =
+�
+(c+1)! a2!
+�a2+bj0+c
+2
++1
+�
+!
+� a2+c−bj0
+2
+�
+!
+̸= 0.
+Since all terms are linearly independent, we obtain (zv0)1 ̸= 0.
+□
+Proposition 4.5. Let V1 = V (a0) ⊕ V (a1) ⊕ V (a2) and V2 = V (b0) ⊕ . . . ⊕
+V (bℓ), with ℓ ≥ 1, be the socle decomposition of two uniserial
+�
+sl(2) ⋉ hn
+�
+-
+modules, where V1 is standard faithful. Then, S0 = V (a0) ⊗ V (b0) and
+(i) St = 0 for all t > min{2, ℓ}. If St ̸= 0, t = 1, 2, then it is irreducible
+as sl(2)-module and if v is a non-zero highest weight vector in St of
+weight µ, then v = �t
+i=0 vi with vi a non-zero highest weight vector
+in V (ai) ⊗ V (bt−i), of weight µ, for all i = 0, . . . , t.
+(ii) S2 = 0 if V2 is non-faithful.
+(iii) S1(V1, V2) ≃ S1
+�
+E(a0, a1), E(b0, b1)
+�
+.
+(iv) If V2 is also standard faithful, then S2 ̸= 0 if and only if V1 ≃ V2
+(that is ai = bi, i = 0, 1, 2) and in this case S2 ≃ V (0).
+Proof. The proof of this proposition is very similar to that of Proposition
+3.2 in [7]. We fix t > 0 and we assume that there is a non-zero highest
+weight vector u of weight µ,
+u =
+�
+i+j=t
+ui,j ∈
+� �
+i+j=t
+V (ai) ⊗ V (bj)
+�hn ̸= 0,
+ui,j ∈ V (ai) ⊗ V (bj).
+Since V (ai) ⊗ V (bj) is an sl(2)-submodule, it follows that ui,j is either zero
+or a highest weight vector of weight µ. Let
+Iµ
+t = {(i, j) : 0 ≤ i ≤ 2, 0 ≤ j ≤ ℓ, i + j = t and ui,j ̸= 0}.
+Since u ̸= 0, it follows that Iµ
+t ̸= ∅ and
+u =
+�
+(i,j)∈Iµ
+t
+qi,j vai,bj,µ
+0
+for certain non-zero scalars 0 ̸= qi,j ∈ F. Now, it follows from items (i) and
+(ii) in Lemma 4.4 (see the details in [7][Proposition 3.2]) that
+(4.5)
+Iµ
+t = {(0, t), (1, t − 1), . . . , (t, 0)}.
+Now, again, items (i) and (ii) in Lemma 4.4 imply that such a non-zero u
+cannot exist if t > min{2, ℓ} and thus St = 0. This proves (i). Furthermore,
+(ii) follows similarly by applying item (iii) in Lemma 4.4.
+(iii) is clear from the definition of S1.
+Let us prove (iv). Assume that V2 = V (b0) ⊕ V (b1) ⊕ V (b2) is standard
+faithful, and suppose that
+(4.6)
+u = q0,2 va0,b2,µ
+0
++ q1,1 va1,b1,µ
+0
++ q2,0 va2,b0,µ
+0
+̸= 0
+is a highest weight vector of weight µ in S2. We already know that q0,2, q1,1, q2,0 ̸=
+0 and, moreover, we must have
+q0,2 va0,b2,µ
+0
++ q1,1 va1,b1,µ
+0
+∈ S1
+�
+E(a0, a1), E(b1, b2)) ̸= 0
+
+16
+LEANDRO CAGLIERO AND IV´AN G´OMEZ RIVERA
+and
+q1,1 va1,b1,µ
+0
++ q2,0 va2,b0,µ
+0
+∈ S1
+�
+E(a1, a2), E(b0, b1)) ̸= 0.
+Theorems 4.1 and 4.3 imply that it is impossible to have
+S1
+�
+E(a0, a1), E(b1, b2)
+�
+̸= 0
+and
+S1
+�
+E(a1, a2), E(b0, b1)
+�
+̸= 0
+unless (a0, a1, a2) = (b0, b1, b2).
+This follows by considering all the cases
+with (a0, a1, a2) and (b0, b1, b2) running over (see §3.6)
+(i) if n = 1 (that is m = 1)
+(k0, k0 + 1, k0),
+k0 ≥ 0;
+(k0, k0 − 1, k0),
+k0 ≥ 1;
+(ii) if n ≥ 2 (that is m ≥ 3),
+(0, m, 0), (1, m + 1, 1), (1, m − 1, 1);
+(it saves time noticing that E(a0, a1)∗ ≃ E(a1, a2) and E(b0, b1)∗ ≃ E(b1, b2)).
+Finally, let (a0, a1, a2) = (b0, b1, b2). We know, from Theorems 4.1 and
+4.3, that
+S1
+�
+E(a0, a1), E(b1, b2)
+�
+≃ S1
+�
+E(a1, a2), E(b0, b1)
+�
+≃ V (0).
+This implies that there is, up to a scalar, a unique element u as in (4.6)
+such that hn(m)u = 0. This implies that zu = 0 and hence u ∈ S2. This
+completes the proof.
+□
+We are now in a position to prove the main theorems of the paper.
+Theorem 4.6. Let V1 = V (a0)⊕V (a1)⊕V (a2) and V2 = V (b0)⊕. . .⊕V (bℓ)
+be the socle decomposition of two uniserial
+�
+sl(2) ⋉ hn
+�
+-modules, where V1
+is standard faithful and V2 is of type Z. Then
+soc(V1 ⊗ V2) = S0 ⊕ S1
+where, as sl(2)-modules,
+S0 = soc(V1) ⊗ soc(V2) ≃
+min{a0,b0}
+�
+k=0
+V (a0 + b0 − 2k)
+and the following tables describe S1 as sl(2)-modules (recall that m = 2n−1):
+Case n = 1 (m = 1).
+V1\V2
+Z(b0, ℓ)
+Z(bℓ, ℓ)∗
+FU +
+a0
+V (a0 + b0 + m).
+V (b1 − a0),
+if b1 ≥ a0, ℓ ≥ 1;
+0,
+otherwise.
+FU −
+a0
+a0 ≥ 1
+V (a0 − b1),
+if a0 ≥ b1, ℓ ≥ 1;
+0,
+otherwise.
+0.
+Case n > 1 (m ≥ 3).
+
+UNISERIAL REPRESENTATIONS OF THE LIE ALGEBRA sl(2) ⋉ hn
+17
+V1\V2
+Z(b0, ℓ)
+Z(bℓ, ℓ)∗
+FU(a0,a0+m,a0)
+a0 = 0, 1
+V (a0 + b0 + m).
+V (b1 − a0),
+if a0 ≤ b1;
+0,
+if a0 > b1.
+FU(1,m−1,1)
+V (m − 1),
+if b0 = 0;
+0,
+otherwise.
+0.
+Proof. We know from Proposition 4.5 that
+soc(V1 ⊗ V2) ≃ soc(V1) ⊗ soc(V2) ⊕ S1
+(in particular we know that St(V1, V2) = 0 for all t ≥ 2). The decomposition
+of soc(V1) ⊗ soc(V2) follows from the Clebsch-Gordan formula.
+We now
+describe S1 in each case.
+Let us consider the submodules
+U1 = V (a0) ⊕ V (a1) ⊂ V1, and
+U2 = V (b0) ⊕ V (b1) ⊂ V2.
+We know that U1 and U2 are non-faithful uniserial
+�
+sl(2) ⋉ hn
+�
+-modules.
+From Proposition 4.5 item (iii) we know that
+S1(V1, V2) ≃ S1(U1, U2),
+If V1 is FU +
+a0 or FU(0,m,0) or FU(1,1+m,1) then U1 ≃ Z(a0, 1) and S1 is
+obtained from Theorem 4.1. If V1 = FU −
+a0, then U1 = Z(a0 − 1, 1)∗ and S1
+is also obtained from Theorem 4.1.
+Finally, if V1 = FU(1,m−1,1), then U1 = E(1, m − 1) and Theorem 4.3
+implies the remaining cases.
+□
+Theorem 4.7. Let V1 = V (a0) ⊕ V (a1) ⊕ V (a2) and V2 = V (b0) ⊕ V (b1) ⊕
+V (b2) be the socle decomposition of two standard faithful uniserial
+�
+sl(2) ⋉
+hn
+�
+-modules. Then
+soc(V1 ⊗ V2) = S0 ⊕ S1 ⊕ S2
+where
+S0 = soc(V1) ⊗ soc(V2) ≃
+min{a0,b0}
+�
+k=0
+V (a0 + b0 − 2k)
+and the following tables describe S1 and S2 as sl(2)-modules (m = 2n − 1):
+Case n = 1 (m = 1), structure of S1.
+V1\V2
+FU(b0,b0+1,b0)
+FU(b0,b0−1,b0)
+b0 ≥ 1
+FU(a0,a0+1,a0)
+V (a0 + b0 + 1).
+V (b0 − a1),
+if a1 ≤ b0;
+0,
+otherwise.
+FU(a0,a0−1,a0)
+a0 ≥ 1
+V (a0 − b1),
+if b1 ≤ a0;
+0,
+otherwise.
+0.
+
+18
+LEANDRO CAGLIERO AND IV´AN G´OMEZ RIVERA
+Case n = 1 (m = 1), structure of S2.
+V1\V2
+FU(b0,b0+1,b0)
+FU(b0,b0−1,b0)
+b0 ≥ 1
+FU(a0,a0+1,a0)
+V (0),
+if a0 = b0;
+0,
+if a0 ̸= b0.
+0.
+FU(a0,a0−1,a0)
+a0 ≥ 1
+0.
+V (0),
+if a0 = b0;
+0,
+if a0 ̸= b0.
+Case n > 1 (m ≥ 3), structure of S1.
+V1\V2
+FU(b0,b0+m,b0)
+b0 = 0, 1
+FU(1,m−1,1)
+FU(a0,a0+m,a0)
+a0 = 0, 1
+V (a0 + b0 + m).
+V (m − 1),
+if a0 = 0;
+0,
+otherwise.
+FU(1,m−1,1)
+V (m − 1),
+if b0 = 0;
+0,
+otherwise.
+V (m − 2).
+Case n > 1 (m ≥ 3), structure of S2.
+V1\V2
+FU(b0,b0+m,b0)
+b0 = 0, 1
+FU(1,m−1,1)
+FU(a0,a0+m,a0)
+a0 = 0, 1
+V (0),
+if a0 = b0;
+0,
+if a0 ̸= b0.
+0.
+FU(1,m−1,1)
+0.
+V (0).
+Proof. As in the previous theorem, we know from Proposition 4.5 that
+soc(V1 ⊗ V2) ≃ soc(V1) ⊗ soc(V2) ⊕ S1 ⊕ S2
+and the decomposition of soc(V1)⊗soc(V2) follows from the Clebsch-Gordan
+formula. We now describe S1 and S2 in each case.
+First, we consider S1. We know from Proposition 4.5 that
+S1(V1, V2) ≃ S1
+�
+E(a0, a1), E(b0, b1)
+�
+.
+If V1 = FU +
+a0 and V2 = FU +
+b0, we know from Theorem 4.1 that
+S1(V1, V2) = S1(Z(a0, 1), Z(b0, 1)) = V (a0 + b0 + m).
+If V1 = FU +
+a0 and V2 = FU −
+b0, we have
+S1(V1, V2) =
+�
+S1(Z(a0, 1), Z(b1, 1)∗),
+if m = 1;
+S1(Z(a0, 1), E(1, 1 − m)),
+if m > 1 and b0 = 1;
+
+UNISERIAL REPRESENTATIONS OF THE LIE ALGEBRA sl(2) ⋉ hn
+19
+and it follows from Theorems 4.1 and 4.3 that
+S1(V1, V2) =
+
+
+
+
+
+V (b1 − a0)
+if m = 1;
+V (m − 1),
+if m > 1, a0 = 0 and b0 = 1;
+0,
+otherwise.
+This completes the description of S1.
+The result for S2 follows from item (iv) in Proposition 4.5.
+□
+5. Intertwining operators
+In this section we obtain, from Theorems 4.6 and 4.7, the space of in-
+tertwining operators between the uniserial representations of g = sl(2) ⋉ hn
+considered in the previous section.
+Recall that, for any pairs of g-modules U1 and U2, we know that
+Homg(U1, U2) ≃ (U ∗
+1 ⊗ U2)g =
+�
+(U ∗
+1 ⊗ U2)hn�sl(2).
+It follows that Homg(U1, U2) is isomorphic to soc(U ∗
+1 ⊗ U2)sl(2) (see (4.1)).
+Thus, we must identify the cases in which St(U ∗
+1 , U2)sl(2) ̸= 0 for t = 0, 1, 2.
+So let V = V (a0) ⊕ V (a1) ⊕ V (a2) (a0 = a2) and W = V (b0) ⊕ . . . ⊕ V (bℓ)
+be the socle decomposition of two uniserial
+�
+sl(2) ⋉ hn
+�
+-modules, where V
+is standard faithful and W is either of type Z or standard faithful. Recall
+that V ∗ ≃ V and that the socle decomposition of W ∗ is V (bℓ) ⊕ . . . ⊕ V (b0).
+• Case Ssl(2)
+0
+̸= 0, W of type Z.
+It follows from Theorem 4.6 that S0(V ∗, W)sl(2) ̸= 0 if and only if
+a0 = b0 (and equal to a2). Also, Theorem 4.6 implies that, in these cases,
+we have dim S0(V ∗, W)sl(2) = 1 and St(V ∗, W)sl(2) = 0, t = 1, 2. Hence
+Homg(V, W) is 1-dimensional and it is described by the following arrow
+V = V (a0) ⊕ V (a1) ⊕ V (a0)
+↓
+W = V (b0) ⊕ · · · ⊕ V (bℓ).
+Similarly, from Theorem 4.6 we know that S0(W ∗, V )sl(2) ̸= 0 if and
+only if a0 = bℓ and in these cases Homg(W, V ) is 1-dimensional and it is
+described by the following arrow
+W = V (b0) ⊕ · · · ⊕V (bℓ)
+↓
+V = V (a0) ⊕ V (a1) ⊕ V (a0).
+• Case Ssl(2)
+0
+̸= 0 or Ssl(2)
+2
+̸= 0, W standard faithful.
+It follows from Theorem 4.7 that S0(V, W)sl(2) ̸= 0 if and only if a0 = b0.
+Hence in what follows, a0 = a2 = b0 = b2. In this case, S1(V, W)sl(2) = 0
+and dim S0(V, W)sl(2) = 1 (note that if V ≃ W, that is a1 = b1, then
+Theorem 4.7 says that dim S2(V, W)sl(2) = 1, see below). The non-zero
+
+20
+LEANDRO CAGLIERO AND IV´AN G´OMEZ RIVERA
+g-morphism corresponding to the 1-dimensional space S0(V, W)sl(2) is de-
+scribed by the following arrow
+V = V (a0) ⊕ V (a1) ⊕ V (a0)
+↓
+W = V (b0) ⊕ V (b1) ⊕ V (b0).
+This is clearly not an isomorphism. In the particular case when V ≃ W,
+the isomorphism is the g-morphism corresponding to the 1-dimensional
+space S2(V, W)sl(2) (see also Proposition 4.5). Thus
+dim Homg(V, W) =
+�
+1,
+if V ̸≃ W (that is a1 ̸= b1);
+2,
+if V ≃ W (that is a1 = b1).
+(Since here V and W are of the same type, we do not need to consider
+Homg(W, V ).)
+• Case Ssl(2)
+1
+̸= 0.
+It follows from Theorems 4.6 and 4.7 that dim Ssl(2)
+1
+= 1 and in all these
+cases, St(V ∗, W)sl(2) = 0, t = 0, 2. Hence Homg(V, W) (or Homg(W, V ))
+is 1-dimensional. We describe all these cases below:
+◦ Cases with W of type Z and Homg(V, W) ̸= 0.
+(i) n = 1, a0 = b1, V = FU −
+a0, W = Z(b0, ℓ), ℓ ≥ 1.
+V = V (a0) ⊕ V (a0 − 1) ⊕ V (a0)
+↓
+↓
+W = V (b0) ⊕ V (b1) ⊕ · · · ⊕ V (bℓ).
+(ii) n = 1, a0 = b1, V = FU +
+a0, W = Z(bℓ, ℓ)∗, ℓ ≥ 1.
+V = V (a0) ⊕ V (a0 + 1) ⊕ V (a0)
+↓
+↓
+W = V (b0) ⊕ V (b1) ⊕ · · · ⊕ V (bℓ).
+(iii) n > 1, a0 = 0, 1, V = FU +
+(a0,a0+m,a0), W = Z(b1, 1)∗.
+V = V (a0) ⊕ V (a0 + m) ⊕ V (a0)
+↓
+↓
+W = V (b0) ⊕ V (b1).
+◦ Cases with W of type Z and Homg(W, V ) ̸= 0.
+(i) n = 1, a0 = bℓ−1, V = FU +
+a0, W = Z(b0, ℓ), ℓ ≥ 1.
+W = V (b0) ⊕ · · · ⊕ V (bℓ−1) ⊕ V (bℓ)
+↓
+↓
+V =V (a0) ⊕ V (a0 + 1) ⊕ V (a0).
+(ii) n = 1, a0 = bℓ−1, V = FU −
+a0, W = Z(bℓ, ℓ)∗, ℓ ≥ 1.
+W = V (b0) ⊕ · · · ⊕ V (bℓ−1) ⊕ V (bℓ)
+↓
+↓
+V =V (a0) ⊕ V (a0 − 1) ⊕ V (a0).
+
+UNISERIAL REPRESENTATIONS OF THE LIE ALGEBRA sl(2) ⋉ hn
+21
+(iii) n > 1, V = FU(a0,a0+m,a0), a0 = 0, 1, W = Z(a0, 1).
+W = V (a0) ⊕ V (a0 + m)
+↓
+↓
+V =V (a0) ⊕ V (a0 + m) ⊕ V (a0).
+◦ Cases with W standard faithful and Homg(V, W) ̸= 0.
+(i) n = 1, a0 = b1, V = FU −
+a0, W = FU +
+b0.
+V = V (a0) ⊕ V (a0 − 1) ⊕ V (a0)
+↓
+↓
+W = V (b0) ⊕ V (b0 + 1) ⊕ V (b0).
+(ii) n = 1, a0 = b1, V = FU +
+a0, W = FU −
+b0.
+V = V (a0) ⊕ V (a0 + 1) ⊕ V (a0)
+↓
+↓
+W = V (b0) ⊕ V (b0 − 1) ⊕ V (b0).
+6. Proof of Theorem 4.3
+Let µ be a possible highest weight in S1. We start with some general
+considerations and next we will work out each case.
+We know from Proposition 4.5 that µ must be highest weight in both
+V (a) ⊗ V (d) and V (b) ⊗ V (c), that is
+(6.1)
+|a − d|, |b − c| ≤ µ ≤ a + d, b + c
+and µ ≡ a+d ≡ b+c mod 2. We also know that µ is indeed highest weight
+in S1 if and only if there is a linear combination
+u0 = q1va,d,µ
+0
++ q2vb,c,µ
+0
+,
+with q1, q2 ̸= 0 (see item (i) in Proposition 4.5), that is annihilated by es
+for all s = 0, . . . , m. Indeed this implies that u0 is also annihilated by z and
+thus in S1 (see (4.1)).
+We now describe esva,d,µ
+0
+and esvb,c,µ
+0
+.
+On the one hand we have (see (2.4))
+(6.2)
+va,d,µ
+0
+=
+�
+i,j
+CG(a
+2, a
+2 − i; d
+2, d
+2 − j | µ
+2, µ
+2 ) va
+i ⊗ vd
+j
+
+22
+LEANDRO CAGLIERO AND IV´AN G´OMEZ RIVERA
+and thus (see (3.2))
+esva,d,µ
+0
+=
+�
+i,j
+CG(a
+2, a
+2 − i; d
+2, d
+2 − j | µ
+2 , µ
+2) va
+i ⊗ esvd
+j
+=
+�
+i,j,k
+(−1)jCG(a
+2, a
+2 − i; d
+2, d
+2 − j | µ
+2, µ
+2 )
+× CG( c
+2, c
+2 − k; d
+2, − d
+2 + j | m
+2 , m
+2 − s) va
+i ⊗ vc
+k
+=
+�
+i,j,k
+(−1)kCG(a
+2, a
+2 − i; d
+2, d
+2 − k | µ
+2 , µ
+2)
+× CG( c
+2, c
+2 − j; d
+2, − d
+2 + k | m
+2 , m
+2 − s) va
+i ⊗ vc
+j.
+(6.3)
+In this sum, if the coefficient of va
+i ⊗ vc
+j is not zero then we must have
+a
+2 − i + d
+2 − k = µ
+2 ,
+c
+2 − j − d
+2 + k = m
+2 − s.
+(6.4)
+On the other hand, we have (see (2.4))
+(6.5)
+vb,c,µ
+0
+=
+�
+i,j
+CG( b
+2, b
+2 − i; c
+2, c
+2 − j | µ
+2 , µ
+2 ) vb
+i ⊗ vc
+j
+and thus (see (3.2))
+esvb,c,µ
+0
+=
+�
+i,j
+CG( b
+2, b
+2 − i; c
+2, c
+2 − j | µ
+2, µ
+2) esvb
+i ⊗ vc
+j.
+=
+�
+i,j,k
+(−1)iCG( b
+2, b
+2 − i; c
+2, c
+2 − j | µ
+2, µ
+2 )
+× CG(a
+2, a
+2 − k; b
+2, − b
+2 + i | m
+2 , m
+2 − s) va
+k ⊗ vc
+j
+=
+�
+i,j,k
+(−1)kCG( b
+2, b
+2 − k; c
+2, c
+2 − j | µ
+2, µ
+2)
+× CG(a
+2, a
+2 − i; b
+2, − b
+2 + k | m
+2 , m
+2 − s) va
+i ⊗ vc
+j.
+(6.6)
+In this sum, if the coefficient of va
+i ⊗ vc
+j is not zero then we must have
+b
+2 − k + c
+2 − j = µ
+2 ,
+a
+2 − i − b
+2 + k = m
+2 − s.
+(6.7)
+Either (6.4) or (6.7) imply
+(6.8)
+i + j = a + c − m − µ
+2
++ s,
+(recall that 0 ≤ i ≤ a and 0 ≤ j ≤ c). Now we consider all the cases.
+(i) The case V2 = E(c, d) with c + d = m and 0 < a ≤ c: Here
+µ = d + a − 2p, 0 ≤ p ≤ min{a, d}
+
+UNISERIAL REPRESENTATIONS OF THE LIE ALGEBRA sl(2) ⋉ hn
+23
+and it follows from (6.8) that
+(6.9)
+0 ≤ i + j = p − d + s.
+The sum (6.3) is
+esva,d,µ
+0
+=
+�
+i,j,k
+(−1)kCG(a
+2, a
+2 − i; d
+2, d
+2 − k | a+d
+2
+− p, a+d
+2
+− p)
+× CG(m−d
+2 , m−d
+2
+− j; d
+2, − d
+2 + k | m
+2 , m
+2 − s) va
+i ⊗ vc
+j.
+In this sum, it follows from (6.4) that
+k = p − i
+j = p − d + s − i.
+The conditions k ≥ 0 and 0 ≤ j ≤ m − d imply p − m + s ≤ i ≤ min{p, p −
+d + s}. Therefore, we obtain (see (2.9) and (2.12))
+esva,d,µ
+0
+=
+min{p,p−d+s}
+�
+i=max{0,p−m+s}
+(−1)p−iCG(a
+2, a
+2 − i; d
+2, d
+2 − p + i | a+d
+2
+− p, a+d
+2
+− p)
+× CG(m−d
+2 , m−d
+2
+− p + d − s + i; d
+2, − d
+2 + p − i | m
+2 , m
+2 − s) va
+i ⊗ vc
+p−d+s−i
+=
+min{p,p−d+s}
+�
+i=max{0,p−m+s}
+(−1)p
+�
+(a + d − 2p + 1)! p! (a − i)! (d − p + i)!
+(a − p)! (d − p)! (a + d − p + 1)! i! (p − i)!
+×
+�
+(m − s)! s! (m − d)! d!
+m! (m − p − s + i)! (p − i)! (d − p + i)! (p − d + s − i)! va
+i ⊗ vc
+p−d+s−i.
+Thus,
+esva,d,µ
+0
+= (−1)p
+�
+(m − d)! (a + d − 2p + 1)! p! d! (m − s)! s!
+(a − p)! (d − p)! (a + d − p + 1)! m!
+wa,d,µ
+s
+with
+wa,d,µ
+s
+=
+min{p,p−d+s}
+�
+i=max{0,p−m+s}
+�
+(a − i)!
+i! (p − i)!2 (m − p − s + i)! (p − d + s − i)! va
+i ⊗vc
+p−d+s−i.
+On the other hand, the sum (6.6) is
+esvb,c,µ
+0
+=
+�
+i,j,k
+(−1)kCG(m−a
+2 , m−a
+2
+− k; m−d
+2 , m−d
+2
+− j | a+d
+2
+− p, a+d
+2
+− p)
+× CG(a
+2, a
+2 − i; m−a
+2 , − m−a
+2
++ k | m
+2 , m
+2 − s) va
+i ⊗ vc
+j.
+In this sum, it follows from (6.7) that
+j = p − d + s − i
+k = m − a − s + i
+
+24
+LEANDRO CAGLIERO AND IV´AN G´OMEZ RIVERA
+and the condition k ≥ 0 implies i ≥ a − m + s, and condition j ≥ 0 implies
+p − d + s ≥ i. Therefore, we obtain (see (2.9) and (2.12))
+esvb,c,µ
+0
+=
+min{a,p−d+s}
+�
+i=max{0,a−m+s}
+(−1)m−a−s+i
+× CG(m−a
+2 , − m−a
+2
++ s − i; m−d
+2 , m−d
+2
+− p + d − s + i | a+d
+2
+− p, a+d
+2
+− p)
+× CG(a
+2, a
+2 − i; m−a
+2 , m−a
+2
+− s + i | m
+2 , m
+2 − s) va
+i ⊗ vc
+p−d+s−i
+=
+min{a,p−d+s}
+�
+i=max{0,a−m+s}
+�
+(a + d − 2p + 1)! (p + m − a − d)! (s − i)! (m − p − s + i)!
+(d − p)! (a − p)! (m − a − s + i)! (p − d + s − i)! (m − p + 1)!
+×
+�
+s! (m − s)! a! (m − a)!
+m! (a − i)! (m − a − s + i)! i! (s − i)! va
+i ⊗ vc
+a−p+s−i.
+Thus
+esvb,c,µ
+0
+=
+�
+(m − s)! s! (a + d − 2p + 1)! (p + m − a − d)! a! (m − a)!
+(d − p)! (a − p)! (m − p + 1)! m!
+wb,c,µ
+s
+where
+wb,c,µ
+s
+=
+min{a,p−d+s}
+�
+i=max{0,a−m+s}
+�
+(m − p − s + i)!
+i! (m − a − s + i)!2 (p − d + s − i)! (a − i)!
+va
+i ⊗vc
+p−d+s−i.
+Now, if a ≤ d and p = a, we have, for all 0 ≤ s ≤ m,
+wa,d,µ
+s
+=
+min{a,a−d+s}
+�
+i=max{0,a−m+s}
+�
+1
+i! (a − i)! (m − a − s + i)! (a − d + s − i)! va
+i ⊗ vc
+p−d+s−i
+= wb,c,µ
+s
+.
+This shows that
+u0 = (−1)a √
+d + 1 va,d,µ
+0
+−
+√
+b + 1 vb,c,µ
+0
+is, indeed, a highest weight vector, of weight µ = d − a = b − c, in S1.
+If p < a then, for s = d, the sum defining wa,d,µ
+d
+has the index i running
+up to i = a while the sum defining wb,c,µ
+d
+has the index i running only up to
+i = p. In both cases, all the coefficients are non-zero, and thus {wa,d,µ
+1
+, wb,c,µ
+1
+}
+is linearly independent. This shows that there is no possible µ in S1 and
+thus S1 = 0. This completes the proof in this case.
+(ii) The case V2 = E(c, d) with d = c+m: Since a < m ≤ d and by equation
+(6.1) we have
+µ = b + c − 2p, 0 ≤ p ≤ min{c, b},
+µ = a + d − 2p′, 0 ≤ p′ ≤ a.
+This implies p′ − p = a and hence p′ = a and p = 0. This yields
+µ = b + c = d − a.
+
+UNISERIAL REPRESENTATIONS OF THE LIE ALGEBRA sl(2) ⋉ hn
+25
+First we prove that, if c = 0, then S1(V1, V2) ≃ V (b). In this case, (6.3)
+becomes
+esva,d,µ
+0
+=
+�
+i,k
+(−1)kCG(m−b
+2 , m−b
+2
+− i; m
+2 , m
+2 − k | b
+2, b
+2)
+× CG(0, 0; m
+2 , − m
+2 + k | m
+2 , m
+2 − s) va
+i ⊗ v0
+0
+(the index j is 0). It follows from (6.4) that
+k = m − s
+i = −b + s.
+Therefore, esva,d,µ
+0
+= 0 if s < b and, for s ≥ b we have (see (2.9) and (2.11))
+esva,d,µ
+0
+= (−1)m−sCG(m−b
+2 , − m−b
+2
++ m − s; m
+2 , − m
+2 + s | b
+2, b
+2)
+× CG(0, 0; m
+2 , m
+2 − s | m
+2 , m
+2 − s) va
+s−b ⊗ v0
+0
+=
+�
+(b + 1) (m − b)! s!
+(m + 1)! (s − b)!
+va
+s−b ⊗ v0
+0.
+On the other hand, since c = 0 and µ = b, (6.6) becomes (see also (3.2)
+or (3.3))
+esvb,c,µ
+0
+= esvb
+0 ⊗ v0
+0
+=
+
+
+
+CG(a
+2, a
+2 − (s − b); b
+2, − b
+2 | m
+2 , m
+2 − s) va
+s−b ⊗ v0
+0,
+if s ≥ b;
+0,
+if s < b.
+=
+
+
+
+�
+a! s!
+(s−b)! m! va
+s−b ⊗ v0
+0,
+if s ≥ b;
+0,
+if s < b.
+This shows that
+u0 =
+√
+b + 1 va,d,µ
+0
+−
+√
+m + 1 vc,d,µ
+0
+is, indeed, a highest weight vector of weight µ = b, in S1.
+Now, suppose that c ̸= 0 and set s = m. Recall that µ = b + c = d − a.
+When we consider the sum (6.3), it follows from (6.4) that
+j = k = a − i.
+
+26
+LEANDRO CAGLIERO AND IV´AN G´OMEZ RIVERA
+The condition 0 ≤ j ≤ c implies a − c ≤ i ≤ a and hence (6.3) becomes (see
+(2.11))
+emva,d,µ
+0
+=
+a
+�
+i=max{0,a−c}
+(−1)a−i CG(a
+2, a
+2 − i; d
+2, d
+2 − (a − i) | d−a
+2 , d−a
+2 )
+× CG( c
+2, c
+2 − (a − i); d
+2, − d
+2 + a − i | m
+2 , − m
+2 ) va
+i ⊗ vc
+a−i
+=
+a
+�
+i=max{0,a−c}
+(−1)i+d−a
+�
+(c + b + 1) (m − b)! (c + b + i)!
+(c + m + 1)! i!
+×
+�
+(m + 1) c! (c + b + i)!
+(c + m + 1)! (c − m + b + i)! va
+i ⊗ vc
+a−i.
+On the other hand, when we consider the sum (6.6), it follows from (6.7)
+that
+j = a − i = −k
+and the condition k ≥ 0 implies that k = j = 0 and i = a = m − b. Thus,
+(6.6) is
+emvb,c,µ
+0
+= CG( b
+2, b
+2; c
+2, c
+2 | c+b
+2 , c+b
+2 ) CG(m−b
+2 , − m−b
+2 ; b
+2, − b
+2 | m
+2 , − m
+2 ) va
+a ⊗ vc
+0
+= va
+a ⊗ vc
+0.
+Since c ̸= 0, the sum in emva,d,µ
+0
+has at least two non-zero terms, while
+the sum in emvc,d,µ
+0
+has a single non-zero term, and thus {emva,d,µ
+0
+, emvc,b,µ
+0
+}
+is linearly independent. This completes the proof in this case.
+(iii) The case V2 = E(c, d) with c = d + m: Since b < m ≤ c, it follows from
+(6.1) that
+µ = b + c − 2p, 0 ≤ p ≤ b
+µ = a + d − 2p′, 0 ≤ p′ ≤ min{a, d}.
+This implies p − p′ = b and hence the only option is p = b, p′ = 0 and this
+yields
+µ = a + d = c − b.
+We compute now esva,d,µ
+0
+. It follows from (6.8) and (6.4) that
+k = −i
+j = s − i,
+and since k ≥ 0, we have k = i = 0 and j = s. Therefore, (6.3) becomes
+(see also (2.9) and (2.10))
+esva,d,µ
+0
+= CG(a
+2, a
+2; d
+2, d
+2 | a+d
+2 , a+d
+2 ) CG(d+m
+2 , d+m
+2
+− s; d
+2, − d
+2 | m
+2 , m
+2 − s) va
+0 ⊗ vc
+s
+=
+�
+(d+m−s)! (m+1)!
+(d+m+1)! (m−s)! va
+0 ⊗ vc
+s.
+As always, the reader should check that all the numbers under the factorial
+sign are non-negative.
+
+UNISERIAL REPRESENTATIONS OF THE LIE ALGEBRA sl(2) ⋉ hn
+27
+We compute now esvb,c,µ
+0
+. It follows from (6.8) and (6.7) that
+j = s − i
+k = m − a − s + i
+and conditions k ≥ 0 and j ≥ 0 imply s ≥ i ≥ s − (m − a). Thus, it follows
+from (6.6), (2.11) and (2.9) that
+esvb,c,µ
+0
+=
+min{a,s}
+�
+i=max{0,s−(m−a)}
+CG(m−a
+2 , − m−a
+2
++ s − i; d+m
+2 , d+m
+2
+− (s − i) | a+d
+2 , a+d
+2 )
+× CG(a
+2, a
+2 − i; m−a
+2 , m−a
+2
+− s + i | m
+2 , m
+2 − s) va
+i ⊗ vc
+s−i
+=
+min{a,s}
+�
+i=max{0,s−(m−a)}
+(−1)s−i�
+(d+m−s+i)! (m−a)! (a+d+1)
+(d+m+1)! (m−a−s+i)!
+×
+�
+a! (m−a)! (m−s)! s!
+i! (s−i)! m! (a−i)! (m−a−s+i)! va
+i ⊗ vc
+s−i
+Again, note that all the numbers under the factorial sign are non-negative.
+For s = m − a = b, the sum giving esvb,c,µ
+0
+has the index i running from
+i = 0 to i = min{a, b} ̸= 0, while the sum giving esva,d,µ
+0
+has the index i
+running only up to i = 0. In both cases, all the coefficients are non-zero,
+and thus {esvb,c,µ
+0
+, esva,d,µ
+0
+} is linearly independent. This shows that there is
+no possible µ in S1 and thus S1 = 0. This completes this case and the proof
+of the theorem.
+References
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+Email address: [email protected]
+

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+1
+Face Attribute Editing with
+Disentangled Latent Vectors
+Yusuf Dalva, Hamza Pehlivan, Oyku Irmak Hatipoglu, Cansu Moran, and Aysegul Dundar
+Abstract—We propose an image-to-image translation framework for facial attribute editing with disentangled interpretable latent
+directions. Facial attribute editing task faces the challenges of targeted attribute editing with controllable strength and disentanglement
+in the representations of attributes to preserve the other attributes during edits. For this goal, inspired by the latent space factorization
+works of fixed pretrained GANs, we design the attribute editing by latent space factorization, and for each attribute, we learn a linear
+direction that is orthogonal to the others. We train these directions with orthogonality constraints and disentanglement losses. To
+project images to semantically organized latent spaces, we set an encoder-decoder architecture with attention-based skip connections.
+We extensively compare with previous image translation algorithms and editing with pretrained GAN works. Our extensive experiments
+show that our method significantly improves over the state-of-the-arts. Project page: https://yusufdalva.github.io/vecgan
+Index Terms—Image translation, generative adversarial networks, latent space manipulation, face attribute editing.
+!
+1
+INTRODUCTION
+Facial attribute editing task has been a popular topic among
+the image translation tasks, and significant improvements
+have been achieved with generative adversarial networks
+(GANs) [4], [5], [20], [26], [36], [40]. Facial attribute editing
+is one of the most challenging translation tasks because only
+one attribute of a face is expected to be modified without
+affecting other attributes, whereas humans are very good at
+detecting if a person’s identity or any other attributes of the
+face change.
+There are two main directions proposed for facial at-
+tribute editing, 1- end-to-end trained image translation
+networks and 2- latent manipulation of pretrained GAN
+networks. For the first one, different image translation ar-
+chitectures are proposed with usually two networks, one
+for encoding style and the other for image editing with
+modified styles are injected into [4], [20]. A style, an at-
+tribute, is usually encoded from another image or sampled
+from a distribution. The attributes both in the encoding
+and editing phases need to be disentangled. To achieve
+such disentanglement, works focus on style encoding and
+progress from a shared style code, SDIT [33], to mixed style
+codes, StarGANv2 [4], to hierarchical disentangled styles,
+HiSD [20]. Among these works, HiSD independently learns
+styles of each attribute, bangs, hair color, and eyeglasses,
+and introduces a local translator which uses attention masks
+to avoid global manipulations. HiSD showcases successes
+on those three local attribute editing tasks and is not tested
+for global attribute editing, e.g., age, smile. Furthermore,
+one limitation of these works is the uninterruptible style
+codes, as one cannot control the intensity of attributes (e.g.,
+blondness) in a straightforward manner.
+The second class of methods builds on well-trained gen-
+erative models, specifically StyleGAN2 models [17], which
+•
+Y. Dalva, H. Pehlivan, I. Hatipoglu, C. Moran, and A. Dundar are with
+the Department of Computer Science, Bilkent University, Ankara, Turkey.
+organize their latent space as disentangled representations
+with meaningful directions in a completely unsupervised
+way. This approach has two steps; in the first step, an input
+image is embedded into the generative model’s latent space
+via additional training of an encoder or latent optimization.
+In the second step, the embedded latent code is modified
+based on the discovered directions such that when the
+edited code is decoded, an attribute is edited in the input
+image. Embedding images in GAN’s space and exploring
+interpretable directions in latent codes have emerged as
+important research endeavors on the fixed pretrained GANs
+[9], [27], [28], [30], [35]. We refer to these models as StyleGAN
+inversion-based methods throughout the manuscript. How-
+ever, these models are not trained end-to-end. Therefore,
+there is no guarantee that the encoded image lies in the
+natural GAN space, which results in the codes with limited
+editability. When encoders are forced to encode images into
+GAN’s natural latent distribution, the results suffer from
+the lack of faithful reconstruction. Additionally, since the
+model is not trained for this task, there is no guarantee that
+interesting attributes would be disentangled (e.g. eyeglasses
+versus age).
+To overcome the challenges of facial attribute editing
+task, our previous conference work proposed an image-
+to-image translation framework with interpretable latent
+directions, VecGAN [5]. The attribute editing directions of
+VecGAN were learned in the latent space and regularized to
+be orthogonal to each other for style disentanglement. With
+this setup, VecGAN also provided a knob for the control-
+lable strength of the change, a scalar value. In this work, we
+improve upon VecGAN [5] with a novel disentanglement
+loss and architectural changes.
+This manuscript extends its conference version [5] with
+the following additions:
+•
+We introduce a disentanglement loss which stabilizes
+the training and improves the results as explained in
+Section 3.5.
+arXiv:2301.04628v1  [cs.CV]  11 Jan 2023
+
+2
+Input
+Smile
+Gender
+Age
+Hair Color
+Eyeglasses
+Bangs
+Fig. 1: VecGAN++ image translation results.
+•
+We augment our generator with attention-based skip
+connections. This way, the network can pass only
+selected information to the decoder. The module is
+explained in Section 3.3. We refer to our final model
+with updated architecture and loss objective as Vec-
+GAN++. Example translation results of VecGAN++
+are given in Fig. 1.
+•
+We compare our method with several state-of-the-art
+image translation methods, especially with popular
+pretrained GAN-based models. We provide results
+with an extensive number of metrics for quality,
+attribute edit accuracy, identity, and background
+preservation (Section 4.2). Our results show the effec-
+tiveness of our framework with significant improve-
+ments over state-of-the-art as provided in Section 4.3.
+•
+We provide a comprehensive analysis of our results
+in Section 5. We report metrics as the strength of
+editing increases for our and competing methods. We
+also analyze the projected style codes and show that
+they can classify the targeted attributes of images,
+e.g. hair color, smile.
+2
+RELATED WORK
+Image to Image Translation. Image-to-image translation
+algorithms aim at preserving a given content from the input
+image while changing targeted attributes. They find a wide
+range of applications from translating semantic maps into
+RGB images [6], [21], [32], RGB images to portrait drawings
+[38] and very popularly to editing faces [2], [4], [7], [11], [20],
+[26], [34], [36], [40]. These algorithms set an encoder-decoder
+architecture and train the models with reconstruction and
+GAN losses [8]. When the translation is designed in a
+unimodal setting, images are processed with an encoder and
+decoder to output translated images from one domain to
+the other [23], [32]. The shortcoming of such a setup is that
+a single input image may correspond to multiple possible
+outputs, which makes the translation problem ambiguous.
+Because of that, multi-modal image translation models are
+proposed in which style is encoded separately from an-
+other image or sampled from a distribution [4], [13]. The
+generator, decoder, receives style and content information.
+This information is either concatenated channel-wise [41], or
+combined with a mask [20] or fed separately to the decoder
+while content goes through the convolutional layers, style
+goes through instance normalization blocks [13], [42]. These
+works use two encoders, one is to extract style, and one
+is to encode content [20], [37]. However, in many cases,
+it is not clear what the content is and what the style is.
+Usually, style is referred to the domain attributes one wants
+to change, and the content is the rest of the attributes, which
+is an ambiguous problem definition. In our work, we design
+the attribute as a learnable linear direction in the latent
+space, and we do not employ separate style and content
+encoders. Instead, we use a single encoder, resulting in a
+more intuitive framework.
+Editing with pretrained GANs (StyleGAN inversion-
+based models). Facial attribute editing is also shown to
+be possible with pretrained GANs. State-of-the-art GAN
+models organize their latent space with interpretable direc-
+tions [16], [17], [39] such that moving along the direction
+only changes one attribute of the image. These directions
+are explored in supervised [27], and unsupervised ways
+[9], [28], [30], [35], and many directions are found for face
+editing, e.g. directions that change the smile, pose, age
+attributes are found, to name a few. To edit a facial attribute
+of an input image, one needs to project the image to a latent
+code in GANs’ latent space [1] such that the generator re-
+constructs the input image from the latent code. There have
+been various architectures [3], [31] and objectives proposed
+to project an image to GAN’s embedding. However, they
+suffer from reconstruction-editability trade-off [29]. That is
+if the image is faithfully reconstructed, it may not lie in the
+true distribution of GANs latent space, and therefore, the
+directions do not work as expected, which prevents editing
+the image. On the other hand, if the projection is close to
+the true distribution, then the reconstruction is poor. We
+
+EXEXEXEXEXEXEX3
+also show this behavior in the Results section 4.3 when
+comparing our method with state-of-the-art editing with
+pretrained GANs methods.
+Even though these methods are not as successful as
+the image translation methods, it is still quite remarkable
+when the generative network is only taught to synthesize
+realistic images, it organizes the use of latent space such that
+linear shifts on them change a specific attribute. Inspired by
+these findings, our image-to-image translation framework is
+designed similarly such that a linear shift in the encoded
+features is expected to change a single attribute of an
+image. Unlike previous works, our framework is trained
+end-to-end for translation task, allowing reference-guided
+attribute manipulation via projection, and does not suffer
+from reconstruction-editability trade-off.
+3
+METHOD
+In this section, we provide an overview of the generator ar-
+chitecture and the training set-up. We follow the hierarchical
+labels defined by [20]. For a single image, its attribute for tag
+i ∈ {1, 2, ..., N} can be defined as j ∈ {1, 2, ..., Mi}, where
+N is the number of tags and Mi is the number of attributes
+for tag i. For example, i can be the tag of hair color, and
+attribute j can take the value of black, brown, or blonde.
+Our framework has two main objectives. As the main
+task, we aim to be able to perform the image-to-image
+translation task in a feature (tag) specific manner. While
+performing this translation, as the second objective, we also
+want to obtain an interpretable feature space that allows us
+to perform tag-specific feature interpolation.
+3.1
+Generator Architecture
+For the image-to-image translation task, we set an encoder-
+decoder based architecture and latent space translation in
+the middle as given in Fig. 2. We perform the translation
+in the encoded latent space, e, which is obtained by e =
+E(x) where E refers to the encoder. The encoded features go
+through a transformation T which is discussed in the next
+section. The transformed features are then decoded by G
+to reconstruct the translated images. The image generation
+pipeline following feature encoding is described in Eq. 1.
+e′ = T(e, α, i)
+x′ = G(e′)
+(1)
+Previous image-to-image translation networks [4], [20],
+[37] set a shallow encoder-decoder architecture to translate
+an image while preserving the content and a separate deep
+network for style encoding. In most cases, the style en-
+coder includes separate branches for each tag. The shallow
+architecture used to translate images prevents the model
+from making drastic changes in the images, which helps
+preserving the person’s identity. Our framework is different
+as we do not employ a separate style encoder and instead
+have a deep encoder-decoder architecture for translation.
+That is because to be able to organize the latent space in
+an interpretable way, our framework requires a full un-
+derstanding of the image and, therefore, a larger receptive
+field which results in a deeper network architecture. A deep
+architecture with decreasing size of feature size, on the
+other hand, faces the challenges of reconstructing all the fine
+details from the input image.
+With the motivation of helping the network to preserve
+tag-independent features such as the fine details from the
+background, we use attention-based skip connections be-
+tween our encoder and decoder as described in Section 3.3.
+The architectural details of the encoder and decoder are
+as follows: For the encoder, following a 1×1 convolution, we
+use 8 successive blocks that perform downsampling, which
+reduces feature map dimensions to 1x1. In our decoder, we
+have an architecture symmetric to the encoder, which is
+composed of 8 successive upsampling blocks. Except for the
+last downsampling block and the first upsampling block, we
+use instance normalization denoted as (+IN). The channels
+increase as {32, 64, 128, 256, 512, 512, 512, 1024, 2048} (for
+output resolution 256 × 256) in the encoder and decrease
+in a symmetric way in the decoder. Each DownBlock and
+UpBlock has a residual block with 3×3 convolutional filters
+followed by a downsampling and upsampling layer, respec-
+tively. For downsampling, we use average pooling; for up-
+sampling, we use nearest-neighbor. We use the LeakyReLU
+activation layer (with slope 0.2) and instance normalization
+layer in each convolutional module.
+3.2
+Translation Module
+To achieve a style transformation, we perform the tag-based
+feature manipulation in a linear fashion in the latent space.
+First, we set a feature direction matrix A, which contains
+learnable feature directions for each tag. In our formulation,
+Ai denotes the learned feature direction for tag i. The di-
+rection matrix A is randomly initialized and learned during
+training.
+Our translation module is formulated in Eq. 2, which
+adds the desired shift on top of the encoded features e
+similar to [30].
+T(e, α, i) = e + α × Ai
+(2)
+We compute the shift by subtracting the target style from
+the source style as given in Eq 3.
+α = αt − αs
+(3)
+Since the attributes are designed as linear steps in the
+learnable directions, we find the style shift by subtracting
+the target attribute scale from the source attribute scale. This
+way, the same target attribute αt can have the same impact
+on the translated images no matter what the attributes were
+of the original images. For example, if our target scale
+corresponds to brown hair, the source scale can be coming
+from an image with blonde or back hair, but since we take
+a step for the difference of the scales, they can both be
+translated to an image with the same shade of brown hair.
+There are two alternative pathways to extract the target
+shifting scale for feature (tag) i, αt. The first pathway, named
+the latent-guided path, samples a z ∈ U[0, 1) and applies a
+linear transformation αt = wi,j · z + bi,j, where αt denotes
+sampled shifting scale for tag i and attribute j. We learn
+linear transformation parameters wi,j and bi,j in training
+time. Here tag i can be hair color, and attribute j can be
+
+4
+E
+A
+i
+Ai
++
+x
+j
+Latent guided
+Reference guided
+Projection
+Ai
+Projection
+Ai
+Removing attribute
+Adding attribute
+A
+i
+x
+-
+Ai
+Skip 
+Network
+x
+x
+Attention-based Skip 
+Connections
+256x64x64
+Encoded 
+Features
+Feature 
+Selection 
+Mask
+Edit-Irrelevant 
+Features
+E
+256x64x64
+G
+Inverse
+e’
+s
+e
+Fig. 2: Our translator is built on the idea of interpretable latent directions. We encode images with an Encoder to a latent
+representation from which we change a selected tag (i), e.g. hair color with a learnable direction Ai and a scale α. To
+calculate the scale, we subtract the target style scale from the source style. This operation corresponds to removing an
+attribute and adding an attribute. To remove the image’s attribute, the source style is encoded and projected from the
+source image. To add the target attribute, the target style scale is sampled from a distribution mapped for the given
+attribute (j), e.g. black, blonde, or encoded and projected from a reference image.
+blonde, brown, or back hair. We learn a different transfor-
+mation module for each attribute, denoted as Mi,j(z). Since
+we learn a single direction for every tag, e.g. hair color,
+this transformation module can put the initially sampled z’s
+into the correct scale in the linear line based on the target
+hair color attribute. As the other alternative pathway, we
+encode the scalar value αt in a reference-guided manner.
+We extract αt for tag i from a provided reference image by
+first encoding it into the latent space, er, and projecting er
+via by Ai as given in Eq. 4.
+αt = P(er, Ai) = er · Ai
+||Ai||
+(4)
+In the reference guidance set-up, we do not use the
+information of attribute j, since it is encoded by the tag i
+features of the image.
+The source scale, αs, is obtained in the same way we
+obtain αt from the reference image. We perform the pro-
+jection for the corresponding tag we want to manipulate, i,
+by P(e, Ai). We formulate our framework with the intuition
+that the scale controls the amount of features to be added.
+Therefore, especially when the attribute is copied over from
+a reference image, the amount of features that will be added
+will be different based on the source image. For this reason,
+we find the amount of shift by subtraction as given in Eq. 3.
+Our framework is intuitive and relies on a single encoder-
+decoder architecture.
+3.3
+Attention-based Skip Connections
+To separate encoded features into two branches based on
+feature relevancy for edits, we include a skip network S in
+our architecture. This network S includes three consecutive
+convolutional layers that apply 1x1 convolutions where in-
+put and output channels are set as 256. These convolutional
+layers are followed by LeakyReLU activations (with a slope
+of 0.2), and a sigmoid activation function follows the last
+one to mask features.
+While encoding our input, we compute this attention
+mask using the encoded features e where feature dimen-
+sions are 256 × 64 × 64. Using this mask, we compute two
+feature tensors, e′ and s, which correspond to edit-relevant
+and edit-irrelevant features, respectively. We provide the
+equation for our attention mechanism in equations 5 and
+6.
+e′ = S(e) ∗ e
+(5)
+s = (1 − S(e)) ∗ e
+(6)
+Following this feature filtering step, we encode the em-
+bedding representing image features at the highest level
+using e′. In our decoding step, we combine edit-irrelevant
+features s with decoded features with a summation of
+64x×64 features. This enables our encoder to focus on face-
+relevant features used in our edits and separate features
+encoding texture details. The attention mechanism in the
+full pipeline is provided in Fig. 2.
+3.4
+Training pathways
+We train our network using two different paths by modify-
+ing the translation paths defined by [20]. For each iteration
+to optimize our model, we sample a tag i for shift direction,
+a source attribute j as the current attribute, and a target
+attribute ˆj.
+
+5
+Non-translation path. To ensure that the encoder-decoder
+structure preserves the images’ details, we reconstruct the
+input image without applying any style shifts. The resulting
+image is denoted as xn as given in Eq. 7.
+xn = G(E(x))
+(7)
+Cycle-translation path. We apply a cyclic translation to
+ensure we get a reversible translation from a latent guided
+scale. In this path, we first apply a style shift by sampling
+z ∈ U[0, 1) and obtaining target αt with Mi,ˆj(z) for target
+attribute ˆj. The translation uses α that is obtained by sub-
+tracting αt from the source style. The decoder generates an
+image, xt, as given in Eq. 8 where e is encoded features from
+input image x, e = E(x).
+xt = G(T(e, Mi,j(z) − P(e, i), i))
+(8)
+Then by using the original image, x, as a reference image,
+we aim to reconstruct the original image by translating xt.
+Overall, this path attempts to reverse a latent-guided style
+shift with a reference-guided shift. The second translation is
+given in Eq. 9 where et = E(xt).
+xc = G(T(et, P(e, i) − P(et, i), i))
+(9)
+In our learning objectives, we use xn and xc for recon-
+struction and xt and xc for adversarial losses, and Mi,j(z)
+for the shift reconstruction loss. Details about the learning
+objectives are given in the next section.
+3.5
+Learning objectives
+Given an input image xi,j ∈ Xi,j, where i is the tag to
+manipulate and j is the current attribute of the image, we
+optimize our model with the following objectives. In our
+equations, xi,j is shown as x.
+Adversarial Objective. We learn a discriminator em-
+ploying an architecture with decreasing resolution and in-
+creasing channel size. Like the generator, we build our
+discriminator with channel sizes of {32, 64, 128, 256, 512,
+512, 512, 1024, 2048}, reducing the feature map dimensions
+to 1x1. We concatenate the extracted style αt from the input
+image to this latent code and apply a 1x1 convolution. This
+final convolution is specific to each tag-attribute pair so that
+the model can use this information.
+During training, our generator performs a style shift
+either in a latent-guided or a reference-guided way, resulting
+in a translated image. In our adversarial loss, we receive
+feedback from the two steps of the cycle-translation path.
+As the first component of the adversarial loss, we feed a
+real image x with tag i and attribute j to the discriminator as
+the real example. To give adversarial feedback to the latent-
+guided path, we use the intermediate image generated in
+the cycle-translation path, xt. Finally, to provide adversarial
+feedback to the reference-guided path, we use the final
+outcome of the cycle-translation path xc. Only x acts as
+a real image; both xt and xc are translated images and
+are treated as fake images with different attributes. The
+discriminator aims to classify whether an image is real or
+fake, given its tag and attribute. The objective is given in Eq.
+10.
+Ladv = 2log(Di,j(x)) + log(1 − Di,ˆj(xt))
++log(1 − Di,j(xc))
+(10)
+Shift
+Reconstruction
+Objective.
+As
+the
+cycle-
+consistency
+loss
+performs
+reference-guided
+generation
+followed by latent-guided generation, we utilize a loss
+function to make these two methods consistent with each
+other [13], [18], [19], [20]. Specifically, we would like to
+obtain the same target scale, αt, both from the mapping and
+the encoded reference image generated by the mapped αt.
+The loss function is given in Eq. 11.
+Lshift = ||Mi,j(z) − P(et, i)||1
+(11)
+Those parameters, Mi,j(z) and P(et, i), are calculated for
+the cycle-translation path as given in Eq. 8 and 9.
+Image Reconstruction Objective. In all of our training
+paths, the purpose is to be able to re-generate the original
+image again. To supervise this desired behavior, we use
+L1 loss for reconstruction loss. In our formulation, xn and
+xc are outputs of the non-translation and cycle-translation
+paths, respectively. Formulation of this objective is provided
+in Eq. 12.
+Lrec = ||xn − x||1 + ||xc − x||1
+(12)
+Orthogonality Objective. To encourage the orthogonal-
+ity between directions, we use soft orthogonality regulariza-
+tion based on the Frobenius norm, which is given in Eq. 13.
+This orthogonality further encourages a disentanglement in
+the learned style directions.
+Lortho = ∥AT A − I∥F
+(13)
+Disentanglement Objective We intend to change the
+scale for the desired semantic in each translation. As a reflec-
+tion of this criteria, we penalize the changes in the attributes
+that are not subjected to any translation. For translated tag
+i, input scales α, and edited scales α′, the disentanglement
+loss is formulated in equation 14. In the formulation, αk
+represents the semantic scale for tag k. Scales are calculated
+based on the projection given in Eq. 4.
+Ldis =
+�
+k̸=i
+||αk − α′
+k||
+(14)
+We find this disentanglement to be complementary to
+our orthogonality objective. When the model is trained with
+additional disentanglement loss, we observe that orthogo-
+nality loss drops to a lower value.
+Full Objective. Combining all of the loss components
+described, we reach the overall objective for optimization
+as given in Eq. 15. Additionally, we add an L1 loss on the
+matrix A parameters to encourage its sparsity.
+min
+E,G,M,Amax
+D λaLadv + λsLshift + λrLrec
++λo(Lortho + Ldis) + λspLsparse
+(15)
+We set the following parameters; λa = 1, λrec = 1.5,
+λs = 1, λo = 1 and λsp = 0.1. We use a learning rate of
+10−4 and train our model for 600K iterations with a batch
+size of 8 on a single GPU.
+
+6
+Input
+Reference
+HiSD
+VecGAN
+Input
+Reference
+Input
+Reference
+VecGAN++
+Fig. 3: Qualitative results of bangs attribute of our final model (VecGAN++ - Ours w/ Attn. Skip + Disen.), VecGAN and
+HiSD. Given reference images, methods extract reference attributes and edit input images accordingly. All methods achieve
+high-quality results. VecGAN++ achieves better edit quality compared to VecGAN. It is important to note that, HiSD learns
+feature-based local translators, which is a successful approach on local edits, e.g. bangs, eyeglasses, and hair color but not
+smile, age, or gender. Our method achieves comparable visual and better quantitative results than HiSD on this local task
+and can also achieve global edits.
+Method
+Lat.
+Ref.
+Avg.
+SDIT [33]
+33.73
+33.12
+33.42
+StarGANv2 [4]
+26.04
+25.49
+25.77
+Elegant [36]
+-
+22.96
+-
+HiSD [20]
+21.37
+21.49
+21.43
+VecGAN [5]
+20.17
+20.72
+20.45
+Ours w/ Disen.
+20.23
+20.57
+20.40
+Ours w/ Attn. Skip + Disen.
+20.15
+20.08
+20.12
+TABLE 1: Quantitative results for Setting I. Lat: Latent
+guided, Ref: Reference guided.
+4
+EXPERIMENTS
+4.1
+Dataset and Settings
+We train our model on CelebA-HQ dataset [22], which con-
+tains 30,000 face images. To extensively compare with state-
+of-the-arts, we follow two training-evaluation protocols as
+follows:
+Setting I. In our first setting, we follow the set-up from
+HiSD [20] to compare our method with end-to-end based
+image translation algorithms. Following HiSD, we use the
+first 3000 images of the CelebA-HQ dataset as the test set
+and 27000 as the training set. These images include annota-
+tions for different attributes from which we use hair color,
+the presence of glass, and bangs attributes for translation
+tasks in this setting. The images are resized to 128 × 128.
+Following the evaluation protocol proposed by HiSD [20],
+we compute FID scores on the bangs addition task. For
+each test image without bangs, we translate them to images
+with bangs with latent and reference guidance. In latent
+guidance, 5 images are generated for each test image by
+randomly sampled scales from a uniform distribution. Then
+this generated set of images is compared with images with
+attribute bangs in terms of their FIDs. FIDs are calculated
+for these 5 sets and averaged. We randomly pick 5 reference
+images for reference guidance to extract the style scale. FIDs
+are calculated for these 5 sets separately and averaged.
+Setting II. We use our second setting to comprehensively
+compare our method with StyleGAN2-based inversion and
+editing methods. For this setting, the training/test split is
+obtained by re-indexing each image in CelebA-HQ back
+to the original CelebA and following the standard split of
+CelebA. This results in 27,176 training and 2,824 test im-
+ages. Our models are trained for hair color, the presence of
+glasses, bangs, age, smiling, and gender attributes. Images
+are resized to 256 × 256 resolution, which is the dimension
+StyleGAN2-based inversion methods use. We evaluate our
+model with smile addition and removal and bangs addi-
+tion attributes. Since the task of smile addition/removal
+requires a high-level understanding of the input face for
+modifying multiple facial components simultaneously, it is
+considered one of the most challenging attributes to edit.
+By benchmarking our model with such an editing task, we
+demonstrate the effectiveness of our framework in terms of
+image understanding.
+4.2
+Metrics
+We mostly build our evaluation on the FID metric as in
+previous works. Additionally, for smile manipulation, we
+evaluate our results on other metrics such as smile clas-
+sification accuracy, Identity Preservation, and Background
+Preservation, described as follows:
+Frechet Inception Distance (FID): For the FID metric
+[10], we calculated the distance between the feature vectors
+of original and generated images, which are obtained using
+the Inception-V3 model.
+Accuracy (Acc): We train an image classification network
+for smile attribute classification on the training split of
+the CelebA-HQ dataset. We use an ImageNet pretrained
+ResNet-50 model and fine-tune it for this task. The model
+achieves 94% accuracy on the validation set for the smile
+attribute. We use this classifier to evaluate the accuracy
+of the generated images to test if the attribute is correctly
+manipulated.
+
+7
+1. Smile (+)
+2. Smile (+)
+3. Smile (+)
+4. Smile (-)
+5. Smile (-)
+6. Bangs (+)
+7. Bangs (+)
+8. Age (+)
+9. Age (+)
+Input
+VecGAN++
+VecGAN
+e4e
+HFGI
+HyperStyle
+StyleTransformer
+StyleRes
+Fig. 4: Qualitative results of our and competing methods. StyleGAN inversion-based methods do not faithfully reconstruct
+input images. They miss many details from the background and foreground. StyleRes achieves better reconstruction results
+compared to others but not VecGAN models. VecGAN++ and VecGAN achieve high fidelity to the originals with only
+targeted attributes manipulated naturally and realistically. We also observe that VecGAN++ significantly improves over
+VecGAN in many examples.
+
+MER
+EXPMER
+EXPMER
+EXPMER8
+Bangs
+Smile
+Method
+FID (+)
+FID (+)
+FID (-)
+Acc (+)
+Acc (-)
+Id (+)
+Id (-)
+BG (+)
+BG (-)
+e4e [29]
+53.62
+35.01
+37.91
+99.9
+99.8
+0.4536
+0.4382
+0.7709
+0.7541
+HyperStyle [3]
+41.37
+25.25
+24.64
+96.8
+98.0
+0.6544
+0.6771
+0.7941
+0.7713
+HFGI [31]
+40.54
+23.49
+26.58
+99.2
+96.6
+0.5522
+0.5290
+0.7320
+0.7324
+StyleTransformer [12]
+44.66
+27.64
+32.71
+99.9
+98.4
+0.5311
+0.5173
+0.7767
+0.7607
+StyleRes [25]
+40.13
+20.53
+21.63
+99.1
+99.2
+0.5647
+0.5606
+0.8547
+0.8720
+VecGAN [5]
+36.47
+17.70
+20.26
+92.7
+65.1
+0.6120
+0.7727
+0.9151
+0.9257
+Ours w/Disen.
+32.40
+17.14
+19.40
+92.5
+89.2
+0.6048
+0.6517
+0.9060
+0.9136
+Ours w/ Attn. Skip + Disen.
+26.92
+17.37
+19.78
+95.2
+76.2
+0.5588
+0.6665
+0.8909
+0.9017
+TABLE 2: Quantitative results for Setting II. (+) and (-) denote the scores for adding and removing an attribute. We
+compare the FID scores for bangs addition. We compare various metrics for smile addition and removal. Explanations
+of the metrics are given in Section 4.2. We achieve better scores than StyleGAN inversion-based methods. We also show
+FID improvements with the disentanglement loss and additional improvements, especially on the addition of bangs with
+attention-based skip connections. We set the attribute strength of StyleGAN inversion-based methods to 2 and 1 for smile
+and bangs attributes, respectively, to report the best FID scores. We provide more analysis on that in Fig. 5.
+65
+70
+75
+80
+85
+90
+95
+100
+Accuracy+
+17.5
+20.0
+22.5
+25.0
+27.5
+30.0
+32.5
+35.0
+37.5
+FID+
+e4e
+Hyperstyle
+HFGI
+StyleTransformer
+StyleRes
+VecGAN
+VecGAN++
+65
+70
+75
+80
+85
+90
+95
+100
+Accuracy+
+0.40
+0.45
+0.50
+0.55
+0.60
+0.65
+0.70
+0.75
+0.80
+ID+
+e4e
+Hyperstyle
+HFGI
+StyleTransformer
+StyleRes
+VecGAN
+VecGAN++
+20
+30
+40
+50
+60
+70
+80
+90
+100
+Accuracy+
+0.73
+0.75
+0.78
+0.80
+0.83
+0.85
+0.88
+0.90
+0.93
+BG+
+e4e
+Hyperstyle
+HFGI
+StyleTransformer
+StyleRes
+VecGAN
+VecGAN++
+20
+30
+40
+50
+60
+70
+80
+90
+100
+Accuracy-
+20.0
+22.5
+25.0
+27.5
+30.0
+32.5
+35.0
+37.5
+40.0
+FID-
+e4e
+Hyperstyle
+HFGI
+StyleTransformer
+StyleRes
+VecGAN
+VecGAN++
+20
+30
+40
+50
+60
+70
+80
+90
+100
+Accuracy-
+0.40
+0.50
+0.60
+0.70
+0.80
+ID-
+e4e
+Hyperstyle
+HFGI
+StyleTransformer
+StyleRes
+VecGAN
+VecGAN++
+20
+30
+40
+50
+60
+70
+80
+90
+100
+Accuracy-
+0.73
+0.75
+0.78
+0.80
+0.83
+0.85
+0.88
+0.90
+0.93
+BG-
+e4e
+Hyperstyle
+HFGI
+StyleTransformer
+StyleRes
+VecGAN
+VecGAN++
+Fig. 5: Plots of FID, ID, and BG metrics as we change the intensity of the attributes and the number of steps to take for
+explored directions. For each intensity, we measure the attribute accuracy in the x-axis. The first row plots present results
+for smile addition, and the second row presents them for smile removal.
+Identity Preservation (Id): To calculate the Id metric, we
+use the CurricularFace model [14] to calculate the similarity
+between the original and generated images. The Curricular-
+Face model uses the ResNet-101 model as a backbone for the
+feature extraction. We calculate the cosine similarity score
+between the features of edited and original images.
+Background Preservation (BG): For the BG metric, we
+first use the facial attribute segmentation masks from the
+CelebAMask-HQ dataset to form background masks. Using
+these masks, we calculate the mean structural similarity
+index between the backgrounds of the original and edited
+images.
+4.3
+Results
+We extensively compare our results with other end-to-end
+image translation methods. In Setting I, as given in Table 1,
+we compare with SDIT [33], StarGANv2 [4], Elegant [36],
+and HiSD [20] models. Among these methods, HiSD learns
+a hierarchical style disentanglement, whereas StarGANv2
+learns a mixed style code. Therefore StarGANv2, when
+translating images, also edits other attributes and does not
+strictly preserve the identity. HiSD achieves disentangled
+style edits. However, HiSD learns feature-based local trans-
+lators, an approach known to be successful on local edits,
+e.g. bangs, and their model is trained for bangs, eyeglasses,
+and hair color attributes. VecGAN achieves significantly
+better quantitative results than HiSD both in latent-guided
+and reference-guided evaluations, even though they are
+compared on a local edit task. We further achieve im-
+provements with disentanglement loss. We also observe
+with the disentanglement loss, the training becomes more
+stable. With attention-based skip connections, both latent
+
+9
+Smile (-)
+Smile (+)
+Smile (-)
+Input
+VecGAN++
+VecGAN
+e4e
+HFGI
+HyperStyle
+StyleTransformer
+StyleRes
+Fig. 6: Generalization results of our and competing methods. StyleGAN inversion-based methods do not faithfully
+reconstruct input images. The reconstruction problem is more severe on these out-of-domain images than the in-domain
+images presented in Fig. 4.
+and reference-based FID scores improve further.
+Fig. 3 shows reference-guided results of our final model,
+VecGAN++, VecGAN, and HiSD. As shown in Fig. 3, meth-
+ods achieve attribute disentanglement, they do not change
+any other attribute of the image than the bangs tag. Vec-
+GAN++ achieves better edit quality compared to VecGAN.
+It is important to note that, HiSD learns feature-based local
+translators, which is a successful approach on local edits,
+e.g. bangs, eyeglasses, and hair color but not smile, age, or
+gender. Our method achieves comparable visual and better
+quantitative results than HiSD on this local task and can also
+achieve global edits.
+In our second set-up of evaluation, we compare our
+method with state-of-the-art StyleGAN2 inversion-based
+methods, e4e [29], HyperStyle [3], HFGI [31], StyleTrans-
+former [12], and StyleRes [25] in Table 2. We compare the
+methods on a local (bangs) and a global attribute (smile)
+manipulation. For the smile attribute, we use the direction
+explored by the InterfaceGAN method [27]. For the bangs
+attribute, we use the direction discovered by the StyleCLIP
+method [24]. The strength attribute is set to 2 to report the
+best FID scores. We provide more analysis on that in Fig.
+5. We also show FID improvements with the disentangle-
+ment loss and additional improvements, especially on the
+addition of bangs with attention-based skip connections.
+Our method and StyleGAN inversion-based methods
+provide a knob to control the editing attribute intensity.
+We obtain plots provided in Fig. 5 by changing the editing
+attribute intensity. As we increase the intensity, edits become
+more detectable. We measure that with a smile classifier.
+Therefore, we plot FID, Id (Identity), and BG (Background
+reconstruction) scores with respect to the attribute inten-
+sity measured by the accuracy of the classifier. Specifically,
+for VecGAN++ and VecGAN, we set αt given in Eq. 3
+to {0.0, 0.33, 0.5, 0.66, 1.0}. For StyleGAN inversion-based
+methods, we set the strength parameter to {1, 2, 3}. These
+models usually set the strength to 3 for successful smile
+edits. As shown in Fig. 5, VecGAN++ and VecGAN achieve
+better FID scores compared to others consistently. With the
+highest strength, where the accuracy of the classifier goes to
+100% for all models, we observe that FID scores for Style-
+GAN inversion-based model scores drastically get worse,
+whereas our results are robust. VecGAN++ achieves better
+FID scores than VecGAN consistently. We find the Id score
+getting worse as the edit strength increases. That results
+from changes in the person and the limitations of the Curric-
+ularFace model. VecGAN++ achieves significantly better BG
+scores than StyleGAN inversion-based models and slightly
+worse than VecGAN. We observe that VecGAN++ achieves
+better edit quality, as reflected in FID scores, than VecGAN.
+On the other hand, VecGAN does smaller edits, resulting in
+better Id and BG scores.
+We provide qualitative comparisons in Fig. 4. StyleGAN
+inversion-based methods do not faithfully reconstruct input
+images. They miss many details from the background and
+foreground. StyleRes achieves better reconstruction results
+compared to others but still worse than ours as measured
+by BG metrics. Also note that even though HyperStyle
+achieves high Id scores that are comparable to ours, their
+reconstructions miss many image details, making the output
+images look unrealistic. VecGAN++ and VecGAN achieve
+high fidelity to the originals with only targeted attributes
+manipulated naturally and realistically. We also observe
+that VecGAN++ visibly improves over VecGAN on many
+examples, especially on the second and third examples for
+smile and on samples with bangs edits. Additionally, Vec-
+GAN++ achieves successful age edits, whereas StyleGAN
+inversion-based methods add eyeglasses very frequently.
+This shows VecGAN++ and VecGAN provide with better
+disentanglement between correlated attributes, e.g. age and
+eyeglasses. This is because our models are trained end-to-
+end with labeled datasets for this task. We provide more
+analysis on these comparisons in the next section.
+
+10
+Scale Frequency Distribution for Smile
+Smiling
+Not Smiling
+(a) Histogram of αt distribution of smile tags of VecGAN.
+Scale Frequency Distribution for Smile
+Smiling
+Not Smiling
+(b) Histogram of αt distribution of smile tags of VecGAN++.
+(c) Training images plotted based on their αt values extracted
+for smiling tag from VecGAN++. Zoom in for details.
+Fig. 7: Analysis of αt for smile tag.
+5
+ANALYSIS AND DISCUSSIONS
+5.1
+Comparing End-to-end Image Translation Networks
+versus StyleGAN Inversion-based Methods
+We propose an end-to-end trained image translation net-
+work in this work and extensively compare our method
+with StyleGAN inversion-based methods. We note the dif-
+ferent advantages and disadvantages of both approaches.
+We observe that end-to-end trained image translation
+networks, especially our proposed framework, do not suffer
+from the reconstruction and editability trade-off. This trade-
+off is pointed out for StyleGAN inversion-based methods
+[29]. That is, when the inversion parameters are optimized
+to reconstruct the input images faithfully, they do not lie in
+the natural StyleGAN distribution space, and therefore the
+edit quality gets poor for those high-fidelity inversions. That
+Scale Frequency Distribution for Hair Color
+Black Hair
+Brown Hair
+Blond Hair
+(a) Histogram of αt distribution of hair color tag of VecGAN.
+Scale Frequency Distribution for Hair Color
+Black Hair
+Brown Hair
+Blond Hair
+(b) Histogram of αt distribution of hair color tag of VecGAN++.
+(c) Training images plotted based on their αt values extracted
+for hair color tag from VecGAN++. Zoom in for details.
+Fig. 8: Analysis of αt for hair color tag.
+is the advantage of our method because it is trained end-to-
+end, and we learn both reconstruction and editing together.
+StyleGAN inversion-based methods enjoy many editing
+capabilities, whereas our framework only achieves pre-
+defined edits for which it is trained. Those methods that
+employ pre-trained StyleGANs rely on StyleGAN’s seman-
+tically rich feature organizations. The editing directions
+are discovered after StyleGAN is trained. Some methods
+discover directions in supervised and unsupervised ways.
+Supervised methods, e.g. InterfaceGAN, require labeled
+datasets the same as ours. On the other hand, with unsuper-
+vised methods and text-based editing methods, directions
+are explored for those that do not have labeled datasets.
+For example, with the GANSpace method [9], editing di-
+
+11
+rections are found for different expressions, and with the
+StyleCLIP method [24], editing directions are found for
+different hairstyles (Mohawk hairstyle, Bob-cut hairstyle,
+Afro hairstyle, e.g.). That is an advantage of StyleGAN
+inversion-based methods.
+Lastly, our method achieves better generalization results
+as presented in Fig. 6. We apply our image translation
+methods and StyleGAN inversion-based methods that are
+trained on high-quality face photographs on Metface images
+[15] without any tuning. Metface dataset [15] includes face
+images extracted from the collection of the Metropolitan
+Museum of Art and exhibits a domain gap with CelebA [22]
+and FFHQ [16] datasets. StyleGAN inversion based methods
+do not reconstruct the input images with high fidelity due
+to the domain gap and output images similar to the style
+of CelebA and FFHQ. Our method has the advantage of
+being robust to domain gaps whereas StyleGAN inversion-
+based methods require fine-tuning the StyleGAN network
+and inversion encoders on new domains.
+5.2
+Analysis of Projected Styles
+We explore the behavior of encoded scales from reference
+images, αt. These scales are supposed to provide informa-
+tion about the attribute of the image (whether a person
+smiles or not) and its intensity (how big the smile is). We
+plot the histograms of αt values from validation images for
+smile tags and use orange and blue colors depending on
+their ground-truth tags from the validation list as shown
+in Fig. 7a and Fig. 7b for VecGAN and VecGAN++, respec-
+tively. For the smiling tag, αt values are mostly disentangled
+with a small intersection. For VecGAN, we remove the
+outliers for visualization purposes. There are some encoded
+scales far away from the clusters. On the other hand, for
+VecGAN++ we do not have such a problem and do not
+remove any data points. Other than that, we find VecGAN
+and VecGAN++ extracted scale attributes to be similar. Next,
+we visualize the samples for the smiling tag for VecGAN++.
+Fig. 7c shows a visualization of validation images plotted
+based on their αt values extracted for the smiling tag. We
+visualize a few samples from each bin from the histogram
+above with the same frequency as the histogram value. The
+visualization shows that αt values encode the intensity of
+the smile. The rightmost samples have large smiles, and
+the leftmost samples look almost angry. On the other hand,
+the images in the middle space are confusing ones. We also
+observe many wrong labeling in the CelebA-HQ dataset by
+going through the middle space.
+We repeat the same analysis for the hair color tag as
+provided in Fig. 8a and Fig. 8b for VecGAN and VecGAN++
+respectively, since hair color tag is a challenging one as it is
+expected to have a continuous scale with no clear separation
+between classes. We observe that VecGAN struggles to sep-
+arate attributes for hair color tag, whereas VecGAN++ does
+a better separation even though it may not be perfect. We
+also visualize the validation images based on their αt values
+extracted for hair color tag in Fig. 8c from VecGAN++. The
+images go from black hair to brown hair to blonde hair. We
+observe that the shade of hair goes lighter, but we also note
+that the extracted scales are not perfect, and there is room
+for improvement.
+6
+CONCLUSION
+This paper introduces VecGAN++, an image-to-image trans-
+lation framework with interpretable latent directions. This
+framework includes a deep encoder and decoder archi-
+tecture with latent space manipulation in between. Latent
+space manipulation is designed as vector arithmetic where
+for each attribute, a linear direction is learned. This design
+is encouraged by the finding that well-trained generative
+models organize their latent space as disentangled repre-
+sentations with meaningful directions in a completely unsu-
+pervised way. Therefore, we also extensively compare our
+method with StyleGAN inversion-based methods and point
+out their advantages and disadvantages compared to our
+method. Each change in the architecture and loss function
+is extensively studied and compared with state-of-the-arts.
+Experiments show the effectiveness of our framework.
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+pages 5104–5113, 2020. 2
+

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+ 
+ 
+Wildfire Smoke Detection by Computer Vision 
+Eldan R., Daniel I. 
+December 26, 2022 
+ 
+[email protected] 
+ 
+Abstract- Wildfires are becoming more frequent and their 
+effects more devastating every day. Climate change has 
+directly and indirectly affected the occurrence of these, as well 
+as social phenomena have increased the vulnerability of 
+people. Consequently, and given the inevitable occurrence of 
+these, it is important to have early warning systems that allow 
+a timely and effective response. 
+Artificial intelligence, machine learning and Computer 
+Vision offer an effective and achievable alternative for 
+opportune detection of wildfires and thus reduce the risk of 
+disasters. YOLOv7 offers a simple, fast, and efficient 
+algorithm for training object detection models which can be 
+used in early detection of smoke columns in the initial stage 
+wildfires. 
+The developed model showed promising results, achieving 
+a score of 0.74 in the F1 curve when the confidence level is 
+0.298, that is, a higher score at lower confidence levels was 
+obtained. This means when the conditions are favorable for 
+false positives. The metrics demonstrates the resilience and 
+effectiveness of the model in detecting smoke columns. 
+ 
+Keywords: 
+Early 
+Warning, 
+Object 
+Detection, 
+Artificial 
+Intelligence, Computer Vision, YOLO. 
+ 
+I. INTRODUCTION 
+ 
+A wildfire is a fire that, whatever its origin and with danger or 
+damage to people, property, or the environment, spreads 
+uncontrolled in rural areas, through woody, bushy or herbaceous 
+vegetation, alive or dead. In other words, it is an unjustified and 
+uncontrolled fire in which the fuels are plants and which, in its 
+propagation, can destroy everything in its path ("Wildfires in Chile 
+- CONAF"). 
+ 
+In the last 10 years there have been 67,567 Wildfires, affecting 
+an area of 1,246,922 hectares of grassland, scrubland, forest 
+plantations, native forest, agricultural land, among others. 
+ 
+Climate change has increased the risk of Wildfires both directly 
+and indirectly (Borunda, A.). Although the causality of fires is 99.7% 
+human, the conditions for the generation of these fires are higher 
+than they would be without climate change. 
+ 
+Given this scenario, it is significant to have early warning 
+systems that, in the event of an inevitable occurrence of a forest fire, 
+make it possible to activate and deploy the necessary resources for 
+its rapid control and extinction, thus preserving the lives of people, 
+their property and the environment. 
+ 
+II. FOREST FIRE DETECTION SYSTEMS 
+ 
+Wildfires are incidents with a high destructive potential and a 
+sudden growth, even more so when weather conditions allow it. 
+Therefore, is very important to apply a rapid firefighting strategy that 
+prevents fires from growing in extent and severity. 
+ 
+The early detection of fires is essential to initiate procedures 
+that culminate in firefighting. Among them is the notification of the 
+start of the fire to the Regional Coordination Center of CONAF 
+(CENCOR) who, in turn, with the respective technical background, 
+analyze the situation and generate the dispatch of relevant land 
+and/or air resources. 
+ 
+A. Mobile Terrestrial Detection 
+ 
+The task consists of moving surveillance people to a given 
+area, either by vehicle or on foot. This practice is quite common in 
+Chile in forestry companies, where it is used to supervise work 
+activities. 
+ 
+B. Fixed Terrestrial Detection 
+ 
+This is the most widely used form of detection in Chile. It 
+consists of having a person observing from metal or wooden towers 
+that are between 15 and 30 meters high, or from lower booths known 
+as detection posts. 
+ 
+C. Airborne Detection 
+ 
+This detection method uses aircraft, usually single-engine 
+high-wing aircraft, to detect fires from the air. The pilot is 
+accompanied by an observer, who oversees doing the observation. 
+This technique makes possible to observe a large amount of area in 
+an abbreviated time and provides accurate and detailed information 
+about the detected fire and the area over which it is flown. However, 
+its operating cost is high. 
+ 
+D. Detection with television systems 
+ 
+This method uses television cameras to transmit their signal via 
+microwaves to screens at a command post, such as in a vehicle in the 
+field or at a coordination center. There, specialists analyze the 
+situation based on what they see on the screen. 
+ 
+E. Satellite Systems 
+ 
+In some parts of the world, due to the lack of forest fire 
+protection organizations or detection systems, the only way to know 
+what is happening is to use low orbit satellite images, such as those 
+provided by the Aqua and Terra satellites. 
+ 
+ 
+ 
+ 
+ 
+ 
+
+ 
+2 
+III. OBJECT DETECTION BY COMPUTER VISION 
+ 
+Computer vision, also known as artificial vision or technical 
+vision, is a scientific discipline that involves techniques for 
+acquiring, processing, analyzing and understanding images of the 
+real world to produce numerical or symbolic information that can be 
+processed by computers (J. Morris, 1995). Just as humans use our 
+eyes and brains to make sense of the world around us, computer 
+vision seeks to create the same effect by allowing a computer to 
+perceive and understand an image or sequence of images and act 
+accordingly given the situation. This understanding is achieved 
+through fields as diverse as geometry, statistics, physics and other 
+disciplines. Data collection is achieved in a variety of ways, such as 
+image sequences viewed from multiple cameras or multidimensional 
+data from medical scanners. 
+ 
+Real-time object detection is a particularly important topic in 
+computer vision, as it is often a necessary component in computer 
+vision systems. Some of its current applications are object tracking, 
+public safety and active surveillance, autonomous vehicle driving, 
+robotics, medical image analysis, among others.  
+ 
+Computing devices that run real-time object detection 
+processes usually use CPUs or GPUs for their tasks, however, 
+nowadays the computational capacity has improved exponentially 
+with the Neural Processing Units (NPU) developed by different 
+manufacturers.  
+ 
+These devices focus on accelerating operations through 
+several types of algorithms, one of the most widely used being the 
+multilayer perceptron or Multilayer Perceptron (MLP), an artificial 
+neural network formed by multiple layers in such a way that it has 
+the ability to solve problems that are not linearly separable. 
+ 
+IV. YOLO 
+ 
+The object detection algorithm used in the present work is 
+YOLO (You only look once), developed by Wang, Chien-Yao et. al, 
+whose latest version was recently released in July 2022. 
+ 
+YOLO is an algorithm that uses neural networks to provide 
+real-time object detection. It is an algorithm known for its speed and 
+accuracy and YOLO is currently used in a variety of applications 
+such as traffic signal detection, people accounting, detection of 
+available spaces in private parking lots, remote animal surveillance, 
+among others. 
+ 
+ 
+A. Operation of YOLO 
+ 
+The YOLO algorithm works by using three techniques: 
+• 
+Intersection over Union (IOU). 
+• 
+Regression of the bounding box. 
+• 
+Residual blocks. 
+ 
+B. Residual blocks 
+ 
+The analyzed image, which can be a frame of a sequence 
+(video), is divided into several grids. Each grid has a dimension SxS.  
+The following image shows an example of grids. 
+Each cell will detect the objects that appear inside them. For 
+example, if an object appears inside a given cell, the cell will 
+perform processing on its own and separately from the others. 
+Fig. 1 - Example of residual block, source: guidetomlandai.com 
+C. Regression of the bounding box 
+ 
+A bounding box is an outline that highlights an object within 
+an image or cell. Each box has a height, a width, a class (what we 
+are looking for: car, dog, traffic light, fire smoke) and a centroid. 
+The following image shows an example of a bounding box. 
+YOLO uses a single bounding box regression to predict the 
+items listed above. 
+Fig. 2 - Example of bounding box, source: appsilondatascience.com 
+D. Intersection over Union (IOU) 
+ 
+Intersection over union is a phenomenon in object detection that 
+describes how blocks overlap in an image, where block is understood 
+as the set of cells where the detected object is located. 
+ 
+YOLO uses IOU to provide an output block surrounding the 
+detected object. Each grid cell is responsible for predicting the 
+bounding boxes and their confidence score.  
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+
+y=(pc,br,b,,bh,bw,C
+b
+b 
+ 
+Fig. 3 - Example of Intersection over Union, source: 
+miro.medium.com 
+E. Output result 
+ 
+YOLO combines the three techniques for accurate detection. 
+First, having the SxS grid of the analyzed image allows to evaluate 
+each section individually and be able to detect the bounding boxes 
+and their respective confidence scores. 
+ 
+For each bounding box, the class of detected object is set and 
+finally, using IOU, the frame is adjusted to ensure that the detection 
+frame covers the entire real object in the output image. 
+ 
+Fig. 4 - Diagram of the YOLO algorithm, source: 
+guidetomlandai.com 
+V. CREATION OF THE MODEL 
+ 
+To detect objects YOLO algorithm requires a model trained 
+with the class or classes of the search elements. For this it is 
+important to establish specifically where, how and when the model 
+will operate to detect fires, for which the following criteria are 
+established: 
+ 
+A. Location of the observer 
+ 
+The analyzed images by the model and used for fire detection 
+were obtained from distant sources, with a wide view of valley areas, 
+forests and/or mountain ranges, above level and with unpredictable 
+atmospheric conditions. 
+ 
+Such conditions of observations are those that we could identify 
+in an observation tower or fire watch. It should be considered that 
+the resolution of these can be varied and not uniform, depending on 
+the capture device used (webcam, HD camera). 
+ 
+B. Type of wildfire to be detected 
+ 
+As the objective of the system is to detect fires in their initial 
+stage, we will discard any images with fire and concentrate on smoke 
+plumes and their development, ideally taken from cameras in 
+different scenarios. 
+Fig. 5- Rodelillo airfield webcam, Valparaíso, December 7, 2020, 
+16:20 hours. 
+C. Redundancy of training images 
+ 
+To generate greater variability and resilience to the model, 
+modifications have been made to part of the image dataset to 
+increase the amount of material for training. 
+ 
+In this regard, the following characteristics were applied to the 
+dataset: 
+ 
+1) 
+Mirror effect: The images were duplicated with a 
+horizontal rotation. This allows to have training material 
+for different wind conditions. 
+ 
+2) 
+Exposure: Duplicate images were generated with changes 
+in exposure between -15% and +15%. This allows 
+improving the visibility of the smoke plume in images that 
+may have been taken with different levels of ambient 
+humidity, which at greater distances distorts the focus and 
+sharpness of the image. 
+ 
+Also, the redundancy modifications and the labeling of the 
+images in the dataset were made in the Roboflow app, a 
+computer vision web software that provides many functions for 
+upload, label, augmentation, export, train and testing models. 
+ 
+VI. SOURCES OF INFORMATION 
+ 
+To increase the effectiveness of the model, it is important to 
+train it with images that are as similar as possible to the scenarios 
+where it will be implemented. In view of the above, different sources 
+of information were selected to obtain images with a wide range of 
+
+Boundingboxes+
++confidence
+SxSqridoninput
+Final detections
+Classprobabilitymap 
+4 
+geographic environments to generate a resilient model that can be 
+implemented in different locations. 
+ 
+A. High Performance Wireless Research and Education 
+Network (HPWREN) 
+ 
+The High-Performance Wireless Research and Education 
+Network is a University of California partnership project led by the 
+San Diego Supercomputing Center and the Institute for Geophysics 
+and Planetary Physics at Scripps Institution of Oceanography. 
+ 
+HPWREN works as a collaborative cyber infrastructure 
+connected to the Internet. The project has a vast network of cameras 
+in the State of California, USA, which have been used for wildfire 
+observation.  
+ 
+In particular, the HPWREN images were obtained from the AI 
+for Mankind project, founded by Wei Shung Chung. 
+ 
+B. Social Networks 
+ 
+Wildfires are high-impact emergencies and are considered by 
+society as public interest events. Therefore, a search for images of 
+Wildfires was made on the Twitter platform using the hashtag 
+"Wildfire" in Spanish, English, Turkish, Greek, Russian and 
+Portuguese. This allowed access to a variety of images with different 
+types of geography and relatively recent, allowing the generation of 
+an updated model training. 
+ 
+C. Images created with Artificial Intelligence 
+ 
+In an innovative way, the well-known artificial intelligences 
+Dall-E from OpenAI and Stable Diffusion from StabilityAI were 
+used to generate images using the following input phrase: "Wildfire 
+smoke in early stage as seen from an observation tower or high and 
+distant point". 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+Fig. 6 - Forest fire smoke image created with Dall-E 
+D. Self-made computer Images 
+ 
+To complement the dataset with smoke columns originating in 
+different places, images were generated by superimposing layers 
+with the Photoshop application. For this purpose, base images of 
+cameras and observation towers without smoke were selected and 
+new images were artificially created with different types of smoke 
+originating from different points. 
+VII. MODEL TRAINING 
+ 
+YOLOv7 is a deep learning-based object detection algorithm 
+that uses a convolutional neural network to detect and classify 
+objects in images and videos. 
+ 
+To train the algorithm, a set of labeled images containing the 
+objects to be detected are needed. The images must be divided in two 
+datasets: a training set and a test set. The training set is used to train 
+the neural network and the test set is used to evaluate the 
+performance of the model once trained. 
+ 
+Training process consists in showing the neural network a set 
+of labeled images and to make it learn to detect and classify the 
+objects in them. To do this, a technique called backpropagation is 
+used, which involves adjusting weights of the neural network based 
+on the errors made in classifying the objects in the images. This 
+process is repeated many times, using different training images each 
+time, until the model reaches an acceptable level of accuracy. 
+ 
+Once trained, the model can be used to detect and classify 
+objects in new images and videos. In general, the larger the training 
+set and the better labeled the images are, the better the model 
+performs in object detection tasks. 
+ 
+The model training dataset contains 1,520 baseline images of 
+smoke plumes in different conditions and viewed from different 
+perspectives, incorporating varied geographic settings to improve 
+model resilience.  
+ 
+Applying the redundancy characteristics, the dataset was 
+strengthened to 2,712 images, distributed as follows: 
+ 
+A. Training Set 
+ 
+Set of 2,405 images to train the neural network of the 
+algorithm to classify the smoke in them. All the images in the dataset 
+contain a bounding box with the exact location of the object to be 
+detected, in this case, the smoke plumes. 
+   
+B. Validation Set 
+ 
+Set of 228 images on which the model is evaluated after 
+training. This set is of relevance for the evaluation metric, as it is the 
+first indicator of model performance during the training. 
+ 
+C. Test Set 
+ 
+Set of 79 images that are unknown to the neural network and 
+were used neither for training nor for validation. It is used to assess 
+the performance of the model against new scenarios. Its metrics are 
+considered the most important because it establishes a performance 
+indicator against the desired scenarios.  
+ 
+VIII. TRAINING PARAMETERS 
+ 
+Model training requires computational power. The higher the 
+computational capacity, faster training process will be done, which 
+in turn will allow a deeper learning process, achieving better 
+performance results. 
+ 
+
+ 
+ 
+The model training process was performed using a pre-trained 
+base model arranged by the YOLOv7 algorithm on the Google Colab 
+platform, using an Nvidia A100-SXM4 GPU with 40 Gb of memory. 
+ 
+A. Batch Size 
+ 
+Batch size is a parameter used in the training process of a 
+machine learning model. It refers to the number of training samples 
+to be processed before updating the model weights. 
+ 
+For example, if the batch size is 32, it means that the model 
+will process 32 training samples at a time and then adjust their 
+weights accordingly. It will then process another batch of 32 samples 
+and adjust the weights again, and so on until all training samples are 
+processed. 
+ 
+Batch size is a parameter that can significantly affect model 
+performance during training. Too small batch size can make training 
+slower as more weight updates are performed, but it can also 
+improve model accuracy. Otherwise, too large batch size can make 
+training faster, but can also reduce model accuracy. Therefore, it is 
+important to choose an appropriate batch size based on the needs of 
+the model and the data set. 
+ 
+The final model is the result of four training phases with 
+different batch sizes. 
+ 
+B. EPOCH or training iterations 
+ 
+An epoch is a complete iteration through the entire training set 
+during the training process of a machine learning model. For 
+example, if the training set has 1,000 samples and the batch size is 
+32, it will take 32 iterations to complete one epoch, since 32 x 32 = 
+1,000~. 
+ 
+During each epoch, the model processes the training samples 
+in batches and adjusts their weights accordingly. At the end of each 
+epoch, the model's performance is evaluated using a test data set and 
+used to assess the model's progress. 
+The number of epochs used during model training is another 
+parameter that can significantly affect model performance. Too 
+small number of epochs can result in an under-fitted model, while 
+too large number can result in an over-fitted model. Therefore, it is 
+important to choose an appropriate number of epochs based on the 
+needs of the model and the data set, the available resources and time. 
+ 
+The smoke detection model was trained in four sessions of 300 
+epochs and a final session of 500 epochs, with a total duration of 
+32.15 hours. 
+ 
+IX. EVALUATION METRICS 
+ 
+A. Mean average precision (mAP) 
+ 
+[email protected] is a performance measure commonly used in object 
+detection tasks that refers to the average detection accuracy mAP 
+(mean Average Precision) for different values of the Intersection 
+over Union (IoU) threshold. 
+ 
+The mAP detection accuracy refers to the average accuracy of 
+an object detection model in correctly detecting and classifying 
+objects in a set of test images. It is calculated by comparing the 
+model predictions with the truth labels of the objects in the test 
+images and measuring the average accuracy across all images. 
+ 
+The IoU threshold refers to the ratio of overlap between the 
+model prediction and the truth label of an object in an image. For 
+example, if the IoU threshold is 0.5, it means that the model 
+prediction is considered correct only if the overlap between the 
+prediction and the truth label is 50% or more. 
+ 
+B. F1 Curve 
+ 
+The F1 curve is a tool commonly used in classification tasks 
+to evaluate the performance of a model. It is used to evaluate the 
+accuracy and recall of a model at different classification thresholds. 
+ 
+Accuracy refers to the proportion of correct model predictions 
+out of the total predictions made. Recall refers to the proportion of 
+correct model predictions over the total number of positive cases in 
+the data set. 
+ 
+The F1 curve is calculated using the formula: 
+𝐹1 = 2 ∗
+(𝐴𝑐𝑐𝑢𝑟𝑎𝑐𝑦 ∗ 𝑅𝑒𝑐𝑎𝑙𝑙 )
+(𝐴𝑐𝑐𝑢𝑟𝑎𝑐𝑦 + 𝑅𝑒𝑐𝑎𝑙𝑙) 
+ 
+This formula combines accuracy and recall in a single measure 
+and is useful when it is important to balance both metrics. 
+ 
+To draw the F1 curve, the classification threshold is varied, 
+and the accuracy and recall are calculated for each threshold. The 
+accuracy and recall values are then plotted on a graph and connected 
+by a line. The result is a curve showing how accuracy and recall vary 
+as the classification threshold changes. The F1 curve is useful for 
+evaluating model performance at different thresholds and for 
+choosing the optimal threshold for the model. 
+ 
+XI. EVALUATION OF THE MODEL 
+ 
+A. Model N° 1 
+Fig. 7 - PR Curve Model No. 1 - Own elaboration 
+The first trained model shows a mean average mAP accuracy 
+of 0.379, that is 37.9% correct on the test set. 
+ 
+Regarding the F1 curve, the model obtained a score of 0.44 
+when the confidence value is set at 0.215. 
+
+1.0
+smoke 0.379
+all classes 0.379 [email protected]
+0.8 -
+0.6
+Precision
+0.4 
+0.2 -
+0.0 +
+0.0
+0.2 
+0.4
+0.6
+0.8
+1.0
+Recall 
+6 
+Fig. 8 - Curve F1 Model No. 1 - Own elaboration 
+ 
+The above results are considered deficient, since their best 
+performance does not exceed 50% effectiveness, and occurs when 
+the confidence value of the model is low, therefore, it has a high 
+tendency to generate false positives. 
+ 
+The confidence level is always a relevant factor in model 
+training, because the lower the confidence level is maintained with 
+good results, it is a sign of resilient learning and resistance to false 
+positives. 
+ 
+B. Model No. 2 
+ 
+Fig. 9 - PR Curve Model No. 2 - Own elaboration 
+The second trained model obtained a mean average mAP 
+accuracy of 0.684, that is 68.4% correct on the test set. The result 
+implies a significant improvement over the first model and is mainly 
+because the weights of the previously trained neural network were 
+used for the new model, collecting the previous learning. 
+ 
+Regarding the F1 curve, the model obtained a score of 0.69 
+when the confidence value is set at 0.313. 
+ 
+This result is much better than the previous one, in that it 
+obtains 69% accuracy even when the confidence value is low, that is 
+when the model is more susceptible to false positives.  
+C. Model No. 3 
+ 
+For the training of Model No. 3, a cleaning of the dataset was 
+performed, eliminating images that were considered ambiguous to 
+the human eye or were far from the objective of what the model is 
+required to learn to detect. This change allowed to improve the 
+training time, however, there were no significant changes in the 
+results, keeping the same values of model N° 2. 
+ 
+D. Model No. 4 
+ 
+Model No. 4 was trained with different parameters than those 
+used previously. For the previous cases, batch sizes of 64 and 32 
+with 300 iterations were used. 
+ 
+For this case a batch size of 16 was used and 500 iterations 
+were performed. This increased the training time considerably and 
+while it improved the results, it was not a significant increase in the 
+first instance. 
+ 
+Fig. 10 - PR Curve Model No. 4 - Own elaboration 
+In relation to the MAP, a score of 0.698 was obtained, only 
+slightly higher than the previous result. 
+Fig. 11 - Curve F1 Model No. 4 - Own elaboration 
+ 
+ 
+
+1.0
+smoke
+all classes 0.44 at 0.215
+0.8 -
+0.6
+0.4 -
+0.2 
+0.0 +
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Confidence1.0
+smoke 0.684
+all classes 0.684 [email protected]
+0.8 
+0.6
+Precision
+0.4
+0.2 
+0.0+
+0.0
+0.2 
+0.4 
+0.6
+0.8
+1.0
+Recall1.0
+smoke 0.698
+all classes 0.698 [email protected]
+0.8 -
+0.6
+Precision
+0.4
+0.2 -
+0.0 +
+0.0
+0.2
+0.4
+0.6 
+0.8
+1.0
+Recall1.0
+smoke
+all classes 0.74 at 0.298
+0.8 -
+0.6 -
+0.4 -
+0.2 
+0.0 +
+0.0
+0.2 
+0.4
+0.6
+0.8
+1.0
+Confidence 
+ 
+However, in relation to the F1 curve, the model showed 
+significantly better results, reaching a score of 0.74 when the 
+confidence level is 0.298, a higher score was obtained and at lower 
+confidence levels, when conditions are advantageous to false 
+positives. This demonstrates the resilience and effectiveness of the 
+model in detecting smoke plumes. 
+ 
+On the other hand, this model proved to make predictions with 
+greater confidence than the previous ones, mainly because it 
+considers the learning from the previous models. 
+Fig. 12 - Test lot Model N° 1 - Own elaboration 
+Fig. 13 - Test lot Model N° 4 - Own elaboration 
+ 
+XII. SYSTEM INSTALLATION AND 
+IMPLEMENTATION 
+ 
+To perform inference, the trained model must be loaded into 
+an inference application: The first step is to load the trained model 
+into an inference application, such as TensorFlow or PyTorch. This 
+requires providing the path to the model file and loading it into 
+memory. 
+ 
+Then, if the input image differs from the parameters expected 
+by the model it is necessary to preprocess the input image. This may 
+include resizing the image to the dimension expected by the model, 
+normalizing the pixel values, and converting the image to a tensor. 
+ 
+Once the input image is ready, you can run the model using the 
+model inference method and provide the input image as input. This 
+will return the model predictions in the form of a tensor. 
+Model predictions are often in tensor form and can be difficult 
+to interpret directly. Therefore, it is necessary to process the 
+predictions to obtain useful information, such as the coordinates of 
+the bounding boxes of the detected objects and the corresponding 
+object classes. 
+ 
+Once the predictions have been processed, it is possible to 
+visualize them by overlaying the object labels on the input image or 
+by displaying the predictions in tabular form. This can help to 
+evaluate the performance of the model and to understand how it 
+works. 
+ 
+A tensor is a mathematical object used in the field of artificial 
+intelligence and object detection to represent and manipulate 
+multidimensional data. Tensors are fundamental elements in data 
+processing and are widely used in machine learning and data 
+analysis. 
+ 
+A tensor can be viewed as a generalization of a matrix, which 
+is a two-dimensional data structure used to represent and manipulate 
+data sets. Like a matrix, a tensor can have more than one dimension, 
+and each dimension is known as an axis. Tensor can be used to 
+represent data in many different forms, such as images, videos, 
+audios and texts. 
+ 
+In the area of artificial intelligence and object detection, 
+tensors are used to process and analyze large amounts of input data, 
+such as images or videos, and to produce output results, such as class 
+labels or predictions. Tensors are also used in natural language 
+processing and machine translation, among other applications. 
+ 
+Fig. 14 - Model No. 4 applied to smoke image with 91% success rate 
+To use the model in video cameras, either in real time or by 
+obtaining images from them, the capture device must be connected 
+to a processing device. This can be a computer or a Raspberry Pi. 
+ 
+It is important to point out that the model does not need to be 
+implemented in the same device that captures the images from the 
+camera, since the architecture designed to meet the objectives of the 
+model is built using the client-server mode, where the clients 
+correspond to one or several sources of information while the server 
+
+Humo:0.91 
+8 
+corresponds to the source where the model is executed and the 
+inferences are made. 
+ 
+For the test model, a home computer with an Nvidia RTX 3060 
+graphics processing card with 16GB of memory was used, using 
+Windows 11 operating system with the Anaconda data analytics 
+environment installed.  
+The PyTorch library package was installed on the computer 
+and through a FrontEnd designed with Flask in Python, a web site 
+was generated to capture free access images from Chilean airfield 
+cameras through scraping to perform tests. 
+ 
+XIII. MODEL IMPROVEMENT 
+ 
+Although the trained model presents an acceptable result, the 
+latest tests indicated that to improve it, it is necessary to make a 
+series of changes, which are detailed below: 
+ 
+1) 
+Use a larger and better labeled training set: often, the 
+larger the training set and the better labeled the images are, 
+the better the performance of the model. 
+ 
+2) 
+Adjust model hyperparameters: there are several 
+hyperparameters that can affect model performance, such 
+as the batch size and the number of epochs used during 
+training. Adjusting these hyperparameters can improve 
+model performance. 
+ 
+3) 
+Use a more complex neural network architecture: using a 
+neural network with more layers or with more units in each 
+layer can improve model performance, but it can also 
+increase training time and the need for more training data. 
+ 
+4) 
+Use regularization techniques: Regularization is a 
+technique used to avoid overfitting the model and improve 
+its 
+generalization. 
+Some 
+common 
+regularization 
+techniques include L1 and L2 regularization, dropout and 
+early stopping. 
+ 
+5) 
+Use advanced optimization techniques: There are several 
+advanced optimization techniques that can improve model 
+performance, such as stochastic gradient descent (SGD), 
+Adam and Adagrad. Using these techniques can improve 
+training speed and accuracy. 
+ 
+XIV. CONCLUSIONS 
+ 
+Undoubtedly, the phenomenon of Wildfires will increase. On 
+the one hand, due to climate change and, on the other, to social 
+phenomena such as migration, the displacement of families from the 
+city to the countryside, and intentionality, among others, which will 
+significantly increase vulnerability to this type of anthropogenic 
+event, both in terms of occurrence and severity. 
+ 
+Given this scenario, it is important that authorities, civil 
+society, and people in general become aware of the seriousness of 
+this situation and adopt preventive behaviors that contribute to 
+mitigating the effects of fires through self-care practices such as 
+preventive forestry. 
+ 
+On the other hand, in the face of the inevitable occurrence of 
+forest emergencies, having early warning systems in place will help 
+reduce response times and thus ensure that forest emergencies can 
+be controlled by first responders in less time, thereby reducing their 
+effects on people, their property and the environment. 
+ 
+The present model offers an alternative that complements early 
+warning systems, both at the state and private levels, through science 
+and technology, using the tools that Artificial Intelligence offers and 
+that can be implemented in a simple way and with minimal 
+knowledge of computer science and programming. 
+ 
+Although the current model has an acceptable performance, to 
+improve it, it is necessary to have a larger and better labeled training 
+set that allows the neural network to learn more and better scenarios 
+of forest fire occurrence in the initial stage. Likewise, it is necessary 
+to rescue the learning of the previous models in the training process, 
+adjusting the parameters so in each learning cycle the efficiency is 
+maximized. 
+ 
+The resources required by the system are fully achievable by 
+the organizations with a low cost vs. benefit, it does not require a 
+large number of people in its use since it works mainly in an 
+automated way and the investment in infrastructure (cameras, 
+internet, towers, masts, etc.), is quickly amortized if compared 
+against the cost of maintenance of conventional systems 
+(observation towers with their respective towers, respectively). 
+ 
+Although this technology is not intended to replace the role of 
+human beings in the detection of Wildfires, it does seek to position 
+itself as an important support element in the efforts to prevent and 
+mitigate the adverse effects that may be generated. 
+ 
+ACKNOWLEDGMENTS 
+ 
+The present work would not be possible without the support of 
+Dwyer, B., Nelson, J. from Roboflow Computer Vision, who trusted 
+in this project and sponsored it, granting in their platform features 
+that allowed to build a bigger dataset, of better quality, applying 
+preprocessing tasks and increasing features, thank you very much. 
+ 
+REFERENCES 
+ 
+[1] National Forestry Corporation (14 December 2022). CONAF. 
+Retrieved from Estadística de Ocurrencia de Incendios Forestales: 
+https://www.conaf.cl/wp-
+content/files_mf/1662998364TABLA1_TEMPORADA2021_01_12
+.09.22_version2022.xls 
+ 
+[2] Borunda, A. (September 21, 2020). National Geographic. Retrieved 
+from 
+Science: 
+https://www.nationalgeographicla.com/ciencia/2020/09/cual-es-la-
+relacion-entre-los-incendios-forestales-y-el-cambio-climatico 
+ 
+[3] Ortega, M. (2013). In Chile Forestal (pp. 37, 38). Corporación 
+Nacional Forestal. 
+ 
+[4] Morris, J. (1995). Computer vision in robotics. Computer Vision in 
+Robotics, 1-20. 
+ 
+[5] Dwyer, B., Nelson, J. (2022). Roboflow (Version 1.0) [Software]. 
+Available from https://roboflow.com. computer vision. 
+ 
+[6] Wang, Chien-Yao, Bochkovskiy, Alexey and Liao, Hong-Yuan 
+Mark (2022). "YOLOv7: Trainable bag-of-freebies sets new state-
+of-the-art for real-time object detectors. " ("YOLOv7: Trainable bag-
+of-freebies sets new state-of-the-art for real ...") arXiv preprint 
+arXiv:2207.02696. https://arxiv.org/abs/2207.02696 
+
+ 
+ 
+[7] Karimi, G. (April 15, 2021). Introduction to YOLO algorithm for 
+object detection. Section.io. https://www.section.io/engineering-
+education/introduction-to-yolo-algorithm-for-object-detection/ 
+ 
+[8] Aiformankind (August 28, 2020). Wildfire smoke detection research. 
+https://github.com/aiformankind/wildfire-smoke-detection-research. 
+ 
+[9] S. Abdullah, S. Bertalan, S. Masar, A. Coskun and I. Kale, "A 
+wireless sensor network for early forest fire detection and monitoring 
+as a decision factor in the context of a complex integrated emergency 
+response system," 2017 IEEE Workshop on Environmental, Energy, 
+and Structural Monitoring Systems (EESMS), 2017, pp. 1-5, doi: 
+10.1109/EESMS.2017.8052688. 
+ 
+[10] Hohberg, S. (09/20/2015). Wildfire Smoke Detection using 
+Convolutional Neural Networks. Berlin University of Technology. 
+https://www.inf.fu-berlin.de/inst/ag-
+ki/rojas_home/documents/Betreute_Arbeiten/Master-Hohberg.pdf 
+ 
+[11] Andreasson, H., & Persson, M. (2016). Wildfire smoke detection 
+based on local extremal region segmentation and surveillance. Forest 
+Ecology 
+and 
+Management, 
+379, 
+330-342. 
+https://www.sciencedirect.com/science/article/abs/pii/S0379711216
+301059. 
+

3tE4T4oBgHgl3EQfbQwY/content/tmp_files/load_file.txt

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf,len=438
+page_content='Wildfire Smoke Detection by Computer Vision Eldan R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=', Daniel I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' December 26, 2022  deldanr@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='com  Abstract- Wildfires are becoming more frequent and their effects more devastating every day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' Climate change has directly and indirectly affected the occurrence of these, as well as social phenomena have increased the vulnerability of people.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' Consequently, and given the inevitable occurrence of these, it is important to have early warning systems that allow a timely and effective response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' Artificial intelligence, machine learning and Computer Vision offer an effective and achievable alternative for opportune detection of wildfires and thus reduce the risk of disasters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' YOLOv7 offers a simple, fast, and efficient algorithm for training object detection models which can be used in early detection of smoke columns in the initial stage wildfires.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' The developed model showed promising results, achieving a score of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='74 in the F1 curve when the confidence level is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='298, that is, a higher score at lower confidence levels was obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' This means when the conditions are favorable for false positives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' The metrics demonstrates the resilience and effectiveness of the model in detecting smoke columns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='  Keywords:  Early  Warning,  Object  Detection,  Artificial  Intelligence, Computer Vision, YOLO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' INTRODUCTION  A wildfire is a fire that, whatever its origin and with danger or damage to people, property, or the environment, spreads uncontrolled in rural areas, through woody, bushy or herbaceous vegetation, alive or dead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' In other words, it is an unjustified and uncontrolled fire in which the fuels are plants and which, in its propagation, can destroy everything in its path ("Wildfires in Chile - CONAF").' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='  In the last 10 years there have been 67,567 Wildfires, affecting an area of 1,246,922 hectares of grassland, scrubland, forest plantations, native forest, agricultural land, among others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='  Climate change has increased the risk of Wildfires both directly and indirectly (Borunda, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' Although the causality of fires is 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='7% human, the conditions for the generation of these fires are higher than they would be without climate change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='  Given this scenario, it is significant to have early warning systems that, in the event of an inevitable occurrence of a forest fire, make it possible to activate and deploy the necessary resources for its rapid control and extinction, thus preserving the lives of people, their property and the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' FOREST FIRE DETECTION SYSTEMS  Wildfires are incidents with a high destructive potential and a sudden growth, even more so when weather conditions allow it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' Therefore, is very important to apply a rapid firefighting strategy that prevents fires from growing in extent and severity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='  The early detection of fires is essential to initiate procedures that culminate in firefighting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' Among them is the notification of the start of the fire to the Regional Coordination Center of CONAF (CENCOR) who, in turn, with the respective technical background, analyze the situation and generate the dispatch of relevant land and/or air resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' Mobile Terrestrial Detection  The task consists of moving surveillance people to a given area, either by vehicle or on foot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' This practice is quite common in Chile in forestry companies, where it is used to supervise work activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' Fixed Terrestrial Detection  This is the most widely used form of detection in Chile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' It consists of having a person observing from metal or wooden towers that are between 15 and 30 meters high, or from lower booths known as detection posts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' Airborne Detection  This detection method uses aircraft, usually single-engine high-wing aircraft, to detect fires from the air.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' The pilot is accompanied by an observer, who oversees doing the observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' This technique makes possible to observe a large amount of area in an abbreviated time and provides accurate and detailed information about the detected fire and the area over which it is flown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' However, its operating cost is high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' Detection with television systems  This method uses television cameras to transmit their signal via microwaves to screens at a command post, such as in a vehicle in the field or at a coordination center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' There, specialists analyze the situation based on what they see on the screen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' Satellite Systems  In some parts of the world, due to the lack of forest fire protection organizations or detection systems, the only way to know what is happening is to use low orbit satellite images, such as those provided by the Aqua and Terra satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='  2 III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' OBJECT DETECTION BY COMPUTER VISION  Computer vision, also known as artificial vision or technical vision, is a scientific discipline that involves techniques for acquiring, processing, analyzing and understanding images of the real world to produce numerical or symbolic information that can be processed by computers (J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' Morris, 1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' Just as humans use our eyes and brains to make sense of the world around us, computer vision seeks to create the same effect by allowing a computer to perceive and understand an image or sequence of images and act accordingly given the situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' This understanding is achieved through fields as diverse as geometry, statistics, physics and other disciplines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' Data collection is achieved in a variety of ways, such as image sequences viewed from multiple cameras or multidimensional data from medical scanners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='  Real-time object detection is a particularly important topic in computer vision, as it is often a necessary component in computer vision systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' Some of its current applications are object tracking, public safety and active surveillance, autonomous vehicle driving, robotics, medical image analysis, among others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='  Computing devices that run real-time object detection processes usually use CPUs or GPUs for their tasks, however, nowadays the computational capacity has improved exponentially with the Neural Processing Units (NPU) developed by different manufacturers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='  These devices focus on accelerating operations through several types of algorithms, one of the most widely used being the multilayer perceptron or Multilayer Perceptron (MLP), an artificial neural network formed by multiple layers in such a way that it has the ability to solve problems that are not linearly separable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' YOLO  The object detection algorithm used in the present work is YOLO (You only look once), developed by Wang, Chien-Yao et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' al, whose latest version was recently released in July 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='  YOLO is an algorithm that uses neural networks to provide real-time object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' It is an algorithm known for its speed and accuracy and YOLO is currently used in a variety of applications such as traffic signal detection, people accounting, detection of available spaces in private parking lots, remote animal surveillance, among others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' Operation of YOLO  The YOLO algorithm works by using three techniques: • Intersection over Union (IOU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' • Regression of the bounding box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' • Residual blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' Residual blocks  The analyzed image, which can be a frame of a sequence (video), is divided into several grids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' Each grid has a dimension SxS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' The following image shows an example of grids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' Each cell will detect the objects that appear inside them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' For example, if an object appears inside a given cell, the cell will perform processing on its own and separately from the others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' 1 - Example of residual block, source: guidetomlandai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='com C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' Regression of the bounding box  A bounding box is an outline that highlights an object within an image or cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' Each box has a height, a width, a class (what we are looking for: car, dog, traffic light, fire smoke) and a centroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' The following image shows an example of a bounding box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' YOLO uses a single bounding box regression to predict the items listed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' 2 - Example of bounding box, source: appsilondatascience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='com D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' Intersection over Union (IOU)  Intersection over union is a phenomenon in object detection that describes how blocks overlap in an image, where block is understood as the set of cells where the detected object is located.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='  YOLO uses IOU to provide an output block surrounding the detected object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' Each grid cell is responsible for predicting the bounding boxes and their confidence score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='  y=(pc,br,b,,bh,bw,C  b  b  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' 3 - Example of Intersection over Union, source: miro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='com E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' Output result  YOLO combines the three techniques for accurate detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' First, having the SxS grid of the analyzed image allows to evaluate each section individually and be able to detect the bounding boxes and their respective confidence scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='  For each bounding box, the class of detected object is set and finally, using IOU, the frame is adjusted to ensure that the detection frame covers the entire real object in the output image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' 4 - Diagram of the YOLO algorithm, source: guidetomlandai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='com V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' CREATION OF THE MODEL  To detect objects YOLO algorithm requires a model trained with the class or classes of the search elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' For this it is important to establish specifically where, how and when the model will operate to detect fires, for which the following criteria are established:  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' Location of the observer  The analyzed images by the model and used for fire detection were obtained from distant sources, with a wide view of valley areas, forests and/or mountain ranges, above level and with unpredictable atmospheric conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='  Such conditions of observations are those that we could identify in an observation tower or fire watch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' It should be considered that the resolution of these can be varied and not uniform, depending on the capture device used (webcam, HD camera).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' Type of wildfire to be detected  As the objective of the system is to detect fires in their initial stage, we will discard any images with fire and concentrate on smoke plumes and their development, ideally taken from cameras in different scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' 5- Rodelillo airfield webcam, Valparaíso, December 7, 2020, 16:20 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' Redundancy of training images  To generate greater variability and resilience to the model, modifications have been made to part of the image dataset to increase the amount of material for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='  In this regard, the following characteristics were applied to the dataset:  1) Mirror effect: The images were duplicated with a horizontal rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' This allows to have training material for different wind conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='  2) Exposure: Duplicate images were generated with changes in exposure between -15% and +15%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' This allows improving the visibility of the smoke plume in images that may have been taken with different levels of ambient humidity, which at greater distances distorts the focus and sharpness of the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='  Also, the redundancy modifications and the labeling of the images in the dataset were made in the Roboflow app, a computer vision web software that provides many functions for upload, label, augmentation, export, train and testing models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' SOURCES OF INFORMATION  To increase the effectiveness of the model, it is important to train it with images that are as similar as possible to the scenarios where it will be implemented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' In view of the above, different sources of information were selected to obtain images with a wide range of  Boundingboxes+ +confidence SxSqridoninput Final detections Classprobabilitymap 4 geographic environments to generate a resilient model that can be implemented in different locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' High Performance Wireless Research and Education Network (HPWREN)  The High-Performance Wireless Research and Education Network is a University of California partnership project led by the San Diego Supercomputing Center and the Institute for Geophysics and Planetary Physics at Scripps Institution of Oceanography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='  HPWREN works as a collaborative cyber infrastructure connected to the Internet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' The project has a vast network of cameras in the State of California, USA, which have been used for wildfire observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='  In particular, the HPWREN images were obtained from the AI for Mankind project, founded by Wei Shung Chung.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' Social Networks  Wildfires are high-impact emergencies and are considered by society as public interest events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' Therefore, a search for images of Wildfires was made on the Twitter platform using the hashtag "Wildfire" in Spanish, English, Turkish, Greek, Russian and Portuguese.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' This allowed access to a variety of images with different types of geography and relatively recent, allowing the generation of an updated model training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' Images created with Artificial Intelligence  In an innovative way, the well-known artificial intelligences Dall-E from OpenAI and Stable Diffusion from StabilityAI were used to generate images using the following input phrase: "Wildfire smoke in early stage as seen from an observation tower or high and distant point".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' 6 - Forest fire smoke image created with Dall-E D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' Self-made computer Images  To complement the dataset with smoke columns originating in different places, images were generated by superimposing layers with the Photoshop application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' For this purpose, base images of cameras and observation towers without smoke were selected and new images were artificially created with different types of smoke originating from different points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' MODEL TRAINING  YOLOv7 is a deep learning-based object detection algorithm that uses a convolutional neural network to detect and classify objects in images and videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='  To train the algorithm, a set of labeled images containing the objects to be detected are needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' The images must be divided in two datasets: a training set and a test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' The training set is used to train the neural network and the test set is used to evaluate the performance of the model once trained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='  Training process consists in showing the neural network a set of labeled images and to make it learn to detect and classify the objects in them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' To do this, a technique called backpropagation is used, which involves adjusting weights of the neural network based on the errors made in classifying the objects in the images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' This process is repeated many times, using different training images each time, until the model reaches an acceptable level of accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='  Once trained, the model can be used to detect and classify objects in new images and videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' In general, the larger the training set and the better labeled the images are, the better the model performs in object detection tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='  The model training dataset contains 1,520 baseline images of smoke plumes in different conditions and viewed from different perspectives, incorporating varied geographic settings to improve model resilience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='  Applying the redundancy characteristics, the dataset was strengthened to 2,712 images, distributed as follows:  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' Training Set  Set of 2,405 images to train the neural network of the algorithm to classify the smoke in them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' All the images in the dataset contain a bounding box with the exact location of the object to be detected, in this case, the smoke plumes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' Validation Set  Set of 228 images on which the model is evaluated after training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' This set is of relevance for the evaluation metric, as it is the first indicator of model performance during the training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' Test Set  Set of 79 images that are unknown to the neural network and were used neither for training nor for validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' It is used to assess the performance of the model against new scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' Its metrics are considered the most important because it establishes a performance indicator against the desired scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='  VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' TRAINING PARAMETERS  Model training requires computational power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' The higher the computational capacity, faster training process will be done, which in turn will allow a deeper learning process, achieving better performance results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='  The model training process was performed using a pre-trained base model arranged by the YOLOv7 algorithm on the Google Colab platform, using an Nvidia A100-SXM4 GPU with 40 Gb of memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' Batch Size  Batch size is a parameter used in the training process of a machine learning model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' It refers to the number of training samples to be processed before updating the model weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='  For example, if the batch size is 32, it means that the model will process 32 training samples at a time and then adjust their weights accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' It will then process another batch of 32 samples and adjust the weights again, and so on until all training samples are processed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='  Batch size is a parameter that can significantly affect model performance during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' Too small batch size can make training slower as more weight updates are performed, but it can also improve model accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' Otherwise, too large batch size can make training faster, but can also reduce model accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' Therefore, it is important to choose an appropriate batch size based on the needs of the model and the data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='  The final model is the result of four training phases with different batch sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' EPOCH or training iterations  An epoch is a complete iteration through the entire training set during the training process of a machine learning model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' For example, if the training set has 1,000 samples and the batch size is 32, it will take 32 iterations to complete one epoch, since 32 x 32 = 1,000~.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='  During each epoch, the model processes the training samples in batches and adjusts their weights accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=" At the end of each epoch, the model's performance is evaluated using a test data set and used to assess the model's progress." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' The number of epochs used during model training is another parameter that can significantly affect model performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' Too small number of epochs can result in an under-fitted model, while too large number can result in an over-fitted model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' Therefore, it is important to choose an appropriate number of epochs based on the needs of the model and the data set, the available resources and time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='  The smoke detection model was trained in four sessions of 300 epochs and a final session of 500 epochs, with a total duration of 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='15 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='  IX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' EVALUATION METRICS  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' Mean average precision (mAP)  mAP@.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='5 is a performance measure commonly used in object detection tasks that refers to the average detection accuracy mAP (mean Average Precision) for different values of the Intersection over Union (IoU) threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='  The mAP detection accuracy refers to the average accuracy of an object detection model in correctly detecting and classifying objects in a set of test images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' It is calculated by comparing the model predictions with the truth labels of the objects in the test images and measuring the average accuracy across all images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='  The IoU threshold refers to the ratio of overlap between the model prediction and the truth label of an object in an image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' For example, if the IoU threshold is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='5, it means that the model prediction is considered correct only if the overlap between the prediction and the truth label is 50% or more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' F1 Curve  The F1 curve is a tool commonly used in classification tasks to evaluate the performance of a model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' It is used to evaluate the accuracy and recall of a model at different classification thresholds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='  Accuracy refers to the proportion of correct model predictions out of the total predictions made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' Recall refers to the proportion of correct model predictions over the total number of positive cases in the data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='  The F1 curve is calculated using the formula: 𝐹1 = 2 ∗ (𝐴𝑐𝑐𝑢𝑟𝑎𝑐𝑦 ∗ 𝑅𝑒𝑐𝑎𝑙𝑙 ) (𝐴𝑐𝑐𝑢𝑟𝑎𝑐𝑦 + 𝑅𝑒𝑐𝑎𝑙𝑙)  This formula combines accuracy and recall in a single measure and is useful when it is important to balance both metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='  To draw the F1 curve, the classification threshold is varied, and the accuracy and recall are calculated for each threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' The accuracy and recall values are then plotted on a graph and connected by a line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' The result is a curve showing how accuracy and recall vary as the classification threshold changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' The F1 curve is useful for evaluating model performance at different thresholds and for choosing the optimal threshold for the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='  XI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' EVALUATION OF THE MODEL  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' Model N° 1 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' 7 - PR Curve Model No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' 1 - Own elaboration The first trained model shows a mean average mAP accuracy of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='379, that is 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='9% correct on the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='  Regarding the F1 curve, the model obtained a score of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='44 when the confidence value is set at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='215.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='0 smoke 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='379 all classes 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='379 mAP@0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='8 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='6 Precision 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='2 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='0 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='0 Recall 6 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' 8 - Curve F1 Model No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' 1 - Own elaboration  The above results are considered deficient, since their best performance does not exceed 50% effectiveness, and occurs when the confidence value of the model is low, therefore, it has a high tendency to generate false positives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='  The confidence level is always a relevant factor in model training, because the lower the confidence level is maintained with good results, it is a sign of resilient learning and resistance to false positives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' Model No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' 2  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' 9 - PR Curve Model No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' 2 - Own elaboration The second trained model obtained a mean average mAP accuracy of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='684, that is 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='4% correct on the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' The result implies a significant improvement over the first model and is mainly because the weights of the previously trained neural network were used for the new model, collecting the previous learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='  Regarding the F1 curve, the model obtained a score of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='69 when the confidence value is set at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='313.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='  This result is much better than the previous one, in that it obtains 69% accuracy even when the confidence value is low, that is when the model is more susceptible to false positives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' Model No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' 3  For the training of Model No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' 3, a cleaning of the dataset was performed, eliminating images that were considered ambiguous to the human eye or were far from the objective of what the model is required to learn to detect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' This change allowed to improve the training time, however, there were no significant changes in the results, keeping the same values of model N° 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' Model No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' 4  Model No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' 4 was trained with different parameters than those used previously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' For the previous cases, batch sizes of 64 and 32 with 300 iterations were used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='  For this case a batch size of 16 was used and 500 iterations were performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' This increased the training time considerably and while it improved the results, it was not a significant increase in the first instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' 10 - PR Curve Model No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' 4 - Own elaboration In relation to the MAP, a score of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='698 was obtained, only slightly higher than the previous result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' 11 - Curve F1 Model No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' 4 - Own elaboration  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='0 smoke all classes 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='44 at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='215 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
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+page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='0 Confidence1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='0 smoke 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='684 all classes 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='684 mAP@0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
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+page_content='6 Precision 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='0+ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
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+page_content='0 Confidence  However, in relation to the F1 curve, the model showed significantly better results, reaching a score of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='74 when the confidence level is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='298, a higher score was obtained and at lower confidence levels, when conditions are advantageous to false positives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' This demonstrates the resilience and effectiveness of the model in detecting smoke plumes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='  On the other hand, this model proved to make predictions with greater confidence than the previous ones, mainly because it considers the learning from the previous models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' 12 - Test lot Model N° 1 - Own elaboration Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' 13 - Test lot Model N° 4 - Own elaboration  XII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' SYSTEM INSTALLATION AND  IMPLEMENTATION  To perform inference, the trained model must be loaded into an inference application: The first step is to load the trained model into an inference application, such as TensorFlow or PyTorch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' This requires providing the path to the model file and loading it into memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='  Then, if the input image differs from the parameters expected by the model it is necessary to preprocess the input image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' This may include resizing the image to the dimension expected by the model, normalizing the pixel values, and converting the image to a tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='  Once the input image is ready, you can run the model using the model inference method and provide the input image as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' This will return the model predictions in the form of a tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' Model predictions are often in tensor form and can be difficult to interpret directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' Therefore, it is necessary to process the predictions to obtain useful information, such as the coordinates of the bounding boxes of the detected objects and the corresponding object classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='  Once the predictions have been processed, it is possible to visualize them by overlaying the object labels on the input image or by displaying the predictions in tabular form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' This can help to evaluate the performance of the model and to understand how it works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='  A tensor is a mathematical object used in the field of artificial intelligence and object detection to represent and manipulate multidimensional data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' Tensors are fundamental elements in data processing and are widely used in machine learning and data analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='  A tensor can be viewed as a generalization of a matrix, which is a two-dimensional data structure used to represent and manipulate data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' Like a matrix, a tensor can have more than one dimension, and each dimension is known as an axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' Tensor can be used to represent data in many different forms, such as images, videos, audios and texts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='  In the area of artificial intelligence and object detection, tensors are used to process and analyze large amounts of input data, such as images or videos, and to produce output results, such as class labels or predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' Tensors are also used in natural language processing and machine translation, among other applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' 14 - Model No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' 4 applied to smoke image with 91% success rate To use the model in video cameras, either in real time or by obtaining images from them, the capture device must be connected to a processing device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' This can be a computer or a Raspberry Pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='  It is important to point out that the model does not need to be implemented in the same device that captures the images from the camera, since the architecture designed to meet the objectives of the model is built using the client-server mode, where the clients correspond to one or several sources of information while the server  Humo:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='91 8 corresponds to the source where the model is executed and the inferences are made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='  For the test model, a home computer with an Nvidia RTX 3060 graphics processing card with 16GB of memory was used, using Windows 11 operating system with the Anaconda data analytics environment installed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' The PyTorch library package was installed on the computer and through a FrontEnd designed with Flask in Python, a web site was generated to capture free access images from Chilean airfield cameras through scraping to perform tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='  XIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' MODEL IMPROVEMENT  Although the trained model presents an acceptable result, the latest tests indicated that to improve it, it is necessary to make a series of changes, which are detailed below:  1) Use a larger and better labeled training set: often, the larger the training set and the better labeled the images are, the better the performance of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='  2) Adjust model hyperparameters: there are several hyperparameters that can affect model performance, such as the batch size and the number of epochs used during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' Adjusting these hyperparameters can improve model performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='  3) Use a more complex neural network architecture: using a neural network with more layers or with more units in each layer can improve model performance, but it can also increase training time and the need for more training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='  4) Use regularization techniques: Regularization is a technique used to avoid overfitting the model and improve its generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' Some common regularization techniques include L1 and L2 regularization, dropout and early stopping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='  5) Use advanced optimization techniques: There are several advanced optimization techniques that can improve model performance, such as stochastic gradient descent (SGD), Adam and Adagrad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' Using these techniques can improve training speed and accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='  XIV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' CONCLUSIONS  Undoubtedly, the phenomenon of Wildfires will increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' On the one hand, due to climate change and, on the other, to social phenomena such as migration, the displacement of families from the city to the countryside, and intentionality, among others, which will significantly increase vulnerability to this type of anthropogenic event, both in terms of occurrence and severity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='  Given this scenario, it is important that authorities, civil society, and people in general become aware of the seriousness of this situation and adopt preventive behaviors that contribute to mitigating the effects of fires through self-care practices such as preventive forestry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='  On the other hand, in the face of the inevitable occurrence of forest emergencies, having early warning systems in place will help reduce response times and thus ensure that forest emergencies can be controlled by first responders in less time, thereby reducing their effects on people, their property and the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='  The present model offers an alternative that complements early warning systems, both at the state and private levels, through science and technology, using the tools that Artificial Intelligence offers and that can be implemented in a simple way and with minimal knowledge of computer science and programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='  Although the current model has an acceptable performance, to improve it, it is necessary to have a larger and better labeled training set that allows the neural network to learn more and better scenarios of forest fire occurrence in the initial stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' Likewise, it is necessary to rescue the learning of the previous models in the training process, adjusting the parameters so in each learning cycle the efficiency is maximized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='  The resources required by the system are fully achievable by the organizations with a low cost vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' benefit, it does not require a large number of people in its use since it works mainly in an automated way and the investment in infrastructure (cameras, internet, towers, masts, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' ), is quickly amortized if compared against the cost of maintenance of conventional systems (observation towers with their respective towers, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='  Although this technology is not intended to replace the role of human beings in the detection of Wildfires, it does seek to position itself as an important support element in the efforts to prevent and mitigate the adverse effects that may be generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='  ACKNOWLEDGMENTS  The present work would not be possible without the support of Dwyer, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=', Nelson, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' from Roboflow Computer Vision, who trusted in this project and sponsored it, granting in their platform features that allowed to build a bigger dataset, of better quality, applying preprocessing tasks and increasing features, thank you very much.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='  REFERENCES  [1] National Forestry Corporation (14 December 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' CONAF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' Retrieved from Estadística de Ocurrencia de Incendios Forestales: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='conaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='cl/wp- content/files_mf/1662998364TABLA1_TEMPORADA2021_01_12 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='09.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='22_version2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='xls  [2] Borunda, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' (September 21, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' National Geographic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' Retrieved from Science: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='nationalgeographicla.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='com/ciencia/2020/09/cual-es-la- relacion-entre-los-incendios-forestales-y-el-cambio-climatico  [3] Ortega, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' In Chile Forestal (pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' 37, 38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
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+page_content='0) [Software].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' Available from https://roboflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='com.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content=' computer vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
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+page_content=' " ("YOLOv7: Trainable bag- of-freebies sets new state-of-the-art for real .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
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+page_content=' https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
+page_content='org/abs/2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQfbQwY/content/2301.05070v1.pdf'}
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+Thermal annealing of sputtered Nb3Sn and V3Si thin
+films for superconducting radio-frequency cavities
+Katrina Howard1,2, Zeming Sun1, and Matthias U. Liepe1
+1 Cornell Laboratory for Accelerator-Based Sciences and Education, Cornell
+University, Ithaca, NY 14853, USA
+2 Department of Physics, University of Chicago, Chicago, IL 60637, USA
+E-mail: [email protected] (K.H.); [email protected] (Z.S.);
+[email protected] (M.U.L.)
+December 2022
+Abstract.
+Nb3Sn and V3Si thin films are promising candidates as thin films for
+the next generation of superconducting radio-frequency (SRF) cavities.
+However,
+sputtered films often suffer from stoichiometry and strain issues during deposition
+and post annealing. In this study, we explore the structural and chemical effects of
+thermal annealing, both in-situ and post-sputtering, on DC-sputtered Nb3Sn and V3Si
+films of varying thickness on Nb or Cu substrates, extending from our initial studies
+[1]. Through annealing at 950 °C, we successfully enabled recrystallization of 100 nm
+thin Nb3Sn films on Nb substrate with stoichiometric and strain-free grains. For 2 µm
+thick films, we observed the removal of strain and a slight increase in grain size with
+increasing temperature. Annealing enabled a phase transformation from unstable to
+stable structure on V3Si films, while we observed significant Sn loss in 2 µm thick Nb3Sn
+films after high temperature anneals. We observed similar Sn and Si loss on films atop
+Cu substrates during annealing, likely due to Cu-Sn and Cu-Si phase generation and
+subsequent Sn and Si evaporation. These results encourage us to refine our process to
+obtain high-quality sputtered films for SRF use.
+Keywords:
+thermal annealing, A15 superconductors, sputtering, thin film, SRF
+Submitted to: Superconductor Science and Technology
+arXiv:2301.00756v1  [cond-mat.mtrl-sci]  2 Jan 2023
+
+Thermal annealing of sputtered Nb3Sn and V3Si thin films for superconducting RF cavities
+2
+1. Introduction
+Nb3Sn and V3Si thin films are of increas-
+ing interest to the superconducting radio-
+frequency community owing to the quest of
+achieving high accelerating gradient and ef-
+ficiency.
+As niobium-based superconducting
+radio-frequency (SRF) cavities are reaching
+the theoretical limits [2], alternative materials
+are of great interest to continue the quest of
+increasing quality factors, accelerating gradi-
+ents, and efficiency [3]. A15 superconductors
+Nb3Sn and V3Si are promising candidates for
+this role, used as thin films inside either Nb
+or Cu cavities [3, 4].
+Both candidates have
+relatively high critical temperatures (Tc,Nb3Sn
+= 18.3 K and Tc,V3Si = 17.1 K), and Nb3Sn
+is predicted to yield a superheating field of ∼
+440 mT that doubles the Nb limit of ∼ 240 mT
+[3, 5, 6, 7, 8]. These properties could allow cav-
+ity operation at an elevated temperature of ∼ 4
+K and the potential for increased accelerating
+gradients [9]. This higher operating temper-
+ature allows for reduced cryogenic costs and
+simpler infrastructure for particle accelerators
+and their applications [3]. Due to their brittle
+nature and low thermal conductivity, Nb3Sn
+and V3Si are best suited for use as a thin film
+inside a host cavity with better thermal con-
+ductivity, such as Nb or Cu [3, 10, 11].
+Nb3Sn thin films have been achieved
+through vapor diffusion, sputtering, electro-
+plating, and chemical vapor deposition [12, 13,
+14, 15, 16, 17]. In the state-of-the-art vapor
+diffusion, a niobium cavity is placed in a high-
+temperature vacuum furnace, and then tin or
+tin chloride sources are vaporized and allowed
+to diffuse into the niobium surface for alloy-
+ing [3, 9, 13, 18, 19, 20].
+In contrast, sput-
+tering utilizes high-energy plasma to directly
+eject target materials onto a substrate at low
+temperatures [4, 6, 12, 21, 22]. Alternatively,
+Nb3Sn films are fabricated via electroplating in
+aqueous solutions working at near-room tem-
+peratures and atmospheric pressure followed
+by heat treatment [14, 15, 16, 23], or via chem-
+ical vapor deposition that takes advantage
+of reactions between volatile precursors [24].
+Nb3Sn has been successfully vapor-diffused in-
+side cavities, where a single-cell reached gradi-
+ents of 24 MV/m, while Nb3Sn 9- and 5-cells
+reached 15 MV/m, both with Q0’s on the order
+of 1010 at operating temperature 4.4 K [19, 20].
+In cavity tests, maximum surface fields of 120
+mT (pulsed operation) and 80 – 100 mT (CW)
+have been achieved, showing that Nb3Sn cav-
+ities can be operated reliably in a flux-free
+metastable state above the lower critical field
+of this material (around 40 mT) [7, 25].
+In the sputtering process, the film prop-
+erties are tailored by controlling the Ar/Kr
+plasma pressure, substrate temperature, sput-
+tering voltage, sputtering current, rate of de-
+position, and post-sputtering anneal temper-
+ature/duration. In literature [4, 6, 9, 12, 26],
+sputtered Nb3Sn films have been demonstrated
+on Nb and Cu surfaces by using a stoichio-
+metric Nb3Sn target, by co-sputtering with Nb
+and Sn targets, or through annealing a sput-
+tered Nb/Sn multilayer. A stoichiometric tar-
+get allows for a design where only a single tar-
+get is used [4, 6, 26, 27].
+Co-sputtering in-
+volves the use of separate Nb and Sn targets
+that are sputtering at the same time, allowing
+for tuning of the power applied to each target
+[21, 28]. Multilayer sputtering also uses sepa-
+rate Nb and Sn targets but alternates the use
+of each target to create many ultrathin layers
+of each material [11, 12, 22]. Tc’s above 17.8 K
+have been observed for single-target and mul-
+tilayer sputtering [6, 11, 12].
+V3Si films have been attempted by ther-
+mal diffusion, magnetron sputtering, and high-
+power impulse magnetron sputtering (HiP-
+
+Thermal annealing of sputtered Nb3Sn and V3Si thin films for superconducting RF cavities
+3
+IMS) [10, 11, 27, 28]. In thermal diffusion, a
+vanadium layer on a silicon-on-insulator sub-
+strate is annealed at high temperature to pro-
+duce V3Si [28]. In the HiPIMS method, power
+is applied as a set of discrete high-energy pulses
+at a low-duty cycle, which can be used to ion
+bombard the substrate, recrystallizing films at
+a low temperature and allowing more control
+of the stoichiometry; this method of deposit-
+ing V3Si films on Cu substrates produced Tc
+up to 10 K [10]. CERN’s magnetron sputtered
+V3Si films on a silver buffer layer upon a Cu
+substrate have reached Tc of 11.2K [27].
+Thermal annealing of the sputtered films,
+either in situ or post-deposition, is required
+to minimize the internal stress induced by
+the
+sputtering
+process
+and
+improve
+the
+stoichiometry and grain structures, which are
+important for their critical temperature and
+cavity RF performance [4, 6, 12].
+However,
+during annealing of sputtered Nb3Sn or Nb/Sn
+multilayers, the films suffer from issues such
+as Sn loss, Cu incorporation into the film
+from Cu substrates, high strain, and interface
+issues at the substrate-film boundary [4, 6,
+12].
+Sn loss is a critical issue because of
+the dependence of Tc on Sn concentration
+[3]. While annealing is frequently performed
+on Nb3Sn films, these high temperatures have
+led to Sn loss in the furnace and Nb-rich
+films with reduced Tc [6, 12], which motivates
+us to mechanistically understand the phase
+transformation associated with annealing. Cu
+incorporation can occur during annealing,
+which lowers the Tc [4].
+This issue can be
+addressed by using a barrier layer such as
+tantalum to reduce the interdiffusion [27]. The
+interface between Nb3Sn and Cu also suffers
+from strain because of their different thermal
+expansion coefficients and lattice mismatch,
+which can cause cracking in the film [4].
+Cracking can release high initial strain in the
+lattice, but does not relieve microstrain and
+increases surface roughness while decreasing
+the uniformity of the film [4, 29]. Currently, no
+sputtered Nb3Sn cavity test has been reported.
+Moreover, V3Si is much less studied than
+Nb3Sn, and there has been no RF test to date
+[10, 11, 27].
+One goal of this work is to optimize
+the sputtering capability of these alternative
+SRF materials at Cornell and compare our
+results with existing efforts in the SRF field.
+Most importantly, we aim to systematically
+investigate the effect of thermal annealing on
+the sputtered Nb3Sn and V3Si thin films in
+order to better understand these observed
+issues and design an optimal process for
+SRF use.
+By understanding the impacts
+of deposition and annealing parameters, our
+goal is to find the root of the issues in
+stoichiometry and strain of thin films. With
+such knowledge, we hope to provide insights
+for the development of sputtered Nb3Sn and
+V3Si cavities.
+In this study, we investigate
+Nb3Sn and V3Si films of different thicknesses
+on both Nb and Cu substrates to optimize
+the
+best
+conditions
+that
+minimize
+strain
+while producing required stoichiometry and
+superconducting properties.
+2. Methods
+Nb3Sn and V3Si thin films were deposited
+using a DC-sputtering system at the Cornell
+Center for Materials Research. A high vacuum
+of 10−6 torr base pressure was achieved using
+a cryo-pumped system. All depositions were
+performed at 5 mTorr Ar pressure. A rotating
+stage was used, when possible, to ensure
+uniformity during deposition.
+As summarized in Table 1, the sputtering
+parameters varied were the film material
+(Nb3Sn vs.
+V3Si), substrate material (Nb
+
+Thermal annealing of sputtered Nb3Sn and V3Si thin films for superconducting RF cavities
+4
+Table 1. Sputtering parameters for Nb3Sn and V3Si film deposition.
+Film
+Substrate Substrate
+holder
+Temperature
+(°C)
+Voltage
+(V)
+Current
+(A)
+Nominal
+thickness
+Nb3Sn
+Nb
+Rotating
+25
+596
+0.15
+100 nm
+Nb3Sn
+Nb
+Rotating
+> 25
+589
+0.26
+2 µm
+Nb3Sn
+Cu
+Heated
+550
+466
+0.214
+300 nm
+V3Si
+Nb
+Rotating
+> 25
+811
+0.196
+2 µm
+V3Si
+Cu
+Heated
+550
+819
+0.222
+300 nm
+vs.
+Cu),
+deposition
+temperature
+(room
+temperature vs. 550 °C in situ heating), and
+film thickness (100 nm, 300 nm, and 2 µm).
+Bulk Nb3Sn and V3Si targets were used, and
+they were purchased from ACI alloy, Inc. The
+impurity concentrations as received were 0.01
+at.%.
+Nb and Cu squared substrates of 1 cm2
+area were used in order to provide insights
+for applications in Nb and Cu substrate
+cavities.
+Before deposition, Nb substrates
+were electropolished, and Cu substrates were
+chemically
+polished
+to
+ensure
+a
+smooth
+surface.
+The Nb3Sn and V3Si films were designed
+to have thicknesses of 100 nm and 2 µm on
+Nb substrates and 300 nm on Cu substrates.
+The deposition rate was 2.5 ˚A/s for all samples
+except for the V3Si film on Cu substrate which
+was 1.8 ˚A/s (as there was difficulty lighting
+the plasma). The deposition temperature for
+the thick 2 µm samples is subject to error
+because the temperature is uncontrolled upon
+the rotating stage and increased through the
+133-minute deposition.
+Subsequently, a 550
+℃ heating stage was applied to investigate the
+effect of in situ heating during deposition.
+After the sputtering process, films were
+annealed in a series of elevated temperatures
+at 600 ℃, 700 ℃, 800 ℃, and 950 ℃, each
+for 6 hours, in a Lindberg high-vacuum (3 ×
+10−7 Torr) furnace. The heating rate was 10
+℃ per minute, and the annealing was followed
+by furnace cooling.
+Structural and chemical analyses were
+conducted between anneals to characterize the
+films. These analysis methods included scan-
+ning electron microscope (SEM) to observe
+the grain structure and size, energy disper-
+sive X-ray (EDS) and X-ray photoelectron
+(XPS) spectroscopies to determine the atomic
+composition, and X-ray diffraction (XRD) to
+gain insight into the crystal structure of the
+film and calculate the strain.
+In this anal-
+ysis, the key features are the quality of the
+film surfaces (smoothness, uniformity, grain
+shape/size), the stoichiometry of the films, and
+the existence and strain of Nb3Sn and V3Si
+diffraction planes. Note that EDS results were
+calibrated with regard to the electron penetra-
+tion depth in each material and the film thick-
+ness.
+Finally, on the 100 nm thin Nb3Sn film
+that yields the best performance, we verified
+its critical temperature using a quantum
+design physical property measurement system
+(PPMS) and quantified the surface roughness
+using atomic force microscopy (AFM).
+
+Thermal annealing of sputtered Nb3Sn and V3Si thin films for superconducting RF cavities
+5
+3. Results and Discussion
+In this section, we first analyze the recrystal-
+lization behavior observed in the 100 nm thin
+Nb3Sn films annealed and discuss the super-
+conducting, composition, and surface proper-
+ties of these films. Next, we show the compo-
+sition and strain evolutions as a function of an-
+nealing temperature in the 2 µm thick Nb3Sn
+and V3Si films (on Nb) and attempt to under-
+stand the Sn loss and strain relief mechanisms.
+Finally, we show the ternary phase transforma-
+tion upon annealing in the 300 nm thick Nb3Sn
+and V3Si films that were deposited on Cu sub-
+strates.
+Representative surface morphologies
+of samples upon deposition and after 700 °C
+and 950 °C anneals are shown in figure S1.
+3.1. Thin Nb3Sn film: Recrystallization
+3.1.1.
+Recrystallization behavior Figure 1
+shows the evolution of surface morphology
+with increasing temperature for the 100 nm
+thin Nb3Sn film on Nb substrate.
+We ob-
+served evident grain recrystallization at 950 ℃
+anneals. The grain size increased from a few
+nanometers as deposited (figure 1a) to approx-
+imately 300 nm after annealing (figure 1d).
+Recrystallization occurs through the release of
+strain energy during annealing and the subse-
+quent migration of grain boundaries [30].
+Here, we discuss the driving force and
+boundary mobility for thermodynamic con-
+siderations of this recrystallization annealing.
+The stored energy per unit volume (Es) at a
+strain level (ϵ) of 0.2, the maximum strain
+measured from our sputtered films, is 2.7 ×
+109 J/m3, based on a 1-dimensional elastic as-
+sumption, Es = 1/2 Eϵ2, where E is Young’s
+modulus and the value for Nb3Sn at 300 K is
+13.7 × 1011 dyn/cm2 [31]. Indeed, this value
+is dramatically larger than the typical lightly-
+Figure 1. Surface SEM images for 100 nm thin Nb3Sn
+films on Nb substrates: (a) as-deposited (a), and (b-d)
+after annealing: (b) 600 ℃, (c) 800 ℃, and (d) 950 ℃.
+
+a
+um
+(c)Thermal annealing of sputtered Nb3Sn and V3Si thin films for superconducting RF cavities
+6
+Figure 2. XRD patterns taken from the 100 nm thin Nb3Sn film as a function of annealing temperature: (a)
+as-deposited, (b) 600 ℃, (c) 700 ℃, (d) 800 ℃, (e) 950 ℃. Observed Nb3Sn diffraction planes are labeled at the
+top.
+deformed energy of 105 J/m3 for driving recrys-
+tallization in metals [32]. This suggests a suffi-
+cient driving force from the sputtering-induced
+strain within the film to enable the recrystal-
+lization annealing.
+Our X-ray diffraction data (figure 2)
+shows the Nb3Sn phase is consistent in terms of
+grain orientation at all annealing temperatures
+including
+950
+℃
+recrystallizations.
+We
+find Nb3Sn peaks near the known powder
+diffraction peaks at 2θ = 33.6°, 37.7°, 41.5°,
+62.8°, 65.6°, 70.6°, and 82.9°[34]. Due to the
+large penetration depth of the X-ray probe,
+strong Nb substrate diffractions are seen at 2θ
+= 38.4°, 53.3°, and 69.3°. A complete list of
+known peak locations is shown in table S1.
+As the annealing temperature was increased
+to 950 ℃, the grain orientations of Nb3Sn
+remained while the growth of grain size was
+significant.
+We assume this recrystallization follows a
+boundary migration mechanism and evaluate
+the
+boundary
+mobility
+by
+the
+Arrhenius
+law, d = A × exp (- Ea / RT), where d is the
+equilibrium grain size, A is the pre-exponential
+factor, Ea is the activation energy, and T is
+annealing temperature. The apparent values
+of the pre-exponential factor and activation
+
+100 nm Nb.Sn XRD
+(200) (210) (211)
+(320) (321)(400)
+(421)
+a) as-deposited
+(
+b) i600 ℃
+c)i700 ℃C
+d) i800 ℃
+e) 950 ℃
+30
+50
+40
+60 
+70
+80
+90
+20 (degrees)Thermal annealing of sputtered Nb3Sn and V3Si thin films for superconducting RF cavities
+7
+energy were determined to be 2.59 × 105 and
+63 ± 2 kJ/mol, respectively, by Schelb [33].
+At an annealing temperature of 950 ℃, the
+maximum attainable grain size is in the range
+of 434 – 643 nm. The observed ∼ 300 nm grain
+size in our work is reasonable considering the
+influence from annealing time (6 h in our work
+versus up to 200 h in Schelb’s work).
+In
+summary,
+recrystallization
+anneal
+above 800 ℃ is effective in relieving the built-in
+strain from sputtering and thus forming stoi-
+chiometric Nb3Sn along with grain coarsening.
+3.1.2. Film properties
+Superconducting prop-
+erties, atomic composition, and surface rough-
+ness were investigated on the 100 nm Nb3Sn
+sample after the 950 ℃ annealing for 6 hours.
+As shown in figure 3, the critical temperature
+is determined to be 17.5 K, while the Nb/Sn
+stoichiometry is 3/1 after sputtering away the
+surface oxides. Similarly, Sayeed et al. [6] re-
+ported Tc values of 17.68 – 17.83 K for 350
+nm Nb3Sn sputtered films that were annealed
+at 800 ℃ for 24 hours and 1000 ℃ for 1 hour
+with low Sn loss.
+They observed significant
+degradation of Tc down to 10.95 K as a con-
+sequence of the dramatic Sn loss down to 4%
+after annealing for 24 h at 1000 ℃. In contrast,
+we did not observe the Sn loss in the 100 nm
+thin films after annealing. We infer recrystal-
+lization plays a major role in retaining the Sn
+ratio as well as maintaining Tc ∼ 17.5 K in the
+100 nm thin films, with the relatively short an-
+nealing time helping to retain the film proper-
+ties. However, our 2 µm thick films as detailed
+in Section 3.2 showed similar Sn loss behavior
+at increasing annealing temperatures as com-
+pared to the 350 nm thick films in Sayeed’s
+work [6], which indicates the importance of a
+recrystallization process to obtain stoichiomet-
+ric Nb3Sn films with Tc 17.5 K. Additionally,
+the atomic force microscopy (AFM) result is
+Figure 3.
+Film properties for the 100 nm thin
+Nb3Sn film on a Nb substrate after 950 ℃ annealing.
+(a) Resistive transition and critical temperature, (b)
+XPS spectrum showing the atomic composition after
+sputtering away the surface 20 nm layer, and (c) AFM
+image showing low surface roughness.
+shown in figure 3c. The film shows low surface
+roughness, with an average roughness of 18.3
+nm, RMS roughness of 25.3 nm, and a maxi-
+mum height difference of 600 nm. In contrast,
+Nb substrates used have an average roughness
+of ∼ 70 nm.
+
+a
+5.0E-07
+4.0E-07
+Resistivity [Ohm-cm]
+3.0E-07
+2.0E-07
+1.0E-07
+0.0E+00
+0
+5
+10
+15
+20
+25
+Temperature [K]
+(b
+6.0E+04
+Nb: 75.2 at.%
+Sn 3d
+units]
+Sn: 24.8 at.%
+5.0E+04
+I Nb 3p
+[arbitrary 
+4.0E+04
+Nb 3s
+Intensity
+3.0E+04
+2.0E+04
+300
+350
+400
+450
+500
+550
+600
+Binding energy [eV]
+0
+(c)
+nm
+40
+20
+un
+5
+0
+-20
+-40
+0
+5
+10
+μmThermal annealing of sputtered Nb3Sn and V3Si thin films for superconducting RF cavities
+8
+3.2. Thick Nb3Sn and V3Si films: relation of
+strain and composition change
+Thermal annealing was performed on 2 µm
+thick
+Nb3Sn
+and
+V3Si
+films
+on
+the
+Nb
+substrate.
+The initial thickness of the films
+greatly altered the annealing behaviors as
+compared to results from 100 nm thin films.
+3.2.1.
+2 µm thick Nb3Sn films The Nb3Sn
+grains nucleated in a triangular shape and
+remained in that shape at all annealing
+temperatures studied (figure 4a and 4b). We
+speculate that these small triangular-shaped
+grains with 100 – 200 nm in size were induced
+by the high built-in stress and subsequent
+plastic deformation during deposition.
+In-
+plane
+stress
+is
+typical
+in
+physical
+vapor
+sputtering, and small grains with angular
+shapes are favored under the stress [35]. This
+argument is supported by the in situ stress
+versus grain size relationship in Leib’s work
+[36].
+Upon annealing, as shown in figure 5a, the
+2 µm thick Nb3Sn films experienced significant
+Sn loss from the as-deposited ∼ 24% down
+to 21% after the initial anneal at 600 ℃
+and further down to nearly 2% after the 950
+℃ anneal.
+Conversely, Nb3Sn phases were
+barely observed in the X-ray diffraction until
+Nb3Sn peaks appeared at 800 ℃ and 950 ℃
+anneals. The strain (ϵ) for a given plane was
+calculated by ϵ = (aT - a0) / a0, where aT is the
+measured lattice constant from Nb3Sn plane
+diffraction and a0 is the lattice parameter from
+database [34]. (Internal strains calculated for
+all samples are summarized in Table S2.) The
+relative strain (∆ϵ), shown in figure 6a, was
+obtained by normalizing strain to the high-
+temperature anneal limit where we observe
+negligible strains.
+Here,
+we
+analyze
+the
+effect
+of
+film
+Figure 4. Surface SEM images for 2 µm thick Nb3Sn
+(a, b) and V3Si (c, d) films on Nb substrates: (a, c)
+as-deposited and (b, d) after 950 ℃ annealing.
+
+umThermal annealing of sputtered Nb3Sn and V3Si thin films for superconducting RF cavities
+9
+thickness on the strain.
+The internal strain
+(ϵ) in a biaxial thin film system where in-
+plane stresses are equal (δ = δ11 = δ22) can
+be
+described
+by
+a
+linear
+relationship
+as
+ϵ = (2 S1 + 1/2 S2 sin2φ) δ, where φ is the angle
+from
+the
+film
+normal
+to
+the
+diffraction
+plane normal, and S1 and S2 are the X-
+ray elastic constants that are determined,
+in an elastic isotropic scenario, by Young’s
+modulus (E) and Poisson’s ratio (υ) and
+given by – υ / E and (1 + υ) / E, respectively
+[36].
+The
+built-in
+stress
+increases
+with
+film thickness (or deposition time at a fixed
+deposition
+rate)
+in
+a
+polycrystalline
+film
+system when the deposition goes beyond the
+initial instantaneous stress stage (< 10 nm
+thickness) [35].
+This positive correlation,
+although slightly affected by the growth-
+interrupt
+stress
+relaxation
+effect
+and
+the
+heating effect, suggests high strain in the 2
+µm thick film; however, the high in-plane
+stress during thicker film sputtering results
+in plastic deformation as indicated by the
+observation of small grain sizes and high
+density of boundaries (figure 4a).
+Furthermore, we cannot fully explain the
+Sn loss in the course of annealing the sputtered
+Nb3Sn films, i.e., the decrease of Sn/(Nb+Sn)
+atomic ratios with the increasing annealing
+temperature (figure 5a).
+Note that this
+Sn loss behavior is repeatedly observed in
+previous Nb3Sn sputtering work [6, 12].
+At
+the annealing temperatures studied, pure Sn
+phases are not expected due to their low
+vaporization temperatures (e.g., 800 ℃ at 10−6
+Torr), so we primarily consider Nb-Sn alloy
+phases in the film.
+The as-deposited film
+showed a 23 – 25% Sn atomic ratio which
+suggests minimal Sn-rich phases (Nb6Sn5 and
+NbSn2) based on the Nb-Sn phase diagram [3];
+these Sn-rich phases were also not observed
+in the X-ray diffraction.
+Without Sn or Sn-
+Figure 5.
+(a) Sn/[Sn+Nb] or Si/[Si+V] ratios in
+the Nb3Sn and V3Si films, respectively, as a function
+of annealing temperature for the 2 µm thick films
+sputtered on Nb substrates and the 300 nm thick films
+on Cu substrates (discussed in Section 3.3). Note that
+the high Si ratios for Cu substrate samples are due to
+exclusion of Cu signals for calculation. As-deposited
+Sn/Si composition on 2 µm thick films are 23 – 25 %.
+(b) Example of the EDS spectrum taken on the 2 µm
+thick V3Si film for generating the composition dataset.
+rich phases, merely Nb3Sn is expected in this
+study as indicated by the 23 – 25% Sn ratio,
+but XRD did not show any detectable Nb3Sn
+diffractions upon deposition; the crystalline
+Nb3Sn
+phase
+has
+an
+extremely
+high
+(>
+2100 ℃) phase transformation temperature,
+making it unlikely to explain the Sn loss.
+We,
+therefore,
+suspect
+the
+generation
+of
+
+45
+42
+42
+(a)
+39
+40
+35
+29
+30
+24
+25
+23
+23
+23
+22
+20
+21
+20
+15
+13
+10
+10
+4
+5
+0
+500
+600
+700
+800
+900
+1000
+Annealing Temperature (°C)
+2 μm V3Si
+i 2 μm Nb3Sn -300 nm Nb3Sn
+300 nm V3Si
+8000
+(b)
+Si
+V
+7000
+V
+6000
+5000
+eV
+cps/
+4000
+3000
+2000
+V
+1000
+0
+0
+1
+2
+3
+4
+5
+6
+keVThermal annealing of sputtered Nb3Sn and V3Si thin films for superconducting RF cavities
+10
+Figure 6. (a) Relative strain that is normalized to
+the high-temperature anneal limit, as a function of
+annealing temperature for the 100 nm and 2 µm Nb3Sn
+films on Nb substrates together with the 300 nm Nb3Sn
+film on Cu substrates.
+(b) Temperature-dependent
+strain diagram calculated from stable (s) and unstable
+(u) (220) diffraction peak shifting for 2 µm and 300 nm
+V3Si films on the Nb and Cu substrates, respectively.
+(c) Example of the XRD patterns taken on the 2 µm
+thick V3Si film showing the stable and unstable (220)
+diffraction peaks for generating the strain diagram.
+amorphous Nb3Sn phases in the film.
+Such
+amorphous phases were reported when using
+non-equilibrium processing techniques [38, 39].
+This could cause the loss of Sn alloys via
+the generation of α-Nb and also explain the
+appearance of Nb3Sn diffraction for anneals
+above 800 ℃, which likely corresponds to
+the crystallization temperature. This requires
+further investigation.
+3.2.2. 2 µm thick V3Si films
+Different from
+thick Nb3Sn films, the as-deposited 2 µm
+thick V3Si film (figure 6b) exhibits a high
+strain of 15%, which supports the positive
+relationship between strain and film thickness
+in an elastic scenario for thick (> 10 nm)
+polycrystalline films. The initial film shows a
+near-stoichiometric value of Si (∼ 23%) shown
+in figure 5b.
+Upon annealing, the V3Si film shows a
+constant Si concentration for all temperatures;
+see figure 5a.
+In contrast, the strain within
+the film is significantly relieved together with
+a transition from the unstable V3Si structure
+to the stable structure between 800 ℃ and 950
+℃ (figure 6b). The structural transformation
+is observed through the shifting of the (220)
+and (222) diffraction peaks in figure 6c.
+These behaviors demonstrate that thermal
+annealing contributes to strain reduction and
+structural stabilization in a thick sputtered
+film.
+However,
+as shown in figure 4d,
+large cracks begin to appear on the film
+after the first anneal at 600 ℃, coinciding
+with a shift toward a more angular grain
+shape with increasing temperature. The high
+strain induced by the sputtering deposition is
+responsible for the cracks although thermal
+relaxation has reduced a significant amount of
+lattice strain.
+
+0.015
+(a)
+0.01
+0.005
+(unitless)
+0
+300
+500
+700
+900
+-0.005
+ Strain
+-0.01
+-0.015
+-0.02
+-0.025
+Temperature (°C)
+100 nm Nb3Sn
+2 μum Nb3Sn --300 nm Nb3Sn
+0.2
+(b)
+Unstable with high strain
+0.15
+Strain (unitless)
+Unstable →> Stable
+0.1
+transition
+0.05
+Stable with low strain
+0
+300
+400
+500
+600
+700
+800
+900
+1000
+-0.05
+Temperature (°C)
+-→--2 μm 220u
+-2 μm 220s
+-蒙--2 μm 222u
+2 μm 222s
+300 nm 220s -
+- 300 nm 222u
+(c)
+ Inensity
+Shift at 800 °C
+Relative J
+52.5
+53
+53.5
+54
+54.5
+55
+20 (degrees)
+upon deposition :
+600 °C
+800°℃C
+950°CThermal annealing of sputtered Nb3Sn and V3Si thin films for superconducting RF cavities
+11
+3.3. Nb3Sn and V3Si films on Cu substrates:
+ternary alloy systems
+By studying the temperature-atomic percent-
+age phase diagrams of Nb-Sn [3] and V-Si [45],
+as well as the three-element composition phase
+diagrams of Cu-Nb-Sn [40, 41, 42] and Cu-V-
+Si [43, 44], we can gain insight into the phase
+transformations our films undergo during the
+annealing process.
+As shown in figures 7a and 7c, 300 nm
+thick Nb3Sn and V3Si films were sputtered on
+Cu substrates using a 550 ℃ in situ heating
+stage.
+Upon annealing, both films undergo
+dramatic grain structure changes due to the
+generation of Cu-Sn or Cu-Si phases. Nb3Sn
+grains start with rounded grains collecting in
+finger-like formations as deposited on a Cu
+surface (figure 7a) and they remelt into small
+angular grains collecting in regions of differing
+densities after 950 ℃ anneal (figure 7b).
+In
+contrast, V3Si grains begin with a finger-like
+pattern after deposition (figure 7c) and end
+with small angular grains and large artifacts
+scattered across the surface after 950 ℃ anneal
+(figure 7d). Overall, there is a trend of grain
+angularization and pattern restructuring with
+increasing temperature.
+The ternary phase transformation that
+includes
+Cu-alloy
+in
+the
+films
+primarily
+determines the film properties. As shown in
+figure 5a, the 300 nm Nb3Sn films on Cu
+substrates suffer from the Sn loss similar to
+2 µm thick films on Nb substrates, but the
+mechanism is different. According to the Nb-
+Sn-Cu phase diagram [40, 41, 42], Cu-Sn and
+Nb-Sn phases generate at low temperatures
+(e.g., 675 ℃ [40]) and these Cu-Sn transform
+into liquid at 800 ℃ under atmospheric
+pressure [41], and at high temperatures (e.g.,
+1000 ℃ [42]), only Nb3Sn and Cu exist.
+In
+our study, high-vacuum annealing vaporized
+Figure 7.
+Surface SEM images for 300 nm Nb3Sn
+(a, b) and V3Si (c, d) films on Cu substrates: (a, c)
+as-deposited and (b, d) after 950 ℃ annealing.
+
+μmThermal annealing of sputtered Nb3Sn and V3Si thin films for superconducting RF cavities
+12
+the Cu-Sn phases leading to a continuous loss
+of Sn with increasing temperature.
+Also,
+we observed low-intensity Nb3Sn diffraction
+at all temperatures whereas convoluted XRD
+peaks that are possibly from Cu-Sn and other
+Nb-Sn phases appeared at low temperatures.
+These observations match with the existence
+of Nb3Sn in the phase diagram at high
+temperatures although the majority of the film
+was evaporated.
+Different from Nb-Sn-Cu, the V-Si-Cu
+phase diagram [43, 44] shows Cu-Si and V-
+Si phases at low temperatures (e.g., 700 ℃
+[43]), but there is no liquid phase at high
+temperatures (e.g., 800 ℃ [44]). Instead, these
+phases transform into V-Si and Cu phases. In
+our study, as shown in figure 5a, the Si/(Si+V)
+ratio begins with a high value of 42% due to
+the presence of Cu-Si phases generated during
+the 550 ℃ in situ heated deposition; note that
+Cu signal is evident, but is not included in the
+calculation.
+After annealing, the Si/(Si+V)
+ratio drops to 20% at 950 ℃. This phenomenon
+strongly supports the disappearance of Cu-
+Si phases in the phase diagram, and only
+V-Si phases together with some Cu metallic
+inclusions are expected in the annealed films.
+Our diffraction data suggest these V-Si phases
+include the stable (220) and unstable (222)
+V3Si structures.
+4. Conclusions and Outlook
+In this study,
+we have demonstrated the
+capability of annealing the sputtered thin films
+to produce successful Nb3Sn and V3Si surfaces
+that have the potential for use inside SRF
+cavities. We observe that annealing is required
+to release the strain in the film and promote
+grain growth.
+For our Nb3Sn samples, the
+best results are found on the recrystallized 100
+nm film, where large grains form at 950 ℃
+anneals.
+These films are also smooth and
+have minimal surface defects.
+The 2 µm
+Nb3Sn films are not able to overcome the
+built-in stress and plastic deformation during
+sputtering, and likely form an amorphous Nb-
+Sn phase that leads to nearly complete Sn loss
+upon annealing. In contrast, the V3Si samples
+retain the stoichiometry at high temperatures,
+along with a transition in the grain shape to
+become more angular.
+Most interesting was
+the behavior of these films with respect to the
+unstable and stable phases of V3Si.
+In the
+2 µm film, there was a complete transition
+from unstable to stable at 800 ℃ along with
+consistent stoichiometry. Because we observe
+this transition and the proper stoichiometry
+at high temperatures, we determine these are
+successful V3Si films.
+For the Cu substrate samples, 550 °C in
+situ heated deposition and the subsequent low-
+temperature anneals produce Cu-Si and Cu-
+Sn phases.
+These phases transform at high
+temperatures, extracting high concentrations
+of Cu inclusions in the film. The Cu impurities
+and Cu-related phases could adversely affect
+the SRF performance of Nb3Sn/V3Si films
+inside Cu cavities. In a future study, we would
+be interested in the use of an ultrathin buffer
+layer between the Cu and the superconducting
+layer to prevent this effect [27].
+In our results, we observed a similar
+Sn loss as in previous studies [6].
+We are
+interested in finding ways to prevent this loss
+such as minimizing strong undercooling and
+avoiding disordered Nb-Sn phases or using
+encapsulation during the annealing process.
+We would like to obtain the benefits of
+annealing such as recrystallization and strain
+removal while avoiding events such as Sn loss
+and cracking.
+Because the 100 nm Nb3Sn
+film was successful, it would be important in
+a future study to further investigate films of
+
+Thermal annealing of sputtered Nb3Sn and V3Si thin films for superconducting RF cavities
+13
+similar thickness to optimize grain growth and
+RF performance.
+Data Availability Statement
+The data that support the findings of this
+study are available upon reasonable request
+from the authors.
+Conflicts of Interest
+The authors declare no competing financial
+interests.
+Acknowledgements
+This work was supported by the U.S. Na-
+tional Science Foundation under Award PHY-
+1549132, the Center for Bright Beams. This
+work made use of the Cornell Center for Ma-
+terials Research Shared Facilities which are
+supported through the NSF MRSEC program
+(DMR-1719875).
+References
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+materials innovation APL Mater. 1 011002 doi:
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+and coatings: Current status, challenges and
+prospects J. Vac. Sci. Technol. A 36 020801 doi:
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+[36] Leib J S Relationships between grain structure
+and stress in thin Volmer-Weber metallic films
+Massachusetts Institute of Technology, 2018.
+[37] Vink
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+strain,
+and
+microstructure in thin tungsten films deposited
+by dc magnetron sputtering J. Appl. Phys. 74
+15 doi: 10.1063/1.354842
+[38] Matsuki K et al. 1988 New amorphous Cu-Nb-
+(Si, Ge or Sn) alloys prepared by mechanical
+alloying of elemental powders Mater. Sci. Eng.
+97 47-51 doi: 10.1016/0025-5416(88)90010-9
+[39] Masumoto T et al. 1980 Superconductivity of duc-
+tile Nb-Based amorphous alloys Trans. Jpn.
+Inst. Met. 21 115-122 doi:
+10.2320/mater-
+trans1960.21.115
+[40] Neijmeijer W L and Kolster B H 1987 The ternary
+system Nb-Sn-Cu at 675 °C Int. J. Mater. Res.
+78 730-737 doi: 10.1515/ijmr-1987-781009
+[41] Pan V M et al. 1980 Phase equilibria and
+superconducting
+properties
+in
+niobium-tin-
+copper alloys USSR
+[42] Hopkins R H et al. 1977 Phase relations and
+diffusion layer formation in the systems Cu-Nb-
+
+Thermal annealing of sputtered Nb3Sn and V3Si thin films for superconducting RF cavities
+15
+Sn and Cu-Nb-Ge Metall. Trans. A 8 91-97 doi:
+10.1515/ijmr-1987-781009
+[43] Reid
+J
+S
+et
+al.
+1992
+Thermodynamics
+of
+(Cr, Mo, Nb, Ta, V, or W)-Si-Cu ternary
+systems
+J.
+Mater.
+Res.
+7
+2424-2428
+doi:
+10.1557/JMR.1992.2424
+[44] Savitskii E M et al. 1979 Influence of copper on
+the structure and superconducting properties of
+transition metals Inorg. Mater. 15 512-515
+[45] Okamoto H et al. 2010 Si-V (Silicon-Vanadium)
+J. Phase Equilib. Diffus. 31 409–410 doi:
+10.1007/s11669-010-9733-5
+
+Thermal annealing of sputtered Nb3Sn and V3Si thin films for superconducting RF cavities
+16
+Appendix
+Figure S1. SEM map of all samples upon deposition, after 700 °C anneal, and after 950 °C anneal.
+
+As-deposited
+700 °℃
+950 °C
+100 nm Nb,Sn
+um
+300 nm Nb,Sn 2 μm Nb3Sn 
+2 μm V3Si
+300 nm V,SiThermal annealing of sputtered Nb3Sn and V3Si thin films for superconducting RF cavities
+17
+Table S1. X-ray diffraction (XRD) peaks of Nb3Sn, V3Si (stable vs. unstable), substrates Nb and Cu, and
+other possibly relevant phases (NbSn2, Nb6Sn5, V5Si3, V6Si5, Nb3Cu, V3Cu Cu15Si4, Cu3Si4, and V (unstable)
+from reference [34]. For NbSn2, Nb6Sn5, V5Si3, and V6Si5 peaks, we only listed the prominent points.
+
+ID
+20
+plane
+ID
+20
+plane
+ID
+20
+plane
+NbSn2
+31.5
+131
+V3Cu-cubic
+43.2
+220
+Nb3Sn
+62.8
+320
+Nb3Sn
+33.6
+200
+Cu15Si4
+43.8
+332
+Nb3Sn
+65.5
+321
+NbSn2
+34.2
+133
+VsSi3
+44.1
+411
+V3Si-unstable
+65.7
+222
+Cu15Si4
+34.6
+321
+V3Si-unstable
+45.1
+211
+V3Si-stable
+69.2
+222
+V3Si-unstable
+36.5
+200
+Cu15Si4
+45.8
+422
+Nb
+69.3
+211
+Nb6Sn5
+37.2
+26
+V3Cu-tetragonal
+46.8
+4
+V-unstable
+69.6
+220
+NbSn2
+37.3
+117
+VsSi
+47.2
+222
+Nb3Sn
+70.6
+400
+Nb3Sn
+37.7
+210
+V3Si-stable
+47.4
+211
+V3Si-unstable
+71.8
+321
+Nb
+38.3
+110
+V-unstable
+47.6
+200
+V3Si-stable
+72.5
+320
+V3Si-stable
+38.3
+200
+Cu15Si4
+47.8
+510
+Nb3Cu
+71.3
+422
+Nb6Sn5
+38.5
+222
+VsCu-tetragonal
+49.4
+200
+Cu
+74
+220
+Nb3Cu
+39.3
+220
+CusSi
+49.5
+4
+V3Si-stable
+75.7
+321
+VsSi3
+39.4
+321
+Cu
+50.4
+200
+V3Si-unstable
+77.6
+400
+NbSn2
+40.9
+224
+CusSi
+50.6
+200
+V-stable
+78.3
+211
+V-unstable
+40.9
+111
+V3Si-unstable
+52.6
+220
+V3Cu-cubic
+79.1
+422
+Nb3Sn
+41.5
+211
+Nb
+53.3
+200
+Nb3Sn
+80.5
+420
+V3Cu-tetragonal
+41.7
+112
+Nb3Sn
+54.4
+310
+V3Si-stable
+82
+400
+VeSis
+42.3
+321
+V3Si-stable
+55.3
+220
+Nb
+82.1
+220
+Cu
+42.3
+111
+Nb3Cu
+56.9
+400
+Nb3Sn
+82.9
+421
+V-stable
+42.7
+110
+V3Si-unstable
+59.4
+310
+V-unstable
+84.1
+311
+VeSis
+42.8
+132
+Nb3Sn
+60.1
+222
+Nb3Sn
+85.3
+322
+V3Si-stable
+43
+210
+V-stable
+62
+200
+V-unstable
+88.7
+222
+VsSi3
+43
+420
+V3Si-stable
+62.5
+310
+V3Si-unstable
+88.9
+420
+CusSi
+43.2
+112
+V3Cu-cubic
+62.7
+400
+Cu
+89.8
+311Thermal annealing of sputtered Nb3Sn and V3Si thin films for superconducting RF cavities
+18
+Table S2. Strain for all detected peaks compared to known peak locations.
+
+2 um V3si
+ peak plane (2theta)
+temperature (C)
+200s (38.3)
+210s (43)
+211s (47.4) 220u/s (52.6/55.3) 222u/s (65.7/69.2)
+320s (72.5)
+321s (75.7)
+400s (82)
+25 
+0.141414
+0.1416167
+-0.0091019
+600
+0.143407
+0.1431387
+-0.0091019
+800
+-0.00371
+0.010709926
+0.002333
+0.0154283
+-0.0056307
+0.0069136
+-0.0091019
+950
+-0.00371
+0.0129835
+0.006355
+0.0119857
+-0.0056307
+0.004912598
+0.0092106
+-0.0091019
+100 nm Nb3Sn
+peak plane (2theta)
+temperature (C)
+200 (33.6)
+210 (37.7)
+211 (41.5)
+320 (62.8)
+321 (65.6)
+400 (70.6)
+421 (82.9)
+25
+-0.03165
+0.007852
+-0.01337
+0.002509
+600
+-0.01792
+-0.01337
+0.000687
+0.004506
+800
+-0.00378
+-0.00833
+-0.01311
+0.003501
+-0.00941
+0.00817
+0.003506
+950
+-0.01232
+-0.00833
+-0.01311
+0.004946
+-0.01073
+0.006914
+0.000522
+2 um Nb3Sn
+peak plane (2theta)
+temperature (C)
+200 (33.6)
+210 (37.7)
+211 (41.5)
+310 (54.4)
+320 (62.8)
+321 (65.6)
+400 (70.6)
+421 (82.9)
+322 (85.3)
+25
+-0.008331347
+-0.02744
+600
+-0.02263
+800
+-0.0066434
+-0.018277662
+-0.0086
+-0.02585
+0.004946
+-0.008075188
+-0.009077
+-0.11031
+950
+-0.009488
+-0.018277662
+-0.0086
+-0.02101
+0.004946
+-0.010732407
+-0.00787
+0.000522439
+-0.11031
+300nmNb3Sn
+peak plane (2theta)
+temperature (C)
+200 (33.6)
+210 (37.7)
+310 (54.4)
+320 (62.8)
+420 (80.5)
+421 (82.9)
+322 (85.3)
+550
+-0.00581
+600
+-0.00949
+-0.00581
+800
+-0.0009
+-0.02073
+0.016175
+0.010778
+0.009809
+0.004506
+-0.12049
+950
+-0.00664
+-0.01084
+0.015205
+0.014072
+-0.11966
+300 nm V3Si
+peak plane (2theta)
+temperature (C)
+200u (36.5)
+200s (38.3)
+210s (43)
+211u (45.1)
+211s (47.4)
+220s (55.3)310u (59.4)
+310s (62.5)222u (65.7) 320s (72.5)400u (77.6)
+400s (82) 420u (88.9)
+550
+600
+-0.0049
+-0.00108
+0.151889
+800
+0.147913
+-0.00121
+-0.0049
+0.15903
+0.014506
+0.020642
+0.171245
+-0.00709
+0.157062
+-0.0058
+0.157251
+-0.00317
+0.158064
+950
+-0.0005
+0.020642
+0.161794
+-0.00016
+0.149852

7NAzT4oBgHgl3EQfEvqd/content/tmp_files/2301.00999v1.pdf.txt

@@ -0,0 +1,2141 @@
+Highly efficient storage of 25-dimensional photonic qudit in a cold-atom-based
+quantum memory
+Ming-Xin Dong,1, 2, 3 Wei-Hang Zhang,1, 2 Lei Zeng,1, 2 Ying-Hao Ye,1, 2 Da-Chuang
+Li,3, ∗ Guang-Can Guo,1, 2 Dong-Sheng Ding,1, 2, † and Bao-Sen Shi1, 2, ‡
+1Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei, Anhui 230026, China.
+2Synergetic Innovation Center of Quantum Information and Quantum Physics,
+University of Science and Technology of China, Hefei, Anhui 230026, China.
+3School of Physics and Materials Engineering, Hefei Normal University, Hefei, Anhui 230601, China.
+(Dated: January 4, 2023)
+Building an efficient quantum memory in high-dimensional Hilbert spaces is one of the funda-
+mental requirements for establishing high-dimensional quantum repeaters, where it offers many
+advantages over two-dimensional quantum systems, such as a larger information capacity and en-
+hanced noise resilience. To date, there have been no reports about how to achieve an efficient
+high-dimensional quantum memory. Here, we experimentally realize a quantum memory that is op-
+erational in Hilbert spaces of up to 25 dimensions with a storage efficiency of close to 60%. The
+proposed approach exploits the spatial-mode-independent interaction between atoms and photons
+which are encoded in transverse size-invariant orbital angular momentum modes.
+In particular,
+our memory features uniform storage efficiency and low cross-talk disturbance for 25 individual
+spatial modes of photons, thus allowing storing arbitrary qudit states programmed from 25 eigen-
+states within the high-dimensional Hilbert spaces, and eventually contributing to the storage of
+a 25-dimensional qudit state. These results would have great prospects for the implementation of
+long-distance high-dimensional quantum networks and quantum information processing.
+Introduction.
+Quantum memories [1, 2] that enable
+quantum state storage and its on-demand retrieval are
+essential requirements for quantum-repeater-based quan-
+tum communication networks [3, 4] and scalable quan-
+tum computation [5]. The storage efficiency exceeding
+the 50% threshold [6–8] is necessary for practical appli-
+cations due to the fundamental requirements of beating
+the quantum no-cloning limit without post-selection [9]
+or realizing error correction in linear optical quantum
+computation [10]. Although quantum memory has been
+widely demonstrated in conventional two-dimensional (or
+qubit) quantum systems, it is highly desirable to realize
+a high-dimensional quantum memory since manipulating
+a photon in a high-dimensional Hilbert space, i.e., qu-
+dit, provides many advantages over the qubit systems in
+terms of practical quantum information processing. For
+example, qudits enable networks to carry more informa-
+tion and increase their channel capacity via superdense
+coding in quantum communication [11–13]; for quantum
+cryptography, it has been shown that qudits can provide
+a more secure flux of information against eavesdroppers
+[14–17] since the upper bound of limited cloning fidelity,
+given by F d
+clon = 1/2+1/(d+1), scales inversely with the
+dimension [11], and they also feature a better resilience
+to noise [18, 19]. Moreover, qudit systems allow the sim-
+plification of quantum logic gates [20], and permit the
+enhanced fault tolerance [21] as well as the efficient dis-
+tillation of resource states [22] in quantum computation.
+In this regard, the capability to sufficiently store the qu-
+dit resources with high efficiency is of crucial importance
+for constituting high-dimensional networks so as to dis-
+tribute high-capacity information in long-distance quan-
+tum communication and facilitate the complex quantum
+computation.
+Qubit memories have been widely demonstrated in
+many schemes that usually encode photons in polariza-
+tion [6, 7, 23–25] degree of freedom (DOF). However, such
+DOF can only support the two-dimensional encodings in-
+volved with the quantum memory operation. To build up
+a qudit memory that can store high-dimensional informa-
+tion, alternative DOFs, such as which-path [26–30], and
+time-bin [31–33], have been proposed in a variety of phys-
+ical systems. In addition, the photonic transverse spatial
+mode, e.g., orbital angular momentum (OAM) mode [34–
+42], has attracted rapidly growing interest because of its
+advance of inherent infinite dimensionality. The storage
+of these spatial qutrit states with an efficiency of 20% us-
+ing the electromagnetically induced transparency (EIT)
+scheme [43] and efficiency of approximately 30% through
+the off-resonant Raman protocol [40, 44, 45] have been
+reported. However, to date, the maximum available di-
+mensionality of quantum memory in experiment is lim-
+ited to d=3 and their efficiencies are far below the 50%
+threshold, largely limiting their practical applications in
+quantum information processing. The implementation of
+quantum memories both having high efficiency and sup-
+porting high dimensions is highly desirable but remains
+an open challenge.
+There are two main challenges to realizing efficient
+high-dimensional quantum memories. The first is to es-
+tablish a uniform light-matter interface to achieve iden-
+tical efficiencies for different spatial modes. The im-
+balanced storage efficiencies in storing different spatial
+modes will significantly degrade the storage fidelity of
+arXiv:2301.00999v1  [quant-ph]  3 Jan 2023
+
+2
+SLM1
+State preparation
+Quantum storage
+Signal
+Control
+SLM2
+To SPCM
+ To ICCD camera
+MOT
+State analyser
+λ/4
+λ/2 Lens
+PBS
+4-f  imaging system
+L1
+f
+f
+L2
+L3
+L4
+L5
+L6
+S
+C
+4-f  imaging system
+3
+1
+2
+FIG. 1.
+Schematic experimental setup.
+The qudit signal,
+encoded in POV mode via SLM 1 and lens L1, is mapped
+into the atomic ensemble for subsequent storage. Here, the
+signal and control fields are both circularly polarized (σ+),
+and the control field is beam expanded to have a waist of
+4 mm to completely cover the signal field at the centre of
+medium.
+the qudit state with the increase of dimensionality. Tak-
+ing the experiment using Laguerre-Gaussian (LG) mode
+as a case in point, the rapidly scaling of the mode waist
+in √m (m is the number of modes) [39] will lead to sig-
+nificant differences in light-matter interactions for dif-
+ferent modes, thus largely limiting its applicability in
+higher-dimensional quantum storage. The second chal-
+lenge is to constitute a highly efficient storage medium
+capable of storing multiple modes as many as possible
+[46]. To achieve this, one needs to take into account sev-
+eral physical parameters simultaneously in the storage
+process, including the transverse spatial extent of the
+storage medium, the waist size of the input modes, and
+the optical depth (OD) of the medium [7, 47]. Therefore,
+the uniform and efficient storage of a large number of
+modes is technically challenging.
+Here, we demonstrate a high-dimensional quantum
+memory working up to a 25-dimensional Hilbert space
+with a storage efficiency of close to 60%, using the
+EIT protocol [48–53] in a laser-cooled atomic ensem-
+ble. Through constituting a highly efficient spatial-mode-
+independent light-matter interface where photons are en-
+coded in a unique perfect optical vortex (POV) mode
+[54] with invariant transverse size, we are able to store
+a 25-dimensional qudit by mapping it onto the 25 bal-
+anced spatial modes at the centre of the storage medium,
+and coherently retrieve these components with identical
+efficiencies via a control laser. The demonstrated high-
+dimensional quantum memory with high efficiency herein
+is promising for high-capacity quantum communication
+and high-dimensional quantum information processing.
+Model and experimental setup.
+Our memory scheme
+based on spatial-mode-independent light-matter interac-
+tion is involved with a three-level Λ-type atomic system,
+where the signal field (with a Rabi frequency Ωp) drives
+the level |1⟩ to |3⟩ and the control field (with a Rabi
+frequency Ωc) drives the level |2⟩ to |3⟩ (Fig. 1, dashed
+circle). The dynamical evolution of the probe field un-
+der the slowly-varying envelope approximation can be de-
+scribed by the Maxwell equation as follows:
+�1
+c
+∂
+∂t + ∂
+∂z
+�
+Ωp = iDeffΓ
+2L σ31
+(1)
+where Γ denotes the decay rate of |3⟩, L is the length
+of medium, and σ31 represents the atomic coherence be-
+tween levels |1⟩ and |3⟩. Deff ∝ Ntrg31L represents the
+effective OD of an atomic ensemble, where we define an
+effective atomic density Ntr while considering a struc-
+tured light field interacts with the storage medium in
+the transverse orientation.
+g31 represents the photon-
+atom coupling coefficient between |1⟩ and |3⟩. It can be
+observed from Eq. (1) that Deff significantly affects the
+performance of storage, and we derive the numerical re-
+lation between the storage efficiency and OD by solving
+the Maxwell-Bloch equations [54].
+For a spatial multi-mode quantum memory, it is nec-
+essary to take into account the effective light-matter in-
+teraction volume for different spatial modes. Here, we
+focus on the coupling of the structure field with the
+storage medium in the cross section, because the trans-
+verse extent of the storage medium is a crucial parame-
+ter in determining the capacity of multi-mode memory
+[46].
+We assume the atomic ensemble with a Gaus-
+sian distribution of the density in the radial direction
+Ntr(r) = N0 exp[−r2/(2σ2
+r)].
+N0 refers to the mean
+atomic density, and σr represents the half width of the
+atomic ensemble [54]. In this work, we propose a scheme
+to establish a uniform light-matter interface for the mem-
+ory of a variety of modes via interacting the photons
+encoded in POV mode with the storage medium. Theo-
+retically, such spatial modes feature identical transverse
+sizes for different m, and thus they are subject to the
+same Ntr(r) of atoms when they undergo the storage
+process.
+The interaction strength between the desired
+POV modes and medium is uniform, which manifests as
+the same Deff and ultimately contributes to the same
+storage efficiency for different m. Based on this mech-
+anism, we constitute a spatial-mode-independent quan-
+tum memory for the further implementation of storage
+of high-dimensional quantum states.
+The experimental set-up for a high-dimensional quan-
+tum memory is schematically depicted in Fig. 1.
+The
+qudits encoded in each spatial mode are formed on the
+basis of the POV eigenstates |ℓ⟩ (ℓ is chosen from -12 to
+12), which is accomplished by means of a Fourier trans-
+formation of the Bessel-Gaussian (B-G) state.
+In this
+regard, we initially prepare the B-G states by project-
+ing the attenuated coherent states at the single-photon
+level onto a phase-only spatial light modulator (SLM1) to
+shape the wave-fronts of photons (Fig. 1, top left). The
+phase patterns loaded on the SLM are programmed by a
+
+2 /4 2 /2
+Lens
+PBSO3
+80
+60
+40
+20
+  0
+80
+60
+40
+20
+  0
+80
+60
+40
+20
+  0
+80
+60
+40
+20
+  0
+80
+60
+40
+20
+ 0
+ 0
+ 0.5
+1.5
+ 2
+1
+Counts (/1500 s)
+Time (μs)
+Intensity distribution
+Input
+Retrieval
+ (MHz)
+Retrieval
+Memory
+Input
+Control 
+= -12
+
+= -6
+
+= 0
+
+= 6
+
+= 12
+
+= -6
+
+OD=198
+= 0
+
+OD=205
+= -12
+
+OD=194
+= 6
+
+OD=202
+= 12
+
+OD=194
+Transmission
+Experiment
+Fitting
+η=60.2%
+η=55.7%
+η=59.6%
+η=60.4%
+η=59.3%
+(a)
+(b)
+(c)
+(d)
+= -12
+
+= -6
+
+= 0
+
+= 6
+
+= 12
+
+-12
+-12
+-10
+-10
+-8
+-8
+-6
+-6
+-4
+-4
+-2
+-2
+0
+0
+2
+2
+4
+4
+6
+6
+8
+8
+10
+10
+12
+12
+Input modes
+Quanta of POV
+Output modes
+ Efficiency
+0.6
+0.48
+0.36
+0.24
+0.12
+0
+(e)
+FIG. 2. Performance of spatial multi-mode quantum storage. (a) Measured absorption spectra for various spatial modes versus
+the signal detuning from the atomic resonance |1⟩ → |3⟩, where the relative computer-controlled holograms loaded on the
+surface of SLM1 are illustrated in the top right. (b) Transverse intensity distributions of various modes recorded at the imaging
+plane of the second 4-f imaging system before (left) and after (right) storage. (c) Temporal waveforms of input (blue) and
+retrieved (red) pulses with temporal lengths of about 500 ns for different modes. (d) Storage efficiencies versus the quanta of
+POV mode. The shaded area represents the maximum fitted value that has been expected, with a span of 1 sigma. (e) 25×25
+input-retrieved cross-talk matrix formed by the basis set from ℓ = −12 to 12.
+combination of Bessel and Gaussian functions. Lens L1
+acting as a Fourier transformer is then used to transform
+the B-G states to the POV states, which are subsequently
+mapped into the centre of the atomic medium for stor-
+age with the assistance of a carefully aligned 4-f imaging
+system.
+We next store and retrieve the POV states via the EIT
+storage protocol in a rubidium medium. To ensure a high
+storage efficiency of quantum memory, it is essential to
+prepare an optically thick atomic ensemble with a large
+OD, which is implemented by using a two-dimensional
+dark-line magneto-optical trap (MOT) technique in our
+work.
+After a programmable storage time, the signal
+photons are retrieved from the memory and sent into
+a qudit state analyser, including the other 4-f imaging
+system consisting of lenses L4 and L5, a Fourier lens L6,
+as well as a spatial-mode projector based on SLM2, a
+single-mode fiber (SMF) and a single-photon counting
+module (SPCM), to fully characterize the output states;
+see the right panel of Fig. 1.
+Performance of multi-mode quantum memory.
+The
+key to achieving multi-mode storage in our scheme is to
+exploit the mode-independent light-matter interaction.
+To confirm the accomplishment of this particular photon-
+atom interface, we first measure the absorption spectra
+for a variety of spatial modes, i.e. ℓ ∈ {−12, −6, 0, 6, 12}
+by scanning the detuning of signal from −2π × 30 to
++2π ×30 MHz, as depicted in Fig. 2(a). The nearly iden-
+tical OD (∼200) for various |ℓ⟩ indicates that the inter-
+actions between POV photons and atoms have hardly
+〉
+|ψ1
+〉
+|ψ2
+〉
+|ψ3
+〉
+|ψ4
+〉
+|ψ5
+〉
+|ψ6
+(a)
+(c)
+(b)
+(d)
+1
+rk
+rk
+rk
+=
+5
+=
+10
+=
+0
+20
+40
+60
+80
+100
+FIG. 3. Characteristics of high-dimensional storage. (a) Dis-
+tributions of storage mode bandwidth for different radial wave
+vectors kr = 1, 5, 10, where the kr of 5 is used in the context.
+(b) Qudit states with d = 2, 5, 10, 15, 20 and 25 (see par-
+ticular expressions in Ref [54]) versus storage efficiency. (c)
+Numerical simulation of two-dimensional fidelity as a function
+of storage-efficiency-uniformity κ1. (d) Theoretical analysis of
+fidelity versus κ1 and κ2 in the case of qudit with d = 3.
+any correlation with their mode number, thus allowing
+our memory to be capable of carrying multiple spatial
+modes simultaneously. As shown in Fig. 2(b), the spa-
+tial profiles of POV eigenstates with a mean photon
+number of n=0.5 in the transverse orientation are de-
+
+4
+〉
+〉 〉
+|5
+|6
++
+〉 〉
+-
+|5
+|6
+i
+Re[  ]
+χ
+Im[  ]
+χ
+-0.5
+1.0
+〉
+|0
+〉
+|0
+〉
+|12
+〉
+|
+12
+-0.5
+1.0
+-0.5
+1.0
+-0.5
+1.0
+-0.5
+1.0
+-0.5
+1.0
+-0.5
+1.0
+-0.5
+1.0
+〉
+|0
+〉
+|12
+〉
+|0
+〉
+|12
+〉
+|0
+〉
+|12
+〉
+|5
+〉
+|5
+〉
+|6
+〉
+|
+6
+〉
+|5
+〉
+|
+6
+〉
+|5
+〉
+|
+6
+〉
+|5
+〉
+|
+6
+〉
+|5
+〉
+|6
+〉
+|5
+〉
+|6
+〉
+|5
+〉
+|6
+〉 〉
+|0
+|12
++
+〉 〉
+-
+|0
+|12
+i
+Re[  ]
+χ
+-1.0
+1.0
+0.0
+Re[  ]
+χ
+-1.0
+1.0
+0.0
+Re[  ]
+χ
+-1.0
+1.0
+0.0
+Re[  ]
+χ
+-1.0
+1.0
+0.0
+〉
+|1
+〉
+|5
+〉
+|9
+〉
+|1
+〉
+|5
+〉
+|9 Im[  ]
+χ
+-1.0
+1.0
+0.0
+Im[  ]
+χ
+-1.0
+1.0
+0.0
+Im[  ]
+χ
+-1.0
+1.0
+0.0
+Im[  ]
+χ
+-1.0
+1.0
+0.0
+〉
+|2
+〉
+|
+〉
+6
+|10
+|
+2
+〉
+|
+6
+〉
+|
+10
+〉
+|
+-3
+〉
+|
+-7
+〉
+|
+-11
+〉
+|-4
+〉
+|
+-4
+〉
+|-8
+〉
+|
+-8
+〉
+|-12
+〉
+|
+-12
+〉
+|0
+〉
+|
+12
+〉
+|0
+〉
+|
+12
+〉
+|0
+〉
+|
+12
+(a)
+(b)
+(c)
+|
+Input
+Retrieval
+Input
+Retrieval
+Input
+Retrieval
+Input
+Retrieval
+Input
+〉
+|(               )
+0
+〉
+|12
+2
++
+/ 
+〉
+|0
+|
+〉
+(             )
+〉
+5
+| 6
+2
++
+/ 
+-             
+(              )
+〉 i
+|
+〉
+5
+| 6 
+2
+/ 
+(                    )
+〉 
++             
++             
+|
+〉
+1
+〉 
+|9
+| 5 
+3
+/ 
+(                      )
+〉 
++             
++             
+|
+〉
+2
+〉 
+|10
+| 6 
+3
+/ 
+(                          )
+〉 
++             
++             
+|
+〉
+-3
+〉 
+|-11
+| -7 
+3
+/ 
+(                          )
+〉 
++             
++             
+|
+〉
+-4
+〉 
+|-12
+| -8 
+3
+/ 
+〉
+|1
+〉
+|5
+〉
+|9
+〉
+|1
+〉
+|5
+〉
+|9
+〉
+〉
+|2
+〉
+|6
+|
+2
+〉
+|
+6
+〉
+|
+10
+〉
+|10
+〉
+|-3
+〉
+|-7
+〉
+|-11
+〉
+|-3
+〉
+|-7
+〉
+|-11
+〉
+|
+-3
+〉
+|
+-7
+〉
+|
+-11
+〉
+|
+-4
+〉
+|
+-8
+〉
+|
+-12
+〉
+|-4
+〉
+|-8
+〉
+|-12
+FIG. 4. Demonstration of the storage of quantum states pro-
+grammed by arbitrary quanta.
+(a) Reconstructed real and
+imaginary parts of density matrices of the retrieved arbitrary
+quantum states with d=2 in different subspaces. (b) Single-
+photon interference fringes for different states.
+(c) Recon-
+structed density matrices of the retrieved qudits with d=3
+in arbitrarily selected subspaces. The upper panel illustrates
+the spatial profiles of the corresponding quantum states be-
+fore and after storage. The mean number of photons per pulse
+here is n = 0.5.
+tected by an ICCD camera (iStar 334T series, Andor)
+working at the single-photon level. The calculated high
+values S of similarity [36] between input and retrieved
+states are 99.65%, 99.63%, 99.65%, 99.61% and 99.54%
+for ℓ = −12, −6, 0, 6, 12 respectively, implying a faithful
+quantum storage for POV states.
+Figure 2(c) shows the temporal waveforms of the in-
+put (blue) and retrieved pulses (red) after a one-pulse-
+delay storage time for various spatial modes.
+As can
+be seen, the retrievals have almost the same waveforms
+for different inputs, providing a clear evidence that our
+memory exhibits identical characteristics for different
+POV modes. To fully analyze the capacity of this spa-
+tial multi-mode quantum memory, we investigate the
+memory efficiencies of POV eigenstates across the en-
+tire range (from -12 to 12) with a step of ∆ℓ = 1; see
+Fig. 2(d). Their approximately the same values at around
+57% clearly illustrate that our memory enables 25 spatial-
+mode storage with efficiency beyond 50%. Note that the
+overall storage-efficiency distributions for different radial
+wave vector kr [54] in a wider mode range are shown
+in Fig. 3(a). Figure 2(e) gives the experimental cross-
+talk between the 25 orthogonal bases after retrieval. The
+average contrast [54], given by C = 1/25 �
+m Cm, is esti-
+mated to be 92.4±1.6%, thereby revealing a low overlap
+noise between orthogonal spatial modes.
+In multi-mode memory, the uniform storage efficiency
+for each POV eigenstate plays a crucial role in high-
+dimensional storage. We consider a high-dimensional
+quantum superposition state with the dimensional-
+ity of d,
+i.e. the so-called qudit state |ψ⟩Input
+=
+1/
+√
+d (|ℓ1⟩ + |ℓ2⟩ + · · · + |ℓd⟩) as input. The retrieved
+state after storage can be written as
+|ψ⟩Retrieval = 1/
+��d
+m=1 η2m(η1 |ℓ1⟩ + η2 |ℓ2⟩ + · · ·
++ ηd |ℓd⟩)
+(2)
+where η1, · · · , ηd denote the storage efficiency for the cor-
+responding eigenmodes. |ψ⟩Retrieval can be further sim-
+plified to η/
+√
+d (|ℓ1⟩ + |ℓ2⟩ + · · · + |ℓd⟩) if η1, · · · , ηd are
+all equal to a constant represented by η. In this case, the
+storage efficiency of the qudits has no dependence on the
+dimensionality d, as displayed by the results in Fig. 3(b).
+Thus, our memory allows storing arbitrarily dimensional
+qudits with the same efficiency even when d is up to 25.
+Storage fidelity is a critical performance parameter
+that has to be taken into account in quantum mem-
+ory. For the storage of qudit in terms of multiple spatial
+modes, its fidelity is extremely sensitive to the unifor-
+mity of the storage efficiency for the internal orthogonal
+states. For simplicity, we consider the case of quantum
+states with d = 2, as shown in Fig. 3(c), where a pa-
+rameter κ1 is defined as the ratio of storage efficiencies
+between |ℓ1⟩ and |ℓ2⟩, i.e. κ1 = η2/η1. It can be found
+that the imbalanced atomic storage (κ1 ≪ 1) would
+largely reduce the fidelity, as estimated by the formula
+F =
+�
+Tr
+��√ρTρretrieval√ρT
+��2, where ρT and ρretrieval
+represent the density matrices corresponding to the tar-
+get and retrieval states. In Fig. 4(a), we reconstruct the
+retrieved density matrices using the quantum state to-
+mography (QST) method for a set of qubit states con-
+stituted by arbitrary eigenstates (e.g. |0⟩, |12⟩, |5⟩, |6⟩
+are chosen herein) after storage. The average fidelity of
+95.8% without any corrections is in good agreement with
+the theoretical expectation, and the measured single-
+photon interference fringes [Fig. 4(b)] with an average
+visibility of 92.3% demonstrate that the coherence be-
+tween two components of the qubits is well preserved
+during storage.
+In analogy to the case of d = 2, Fig. 3(d) illustrates the
+effect of efficiency-uniformity between internal modes on
+the fidelity for d = 3, where κ2 is defined as η3/η1. To
+obtain a high fidelity, κ1 and κ2 should both approach
+unity. In Fig. 4(c), we randomly choose three eigenvec-
+tors in the range from |−12⟩ to |12⟩ to prepare the high-
+dimensional states for storage. The high mean fidelity
+is measured to be 96.4% owing to κ1 ≈ κ2 ≈ 1.
+Note
+that these results can hardly be obtained in those exper-
+iments [39] using conventional vortex modes (e.g., LG
+mode) because of the inevitable non-uniform efficiency
+for different spatial modes.
+Moreover, we characterize
+the retrieved state of |ψ2⟩ for d =5, and the raw fi-
+delity reaches 90.7±0.7% (the error bar is estimated from
+Poissonian statistics and using Monte Carlo simulations),
+as shown in the right panel of Fig. 5. All these experi-
+mental results indicate our memory capability of storing
+
+1
+11
+-
+1
+1
+1
+1
+1
+1
+1
+1
+1-
+1
+1
+-
+-
+/
+1
+-
+1
+1
+-
+-
+1
+-
+1
+1
+1
+1
+1
+1
+1-
+-
+-
+1
+-
+1
+1
+1
+-
+1
+1
+1
+1
+1
+1
+11
+11
+1
+1-
+1
+-
+1
+1
+-
+1
+-
+-
+1
+1
+1
+1
+1-
+1
+-
+-
+1
+1
+-
+1
+1
+-
+1
+1
+1
+1
+1
+1CC-
+-
+1
+-
+1
+1
+1
+1
+1
+1
+-
+1
+1
+1
+-
+1
+1
+1
+1
+-
+1
+1
+1
+1
+-
+1
+1
+1
+<
+1
+1一
+1
+1
+1
+1
+1
+1
+1
+1
+1
+1
+1
+1
+1一
+-
+1
+1
+-
+1
+1
+1
+1
+-
+-
+1
+1
+1
+1
+1
+1
+1
+-
+1
+1
+1
+1
+-
+1
+1
+1
+<
+1
+11
+1
+1
+1
+1
+1
+1
+1
+11
+/ /
+-
+1
+1
+1
+-
+1
+1
+1
+1
+1
+-
+1
+1
+1
+-
+1
+1
+1
+1
+1
+1
+1
+1
+-
+1
+1
+1
+<
+1
+11
+1
+1
+1
+1
+1
+1
+1
+1
+1
+1
+1
+1
+1
+1
+11
+1
+1
+1
+1
+1
+1
+1
+1
+1
+1
+1
+1
+1
+1
+1
+1
+1
+11
+1
+1
+1
+1
+15
+〉
+|-2
+〉
+|
+-2
+〉
+|-1
+〉
+|
+-1
+〉
+|0
+〉
+|
+0
+〉
+|1
+〉
+|1
+|2
+〉
+|2
+-1.0
+-0.5
+ 0.0
+0.5
+1.0
+Re[   ]
+χ
+〉
+|-2
+〉
+|
+-2
+〉
+|-1
+〉
+|
+-1
+〉
+|0
+〉
+|
+0
+〉
+|1
+〉
+|1
+〉
+|2
+〉
+|2
+-1.0
+-0.5
+ 0.0
+0.5
+1.0
+Im[   ]
+χ
+〉
+FIG. 5. Storage in high-dimensional space exceeding the clas-
+sical benchmark. The measured fidelities as a function of the
+mean photon number per pulse n. The purple/yellow points
+are experimental data without/with background subtraction.
+The blue solid line is the classical limit after considering the
+finite storage efficiency and Poissonian statistics of the input.
+arbitrary-mode-encoded qudit states programmed from
+25 eigenvectors.
+To further prove the quantum nature of the mem-
+ory, we compare the fidelities obtained in our experi-
+ment with the maximum available fidelities in a classical
+memory device based on a completely classical strategy
+[6, 24, 37, 38]. After considering the Poissonian statistics
+of photon number for a coherent state, the classical fi-
+delity threshold for a state with a fixed photon number
+N can be written as
+Fclass(n) =
+∞
+�
+N=1
+�N + 1
+N + 2
+�
+e−nnN
+(1 − e−n)N!
+(3)
+where n is the mean photon number per pulse. As pre-
+sented in Fig. 5, the solid line is the theoretically classical
+limit after taking η = 0.57 in our work. We observe that
+all the experimental points exceed the classical bench-
+mark for different mean photon numbers, which confirms
+the quantum character of our device.
+We now turn to study the capability of our memory to
+store a 25-dimensional quantum state. The main chal-
+lenge to achieving the storage of a 25-dimensional qudit
+state is to preserve the identical memory efficiency for
+each mode, thus preventing the decay of coherence be-
+tween 25 spatial modes during the storage process. Here,
+a 25-dimensional qudit state |Ψ⟩ given by a coherent su-
+perposition of 25 individual spatial modes from |−12⟩ to
+|12⟩ is prepared for the demonstration of 25-dimensional
+qudit storage, which is represented as
+|Ψ⟩ =
+1
+√
+25
++12
+�
+ℓ=−12
+|ℓ⟩
+(4)
+To fully characterize the retrieved state, we perform
+the high-dimensional QST [54, 55], where the real and
+imaginary parts of the reconstructed density matrix with-
+out (with) background correction are plotted in the log-
+ical basis of {|−12⟩ ,|−11⟩, |−10⟩, · · · , |12⟩}, as shown
+〉
+|-12
+〉
+|12
+〉
+|0
+〉
+|
+-12
+〉
+|
+0
+|
+-12
+〉
+〉
+|-12
+〉
+|12
+〉
+|0
+〉
+|
+-12
+〉
+|
+0
+|
+-12
+〉
+〉
+|-12
+〉
+|12
+〉
+|0
+〉
+|
+-12
+〉
+|
+0
+|
+-12
+〉
+〉
+|-12
+〉
+|12
+〉
+|0
+〉
+|
+-12
+〉
+|
+0
+|
+-12
+〉
+-0.1
+0.1
+-0.1
+0.1
+-0.1
+0.1
+-0.1
+0.1
+Re(      )
+ρraw
+Re(      )
+ρcorr
+Im(      )
+ρcorr
+Im(      )
+ρraw
+(a)
+(c)
+(b)
+(d)
+FIG. 6.
+Experimental realization of 25-dimensional qudit
+storage.
+The characterization of the retrieved qudit state
+|ψ6⟩ after the storage process by performing QST. (a)/(c) and
+(b)/(d) are the real and imaginary parts of the reconstructed
+density matrices for retrieved state |ψ6⟩ without/with back-
+ground correction, respectively.
+in Fig. 6(a,b) (Fig. 6(c,d)), respectively.
+The raw fi-
+delity between the retrieved states and ideal state is esti-
+mated to be 72.8 ± 0.6%, where the imperfection fidelity
+is mainly caused by the dark counts of the detector and
+residual control laser leakage. After the subtraction of
+the background, the fidelity reaches 90.3 ± 0.6%, far ex-
+ceeding the classical limit of 70.2% for mean photon num-
+ber n = 0.5, where the memory efficiency of state |ψ6⟩
+equals to 60% is taken into account. Note that the resid-
+ual fidelity is primarily due to the imperfections in the qu-
+dit preparation and measurement. All the above results
+clearly beat the classical benchmark, thus demonstrating
+the quantum character of our 25-dimensional memory
+implementation.
+Conclusion.
+In summary,
+we have experimentally
+demonstrated the efficient quantum storage for high-
+dimensional quantum states with d up to 25 using the
+POV modes of photons. The reported high-dimensional
+quantum memory achieves a storage efficiency of >50%,
+exceeding the threshold value for practical quantum in-
+formation applications. Remarkably, the dimensionality
+of this memory is scalable to as high as 100 through
+further optimization of the waist of POV modes [54],
+thus presenting a clear route to the scalability of di-
+mensions. In addition, our multi-mode memory is also
+promising for the compatibility with fiber-based quan-
+tum information transfer systems, which are capable of
+spatially-structured photon transmission [56, 57].
+The
+high-dimensional quantum memory demonstrated herein
+gives a great perspective for the practical high-capacity
+and long-distance quantum communication networks.
+This work was supported by National Key R&D Pro-
+
+1
+1-
+-
+1
+1
+1
+1
+1
+1
+1
+-
+1
+1
+1
+1
+1
+1
+1
+-
+1
+1
+1
+1
+10.10
+0.05
+0.00
+-0.05
+-0.101
+1
+0.10
++
+4
++
+-
+0.05
+:
+71
+0.00
+1
++
+中
++
++
+-0.05
+4
++
+T
++
+4
++
+1
++
+*
++
+1
++
+-0.10
+1
+1
+40.10
+0.05
+0.00
+-0.05
+-0.101
+0.10
+-
+7
+4
+7
+0.05
+/
+T
+T
+0.00
+4
+T
+1
++
+4
+5
+1
+T
+-0.05
+1
+:
+1
+1
+-0.10
+1
+T
+T
+16
+gram of China (Grants No.
+2017YFA0304800), Anhui
+Initiative in Quantum Information Technologies (Grant
+No. AHY020200), the National Natural Science Founda-
+tion of China (Grants No. U20A20218, No. 61722510,
+No. 11934013, No. 11604322, No. 12204461), and the In-
+novation Fund from CAS, and the Youth Innovation Pro-
+motion Association of CAS under Grant No. 2018490.
+∗ [email protected]
+† [email protected]
+‡ [email protected]
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+
+Supplemental Material for Highly efficient storage of 25-dimensional photonic qudit in
+a cold-atom-based quantum memory
+Ming-Xin Dong,1, 2, 3 Wei-Hang Zhang,1, 2 Lei Zeng,1, 2 Ying-Hao Ye,1, 2 Da-Chuang
+Li,3, ∗ Guang-Can Guo,1, 2 Dong-Sheng Ding,1, 2, † and Bao-Sen Shi1, 2, ‡
+1Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei, Anhui 230026, China.
+2Synergetic Innovation Center of Quantum Information and Quantum Physics,
+University of Science and Technology of China, Hefei, Anhui 230026, China.
+3School of Physics and Materials Engineering, Hefei Normal University, Hefei, Anhui 230601, China.
+(Dated: January 4, 2023)
+arXiv:2301.00999v1  [quant-ph]  3 Jan 2023
+
+I. EXPERIMENT DETAILS
+As depicted in Fig. 1, the relevant quantum states |1⟩, |2⟩ and |3⟩ correspond to the 85Rb atomic hyperfine levels
+��5S1/2, F = 2
+�
+,
+��5S1/2, F = 3
+�
+and
+��5P1/2, F = 3
+�
+respectively. Here, the POV signal is resonant with the atomic
+transition of |1⟩ ↔ |3⟩, and the auxiliary control field with Rabi frequency of 2π × 20.07 MHz dynamically drives
+the transition of |2⟩ ↔ |3⟩ to achieve reversible transfer between photonic states and atomic collective spin excitation
+during the storage process. The control field entering into the atomic medium has an angle of 3° with respect to the
+signal to write and read the photonic qudit state. After an on-demand storage time, the retrieved state is characterized
+by a state analyser. The qudit analyser is set for implementing related projective measurements of the retrieved states
+into distinct bases. Before signal photons being measured by the detector, two homemade Fabry-Perot etalons (with
+a total transmittance of 64%, an isolation rate of 46 dB and a bandwidth of 500 MHz), acting as frequency filters,
+are inserted into the path (not shown in Fig. 1) to reduce the noise scattered from the control field. The outputs
+are finally detected by the SPCM (avalanche diode, PerkinElmer SPCM-AQR-16-FC; 60% efficiency, maximum dark
+count rate of 25/s) for recording the relative measurement counts.
+4-f imaging system.
+The first 4-f imaging system shown in Fig. 1 is used to map the photonic POV modes onto
+the centre of the storage medium, and mainly consists of lenses L2 and L3, which are separated by a distance of
+f2 + f3. Here, f2 and f3 refer to the focal lengths of lenses L2 and L3. The focal lengths of lenses L1, L2, L3, L4, L5
+and L6 are 75, 500, 300, 300, 500 and 75 mm, respectively. The POV spatial structure is located in the front focal
+plane of L2, and its image is obtained at the back focal plane of L3. In addition, this shrunken 4-f imaging system
+is further exploited to reduce the waist of the input photonic modes, with a scaling ratio of f3/f2, to better match
+with the transverse size of the storage medium.
+High-d state expressions.
+In Fig. 3(b) of main text, the used six quantum states are |ψ1⟩ = (|0⟩ + |12⟩) /
+√
+2, |ψ2⟩ =
+1/
+√
+5 �ℓ=+2
+ℓ=−2 |ℓ⟩, |ψ3⟩ = 1/
+√
+10
+��ℓ=−1
+ℓ=−5 |ℓ⟩ + �ℓ=+5
+ℓ=+1 |ℓ⟩
+�
+, |ψ4⟩ = 1/
+√
+15 �ℓ=+7
+ℓ=−7 |ℓ⟩, |ψ5⟩ = 1/
+√
+20(�ℓ=−1
+ℓ=−10 |ℓ⟩ +
+�ℓ=+10
+ℓ=+1 |ℓ⟩), |ψ6⟩ = 1/
+√
+25 �ℓ=+12
+ℓ=−12 |ℓ⟩, respectively.
+II. HIGH-OPTICAL-DEPTH COLD ATOMIC ENSEMBLE
+We trap the cold atoms of Rubidium 85 (85Rb) in a two-dimensional dark-line MOT. Each trapping laser beam
+has a power of 36 mW with a beam waist of 2 cm. Two vertically oriented repump laser beams have a total power of
+60 mW with a beam waist of 2 cm, with two copper bars located in their central positions to prepare two dark-line
+images [1, 2]. These images are overlapped at the centre of the MOT along the longitudinal axis by using two 4-f
+imaging systems, and thus constituting a dark-line volume. Here, the diameters of the copper bars are 1.5 mm. The
+experimental repetition rate is 100 Hz, and the experimental operation window is 1.3 ms, during which the MOT
+magnetic field is switched off completely. Initially, all of the atoms are prepared in the ground state |1⟩ by turning off
+the repump laser 500 µs earlier than the trapping laser. Thanks to the high OD herein, we have achieved a storage
+efficiency of 72.3% for a single photon in Gaussian mode, as shown in Fig. S1.
+Figure S1 shows the temporal waveforms of input (blue) and retrieved pulses (red) after a one-pulse-delay storage
+time for Gaussian mode. The storage efficiency of quantum memory can be defined as
+η =
+�
+|ψretrieval(t)|2 dt
+�
+|ψinput(t)|2 dt
+(1)
+where ψinput(t) and ψretrival(t) represent the input and retrieved quantum states.
+The storage efficiency for the
+Gaussian mode is estimated to be 72.3% by integrating the counts of photons in their entire temporal waveforms.
+III. PREPARATION AND ANALYSIS OF POV MODES
+The structured light beams carrying OAM are of many interests due to their wide applications in micromanipulation
+[3], optical imaging [4] and quantum information processing [5, 6]. The photons carrying OAM are described by a
+helical phase factor eiℓφ, with ℓ being the topological charge indicating the OAM of ℓℏ per photon and φ is the
+azimuthal angle.
+Here, ℓ takes any integer values, therefore the available Hilbert-space dimension is infinite in
+principle, which provides a huge information capacity for classical and quantum information systems. However, the
+2
+
+Counts (/1500 s)
+Time (μs)
+ 0
+ 0.5
+1.5
+ 2
+1
+120
+160
+80
+40
+0
+Memory
+Control 
+Retrieval
+Input
+FIG. S1. The temporal waveform of input (blue) and retrieved (red) pulses for Gaussian mode.
+conventional optical vortices, e.g., LG beams, always exhibit a strong dependence of their transverse size on the
+topological charge number, thus limiting their further applications in some circumstances, such as optical trapping
+and tweezing, multiplexed optical communication using a single fibre with a fixed annular index profile [7], as well
+as the high-dimensional quantum information processing [8]. To overcome these limitations, the concept of a perfect
+optical vortex [9, 10] has been proposed whose radius is independent of ℓ. The POV mode is defined as the Fourier
+transformation of a B-G function, and the complex field amplitude is given by
+Eℓ
+POV(r, φ) = iℓ−1 ωg
+ω0
+exp(iℓφ) × exp(−r2 + r2
+r
+ω2
+0
+)Iℓ(2rrr
+ω2
+0
+)
+(2)
+where ωg is the beam waist of the Gaussian mode, ω0= 2f/kωg denotes the beam waist of the Gaussian mode at the
+focal plane with a wave vector k=2π/λ, in which f is the focal length of the Fourier lens. rr=krf/k is the radius of
+POV mode (kr is the radial wave vector). For the case of large kr and small ω0, Eq. (4) can be further reduced to
+EPOV (r, φ) ∝ iℓ−1/krδ (r − rr) exp (iℓφ)
+(3)
+where δ(r) represents the Dirac delta function. We can clearly observe that this ideal POV mode has a transverse
+radius of rr which is independent of ℓ. In the experiments, one can obtain the POV mode at the Fourier plane of the
+B-G phase. In addition, the radius of the perfect vortex could be controlled by varying the radial wave vector kr of
+the B-G beam.
+IV. THEORETICAL ANALYSIS OF THE MODE-INDEPENDENT QUANTUM STORAGE
+We consider a three-level atom model in the storage process, and the dynamics of the laser-driven atomic system
+can be described by the master equation as follows
+∂ˆρ
+∂t = − i
+ℏ[Hint, ˆρ] − 1
+2
+�
+ˆΓ, ˆρ
+�
+(4)
+where Hint is the interaction Hamiltonian that describes the light-atom coupling, and the second term attributes to
+the atomic relaxation that describes the radiative decay and the decoherence processes of the excited state and ground
+state. Under the rotating-wave approximation [11], Hint is given by
+Hint = −ℏ
+2
+�
+�
+0
+0
+Ωp
+0
+−2(∆ωp − ∆ωc)
+Ωc
+Ωp
+Ωc
+−2∆ωp
+�
+�
+(5)
+3
+
+rr
+σ
+r
+σ
+1
+rk
+rk
+rk
+=
+5
+=
+10
+=
+FIG. S2. (left) Schematic of the cross-section distribution of the POV mode and atomic ensemble with cylindrical symmetry.
+(Right) The measured storage efficiency η as a function of quanta of POV for several radial wave vectors kr = 1, 5, 10. The
+symbols are experimental data and the solid lines are the theoretically simulated curves. The fitting parameters {Tp, σr, κ}
+are {300 ns, 0.275 ± 0.025 mm, 1.1}, respectively. It can be seen that the theoretical calculations are in good agreement with
+the experimental results.
+where Ωp(c) and ∆ωp(c) denote the Rabi frequency and detuning of the signal (control) field, respectively.
+The
+Maxwell-Bloch equations can then be written as
+∂tσ31 = (i∆ωp − γ31)σ31 + i
+2Ωcσ21 + i
+2Ωp
+∂tσ21 = [i(∆ωp − ∆ωc) − γ21] σ21 + i
+2Ωcσ31
+(1/c∂t + ∂z)Ωp = i DeffΓ
+2L σ31
+(6)
+Here, σij represents the atomic coherence between levels |i⟩ and |j⟩. As referred to Ref. [12], we can obtain the
+numerical relation between the effective atomic optical depth Deff and storage efficiency η by taking the Fourier
+transform, which is given by
+η = exp(−2γ21DeffΓ/Ω2
+c)
+�
+1 + 32 ln 2 γ31DeffΓ
+(TpΩ2c)2
+× 1
+2
+�
+�erf(2
+√
+ln 2κ) + erf(2
+√
+ln 2 DeffΓ/(TpΩ2
+c) − κ
+�
+1 + 32 ln 2 γ31DeffΓ
+(TpΩ2c)2
+)
+�
+�
+(7)
+where γ21 is the ground-state decoherence rate between levels |2⟩ and |1⟩. γ31 = Γ/2 is the decay rate of |3⟩ ↔|1⟩. Tp
+denotes the full width at half maximum (FWHM) of signal pulse duration. κ is the proportionality of the time span,
+when the control field is switched off, to the Tp.
+As depicted in the main text, we take into account that the atomic ensemble has a Gaussian distribution of the
+density in the radial direction Ntr(r) = N0 exp[−r2/(2σ2
+r)] (Fig. S2, left). rr=krf/k represents the transverse size
+of the POV mode of signal. The same value of rr for different quanta ℓ results in a mode-independent light-matter
+interaction, which is the key to realizing high-dimensional quantum storage in our work.
+V. STORAGE-EFFICIENCY DISTRIBUTION FOR A WIDE MODE SPECTRUM
+The left panel of Fig. S2 depicts the transverse distribution of POV photons and atomic ensemble. The radius size
+of the POV mode with kr = 5 at the centre of the MOT is theoretically estimated to be rr = 218 µm, which is very
+close to the measured value of 222 µm detected by the ICCD camera.
+The POV mode can be derived from the Fourier transformation of a Bessel function. As the generation of an ideal
+Bessel beam is difficult in the experiment, we turn to its finite-energy approximation, i.e., the Bessel-Gaussian beam,
+4
+
+0
+-50
+50
+100
+-100
+rk
+5
+7
+9
+11
+OD
+120
+200
+280
+360
+
+0
+-50
+50
+100
+0.8
+0.6
+0.4
+0.2
+0
+-100
+
+(a)
+(b)
+FIG. S3.
+Theoretical simulations of the multi-mode storage performance.
+a, The distribution of storage efficiency η as a
+function of kr and ℓ. b, The storage efficiency η versus OD and ℓ with kr = 13.
+to prepare POV. When the quanta of POV is invloved in a large range, it is worthwhile to consider a second-moment
+width of the beam profile in the practical physical process, which is defined as [13]:
+ωℓ = ω0
+√
+ℓ + 1 + rr
+�
+1 + Iℓ+1(r2r/ω2
+0)/Iℓ(r2r/ω2
+0)
+(8)
+From this, we can find that the ratio of rr and ω0 is a crucial parameter that could evaluate the quality of POV,
+where rr/ω0 ≫ 1 is the ideal case. As shown in Fig. S2 (right), the storage efficiency of POV decreases as ℓ increases
+in the case of large quanta, which agrees well with the theoretical result obtained by combining Eqs. (7) and (8).
+To achieve mode-independent quantum storage over a wider range, one can control the Bessel parameters, e.g. kr,
+whereas the limitation of the radius size it causes in the atomic ensemble should be taken into account. This difficulty
+can be overcome by adjusting the scaling ratio of the controllable 4-f imaging system used in our work. The capacity
+of quantum memory allows for scaling in our scheme, which has a great prospect to achieve a higher-dimensional
+quantum memory by means of further optimization of several physical parameters, such as kr and OD. To clearly
+display it, we theoretically investigate these parameters for improving the storage efficiency and uniform-efficiency-
+mode range, as shown in Fig. S3. The uniform-efficiency-mode range increase with the increase of kr whereas the
+uniform efficiency is decreased (Fig. S3(a)). This limitation can be overcome by further improving the OD of the
+storage medium, as shown in Fig. S3(b). Therefore, we can achieve a quantum memory enabling higher mode and
+having higher storage efficiency based on our scheme.
+VI. DEFINITIONS OF SIMILARITY AND CROSS-TALK
+To quantitatively assess the preservation of spatial structures for input and retrieved states, we calculate the
+similarity S = �
+i
+�
+j AijBij/
+���
+i
+�
+j A2
+ij
+� ��
+i
+�
+j B2
+ij
+�
+, where A and B denote the grey-scale matrices of two
+images [5], and the subscripts i and j represent different pixels. The cross-talk disturbance in our work is quantified
+by defining a contrast Cm = (Emm − Emn,max) /Emm, where m, n refer to the input and projected mode numbers
+chosen from -12 to 12, and Emm, Emn,max represent the diagonal coefficients and maximal off-diagonal coefficients
+inside the 25×25 matrix [Fig. 2(e)], and the average contrast is estimated by C = 1/25 �
+m Cm.
+VII. STORAGE OF COHERENT SUPERPOSITION STATES WITH SPECIFIC MODES
+We access the storage of coherent superposition states, |ψ±ℓ⟩ = (|− |ℓ|⟩ + |+ |ℓ|⟩) /
+√
+2, which are encoded in POV
+modes with quanta from low-order alphabets to high-order alphabets. The correspondingly coherent interference
+5
+
++
+-1
+1
++
+-2
+2
++
+-3
+3
++
+-4
+4
++
+-5
+5
++
+-6
+6
++
+-7
+7
++
+-8
+8
++
+-9
+9
++
+-10
+
+10
+
++
+-11
+
+11
+
++
+-12
+
+12
+
+Input
+Retrieval
+FIG. S4. The intensity distributions for input (upper layer) and retrieved (lower layer) superposition states.
+patterns are shown in Fig. S4 when we chose ℓ from 1 to 12. The similarities between inputs (top) and outputs
+(bottom) are estimated to be 99.74%, 99.64%, 99.55%, 99.70%, 99.40%, 99.70%, 99.69%, 99.70%, 99.66%, 99.60%,
+99.62%, 99.63%, indicating the high-fidelity storage for various superposition states. Meanwhile, the calculated storage
+efficiencies for these states are 60.6%, 62.1%, 62.1%, 64.8%, 62.2%, 62.6%, 60.9%, 62.8%, 61.2%, 65.5%, 60.7%, 60.2%,
+manifesting approximately the same value for different ℓ, which agrees well with the theoretical expectation.
+VIII. DIMENSION VERSUS STORAGE TIME
+We have measured the storage time of memory for qudits with different dimensions, as shown in Fig. S5. Here,
+the input qudit states of �ℓ=+1
+ℓ=−1 |ℓ⟩ /
+√
+3, �ℓ=+2
+ℓ=−2 |ℓ⟩ /
+√
+5, �ℓ=+5
+ℓ=−5 |ℓ⟩ /
+√
+11, �ℓ=+12
+ℓ=−12 |ℓ⟩ /
+√
+25 with dimensions of d =
+3, 5, 11, 25 respectively, are used to display the relation of storage time with dimensions. It can easily be observed
+that our quantum memory is robust to the dimensions of quantum states since the storage time is almost the same
+for different dimensional qudits. Our memory features identical storage characters (e.g. storage efficiency and storage
+time) for 25 POV modes with l from -12, -11, -10 · · · 10, 11, 12. Thus, these same quantum storage characters for 25
+individual eigenstates allow us to store arbitrary qudits with dimensions d between 1 to 25, which shows the strong
+programmability of our memory. In conclusion, it has great prospects for building a higher-dimensional quantum
+memory using our scheme.
+FIG. S5. The storage time for qudits with different dimensions.
+6
+
+IX. HIGH-DIMENSIONAL QST
+QST is an efficient and robust technique for accurately characterizing the measured quantum states. Thus, to
+determine the full states of the retrieval in the storage process, we use a high-dimensional QST to reconstruct the
+density matrix. The d-dimensional density matrix ρd can be described by
+ˆρd = 1
+d
+d2−1
+�
+j=0
+rjˆλj
+(9)
+where ˆλj represents the operator for a SU(d) system, and rj =
+�
+ˆλj
+�
+= Tr[ˆρˆλj] is the expectation value of the
+operators. In practical measurements, an arbitrary but complete set of basis states {|ψi⟩} is exploited to implement the
+related projection operations with operators {ˆµi = |ψi⟩ ⟨ψi|}. Owing to the completeness of ˆµi, this can be written as
+ˆµi = �
+j Aj
+i ˆλj. Here, we take the corresponding measurement values ni = N ⟨ψi|ˆρ|ψi⟩ = NTr[ˆρˆµi]=N �
+j Aj
+iTr[ˆρˆλj] =
+N �
+j Aj
+irj (N is a constant of proportionality, which is dependent on the detector efficiency and count of photons).
+The density matrix can finally be written as ˆρd = N −1 �
+i,j
+�
+Aj
+i
+�−1
+niˆλj/d.
+Ultimately, we use the maximum-
+likelihood estimation technique with the combination of 625 individually projected measurement values to derive a
+physical density matrix for a 25-d qudit, as shown in Fig. 6 of main text.
+∗ [email protected]
+† [email protected]
+‡ [email protected]
+[1] Y. Wang, et al., Efficient quantum memory for single-photon polarization qubits. Nat. Photon. 13, 346–351 (2019).
+[2] S. Zhang, et al., A dark-line two-dimensional magneto-optical trap of 85Rb atoms with high optical depth. Rev. Sci.
+Instrum. 83, 073102 (2012).
+[3] D. G. Grier, A revolution in optical manipulation. Nature 424, 810–816 (2003).
+[4] N. Uribe-Patarroyo, A. Fraine, D. S. Simon, O. Minaeva, A. V. Sergienko, Object identification using correlated orbital
+angular momentum states. Phys. Rev. Lett. 110, 043601 (2013).
+[5] D.-S. Ding, Z.-Y. Zhou, B.-S. Shi, G.-C. Guo, Single-photon-level quantum image memory based on cold atomic ensembles.
+Nat. Commun. 4, 2527 (2013).
+[6] A. Nicolas, et al., A quantum memory for orbital angular momentum photonic qubits. Nat. Photon. 8, 234–238 (2014).
+[7] H. Yan, E. Zhang, B. Zhao, K. Duan, Free-space propagation of guided optical vortices excited in an annular core fiber.
+Opt. Express 20, 17904–17915 (2012).
+[8] W. Zhang, et al., Experimental realization of entanglement in multiple degrees of freedom between two quantum memories.
+Nat. Commun. 7, 1–7 (2016).
+[9] P. Vaity, L. Rusch, Perfect vortex beam: Fourier transformation of a bessel beam. Opt. Lett. 40, 597–600 (2015).
+[10] M. Liu, et al., Broadband generation of perfect poincar´e beams via dielectric spin-multiplexed metasurface. Nat. Commun.
+12, 1–9 (2021).
+[11] M. Fleischhauer, A. Imamoglu, J. P. Marangos, Electromagnetically induced transparency: Optics in coherent media. Rev.
+Mod. Phys. 77, 633 (2005).
+[12] Y.-F. Hsiao, et al., Highly efficient coherent optical memory based on electromagnetically induced transparency. Phys.
+Rev. Lett. 120, 183602 (2018).
+[13] J. Pinnell, V. Rodr´ıguez-Fajardo, A. Forbes, How perfect are perfect vortex beams? Opt. Lett. 44, 5614–5617 (2019).
+7
+

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+arXiv:2301.03984v1  [hep-th]  10 Jan 2023
+Probing Pole Skipping through Scalar-Gauss-Bonnet coupling
+Banashree Baishya∗ and Kuntal Nayek†
+Department of Physics,
+Indian Institute of Technology Guwahati,
+Guwahati 781039, India
+(Dated: January 11, 2023)
+The holographic phenomena of pole skipping have been studied in the presence of scalar-Gauss-
+Bonnet interaction in the four-dimensional Anti-de Sitter-Schwarzchild black hole background. Pole
+skipping points are special points in phase space where the bulk linearised differential equations
+have multiple ingoing solutions. Those special points are claimed to be connected to chaos. In this
+paper, we initiated a novel study on understanding the response of those special points under the
+application of external sources. The source is identified with the holographic dual operator of the
+bulk scalar field with its non-normalizable solutions. We analyze in detail the dynamics of pole
+skipping points in both sound and shear channels, considering linear perturbation in bulk. In the
+perturbative regime, characteristic parameters for chaos, namely Lyapunov exponent and butterfly
+velocity, remain unchanged. However, the diffusion coefficient has evolved non-trivially under the
+external source.
+∗ [email protected]
+† [email protected]
+
+2
+CONTENTS
+1. Introduction
+2
+2. Holographic Gravity Background
+3
+3. Scalar field perturbation
+5
+4. Metric perturbations
+7
+4.1. Shear Channel
+7
+4.2. Sound Channel
+10
+5. Analysis of chaos
+12
+5.1. From vv component of linearised Einstein equation
+12
+5.2. From the master equation
+12
+6. Discussions
+13
+A. Coefficient of Master Equation: Shear Channel
+14
+B. Coefficient of Master Equation: Sound Channel
+14
+Acknowledgements
+15
+References
+15
+1.
+INTRODUCTION
+Chaos, at the classical level, explains various macroscopic phenomena of hydrodynamics from a microscopic view-
+point. These phenomena are local criticality, zero temperature entropy, diffusion transport, Lyapunov exponent, and
+butterfly velocity. At the quantum level, chaos is similarly essential to studying those phenomena [1–3]. Recently,
+chaos in many body systems has drawn tremendous interest. It can be observed from the energy density two-point
+function. Since the holographic tools have an extensive advantage in studying those two-point functions from gravity
+theory, nowadays, the AdS/CFT correspondence [4, 5] is being used to describe the chaotic behavior in many-body
+quantum system [6–9]. However, using the holographic description, the microscopic behavior of quantum chaos was
+first established in [10]. The two-point energy density function can be described with the four-point out-of-time-
+ordered correlator(OTOC).
+⟨V (t, ⃗x)W(0)V (t, ⃗x)W(0)⟩β0 ≈ eλL(t−|⃗x|/vB)
+(1.1)
+where λL is the Lyapunov exponent and vB is the butterfly velocity related to chaos. For a chaotic system, the
+two-point energy density function shows non-uniqueness around some special points in momentum space (ω, k).
+Holographically, the OTOC is non-uniquely defined at those points. These points are where the poles and zeros of
+the energy density function overlap. They are marked as the Pole-skipping (PS) points. For example, the boundary
+two-point (Green’s) function is a the ratio of the normalized mode to the non-normalized mode of the bulk field Φ,
+which generally takes the form as GR ∝ Φb(ω,k)
+Φa(ω,k), At the pole-skipping point, Φb(ω∗, k∗) = Φa(ω∗, k∗) = 0 and makes
+the Green’s function ill-defined. The line of poles is defined by Φa(ω∗, k∗) = 0 whereas the line of zeros is given by
+Φb(ω∗, k∗) = 0. Thus the pole-skipping points are some special locations in the ω − k plane. By analyzing the shock
+waves in an eternal black-hole background, chaos parameters are related to OTOC [11]. At the above special points
+(ω∗, k∗) of energy-density two-point function, one can relate the parameters of chaos as,
+ω∗ = iλL,
+k∗ = iλL
+vB
+(1.2)
+where λL and vB are the Lyapunov exponent and butterfly velocity associated with the considered chaotic system.
+However, the behavior of the energy density function is universal for maximally chaotic systems. The microscopic
+dynamics of various hydrodynamic quantities are deeply related to the near-horizon analysis of holographic gravity.
+Indeed, the pole-skipping points can be identified from the in-going bulk field near the horizon. At those special
+
+3
+points, the bulk field leads to the multi-valued Green’s function at the boundary[12]. In simple words, there is no
+unique in-going solution at the horizon for those pole-skipping points. This holographic study has been performed for
+various bulk theories [9, 13–21]. In [12, 22], the pole-skipping points have been found for the BTZ background. They
+have shown the intersection of the lines of poles and zeros and the existence of two regular in-going solutions near the
+horizon. The pole-skipping has been also studied with finite coupling correction [23], with higher curvature correction
+[24] and also in the case of zero temperature [25]. Hydrodynamics transport phenomena have been studied with the
+pole-skipping [26–28]. Similar pole-skipping points have been also evaluated for the fermionic models [22, 29]. In
+the above articles, we have seen the pole-skipping points in the ω − k plane located at Im(ω) are related to chaos.
+However, they follow the chaos bound [30]. We have also seen that these special points describe various hydrodynamic
+mechanisms apart from chaos, e.g., the momentum density two-point function gives shear viscosity, diffusion modes,
+etc.
+Higher curvature corrections and stringy correction to the pole-skipping have been explicitly studied [23, 24]. Due
+to the effect of these corrections, the Lyapunov exponent and butterfly velocity have been modified. In this article,
+we discuss the effect of the higher order Gauss-Bonnet curvature term coupled with a scalar functional ζ(φ) ∼ φp of
+a scalar field φ, where p is an integer. However, the effect of this coupling is considered to be so trivial that no back-
+reaction is included in the bulk solution. In the bulk theory, we take the standard four-dimensional Schwarzchild-Anti
+de-Sitter metric which asymptotically reduces to pure AdS. So, on the boundary, we have a Conformal Field Theory at
+a finite temperature which is maximally chaotic in nature. Therefore without modification (due to back-reaction) the
+chaos profile remains unaffected. In this background, we have studied the pole-skipping points for scalar and metric
+perturbations. We expect the effect of interaction on the pole-skipping points. We show this effect with respect to
+the variation of the source of the scalar field located on the boundary. We plot those effects for different powers p. In
+the sound channel, the flow and decay of energy density are expected to be affected by this interaction. Unlike the
+interaction-free background, we find decay in momentum density in the shear channel at a higher value of p. Here
+we have pointed out the variation of the diffusion coefficient with the scalar source. It also shows consistent behavior
+with the effect of interaction.
+We briefly mention the result of this work as follows. We have noticed the effect of the interaction on the solution
+of the scalar field. To show this, we have plotted the values of the scalar φ at the boundary against its value on the
+boundary, i.e., scalar source Os. The relation between these two quantities has shown non-linearity for higher power
+p of scalar. However, for a low regime of source value, it remains linear. Similarly in the pole-skipping points of the
+scalar field, we find an additional correction term in k due to the interaction. Because of this correction, the imaginary
+value of k decreases. As we are interested in the perturbative regime, we will not allow the scalar source to increase
+much. In all of the plots, we will take the maximum value of the scalar source in O(100). In the shear channel, we
+find a similar effect on k. However, for p > 3, we find imaginary k which implies the exponential decay or growth of
+the corresponding density function. Here we calculate the diffusion coefficient from the lowest point. It shows that
+the rate of diffusion decreases with the increase of scalar source and it is always below 1/4πT for p > 3. On the
+other hand, in the sound channel, we find the effect of interaction for all p > 1 are similar. In this channel, without
+interaction, k4 has pure real (< 0) values. Due to interaction, it encounters an imaginary part which increases with
+the effect of the scalar source. As the real k4 < 0 gives k with equal real and imaginary parts indicating the energy
+transport and decay/growth respectively. With the effect of interaction, the real and imaginary parts of k become
+unequal. Thus one can conclude this is a result of the variation of thermal transport due to interaction.
+We have organized the paper as follows. In section 2, we briefly describe our model, showing Einstein’s equation
+and background metric. We have also talked about the behaviour of the background scalar field and calculated the
+source and condensation values. In section 3, we have studied pole-skipping for scalar field perturbation. The metric
+perturbations – shear and sound modes – have been discussed in section 4. In the following section, we have calculated
+the chaos-related parameters, first, from the perturbed vv component and then from the master equation. Finally,
+we concluded our results with a brief overview of the paper in section 6.
+2.
+HOLOGRAPHIC GRAVITY BACKGROUND
+Now, in the holographic model, as we want to study pole-skipping at finite temperatures, we need to use a black
+hole solution in bulk. We consider a four-dimensional Anti-de Sitter Schwarzchild black hole. Holographically, the
+boundary theory is three-dimensional gauge theory. The bulk metric asymptotically gives (3 + 1) dimensional AdS
+space. So, the corresponding boundary theory is a finite temperature field theory. Initially, we consider pure black
+hole solution and associated Einstein’s action in the bulk theory as,
+SEH =
+�
+d4x√−g (κR + Λ)
+(2.1)
+
+4
+where κ = (16πGN)−1 is a constant related to the four-dimensional Newton’s constant with mass dimensions 2 (here
+we set it to unity.). The associated field equation
+Gµν ≡ Rµν − 1
+2Rgµν = 1
+2κΛgµν
+(2.2)
+gives the 3 + 1 dimensional AdS-Schwarzchild black hole solution
+ds2 = L2 �
+−r2f(r)dt2 +
+dr2
+r2f(r) + h(r)
+�
+dx2 + dy2��
+(2.3)
+f(r) = 1 −
+� r0
+r
+�3 ,
+h(r) = r2
+Where L is the AdS radius. In the Einstein action, R is the Ricci scalar of the background (2.3) and Λ is related to the
+cosmological constant in four dimensions. In our case, Λ = 6κ/L2 and r is the radial coordinate of the black hole with
+the horizon radius r0. The horizon radius is related to the temperature T of the black hole as 4πT = r2
+0f ′(r0) = 3r0,
+where prime denotes derivative w.r.t. r.
+Now in the action (2.1), we have added a perturbative term 1
+2α′ζ(φ)RGB, where α′ is arbitrary coupling constant
+which is very small (≪ 1) real number. It acts as the perturbation parameter. ζ(φ) is a dimensionless real scalar
+functional of the minimally coupled scalar field φ of mass m. In this present study, we have considered ζ(φ) = Lpφp,
+p ∈ Z+. In this present discussion, we will consider L = 1. The term RGB is the higher-ordered Gauss-Bonnet
+curvature term (in 4d), which is coupled to the scalar φ(r) through ζ. Gauss-Bonnet term can be written as,
+RGB = RµνρσRµνρσ − 4RµνRµν + R2.
+With this scalar-Gauss-Bonnet interaction term, the background action takes the following form as
+S =
+�
+d4x√−g
+�
+κR + Λ + α′
+2 ζ(φ)RGB
+�
+.
+(2.4)
+For p = 0, pole-skipping has been exclusively studied previously in the five dimensions [24] and it has considered the
+back-reaction of the higher curvature on the background. In our study, we are interested in p ̸= 0 cases and treating
+α′ as a perturbative parameter, our background will remain unaffected by the back-reaction of the scalar field. Now
+taking the variation of the metric tensor in (2.4), we get the Einstein equation as follows
+(κ − 2α′∇ρ∇ρζ(φ))Gµν − 1
+2gµν
+�
+Λ + 1
+2α′ζ(φ)RGB
+�
++ α′ζ(φ)
+�
+RRµν − 4RρµRρ
+ν + R ρστ
+µ
+Rνρστ
+�
+−α′ �
+R∇(µ∇ν)ζ(φ) − 4Rρ(µ∇ν)∇ρζ(φ) + 2
+�
+gµνRρσ + Rµ(ρσ)ν
+�
+∇ρ∇σζ(φ)
+�
+= 0,
+(2.5)
+where Gµν is the Einstein tensor.
+The aforementioned scalar field φ is a minimally coupled scalar in the black hole background (2.1). In the interaction
+term, the scalar couples with the second-order curvature terms. Taking this curvature coupling into account the Klein-
+Gordon equation of φ becomes,
+1
+√−g ∂µ
+�√−ggµν∂νφ
+�
+− m2φ + α′
+2 RGB
+∂
+∂φζ(φ) = 0
+(2.6)
+Our aim would be to compute the near horizon in going modes and their properties. Therefore, it is fruitful to perform
+our calculations in the ingoing Eddington-Finkelstein co-ordinate. So, we consider v = t + r∗, where v is the null
+co-ordinate and r∗ is the tortoise co-ordinate. The metric (2.3) transforms into,
+ds2 = −r2f(r)dv2 + 2dvdr + r2 �
+dx2 + dy2�
+.
+(2.7)
+The metric (2.3) is singular at r = r0. In this new coordinate, the apparent singularity is removed. The metric has
+rotational symmetry in the (x, y) plane. In the background (2.7),
+R = −12,
+RGB(r) = 12
+�
+2 + r6
+0
+r6
+�
+.
+At horizon, RGB(r0) = 36 and at the boundary RGB(r → ∞) ≈ 24. So, in the action (2.4), the scalar-Gauss-Bonnet
+interaction term can be considered as perturbation if α′ ≪ 1 where the scalar is assumed to be constant of O(1) at
+both ends.
+
+5
+In this background, the Klein-Gordon equation turns out to be,
+r2f(r)φ′′(r) +
+�
+r2f ′(r) + 4rf(r)
+�
+φ′(r) − m2φ(r) + α′
+2 RGB
+∂
+∂φζ(φ) = 0.
+(2.8)
+The asymptotic (r → ∞) behavior of equation (2.8) gives the following
+lim
+r→∞ φ(r) = Osr∆−3 + Ocr−∆.
+(2.9)
+Where, at infinity (where is our boundary), the leading coefficient Os is the source, and the subleading coefficient
+Oc is the condensation of the dual boundary dual operator.
+The scaling dimension of the dual operator ∆ =
+3/2 +
+�
+9/4 + m2. There is a lower bound on the scalar mass called the bound of BF (Breitenlohner and Freedman)
+which states that m2 ≥ −d2/4 for (d + 1) gravitational background. Otherwise, the background solution will be
+unstable. In our case, this bound will be m2 > −9/4. From equation (2.9), we can write,
+lim
+r→∞ rφ′(r) = (∆ − 3)Osr∆−3 − ∆Ocr−∆.
+(2.10)
+Now, we can easily get the source and condensation from equations (2.9) and (2.10) by some algebra as shown in [31]
+as
+Os = lim
+r→∞
+r3−∆ (∆φ(r) + rφ′(r))
+2∆ − 3
+(2.11)
+Oc = lim
+r→∞
+r∆ ((∆ − 3)φ(r) − rφ′(r))
+2∆ − 3
+.
+(2.12)
+Since our background is neutral, the scalar field will not form any condensation. Rather, in the next sections, we will
+mainly see the effect of the source in the channels.
+3.
+SCALAR FIELD PERTURBATION
+In this section, we study the dispersion relation associated with the scalar field φ which is a minimally coupled
+scalar with mass m. This scalar field φ is regular at the horizon and decays in the asymptotic limit. With these
+conditions, the solution of the scalar can be found from the equation (2.8). Now assuming the scalar field is a function
+of the radial coordinate r only, i.e., ζ(φ) = φ(r)p. We take the near-horizon expansion of the field as
+φ(r) =
+∞
+�
+n=0
+φ(n)(r0) × (r − r0)n = φ(r0) + φ′(r0)(r − r0) + φ′′(r0)(r − r0)2 + · · ·
+where, φ(n) ≡ dnφ(r)
+drn |r=r0. From these series, the first three derivatives of φ at r = r0 can be found as,
+φ′ (r0) = m2φ (r0) − 18α′pφ (r0)p−1
+3r0
+φ′′ (r0) = −18α′p
+�
+pm2 − 12
+�
+φ (r0)p−1 + m2 �
+m2 − 6
+�
+φ (r0)
+18r2
+0
+φ′′′ (r0) =
+1
+162r3
+0
+�
+−18α′p
+�
+(2(p − 2)p + 3)m4 + 6(3 − 7p)m2 + 432
+�
+φ (r0)p−1
++m2 �
+(m2 − 6)(m2 − 9) − 3(m2 − 18)
+�
+φ (r0)
+�
+Similarly, we can also find the higher order derivatives in terms of φ(r0). We can solve the scalar field from (2.8)
+numerically by providing some horizon value to the scalar field. From this solution, we can evaluate Os and Oc
+as shown in (2.11). For the near horizon study, the regularity condition of the scalar field on the horizon is very
+important. So, for numerical evaluation of the source Os or to get a consistent solution of φ(r); φ(r0) should be finite
+and small enough so that the near-horizon expansion remains convergent. From the plot of φ(r0) vs Os, we can say
+that at lower values of source Os, the relation between these two quantities is almost linear. But at higher values it
+becomes non-linear and the degree of non-linearity strongly depends on the power (p) of the interaction. Due to this
+fact, in this present work, we will confine our all numerical calculations to the low-value regime of the Os or φ(r0).
+
+6
+0
+1
+2
+3
+4
+5
+-1
+0
+1
+2
+3
+4
+5
+6
+s
+ϕ[r0]
+Figure 1. Left: The plot of Os vs φ(r0) for p = 2 (green color), p = 3 (red color), p = 4 (blue color) and p = 5 (magenta color).
+Here we have taken scalar mass m2 = −2, α′ = 0.001, and r0 = 1.
+▲▲
+▲
+▲
+▲▲
+▲
+▲
+▲
+▲
+▲▲
+-4
+-2
+0
+2
+4
+-3.5
+-3.0
+-2.5
+-2.0
+-1.5
+-1.0
+-0.5
+0.0
+Im[k]
+Im[ω]
+2 π T
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+-1.5
+-1.0
+-0.5
+0.0
+0.5
+1.0
+s
+k1
+2
+Figure 2. Left: The plot of
+Im[ω]
+2πT
+vs Im [k] at α′ = 0.01 for p = 1 (orange circle), p = 2 (green rectangle) and p = 3 (red
+triangle). Right: The plot of k2
+1 vs Os for p = 2 (green dot-dashed line), p = 3 (red dashed line) and p = 4 (blue dotted line).
+Here we have taken scalar mass m2 = −2, α′ = 0.01 and r0 = 1.
+Now to study the dispersion relation of the scalar field, we take the perturbation φ(r) → φ(r)+ e−iωv+ikxϕ(r). The
+linearized equation from (2.8) is
+r2f(r)ϕ′′(r)+
+�
+r2f ′(r) + 4rf(r) − 2iω
+�
+ϕ′(r)+
+�
+6α′(p − 1)p
+�
+f(r)2 − 2f(r) + 3
+�
+φ(r)p−2 − k2 + m2r2 + 2irω
+r2
+�
+ϕ(r) = 0
+(3.1)
+Expanding the solution near the horizon r = r0 and using the matrix method as given in [22], we get the pole skipping
+points (ω, k). We find the lowest order point is ω1 = − 3
+2ir0 = −2iπT and
+k2
+1 + r2
+0
+�
+m2 − 18α′p(p − 1)φ (r0)p−2 + 3
+�
+= 0
+Without any perturbation (α′ = 0), we get the results for pure Schwarzchild black hole k2
+1 = −(3 + m2)r2
+0, i.e., k1 is
+completely imaginary. But, due to the effect of the interaction, k1 can be real after a particular value of Os. Similar
+behaviour is also found for the higher-order pole-skipping points. Though we keep α′ small enough in the perturbative
+regime, k2
+1 becomes positive as the scalar source increases. So k becomes real. From the equation of the perturbed
+scalar (3.1), it is clear to predict that for p = 0 and 1, there is no effect on (ω, k), i.e., we get the values of the black
+hole background without any perturbation. For p ≥ 2, p effects k1 in similar way as α′ does. For the small enough
+Os, we have found k in the imaginary plane which has been plotted in the left panel of Figure 2. Here we have plotted
+first three poles (ω, k) in the complex plane for p = 1, 2 & 3. For p = 1 & 2, we have found 2n-number of points for
+kn, i.e., n number of complex roots for kn. However, for p = 3, we have found one real and n − 1 complex root of
+each kn. Because of these real roots, we have three points on the Im(k) axis. For p = 2, the interaction imposes a
+constant shift in k. But for p ≥ 3 the shift due to the interaction is proportional to the source. So, as the source
+goes to zero, kn becomes the same as the pure AdS black hole. These have been shown in the right panel of the same
+figure. Here, we have presented the variation of k2
+1 with the scalar source Os for p = 2, 3 & 4. It is found that k1
+
+7
+becomes real-valued above a certain value of Os. Now, if we allow only the imaginary values of k1, we need to put a
+cutoff on Os. The same behaviour can be found for the higher order k. However as we go to higher order in poles or
+in interaction, we need to impose a smaller cutoff on the source value to get the pure imaginary root of k.
+4.
+METRIC PERTURBATIONS
+In the pole-skipping phenomena, we study the properties of the stress-energy tensor of the boundary field theory.
+Now with AdS/CFT duality, the bulk fields are mapped to boundary operators. Therefore, the boundary stress-energy
+tensors are associated with the metric perturbation of the bulk. In our bulk, we consider the metric perturbation
+gµν → gµν + e−iωv+ikxδgµν(r),
+(4.1)
+where ω and k are energy and momentum parameters of the fluctuation and the fluctuation propagates radially. So,
+in the boundary field theory, we have the two points correlators which are < Tvv, Tvv >, < Tvv, Tvx >, < Tvv, Txx >,
+< Tvv, Tyy > in longitudinal mode and < Tvy, Tvy >, < Tvy, Txy >, < Txy, Txy > in a transverse mode where Tµν is the
+stress-energy tensor on the boundary. The metric perturbation: δgvv, δgvx, δgxx, δgyy and δgvy, δgxy are associated
+to the above two modes respectively. We impose the radial gauge condition δgrµ = 0 for all µ. We also use the
+traceless perturbation for simplicity, i.e., gµνδgµν = 0 which gives δgyy = −δgxx. However the longitudinal modes
+are actually the scalar modes, it does not couple with a minimally coupled scalar. Therefore we can perturb only
+gµν without effecting φ. Finally, we have three independent perturbations in longitudinal mode and two in transverse
+mode.
+4.1.
+Shear Channel
+As the momentum vector (ω, k) of the metric fluctuation is taken along (v, x)-plane, for shear mode, we consider
+the components coupled to y-direction. Here we take gxy and gvy as the only non-vanishing perturbations and these
+are completely decoupled from the longitudinal perturbations. These are associated with Tvy, Txy on the boundary.
+The linearised Einstein equations will give the dynamics of these fluctuations. At some special values of (ω, k), the
+solution of those equations near the horizon becomes non-unique and gives more than one independent solution. Those
+special points (ω, k) in this holographic gravity background are connected to the coincidence of poles and zeros of the
+boundary Greens function, Gµy,νy where µ, ν = v, x.
+Now we put these perturbations in the metric (2.7) and find the linearised form of the field equation (2.5) with
+only non-vanishing perturbations gxy and gvy. We find that vy, ry and xy components of the linearised equations
+are only non-trivial, whereas other equations are self-satisfied. Out of these three equations, we find two coupled
+second-order differential equations as δg′′
+vy(r) = f1
+�
+δg′
+vy, δgvy, δgxy
+�
+and δg′′
+xy(r) = f2
+�
+δg′
+vy, δgvy, δgxy
+�
+.
+Again,
+under diffeomorphism transformation with the vector field e−iωv+ikxξµ, one can show that δgvy and δgxy will form a
+gauge invariant combination Zsh as,
+Zsh = 1
+r2 (ωδgxy + kδgvy) .
+So, two second-order differential equations (DE) of δgvy and δgxy combine into a single second-order DE of Zsh. The
+final master equation is
+MshZ′′
+sh(r) + PshZ′
+sh(r) + QshZsh(r) = 0.
+(4.2)
+Where, the coefficients Msh, Psh and Qsh are functions of ω, k and φ(r). The details expressions are given in Appendix
+A. There we have considered the coefficients up to α′ order. As α′ = 0 the master equation reduces to the same as
+the pure AdS black hole. The near horizon structure of the master variable is taken as follows.
+Zsh =
+�
+n=0
+Zn × (r − r0)n.
+Now we expand the master equation (4.2) around r = r0. At zeroth order O((r − r0)0), it gives the linear algebraic
+equation of Z0 and Z1. The coefficients of Z0 and Z1 are functions of two primary variables ω and k. So, at a
+particular point, ω = ω1 the vanishing of the coefficient of Z1 indicates that Z1 is arbitrary. Again at the same ω
+value, we find a special value of k = k1 where the coefficient of Z0 vanishes. Therefore at the point (ω1, k1) the
+near horizon solution of Zsh is defined with two arbitrary parameter Z0, Z1 and the solution is combination of two
+
+8
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+0.85
+0.90
+0.95
+1.00
+1.05
+1.10
+1.15
+s
+k12
+3 r02
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+s
+kn
+2
+3 √n r0
+2
+Figure 3. Left: The plot of
+k2
+1
+3r2
+0 vs Os where p = 2 (green color), p = 3 (magenta color), p = 4 (blue color), and p = 5 (red
+color). Right: The plot of
+k2
+n
+3√nr2
+0 vs Os for n = 1 (green color), n = 2 (blue color), n = 3 (magenta color) and for two different
+powers p = 2 (solid line) and p = 5 (dashed line). Here we have taken α′ = 0.001 and m2 = −2.0.
+arbitrary solutions C1(r − r0)Z0 + C2(r − r0)Z1. So we find a non-unique solution at the point (ω1, k1) – which is the
+first order pole-skipping point. Here we find ω1 = − 3
+2ir0 and
+k2
+1 = 3r2
+0
+�
+1 − 3α′φ(r0)p ξ(2ξ2 − ξ − 17)
+2ξ + 1
+�
+(4.3)
+where, ξ = mp2/3 We find ω1 same as the previous result [8] for AdS4 black hole. But k1 contains a non-trivial shift
+due to the interaction. At α′ = 0, it gives the same k2
+1 as given in [8]. With nonzero α′, the shift in momentum
+depends on the details properties of the scalar field and its interaction namely, power p of the interacting field φ, the
+value of the field at horizon φ(r0) and mass of it m. Now the scalar mass m can not be zero to get the nonzero shift.
+Also, we need to maintain the value of α′ in such a way that the shift remains small enough, i.e., the absolute value of
+the correction term inside the square bracket in (4.3) is always less than unity. Next few higher-order pole-skipping
+points are ωn = − 3
+2inr0 for n = 2, 3, . . . and
+k2
+2 = 3
+√
+2r2
+0
+�
+1 −
+3α′ξφ(r0)p
+4(2ξ + 1 −
+√
+2)2
+�
+12ξ4 + 4(21 −
+√
+2)ξ3 + (209 − 74
+√
+2)ξ2 + (134 − 238
+√
+2)ξ + 136 + 20
+√
+2)
+��
+(4.4)
+k2
+3 = 3
+√
+3r2
+0
+�
+1 +
+ξ(5 −
+√
+3)
+66(6ξ − 3 + 2
+√
+3)3
+�
+−3888ξ6 + 54432ξ5 − (32400 − 21528
+√
+3)ξ4 + (976140 − 224964
+√
+3)ξ3
+−(1108017 − 786374
+√
+3)ξ2 + (1134059 − 427507
+√
+3)ξ + 295381
+√
+3 − 222201
+��
+(4.5)
+and so on. In all of these k values, the absolute value of the perturbative correction increases with φ(r0) or the source
+Os but the sign of the term is solely decided by the factor pm2. We find that k2
+n can be both greater or less than
+3√nr2
+0 depending on the value of pm2. For example k2
+1 > 3r2
+0 for 3
+4
+�
+1 −
+√
+137
+�
+≤ m2p < − 3
+2 and − 9
+4 ≤ m2 ≤ − 5
+4.
+However, for other higher mode points k2
+n is always less than 3√nr2
+0. It puts no further restriction on scalar mass.
+In Figure 3, we have plotted
+k2
+1
+3r2
+0 . The ratio has been varied with the scalar source Os for four different order of
+interaction p = 2, 3, 4 & 5 with perturbation parameter α′ = 0.001 and scalar mass m2 = −2. Fig.3 depicts that
+while the source is off the ratio is equal to unity. As the source increases from zero, the ratio deviates from unity
+and increases or decreases according to the power of the Scalar-Gauss-Bonnet interaction term p. At the given mass
+value, for 0 < p ≤ 4, the ratio increases with source, and for p ≥ 5 the ratio decreases from unity. The same has
+been depicted in the left panel of the figure. Whereas on the right panel of the same figure, k2
+1/(3r2
+0), k2
+2/(3
+√
+2r2
+0)
+and k2
+3/(3
+√
+3r2
+0) have been varied with the scalar source for p = 2 and p = 5. k2
+1/(3r2
+0) increases with source Os for
+p = 2 and decreases for p = 5 which is consistent with analytical observations as discussed above. On the other hand,
+k2
+2/(3
+√
+2r2
+0) and k2
+3/(3
+√
+3r2
+0) decrease with source for both p = 2 & 5.
+It has already been observed that for pure Schwarzchild-AdS4 background, the first order pole-skipping point
+obeying the dispersion relation ω = −iDsk2 emerges from the boundary Greens function [8].
+However, the first
+
+9
+
+
+
+
+▲
+▲
+▲
+▲
+■
+■
+■
+■
+-3
+-2
+-1
+0
+1
+2
+3
+-3.0
+-2.5
+-2.0
+-1.5
+-1.0
+-0.5
+0.0
+k
+Im[
+ω
+2 π T
+]
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+1.35
+1.40
+1.45
+1.50
+1.55
+1.60
+s
+4π×sT
+Diffusion Coefficient vs Scalar Source
+Figure 4. Left: The plot of PS points in ω − k plane for α′ = 0 (blue color) and α′ = 0.001 (red color), φ(r0) = 1.1, p = 3 and
+m2 = −2. Three different shapes have been used for three different modes. The solid curve (gray color) is ω = −ik2
+4πT . Right:
+Plot of 4πDsT vs Os for p = 2 (red line), p = 3 (blue line) and p = 4 (green line), m2 = −2.0 and α′ = 0.001.
+pole-skipping point gives an upper bound on the diffusion constant. The diffusion constant is related to the first order
+pole-skipping as Ds = iω1
+k2
+1 . Here, in our case, we find the diffusion constant Ds as
+Ds = iω1
+k2
+1
+=
+1
+2r0
+�
+1 + 3α′φ(r0)p ξ(2ξ2 − ξ − 17)
+2ξ + 1
+�
+(4.6)
+For d + 2 dimensional pure AdS-Schwarzchild black hole, the diffusion constant is bounded as 1 ≤ 4πDsT ≤ d+1
+d
+1. If
+the scalar field follows the BF bound and unitarity condition, the scalar mass follows the bound −2.25 < m2 < −1.25.
+DsT in (4.6) can be found in 1 ≤ 4πDsT ≤ 3
+2 for all p ≤ 6 for the mass ranges given in Table I. In the allowed mass
+range, the diffusion constant violates the bounds for p ≥ 7. In Figure 4, the left panel have shown the plot of the
+pole-skipping points in the ω −k plane. Here we have plotted the standard dispersion relation of the boundary theory
+in a low-frequency regime, ω(k) = −iDsk2 where Ds =
+1
+4πT given in [8]. When α′ = 0 or the perturbative correction
+is very small, the first pole-skipping point falls on the dispersion curve. As the e��ect of interaction increases the
+first pole-skipping point skips the dispersion curve. However, the other pole-skipping points always stay away from
+the dispersion curve. At the right panel of Figure 4, we have plotted the diffusion constant obtained in (4.6). Here
+the 4πDsT have been varied with the scalar source for three different p values. As the source is zero the diffusion
+constants for all 2 ≤ p ≤ 6 become equal to the upper bound
+3
+8πT . In the plot, as the source increases from zero, the
+diffusion constants start falling from the highest bound. For the coupling function ζ(φ) ∼ φ2 and φ3, the diffusion
+constant decreases monotonically. At a particular value of the source, 4πDsT has become equal to unity, and for
+further increase of source value, it has fallen below its lower bound. However, for p = 4, the diffusion constant is found
+to remain very close to its upper bound for a comparatively long range of Os. After that, it started decreasing very
+rapidly and reached below 1. At these higher values of the source, the diffusion constant for p = 4 has two different
+values for a single value of scalar source Os. It seems unusual. So we should control the source not to exceed ∼ 3.
+Again though we know that the diffusion constant should not violate its lower bound, our results are not unphysical.
+Table I. The mass range associated to p to follow the allowed bound of the diffusion coefficient
+Interaction order p (φp)
+mass range
+p = 1
+−2.25 < m2 < −1.5
+p = 2
+−2.25 < m2 < −1.25
+p = 3
+−2.25 < m2 < −1.25
+p = 4
+−2.007 < m2 < −1.25
+p = 5
+−1.605 < m2 < −1.25
+p = 6
+−1.338 < m2 < −1.25
+1 For pure Schwarzchild-AdSd+2, the shear mode diffusion rate is
+1
+4πT , where T = d+1
+4π r0 and r0 is the horizon [32]. So DsT is independent
+of the dimensions of the black hole. The first order pole-skipping point of the shear mode is dimension dependent, ω = − d+1
+2 ir0 and
+k2
+1 = d(d+1)
+2
+r2
+0. Therefore iω1
+k2
+1 =
+1
+d r0 = d+1
+d
+1
+4πT
+
+10
+Since our whole calculation is assumed to be in a perturbative regime, we are free to choose any tiny value of α′
+and any small range of the scalar source for the numerical evaluation. Thus the better estimation in our case always
+makes 1 ≪ 4πDsT ≤ 3
+2 for 1 < p ≤ 6.
+4.2.
+Sound Channel
+The longitudinal components of the metric perturbation are called the scalar or sound modes of the perturbation.
+These are associated with the energy density correlation on the boundary. The corresponding stress-energy tensor in
+this mode are Tvv, Tvx, Txx and Tyy on the boundary field theory. These make the two points correlation functions
+Gvv,vv, Gvv,vx, Gvv,xx and Gvv,yy which are induced by the metric perturbations. In holographic gravity theory the
+required perturbations are δgvv, δgvx and δgxx with the trace-less perturbation, i.e., δgyy = −δgxx. Like the shear
+mode, the metric perturbations also combine into a diffeomorphism invariant master variable Zso.
+Zso = 1
+r2
+�
+k2δgvv + 2ωkδgvx − k2
+2
+�
+2f ′(r) + rf(r) − 4ω2
+k2
+�
+δgxx
+�
+(4.7)
+The second-order differential equations of δgvv(r), δgvx(r) and δgxx(r) are combined into the master equation.
+MsoZ′′
+so(r) + PsoZ′
+so(r) + QsoZso(r) = 0
+(4.8)
+The coefficients of (4.8) are linear in α′ which are given in appendix B. At α′ = 0, the master equation reduces to the
+same for the pure Schwarzchild-AdS4 background. Considering the near horizon structure of Zso similar to Zsh, we
+find the pole-skipping points for various orders.
+Here we find two types of pole-skipping points from this master equation (4.8).
+The denominator of all the
+coefficients of the equation contains a common term 3k2 − 4ω2 + k2f(r). At the near horizon regime, it introduces
+a pole at 3k2 − 4ω2 = 0. Now if we consider 3k2 ̸= 4ω2 we get only ωn = − 3
+2inr0 for n = 1, 2 · · · at the lower-half
+plane of complex ω. But when we impose the condition 3k2 = 4ω2, we can also find ω in the upper-half plane of ω,
+ωn = 3
+2inr0. It will be discussed later. Now we focus on the unequal condition.
+For 3k2 ̸= 4ω2, the first order pole-skipping point is found at ωn = − 3
+2nir0 = −2πnT and first few k4
+n are given as
+k4
+1 + 9r4
+0 − α′(3 + 3i)m2p r4
+0
+�
+m2p + (6 + 6i)
+�
+φ (r0)p = 0
+(4.9)
+k4
+2 + 18r4
+0 + α′ 3m2p r4
+0
+�
+m2p
+��
+5
+√
+2 − 2i
+�
+m2p + 40i − 64
+√
+2
+�
++ 126
+�
+3
+√
+2 − 4i
+��
+φ (r0)p
+5
+√
+2 − 2i
+= 0
+(4.10)
+k4
+3 + 27r4
+0 − α′ 2m2p r4
+0φ (r0)p
+91
+√
+3 + 63i
+��
+37
+√
+3 + 3i
+�
+m6p3 − 21
+�
+61
+√
+3 + 11i
+�
+m4p2 + 63
+�
+306
+√
+3 + 31i
+�
+m2p
+−27
+�
+5369
+√
+3 + 69i
+��
+= 0
+(4.11)
+Higher order k can also be found in the same way. At α′ = 0, we get the Schwarzchild-AdS4 values k4
+1 = −9r4
+0, k4
+2 =
+−18r4
+0, k4
+3 = −27r4
+0 and so on. In (4.9), the imaginary part 3α′ �
+12 + pm2�
+m2pr4
+0φ(r0)p is zero for m2 = − 12
+p which
+is beyond the BF bound − 9
+4 < m2 for p ≤ 4. But for p ≥ 5 we can make k4
+1 real at the above value of m2. A similar
+behaviour is also expected from the higher order k. Here we have compared the position of pole skipping points of
+φ2 interaction with the absence of interaction (α′ = 0) in Figure 5. The real and imaginary parts of k have been
+separately plotted against ω/2iπT . In both cases, the real and imaginary parts are almost equal to each other in each
+mode. For each part, the values have mirror symmetry with respect to the Re[k] = Im[k] = 0 axes. The shift due
+to interaction is very hard to identify in k1. For k2 and k3 on the other hand, one observes a measurable amount of
+shift. It has been depicted in above Figure 5. Without interaction, in each of these three modes, four real numbers
+make four k. With interaction, the same happened for k1. But for k2 and k3 eight real numbers makes four complex
+values of k.
+Again we have numerically shown the variation of k with the source Os of the scalar in Figure 6. At the left plot of
+this figure, we have plotted the real and imaginary parts of k4/(9r4
+0) against the scalar source. Here we have evaluated
+the ratio of our result with the result of pure AdS-Schwarzchild. This ratio has no explicit r0 dependent. It depends
+on scalar mass, interaction order, and scalar value on the horizon. For α′ = 0.001, m2 = −2, and for different p the
+ratio has been evaluated. When the source is off, the imaginary part of the mentioned ratio is zero, whereas the real
+part is −1. Which is consistent with the case without interaction. The imaginary part in k4 is contributed only from
+the interaction. As we have seen at the small value source is linearly proportional to φ(r0), so, φ(r0) also goes to
+zero as the source becomes zero and thus vanishes the correction term in (4.9). For p = 2, 3, 4 & 5 we have found the
+
+11
+-4
+-2
+0
+2
+4
+-3.0
+-2.5
+-2.0
+-1.5
+-1.0
+-0.5
+0.0
+Re[k]
+Im[
+ω
+2 π T
+]
+-4
+-2
+0
+2
+4
+-3.0
+-2.5
+-2.0
+-1.5
+-1.0
+-0.5
+0.0
+Im[k]
+Im[
+ω
+2 π T
+]
+Figure 5. The plot of real part (right panel) and imaginary part (left panel) of k vs
+ω
+2πT for p = 2, m2 = −2, α′ = 0 (solid
+rectangle) and α′ = 0.01 (open circle).
+0
+1
+2
+3
+4
+5
+-1.5
+-1.0
+-0.5
+0.0
+0.5
+Os
+k14
+9 r04
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+-1.5
+-1.0
+-0.5
+0.0
+0.5
+Os
+kn4
+9 n r04
+Figure 6. Left: The plot of real (solid line) and imaginary (dashed line) parts of
+k4
+1
+9r4
+0 vs Os for different order η. For α′ = 0.001,
+m2 = −2, p = 2 (green color), p = 3 (red color), p = 4 (blue color) and p = 5 (magenta color). Right: The plot of real (solid
+line) and imaginary (dashed line) parts of
+k4
+n
+9nr4
+0 vs Os for different pole-skipping points. For α′ = 0.001, m2 = −2, p = 3, n = 1
+(magenta color), n = 2 (blue color) and n = 3 (red color).
+same behaviour as Os → 0. Now if the source is turned on and increased gradually, as long as the source is small
+enough, both the imaginary and real parts change slowly. However, as p increases the rate of change also increases.
+The reason is clear from the presence of φ(r0)p factor in the correction terms. During this change the imaginary part
+of the ratio k4
+1/9r4
+0 shifts from 0 towards −1 and the real part changes in the exact opposite direction. Therefore
+the absolute value of the real (imaginary) part decreases (increases). Thus at some point on Os, real and imaginary
+lines cross each other where their value is exactly equal and lie in between 0 and 1. Again after a certain amount of
+increase in the source, the real part crosses the horizontal axis. At that value Os, k4
+1 becomes a completely imaginary
+number. These two cross-over points highly depend on p, in the given plot, the p = 3 plot has made the first cross-over
+whereas the p = 1 plot has made the last cross-over. As the source value increases further the real (imaginary) values
+become more and more positive (negative). Since we are interested in the perturbative effect, we will not consider
+those high values of k4
+1. At the right panel of the same figure, we have plotted the ratio k4
+n/(9nr4
+0) where n = 1, 2 & 3.
+Here interaction order is fixed at φ3. We have noticed that the behaviour of the real and imaginary parts of the
+ratio is almost identical to the left panel. We have found the two cross-overs for each of the three modes of k. At
+these cross-over points, the behaviour of k2
+n is completely identical to before. For the lowest order pole-skipping, k1,
+the cross-over happened at the highest Os value, and the cross-over points come closer to Os = 0 as the order of
+pole-skipping increases. Therefore the order of interaction and the order of the pole-skipping affect k in the same
+way. Mainly the location of the cross-over points is almost identically affected by these two parameters. However, the
+cross-over points can be found analytically from (4.9)-(4.11). For example, the real and imaginary parts of k4
+1 are
+Re[k4
+1] = −9r4
+0
+�
+1 − 1
+3α′p2m4φ(r0)p
+�
+Im[k4
+1] = 3α′pm2r4
+0
+�
+12 + pm2�
+φ(r0)p
+
+12
+The first cross-over happens at the value of Os corresponding to φ(r0) =
+�
+−4α′m2p
+�−1/p where the real and imaginary
+part of k4
+1 are equal to each other. The (second) cross-over on the Os axis occurs for φ(r0) =
+�
+3
+α′p2m4
+�1/p
+. Here k4
+1
+is completely imaginary 9ir4
+0
+�
+12
+m2p + 1
+�
+. The first cross-over occurs only if m2 < 0. For a moment if we assume that
+m2 > 0, then there is only the second cross-over where the k4
+1 becomes completely imaginary.
+5.
+ANALYSIS OF CHAOS
+5.1.
+From vv component of linearised Einstein equation
+From the shock wave analysis, it is found that the exponential factor of OTOC can be directly observed from
+the δE00 component of the linearized Einstein equation in the in-going Eddington-Finkelstein co-ordinate. In the
+discussed background (2.7), the information about OTOC can be obtained from the vv component of the equation
+(2.5). Considering the metric perturbation coupled with the vv component of the metric (which are actually the
+perturbations associated with the sound mode) one can write the δEvv at r = r0 as follows.
+δgvv(r0)
+�
+k2 − 2ir0ω
+�
++ kδgvx(r0) (2ω − 3ir0) = 0
+(5.1)
+Since it is well-known that at the special points (ω∗, k∗), we have no constraint on the perturbed metric components
+at r = r0 [7]. Therefore in the above equation the coefficients of δgvv(r0) and δgvx(r0) have to zero. Thus we have
+ω∗ = 3ir0
+2
+= 2πiT,
+k2
+∗ = −3r2
+0
+(5.2)
+This (ω∗, k∗) is the zeroth order pole-skipping point which is connected to the Lyapunov exponent and butterfly
+velocity as shown in (1.2). In our model, we get, λL = 2πT and vB =
+√
+3
+2 , which is the exact result[8] as in the case
+of background where the coupling term is not present in the action.
+5.2.
+From the master equation
+In the last section, where we have discussed the pole-skipping of the sound mode perturbation, we took the condition
+that 3k2 ̸= 4ω2. Because we have seen at the horizon the differential equation (4.8) encounters a singularity. Here
+we will discuss that issue. From past works [8, 24], we have seen that 3k2 = 4ω2 had come with a new set of points
+(ω, k) in Im[ω] > 0 plane which was actually related to the chaos parameters. In our case, we can re-arrange the
+master equation (4.8) as
+Z′′
+so(r) + P(r)Z′
+so(r) + Q(r)Zso(r) = 0
+(5.3)
+In this equation, the denominators of both P(r) and Q(r) has a multiplicative factor of (3 + f(r)) k2 − 4ω2 which
+reduces to 3k2 − 4ω2 at r = r0. So to get the regular solution of (5.3) at r = r0, we must impose an extra condition
+on ω or k. Here we will find it.
+First we put k =
+2
+√
+3ω in (5.3) and expand it around r = r0. We find that P(r) and Q(r) process the first and
+second order pole at r = r0.
+P(r) =
+P−1
+(r − r0) + O
+�
+(r − r0)0�
+,
+P−1 = −1 − 2iω
+3r0
+− 144α′ir0ωζ′(r0)
+3r0 − 2iω
+Q(r) =
+Q−2
+(r − r0)2 + O
+�
+(r − r0)−1�
+,
+Q−2 = 1 + 2iω
+3r0
++ 4iα′ω(27r2
+0 + 12ir0ω + 4ω2)ζ′(r0)
+r0(3r0 − 2iω)
+Therefore r = r0 is a regular singular point for the differential equation (5.3). Now, suppose Zso has a series solution
+near the singular point given as
+Zso = (r − r0)l
+�
+n∈[0,Z+)
+Zn(r − r0)n
+(5.4)
+The only condition which makes this solution regular at the horizon is l = 0, 1, 2, · · ·. Therefore the first recursion
+relation coming from (5.3) is
+l2 + l (P−1 − 1) + Q−2 = 0.
+(5.5)
+
+13
+This gives two roots (say, l1 and l2) in the following form.
+l1 = 1 − 6α′(3r0 − 2iω)ζ′(r0)
+l2 = 1 + 2iω
+3r0
++ 6α′ (3r0 + 2iω)2
+3r0 − 2iω ζ′(r0)
+So for arbitrary interaction, the only possible integer roots are l1 = 1 and l2 = 0. This gives only two values of ω as
+± 3
+2ir0. Therefore we get the same values of the chaos parameters as we have already found in the last subsection.
+6.
+DISCUSSIONS
+Here in this article, we have studied the pole-skipping phenomena in non-extremal gravity theory in presence of
+the Gauss-Bonnet-scalar interaction. We have considered a four-dimensional Schwarzchild-AdS black hole solution
+as the holographic bulk theory. On the boundary, we have a finite temperature conformal theory. The interaction
+is sourced by an operator of dimension ∆ of the boundary theory, which is dual to the scalar field φ in the bulk.
+In the Einstein action, the interaction term is added perturbatively (2.1). In the perturbative approximation, this
+external scalar source has no effect on the original bulk solution but made a nontrivial contribution in the linearised
+field equations (2.5). We have found that k of the pole-skipping points (ω, k) corresponding to the scalar field and
+metric perturbation have been affected by the external scalar source Os. Whereas, ω remains unchanged.
+Unlike the unperturbed model, the minimally coupled scalar φ has contained both real and imaginary k in the
+pole-skipping points. As the source is increased, the points of the imaginary k plane have moved into the real k plane.
+We have presented these facts pictorially in Figure 2. In Schwarzchild-AdS4 without external effect [8], k is always
+real in the shear mode. Here we have found that the shear mode k has the possibility to have both real and imaginary
+values depending on the effect of the scalar source. We have analytically found the effect of the interaction on the
+first three poles located at ωn = −2inπT and corresponding k ∼ T which are given in (4.3), (4.4) & (4.5). The first
+order pole-point k2
+1 is always greater than 3r2
+0 for ζ = φ, φ2, φ3 & φ4 and has decreased for other higher powers of φ.
+However, for the second and other higher orders of pole-skipping, k2 has always decreased with the increasing source
+for all positive integer powers of φ in ζ(φ). These have been shown in Figure 3. Here, the increase (or decrease)
+of real k implies a slow (or fast) rate of momentum transportation in shear mode and the imaginary k means the
+exponential decay of the momentum density. As a result, when positive k2
+1 has increased with the increasing source
+Os, the mobility of the corresponding modes has decreased. Thus the decreasing mobility has decreased the value of
+diffusion coefficient Ds. In Figure 4, we have presented this consistent behaviour of diffusion coefficient. At Os → 0,
+k2
+1 is at a minimum value, and therefore, momentum flow is maximum which has given the maximum value of Ds.
+So, due to the effect of the external source, the flow of momentum in shear mode has decreased for η ≤ 4, otherwise,
+it has increased.
+In the sound mode, the first three pole-skipping points have been derived from the master equation as ωn = −2inT
+and corresponding kn ∼ T is given in (4.9), (4.10) & (4.11).
+In the non-perturbative case where either α′ → 0
+or Os → 0, our results have reduced into the pole-skipping points of pure Schwarzchild-AdS4 background [8], i.e,
+k4
+n = −9nr4
+0. It gives a complex value (of equal real and imaginary parts) of k. As the source is turned on, we have
+found that an imaginary part has been added with the negative real part of k4. It means the real and imaginary
+part of k is no more equal. We have shown all of these in Figure 6. However, from the OTOC calculation in the
+last section, we have found the Lyapunov exponent λ = −iω = 2πT and the butterfly velocity vb =
+√
+3
+2
+where
+ω0 = 2iπT and k0 = ± 4
+√
+3iπT . These results have been further verified with a different approach by analyzing the
+power series solution of the sound mode master equation near the horizon. Therefore (ω0, k0) is considered as the
+lowest order pole-skipping point in sound mode instead of (ω1, k1). So the pole-skipping points of sound mode are
+(ω0, k0), (ω2, k2), (ω3, k3) and so on. The pole-skipping points (ω, k) describe the flow of energy density. Here k has
+both the real and imaginary parts. It signifies that the real part is associated with the flow of the energy density in
+longitudinal mode whereas the imaginary part of k is related to the exponential decay of the energy density. Therefore
+with the effect of interaction, when the energy density diffusion has increased the exponential decay has decreased
+and vice-versa. It would be interesting to study these flows and decays quantitatively.
+However, we have found some non-trivial effects of the interaction on the sound mode and shear mode. We have not
+found any effect on the chaotic behaviour. The reason is mainly the perturbative approach to the interaction term.
+If one considers the backreaction of the interaction, the Lyapunov exponent and the butterfly velocity are expected
+to be affected by the interaction. With backreaction, one can expect k0 and k1 to be equal in the sound mode.
+
+14
+Appendix A: Coefficient of Master Equation: Shear Channel
+Three coefficients of the master equation can be written in the linear order of the perturbation parameter α′
+Msh(r) = M(0)
+sh + α′M(1)
+sh + O(α′2)
+Psh(r) = P(0)
+sh + α′P(1)
+sh + O(α′2)
+Qsh(r) = Q(0)
+sh + α′Q(1)
+sh + O(α′2)
+(A.1)
+We have found the above functions as follows.
+M(0)
+sh = r2f(r)
+(A.2)
+P(0)
+sh = ωf(r)
+�
+5rω + 2ik2�
+− 8k2rf(r)2 + ω2(3r − 2iω)
+ω2 − k2f(r)
+(A.3)
+Q(0)
+sh = −10k2r2f(r)2 + f(r)
+�
+k4 + 9ik2rω + 4r2ω2�
++ ω
+�
+k2(−ω − 3ir) + 6rω(r − iω)
+�
+r2 (ω2 − k2f(r))
+(A.4)
+and
+M(1)
+sh = 0
+(A.5)
+P(1)
+sh =
+r2f(r)
+(ω2 − k2f(r))2
+�
+rζ′′(r)
+�
+ω2 − k2f(r)
+� �
+f(r)
+�
+2k2f(r) + ω2�
+− 3ω2�
++ ζ′(r)
+�
+f(r)
+�
+k2f(r)
+�
+4k2f(r) − 6k2
+−11ω2�
++ 24k2ω2 − 2ω4�
+− 9k2ω2�
+− ωF
+�
+(A.6)
+Q(1)
+sh =
+1
+r (ω2 − k2f(r))2
+�
+rζ′′(r)
+�
+ω2 − k2f(r)
+� �
+f(r)
+�
+ω2 �
+4k2 − irω
+�
+− 2k2f(r)
+�
+−3r2f(r) + k2 + 3r2 + irω
+��
++ω3(−2ω + 3ir)
+�
++ ζ′(r)
+�
+f(r)
+�
+f(r)
+�
+−k2f(r)
+�
+−14k2r2f(r) + 3k4 + 4k2r(6r + iω) + 34r2ω2�
++ 3k6
++6k4 �
+3r2 + irω + ω2�
++ k2rω2(72r + 11iω) + 2r2ω4�
++ ω2 �
+−6k4 − 3k2 �
+18r2 + 8irω + ω2�
++2rω(−6r + iω))) + 3ω3 �
+6r2ω + k2(ω + 3ir)
+��
++ ir
+�
+2ω2 − f(r)
+�
+k2 − 2irω
+��
+F
+�
+(A.7)
+where,
+F = 6k2r2ω(f(r) − 1) [r(f(r) − 3)f(r)ζ′′(r) − 3((f(r) − 2)f(r) + 3)ζ′(r)]2 /
+�
+rζ′′(r)
+�
+f(r)
+�
+f(r)
+�
+k2 − 3irω
+�
+−3k2 + 3irω
+�
+− 18irω
+�
+− 3ζ′(r)
+�
+(f(r) − 2)f(r)
+�
+k2 − irω
+�
++ 3
+�
+k2 + 3irω
+��
++ ir3ω(f(r) − 3)f(r)ζ′′′(r)
+�
+Here the ζ function takes its appropriate form.
+Appendix B: Coefficient of Master Equation: Sound Channel
+Three coefficients of the master equation can be written in the linear order of the perturbation parameter α′
+Mso(r) = M(0)
+so + α′M(1)
+so + O(α′2)
+Pso(r) = P(0)
+so + α′P(1)
+so + O(α′2)
+Qso(r) = Q(0)
+so + α′Q(1)
+so + O(α′2)
+(B.1)
+where,
+M(0)
+so = r4f(r)
+(B.2)
+P(0)
+so = r2 �
+f(r)
+�
+11k2rf(r) + 2k2(6r − iω) − 20rω2�
++
+�
+3k2 − 4ω2�
+(3r − 2iω)
+�
+k2f(r) + 3k2 − 4ω2
+(B.3)
+Q(0)
+so = −f(r)
+�
+−25k2r2f(r) + k4 + 12k2r(r + iω) + 16r2ω2�
+− 3k4 + k2(9r + 2iω)(3r − 2iω) − 24rω2(r − iω)
+k2f(r) + 3k2 − 4ω2
+(B.4)
+M(1)
+so = 0
+(B.5)
+
+15
+P(1)
+so =
+r2f(r)
+(2irω + k2) (k2f(r) + 3k2 − 4ω2)3
+�
+r3ζ′′(r)
+�
+f(r)
+�
+k2f(r)
+�
+k2f(r)2 �
+9k4 + 2k2r(−24r + 25iω) + 64r2ω2�
++2f(r)
+�
+−27k6 + 6k4 �
+24r2 − 9irω + 11ω2�
++ 4k2rω2(−72r + 17iω) + 128r2ω4�
++ 12
+�
+3k2 − 4ω2� �
+2k4
++k2 �
+−12r2 − 4irω + 3ω2�
++ 2rω2(8r + 3iω)
+��
++ 2
+�
+3k2 − 2ω2� �
+3k2 − 4ω2�2 �
+k2 + 2irω
+��
+− 3
+�
+3k2
+−4ω2�3 �
+k2 + 2irω
+��
++ 4ζ′(r)
+�
+f(r)
+�
+k2f(r)
+�
+r2f(r)
+�
+−k2f(r)
+�
+5k4 + 3k2r(−12r + 7iω) + 48r2ω2�
++ 9k6
++k4 �
+−180r2 + 33irω − 26ω2�
++ 4k2rω2(96r + 5iω) − 192r2ω4�
++ 3k8 + 4k6 �
+36r2 + 6irω − ω2�
++k4r
+�
+324r3 + 243ir2ω − 324rω2 − 32iω3�
+− 8k2r2ω2 �
+90r2 + 81irω − 34ω2�
++ 48r3ω4(8r + 9iω)
+�
++
+�
+3k2 − 4ω2� �
+3k8 − k6 �
+63r2 + 4ω2�
+− k4r
+�
+108r3 + 171ir2ω − 126rω2 + 8iω3�
++ 8k2r2ω2 �
+18r2 + 27irω
+−ω2�
+− 16ir3ω5��
++ 9k2r2 �
+3k2 − 4ω2�2 �
+k2 + 2irω
+���
+(B.6)
+Q(1)
+so = −
+1
+r (2irω + k2) (f(r)k2 + 3k2 − 4ω2)3
+��
+ω(3r + 2iω)
+�
+2rω − ik2� �
+3k2 − 4ω2�3 + f(r)
+�
+f(r)
+�
+f(r)
+�
+−3k8
++18
+�
+−19r2 − 2iωr + ω2�
+k6 + 4r(3r − iω)
+�
+108r2 + 99iωr − 26ω2�
+k4 − 8r2ω2 �
+360r2 + 234iωr − 85ω2�
+k2
++32r3ω4(48r + 35iω) + f(r)
+�
+3k8 + r(180r + 23iω)k6 − 2r2 �
+576r2 − 108iωr + 257ω2�
+k4 + 64r3ω2(30r
+−iω)k2 − r2 �
+43k4 + 6r(41iω − 40r)k2 + 320r2ω2�
+f(r)k2 − 512r4ω4��
+−
+�
+3k2 − 4ω2� �
+9k6 + 6
+�
+18r2
++3iωr − 7ω2�
+k4 + 8irω
+�
+45r2 + 42iωr − 11ω2�
+k2 + 8r2ω3(ω − 60ir)
+��
+k2 +
+�
+k2 + 2irω
+� �
+3k2 − 4ω2�2 �
+3k4
++
+�
+9r2 − 18iωr + 14ω2�
+k2 − 4irω3���
+ζ′′(r)r3 +
+��
+k2 + 2irω
+� �
+3k6 + 2ω(−3ir − 2ω)k4 − 6r2 �
+9r2 + 6iωr
+−2ω2�
+k2 + 72r4ω2� �
+3k2 − 4ω2�2 + f(r)
+�
+2
+�
+3k2 − 4ω2� �
+3k10 +
+�
+63r2 + 18iωr − 4ω2�
+k8 − r
+�
+189r3 + 18iωr2
++66ω2r + 28iω3�
+k6 + 2r2ω
+�
+−297ir3 + 315ωr2 + 6iω2r + 28ω3�
+k4 + 8ir3ω3 �
+63r2 + 18iωr + 4ω2�
+k2
++32r4ω5(6ir + ω)
+�
++ f(r)
+�
+3k12 +
+�
+−90r2 + 36iωr − 4ω2�
+k10 + 12r
+�
+171r3 + 63iωr2 + 24ω2r − 4iω3�
+k8
++4r2 �
+972r4 + 1971iωr3 − 1863ω2r2 − 186iω3r − 32ω4�
+k6 − 32r3ω2 �
+270r3 + 531iωr2 − 243ω2r − iω3�
+k4
++64r4ω4 �
+72r2 + 132iωr − 13ω2�
+k2 + 2r2f(r)
+�
+3
+�
+−15k8 − 2
+�
+186r2 + 37iωr − 9ω2�
+k6 + 2r
+�
+−396r3
+−417iωr2 + 339ω2r + 50iω3�
+k4 + 8r2ω2 �
+180r2 + 232iωr − 73ω2�
+k2 − 64r3(8r + 13iω)ω4�
++ f(r)
+�
+−3k8
++r(21r − 20iω)k6 + 2r2 �
+684r2 − 24iωr + 43ω2�
+k4 − 8r3(300r + 91iω)ω2k2 + r2 �
+47k4 + 12r(17iω − 30r)k2
++480r2ω2�
+f(r)k2 + 768r4ω4��
+k2 + 256ir5ω7���
+ζ′(r)
+�
+(B.7)
+ACKNOWLEDGEMENTS
+We would like to acknowledge Debaprasad Maity for his useful suggestions.
+[1] Y. Gu, X. L. Qi and D. Stanford, “Local criticality, diffusion and chaos in generalized Sachdev-Ye-Kitaev models,” JHEP
+05 (2017), 125 doi:10.1007/JHEP05(2017)125 [arXiv:1609.07832 [hep-th]].
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+

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+arXiv:2301.00782v1  [math.AP]  2 Jan 2023
+ON MULTIDIMENSIONAL AXISYMMETRIC OSCILLATIONS OF
+A COLLISIONAL COLD PLASMA
+OLGA S. ROZANOVA*, MARIA I. DELOVA
+Abstract. We study the influence of the friction term on the radially sym-
+metric solutions of the repulsive Euler-Poisson equations with a non-zero back-
+ground, corresponding to cold plasma oscillations in many spatial dimensions.
+It is shown that for any arbitrarily small constant non-negative constant fric-
+tion coefficient, there exists a neighborhood of the zero stationary solution in
+the C1 norm such that the solution of the Cauchy problem with initial data
+belonging to this neighborhood remains globally smooth in time. Moreover,
+this solution stabilizes to the zero as t → ∞. This result contrasts with the
+situation of zero friction, where any small deviation from the zero equilibrium
+generally leads to a blow-up. Our method allows to estimate the lifetime of
+smooth solutions. Further we prove that for any initial data one can find such
+friction coefficient that the respective solution to the Cauchy problem keeps
+smoothness for all t > 0 and stabilizes to zero.
+1. Introduction
+We study a frictional version of the repulsive Euler-Poisson equations
+∂n
+∂t + div (nV) = 0,
+∂V
+∂t + (V · ∇) V = k ∇Φ − ν V,
+∆Φ = n − n0,
+(1)
+where the the scalar functions n and Φ are the density and a repulsive (for k > 0)
+force potential, respectively, the vector V is the velocity, they depend on the time
+t and the point x ∈ Rd, d ≥ 1. Here n0 > 0 is the density background, ν > 0 is a
+friction coefficient.
+If we denote ∇Φ = −E, and set n0 = 1, such that
+n = 1 − div E,
+(2)
+we can remove n from (1) and rewrite it as
+∂V
+∂t + (V · ∇) V = −E − νV,
+∂E
+∂t + Vdiv E = V.
+(3)
+In this paper we study the Cauchy problem for (1) or (3) and our main concern
+is to study initial data that guarantee a globally smooth solution.
+System (3) corresponds to the hydrodynamics of “cold” or electron plasma in the
+non-relativistic approximation in dimensionless quantities (see, e.g., [1], [5], [8]). In
+this interpretation the friction coefficient characterizes the intensity of electron-ion
+collisions during plasma oscillations. The cold plasma equations is now very popular
+object of study, may be more popular than the Euler-Poisson equations themselves.
+The reason is that the cold plasma in used in the accelerators of electrons in the
+2020 Mathematics Subject Classification. Primary 35Q60; Secondary 35L60, 35L67, 34M10.
+Key words and phrases. Euler-Poisson equations, quasilinear hyperbolic system, cold plasma,
+blow up.
+1
+
+2
+OLGA S. ROZANOVA*, MARIA I. DELOVA
+wake wave of a powerful laser pulse [11]. From this point of view the initial data
+(i.e. the initial laser pulse) that corresponds to a solution that cannot survive being
+smooth are not applicable technically.
+System (3) in 1D case was studied [13], however, in the multidimensional case
+it is very difficult from both mathematical and physical points of view. Indeed, it
+describes a non-hyperbolic superposition of different types of waves, each of them
+have a tendency to break out in a finite time. Therefore the theoretical results here
+are very scarce.
+The situation is more optimistic if we restrict ourselves to the class of axisym-
+metric solutions. Thus, we consider one-dimensional solutions in space, given on
+the half-line. In [10], [3], [18], [17] it was shown that for n0 = 0, k > 0 and n0 ≥ 0,
+k < 0 in the non-frictional case there is a threshold in terms of the initial data.
+Namely, one can specify exactly the class of initial data corresponding to a globally
+smooth solution, and these data form a neighborhood of the stationary state in the
+C1 -norm. As it has been recently shown [15], for n0 > 0, k > 0, ν = 0 the situation
+is strikingly different: namely, for d ̸= 1 and d ̸= 4 an arbitrarily small pertur-
+bation of the zero stationary state blows up in the general case. The exception
+is the initial data in the form of a simple wave, starting from which the solution
+can remain globally smooth and tend to an affine solution as t → ∞. In any case,
+the initial data corresponding to simple waves form a zero-measure manifold in the
+neighborhood of the stationary state.
+In this paper, we study the effect of constant friction on the blow-up process.
+Namely, we establish that the presence of friction normalizes the situation with the
+threshold for the initial data. Namely, for an arbitrarily small ν > 0 and any d,
+there exists a neighborhood of the zero stationary state in the C1-norm such that
+the corresponding solution of the Cauchy problem preserves smoothness (Theorem
+1). For small ν, Theorem 2 gives sufficient conditions guaranteeing blow-up or non-
+blow-up in terms of initial data, which can be applied to numerical tests. Besides,
+we show that for any initial data, one can find ν such that the corresponding
+solution of the Cauchy problem is globally smooth (Theorem 3). In other words,
+this situation is absolutely analogous to d = 1, and the increase in spatial dimension
+does not lead to any new phenomena.
+Thus, we consider axisymmetric solutions of (3)
+V = F(t, r)r,
+E = G(t, r)r,
+where r = (x1, x2, ..., xd) is the radius-vector, r =
+�
+x2
+1 + x2
+2 + ... + x2
+d.
+The initial data that correspond to these solutions are
+(V, E)|t=0 = (V0(r), E0(r)) = (F0(r)r, G0(r)r),
+(F0(r), G0(r)) ∈ C2(¯R+).
+(4)
+We assume that (V0(r), E0(r)) are bounded together with their derivatives uni-
+formly on r ∈ ¯R+ and denote ∥f∥C1(R+) =
+1�
+i=0
+sup
+r∈R+
+|f (i)(r)|.
+The physically natural condition n|t=0 > 0 dictates divE < 1, see (2).
+The main results of the paper are as follows.
+Theorem 1. 1. For arbitrary small ν > 0 there exists ε(ν) > 0, such that the
+solution of the problem (3) - (4) satisfying
+∥V0(r), E0(r)∥C1(R+) < ε,
+(5)
+
+AXISYMMETRIC OSCILLATIONS OF A COLLISIONAL PLASMA
+3
+keeps C1 - smoothness for all t > 0. Moreover,
+∥V, E∥C1(R+) ≤ const e− ν
+2 t → 0,
+t → ∞.
+(6)
+Let us denote
+u0 = div V0 − dF0,
+v0 = div E0 − dG0,
+H0 = u0,
+H1 =
+�d − 2
+2
+F0 − ν
+2
+�
+− v0,
+φ = −d + 2
+2
+G + (d − 2)νF − (d − 2)(d − 4)
+2
+F 2,
+J+ = 1 − ν2
+4 − d + 2
+2
+G− + ν(d − 2)F+ + (1 − δ3d)(d − 2)(d − 4)
+2
+F 2
++,
+M± =
+�1 − dG∓
+1 − dG0
+� d+2
+2d
+,
+0 < M− < M+,
+where (G, F) is the solution of the problem (12) subject to initial data (G0, F0),
+G− < 0 and G+ > 0, G+ < 1
+d are the left and right roots of equation (15) or (14),
+F = 0 (they depend on (G0, F0)), F+ is given as (33) , δij is the Kronecker symbol.
+The next theorem gives more information about the size of the neighborhood of
+the origin containing globally smooth solutions in the case of small ν.
+Theorem 2. Let ν < 2.
+a) A sufficient condition on initial data (4) that guaranties the smoothness of
+the solution of the problem (3) - (4) for all t > 0 is the following:
+inf
+r∈R+)
+F1(ν, V0(r), E0(r)) < 1,
+(7)
+F1(ν, V0(r), E0(r)) = 2
+ν M+
+�
+H2
+0 +
+�
+1 − ν2
+4
+�−1
+H2
+1
+e
+∞
+�
+0
+|φ(τ|dτ
+.
+b) If there exists T > 0 such that
+inf
+r∈R+)
+F2(T, ν, V0(r), E0(r)) < 1,
+(8)
+F2(T, ν, V0(r), E0(r)) = 2
+ν M+
+�
+H2
+0 +
+�
+1 − ν2
+4
+�−1
+H2
+1
+e
+�
+J+−1+ ν2
+4
+�
+T ,
+then the solution of the problem (3) - (4) preserves smoothness for t ∈ [0, T ].
+c) If the initial data (4) are such that there exists a point r ∈ R+ for which
+condition
+F3(ν, V0(r), E0(r)) ≥ 1,
+(9)
+F3(ν, V0(r), E0(r)) = 2
+ν M−
+�
+H2
+0 + J−1
++ H2
+1,
+H0 ≤ 0,
+H1 < 0
+holds. Then the solution of problem (3) - (4) blows up within t <
+π
+√
+J+ .
+
+4
+OLGA S. ROZANOVA*, MARIA I. DELOVA
+Theorem 3. For arbitrary initial data (4) there exists such ν > 0 that the solution
+of problem (3) - (4) keeps C1 - smoothness for all t > 0 and the asymptotic property
+∥V, E∥C1(R+) ≤ const e− ν−√
+4−ν2
+2
+t → 0,
+t → ∞.
+(10)
+holds.
+Theorems 1, 2 and 3 can be reformulated in the terms of the Euler-Poisson
+equations (1). The stationary stationary state in this case is
+V = 0,
+Φ = const,
+n = 1.
+In this work we use the technique of linearization my means of the Radon lemma,
+the same as in [15]. It turn out to be very convenient for the analysis of the non-
+strictly hyperbolic systems often arising when studying the reduced cold plasma
+equations.
+The paper is organised as follows: Sec.2 is devoted to auxiliary results on the
+behavior of solution and its derivatives, Secs.3, 4 and 5 contain the proofs of The-
+orems 1, 2, and 3, respectively. Sec.6 is devoted to a discussion on the importance
+of the results for physics and the formulation of future problems in this area.
+2. Behavior of solutions along characteristics
+We use the fact that V = F(t, r)r, E = G(t, r)r and get
+∂F
+∂t + Fr∂F
+∂r = −F 2 − G − νF,
+∂G
+∂t + Fr∂G
+∂r = F − dFG.
+(11)
+(V0, E0) = (F(0, r)r, G(0, r)r) = (F0(r)r, G0(r)r),
+(F0(r), G0(r)) ∈ C2(R+).
+2.1. Physical constraints on solution components. Let us fix r0 ∈ ¯R+. Along
+the characteristics ˙r = ∂r
+∂t = Fr, r(0) = r0, of the system (11), the functions F
+and G obey the system of equations
+˙F = −F 2 − G − νF,
+˙G = F − dFG.
+(12)
+Therefore
+dG
+F(1 − dG) = dr
+Fr,
+1 − dG = const · r−d.
+(13)
+Since r ≥ 0, the sign of the expression 1 − dG does not change, i.e. sign(1 − dG) =
+sign(1−dG(0, r0)). Therefore, the motion on the phase plane (G, F) corresponding
+to system (12) occurs either in the half-plane G < 1
+d, or in the half-plane G > 1
+d,
+or on the line G = 1
+d.
+The equilibria of (12) are the following:
+• if ν <
+2
+√
+d, then there exists the only point (F = 0, G = 0), a stable focus;
+• if ν =
+2
+√
+d, then there exist two points: (F = 0, G = 0), a stable focus, and
+(F = − ν
+2, G = 1
+d), a saddle-node.
+• if ν >
+2
+√
+d, then there exist three points: (F = 0, G = 0), a stable focus
+(ν < 2) or a stable node, otherwise, and (F = −
+ν±√
+ν2�� 4
+d
+2
+, G =
+1
+d), a
+saddle and an unstable node.
+
+AXISYMMETRIC OSCILLATIONS OF A COLLISIONAL PLASMA
+5
+We see that there are no equilibria in the domain G >
+1
+d, hence there are no
+bounded trajectories in this region. If the motion on the plane (G, F) starts from
+the point for which G(0, r(0)) >
+1
+d, then the phase trajectory rests in the half-
+plane G >
+1
+d and G(t, r(t)) → +∞ for t → t∗ < ∞. Moreover, due to (13), we
+have r(t) → 0 for t → t∗. In this case, we get a contradiction with the positivity of
+density, since n = 1 − div E = 1 − Grr − dG > 0. On the line G = 1
+d the density is
+zero.
+Thus, we study the problem only in the half-plane G < 1
+d, F ∈ R.
+2.2. Boundedness of the solution. In the half-plane G < 1
+d, F ∈ R system (12)
+has one equilibrium (0, 0). It corresponds to the stationary state V = E = 0 and it
+is stable for any values of the parameters d and ν > 0. Namely, as a linear analysis
+show,
+• if 0 < ν < 2 it is a stable focus;
+• if ν = 2 it is a stable degenerate node;
+• if ν > 2 it is a stable node.
+Lemma 1. There exists δ > 0 such that if the initial data (F0(r0), G0(r0)) belong
+to the δ- neighborhood of the origin, r0 ∈ ¯R+, then any solution to (12) tends to
+zero exponentially as t → +∞.
+The proof follows from the fact that (0, 0) is asymptotically stable for all choices
+of parameters ν, d. □
+Lemma 2. Let Φ(G, F) = Φ(G0, F0) be a closed phase curve corresponding to the
+solution of system (12) for ν = 0 with initial data (F0, G0). Then the phase curve
+corresponding to the solution of system (12) ν > 0 with initial data (F0, G0) lies
+strictly inside the curve Φ(G, F) = Φ(G0, F0).
+Proof. Let us construct the phase curve of (12) at ν = 0, i.e the solution of
+dF
+dG = −
+F 2 + G
+F(1 − dG).
+It implies
+dZ
+dG = −
+2
+1 − dGZ −
+2G
+1 − dG,
+Z(G) = F 2.
+The solution is
+Φ(G, F) = (d − 2)F 2 − 2G + 1
+(d − 2)(1 − dG)
+2
+d
+= Φ(G0, F0) = Cd,
+(14)
+Cd = (d − 2)F 2
+0 − 2G0 + 1
+(d − 2)(1 − dG0)
+2
+d ,
+for d ̸= 2
+Φ(G, F) = 2F 2 + ln(1 − 2G)(1 − 2G) + 1
+2(1 − 2G)
+= Φ(G0, F0) = C2,
+(15)
+C2 = 2F 2
+0 + ln(1 − 2G0)(1 − 2G0) + 1
+2(1 − 2G0)
+,
+for d = 2. As it was shown in [15] The curves given as (15) and (14) are bounded,
+they contain the origin and intersect the axis F = 0 in two points: (G−, 0), G− < 0,
+and (G+, 0), G+ > 0, see the pictures in [15].
+
+6
+OLGA S. ROZANOVA*, MARIA I. DELOVA
+Let us consider V (t) = Φ(G, F) as a Lyapunov function in the half-plane G < 1
+d,
+F ∈ R. The derivative of V (t) due to system (12) is
+dV
+dt = −
+2νF 2
+(1 − dG)
+2
+d ≤ 0.
+(16)
+If we denote (G(t), F(t)) and ( ¯G(t), ¯F(t)) the point on the phase curve of (12) for
+ν > 0 and ν = 0, respectively. and Thus, the distance |(G(t), F(t))| < |( ¯G(t), ¯F(t))|,
+t > 0, since dV
+dt = 0 if and only if F = 0 and F = 0 does not solve (12). □
+Lemma 3. System (12) has no limit cycles in the half-plane G < 1
+d, F ∈ R.
+Proof. We use the Lyapunov function from Lemma 2 to prove the absence of a
+limit cycle by contradiction.
+Assume that a limit cycle (a closed trajectory Γ)
+exists. Then it contains a stable equilibrium (0, 0) inside. We denote as d(Y1, Y2)
+the distance between points Y1(t) = (G1(t), F1(t)), Y2(t) = (G2(t), F2(t)).
+For
+some initial point Y (t0) = (G∗, F∗) on Γ there exists a time t1 > t0 such that
+Y (t1) = Y (t0) and, accordingly, d(Y (t0), Y (t1)) = 0. The curve Γ contains (0, 0)
+inside, so there are two points on this trajectory for which F = 0, they are (G+, 0)
+and (G−, 0), 0 < G+ < 1
+d, G− < 0. At these points dV (G+,0)
+dt
+= dV (G−,0)
+dt
+= 0. At
+other points of Γ we have dV
+dt < 0. Then the function V (t) does not increase along
+Γ. Moreover, dV
+dt = 0 only at two points on Γ, so V (t) strictly decreases along the
+trajectory, i.e., V (t1) − V (t0) < 0 for t1 > t0. For the above points Y (t0), Y (t1),
+therefore d(Y (t0), Y (t1)) > 0. Thus, we obtain a contradiction. □
+Thus, Lemmas 1, 2 and 3 imply the following property of the phase trajectories:
+Lemma 4. Let d ≥ 2. Then the phase curves of system (12) are bounded in the
+half-plane G < 1
+d, F ∈ R and tends to zero as t → +∞.
+Remark 1. Lemma 4 is not valid for d ≥ 1. Indeed, in this case the system (12)
+coincides with the system (17) for the derivatives. As was shown in [13], for any
+ν > 0 there exists a point on the phase plane such that the phase curve starting
+from this point goes to infinity as t → t∗ < ∞.
+2.3. Study of the behavior of derivatives. This section closely follows [15], but
+for the convenience of the reader we give a sketch of the reasonings.
+Let us denote D = div V, λ = div E. Equations (3) imply
+∂D
+∂t + (V · ∇D) = −D2 + 2(d − 1)FD − d(d − 1)F 2 − λ − νD,
+∂λ
+∂t + (V · ∇λ) = D(1 − λ).
+Along the characteristics given as ˙r = Fr the functions D, λ obey
+˙D = −D2 + 2(d − 1)FD − d(d − 1)F 2 − λ − νD,
+˙λ = D(1 − λ).
+(17)
+We introduce new variables u = D − dF, v = λ − dG. Systems (17) and (12) imply
+˙u = −u2 − 2uF − v − νu,
+˙v = −uv + (1 − dG)u − dFv.
+(18)
+System (18) can be linearized my means of the Radon lemma (e.g. [6], [12]).
+
+AXISYMMETRIC OSCILLATIONS OF A COLLISIONAL PLASMA
+7
+Theorem 4. [The Radon lemma] A matrix Riccati equation
+˙W = M21(t) + M22(t)W − WM11(t) − WM12(t)W,
+(19)
+(W = W(t) is a matrix (n×m), M21 is a matrix (n×m), M22 is a matrix (m×m),
+M11 is a matrix (n×n), M12 is a matrix (m×n)) is equivalent to the homogeneous
+linear matrix equation
+˙Y = M(t)Y,
+M =
+�
+M11
+M12
+M21
+M22
+�
+,
+(20)
+(Y = Y (t) is a matrix (n × (n + m)), M is a matrix ((n + m) × (n + m)) ) in the
+following sense.
+Let on some interval J ∈ R the matrix-function Y (t) =
+� Q(t)
+P(t)
+�
+(Q is a
+matrix (n × n), P is a matrix (n × m)) be a solution of (20) with the initial data
+Y (0) =
+�
+I
+W0
+�
+(I is the identity matrix (n × n), W0 is a constant matrix (n × m)) and det Q ̸= 0
+on J . Then W(t) = P(t)Q−1(t) is the solution of (19) with W(0) = W0 on J .
+Let us (18) as (19) with
+W =
+�
+u
+v
+�
+,
+M11 =
+�0�
+,
+M12 =
+�1
+0�
+,
+M21 =
+�0
+0
+�
+,
+M22 =
+�−2 F − ν
+−1
+1 − d G
+−d F
+�
+.
+Then according Theorem 4 the solition of (18) is
+W(t) = P(t)
+Q(t),
+where P(t) = (p1(t), p2(t))T and Q(t) solves the linear system
+� Q
+P
+�·
+= M
+� Q
+P
+�
+,
+M =
+
+
+0
+1
+0
+0
+−2F − ν
+−1
+0
+1 − dG
+−dF
+
+
+(21)
+subject to the initial data
+� Q
+P
+�
+(0) =
+�
+1
+W0
+�
+,
+W0 =
+� u0
+v0
+�
+=
+� div V0 − dF0(r0)
+div E0 − dG0(r0)
+�
+.
+Since the vector function P(t) and the function Q(t) are components of the solution
+of a linear system of differential equations with continuous coefficients, these func-
+tions do not go to infinity for any finite value of t. Hence the functions u, v go to
+infinity along the characteristic starting from the point r0 ∈ R if and only if there
+exists a finite t∗ > 0 such that Q(t∗, r0) = 0. Since u = D − dF, v = λ − dG and
+the functions G, F are bounded, the derivatives of the solution of (3) are bounded
+if and only if the functions u, v are bounded. Thus, the conditions for the bound-
+edness of derivatives coincide with the conditions under which the function Q(t)
+does not vanish for any finite t.
+System (21) implies
+
+8
+OLGA S. ROZANOVA*, MARIA I. DELOVA
+Q(t) = 1 +
+t
+�
+0
+p1(τ)dτ,
+˙p1 = −(2F + ν)p1 − p2,
+˙p2 = (1 − dG)p1 − dFp2.
+It follows
+¨p1 + ((d + 2)F + ν) ˙p1 + (2 ˙F + (1 − dG) + dF(2F + ν))p1 = 0,
+and, taking into account ˙F = −F 2 − G − νF,
+¨p1 + ((d + 2)F + ν) ˙p1 + (2(d − 1)F 2 − (2 + d)G + (d − 2)νF + 1)p1 = 0.
+We change
+p1(t) = H(t)e− ν
+2 te
+− d+2
+2
+t�
+0
+F (τ)dτ
+and obtain
+¨H + JH = 0,
+(22)
+with
+J = 1 − 1
+4ν2 − (d − 2)(d − 4)
+4
+F 2 + (d − 2)νF − (d + 2)
+2
+G.
+Thus,
+Q(t) = 1 +
+t
+�
+0
+p1(τ)dτ = 1 +
+t
+�
+0
+H(τ)e− ν
+2 τe
+− d+2
+2
+τ�
+0
+F (ξ)dξ
+dτ.
+Thus, for the boundedness of derivatives, it is necessary to require that for all
+t > 0 condition
+t
+�
+0
+H(τ)e− ν
+2 τe
+− d+2
+2
+τ�
+0
+F (ξ)dξ
+dτ > −1
+(23)
+holds.
+It is easy to check that
+H(0) = H0 = u0,
+˙H(0) = H1 =
+�d − 2
+2
+F0 − ν
+2
+�
+u0 − v0.
+(24)
+3. Proof of Theorem 1
+First of all, we notice that condition (5) follows from
+sup
+r∈R+
+|(rG0(r), rF0(r), u0(r), v0(r))| < ε,
+(25)
+where |(x1, . . . , xk)| =
+�
+x2
+1 + · · · + x2
+k, k ∈ N.
+Since we are interested in small values of ν, we restrict ourselves to the case of
+ν < 2.
+1. Let us fix r ∈ R+. The matrix of linearization of the system of four equations
+(12), (18), consists of two blocks
+�−ν
+−1
+1
+0
+�
+, its complex conjugate eigenvalues
+are λ1,2 = − ν±ih1
+2
+, where h1 =
+√
+4 − ν2.
+therefore the equilibrium the origin
+
+AXISYMMETRIC OSCILLATIONS OF A COLLISIONAL PLASMA
+9
+on the phase space G, F, u, v is asymptotically stable. This implies that for any
+sufficiently small ε - neighborhood of the origin there exists δ(ε) < ε such that if
+|(rG0, rF0, u0, v0)| < δ, then |(G, F, u, v)| < ε. Moreover,
+|rG, rF, u, v| ≤ C1e− ν
+2 t,
+C1 = const,
+(26)
+the constant C1 depends on ν and ε, [4], Ch.XIII, Sec.1, ε = ε(ν, r).
+Thus, in condition (25) we take ε(ν) = inf
+r∈R+
+ε(ν, r). The asymptotics (6) follows
+immediately.
+4. Proof of Theorem 2
+There is an alternative method to proof Theorem 1. Namely, we can show that
+for an arbitrary small ν > 0 there exists ε(ν) > 0 such that if (25) holds, then
+���
+t
+�
+0
+H(τ)e− ν
+2 τe
+− d+2
+2
+τ�
+0
+F (ξ)dξ
+dτ
+��� < 1
+(27)
+for all t > 0. This condition evidently implies (23), therefore the derivatives of the
+considered solution are bounded, and the solution of (3), (4) keeps smoothness.
+However, detailed estimates of the functions under the sign of integral gives us
+a possibility to obtain more or less practical sufficient condition that guarantees
+smoothness of solutions in terms of initial data.
+1. Let us denote S(t) = e
+− d+2
+2
+t�
+0
+F (ξ)dξ
+. Then ˙S = − d+2
+2 FS, and, as follows from
+(12),
+S =
+��� 1 − dG
+1 − dG0
+���
+d+2
+2d ,
+(28)
+therefore, due to (16),
+0 < M− < S < M+,
+M+ =
+���1 − dG−
+1 − dG0
+���
+d+2
+2d ,
+M− =
+���1 − dG+
+1 − dG0
+���
+d+2
+2d
+(29)
+where G− < 0 is the point, where the curve (14) (or (15)) intersects the axis F = 0,
+see Lemma 2.
+2. To estimate H(t) we use the following result [2] (Th.2, Ch.2): all solution of
+the equation
+¨z + (1 + ϕ(t))z = 0,
+∞
+�
+|ϕ(τ)|dτ < ∞
+are bounded. Moreover, z2 + ˙z2 ≤ (y2 + ˙y2) e
+2
+t�
+0
+|ϕ(τ)|dτ
+, where y is a solution of
+¨y + y = 0 such that z(0) = y(0), ˙z(0) = ˙y(0). It implies
+z2 ≤ (y2 + ˙y2) e
+2
+t�
+0
+|ϕ(τ)|dτ
+= (y2(0) + ˙y2(0)) e
+2
+t�
+0
+|ϕ(τ)|dτ
+.
+(30)
+In (22) we can change the time variable as t1 = h1t, to obtain ¨H + J1H = 0,
+where J1 = 1 + ϕ1(t1),
+ϕ1(t1) = 1
+h1
+�
+−(d − 2)(d − 4)
+4
+F 2 + (d − 2)νF − (d + 2)
+2
+G
+�
+.
+
+10
+OLGA S. ROZANOVA*, MARIA I. DELOVA
+From (26) we can conclude that |ϕ1(t1)| ≤ const e−
+ν
+2h1 t1. Since
+∞
+�
+|ϕ1(τ)|dτ < ∞,
+then |H(t1)| is bounded for all initial data H(0), ˙H(0). Now we go back to the time
+variable t and use the notation of Theorem 2 for φ and J+. Taking into account
+(30) we get
+|H(t)| ≤
+�
+H2
+0 + 4H2
+1
+4 − ν2 e
+∞
+�
+0
+|φ(τ)|dτ
+.
+(31)
+and
+|H(t)| ≤
+�
+H2
+0 + 4 ˙H2
+0
+4 − ν2 e
+t�
+0
+|φ(τ)|dτ
+≤
+�
+H2
+0 + 4H2
+1
+4 − ν2 e
+�
+J+−1+ ν2
+4
+�
+T ,
+(32)
+for every T > 0.
+3. Note that due to Lemma 2, for all t > 0 the points (G, F) lie inside the
+bounded curves (15) or (14), therefore the maximal (positive) G+ and minimal
+(negative) G− values of G, as well as maximum of F 2, denoted as F 2
++, can be found
+from the analytic expression for these curves. Therefore for every (G0, F0) we can
+find J+ = const such that J ≤ J+, where J is given as (23).
+Estimates (29), (31), (27) imply condition (7), whereas (29), (32), (27) imply
+condition (8), if we substitute (24) and use the notation of Theorem 2.
+We can only notice that we do not need to know the value of F+, which appear
+in J+. Indeed, for d = 2 the expression for J+ does not contain F+, for d > 2 the
+value of F+ can be found via G.
+Let us prove the latter statement.
+At the maximum point of (14) we have
+dF
+dG = 0, i.e. F 2 = −G. Therefore, the value of G, at which the extremum is
+reached, can be found from the equation
+−G = 2G − 1
+d − 2 + (1 − dG)
+2
+d Cd,
+Cd = (d − 2)F 2
+0 − 2G0 + 1
+(d − 2)(1 − dG0)
+2
+d ,
+which solution is G = 1
+d
+�
+1 − ((d − 2)Cd)
+d−2
+d
+�
+. Thus,
+F+ = 1
+d
+�
+((d − 2)Cd)
+d−2
+d
+− 1
+�
+.
+(33)
+4. Now we prove (9). Let us denote as H∗ < 0 the value of H(t) at the point t∗
+of a negative minimum. Assume that we know the estimate H∗ ≤ H∗
++ < 0, then
+t
+�
+0
+H(τ)e− ν
+2 τe
+− d+2
+2
+τ�
+0
+F (ξ)dξ
+dτ < 2H∗
++M+
+ν
+< −1,
+is a sufficient condition for the blow-up.
+Let H+(t) be the solution to the Cauchy problem
+¨H+ + J+H+ = 0,
+H+(0) = H(0),
+˙H+(0) = ˙H(0).
+Indeed, it is easy to check that
+d
+dt
+�
+H2 +
+˙H2
+J+
+�
+= 2(J+ − J)
+J+
+H ˙H.
+
+AXISYMMETRIC OSCILLATIONS OF A COLLISIONAL PLASMA
+11
+Since H(0) ≤ 0, ˙H(0) < 0, then for t ∈ (0, t∗) we have H ˙H ≥ 0 and H2 +
+˙H2
+J+ ≥
+H2(0) +
+˙H2(0)
+J+ , and in the point of minimum H2
+∗ ≥ H2(0) +
+˙H2(0)
+J+
+≡ (H∗
++)2. Note
+that H(t) obtains its minimum on the semi-period of H+, i.e. t∗ ≤
+π
+√
+J+ . Now it
+rests to substitute (24).
+Thus, Theorem 2 is proved.
+5. Proof of Theorem 3
+Now we assume ν > 2 and fix r0 ∈ R+.
+1. The eigenvalues of the matrix of linearization of (12) are now real and nega-
+tive: λ1,2 = − ν±h2
+2
+, where h2 =
+√
+ν2 − 4. Therefore ([4], Ch.XIII, Sec.1)
+|(G, F)| ≤ C2e− ν−h2
+2
+t,
+C2 = const > 0,
+(34)
+2. We change the time as t2 = h2
+2 t, and rewrite (22) as ¨H − J2H = 0, where
+J2 = 1 + ϕ2(t),
+ϕ2(t) = − 4
+h2
+2
+�
+−(d − 2)(d − 4)
+4
+F 2 + (d − 2)νF − (d + 2)
+2
+G
+�
+.
+The equation
+u′′ − (1 + ϕ2(τ))u = 0,
+∞
+�
+|ϕ2(τ)|dτ < ∞,
+has two solution such that u(t) ∼ eτ and u(t) ∼ e−τ as τ → ∞ [2]. Moreover, for
+|ϕ2| < 1 the solution is non-oscillating and has at most one root for t2 > 0. Thus,
+(22) has two non-oscillating solutions H(t) ∼ e
+h2
+2 t and H(t) ∼ e− h2
+2 t as t → ∞.
+3. Due to (29), it is enough to prove that
+���
+t
+�
+0
+H(τ)e− ν
+2 τdτ
+��� → 0,
+ν → ∞.
+(35)
+To this aim we perform twice the integration by parts to obtain
+t
+�
+0
+H(τ)e− ν
+2 τdτ = Ψ(H(t), ν) +
+t
+�
+0
+H(τ)e− ν
+2 τR(τ)dτ,
+where
+Ψ(H(t), ν) = ν
+2 (H(0) − H(t)e− ν
+2 t) + ˙H(0) − ˙H(t)e− ν
+2 t,
+R(t) = (d − 2)(d − 4)
+4
+F 2 + (d − 2)νF − (d + 2)
+2
+G.
+Taking into account (24),
+Ψ(H(t), ν) = d − 2
+2
+F0u0 − v0 − ν
+2 H(t)e− ν
+2 t − ˙H(t)e− ν
+2 t.
+4. Let us denote as ¯H the solution of (22), (24) with R = 0, which formally
+corresponds to F = G = 0. This solution can be found explicitly as
+¯H(t) = H(0) cosh h2t
+2 + 2 ˙H(0)
+h2
+sinh h2t
+2 ,
+
+12
+OLGA S. ROZANOVA*, MARIA I. DELOVA
+and
+Ψ( ¯H(t), ν) = u0
+h2
+e− ν
+2 t sinh h2t
+2
++
+�d − 2
+2
+F0u0 − v0
+� �
+1 − e− ν
+2 t
+�
+cosh h2t
+2 + ν
+h2
+sinh h2t
+2
+��
+.
+It can be readily checked that for any fixed t > 0 as ν → ∞ we have
+1
+h2
+e− ν
+2 t sinh 1
+2h2t = 1
+ν + O
+� 1
+ν2
+�
+,
+1 − e− ν
+2 t
+�
+cosh h2t
+2 + ν
+h2
+sinh h2t
+2
+�
+= t
+ν + O
+� 1
+ν2
+�
+,
+therefore Ψ( ¯H(t), ν) → 0 as ν → ∞.
+5. Further we are going to prove that
+H(t) = ¯H(t) + O
+�1
+ν
+�
+,
+˙H(t) = ˙¯H(t) + O(1),
+ν → ∞,
+t > 0.
+(36)
+Indeed, w(t) = H(t) − ¯H(t) is the solution to the non-homogeneous problem
+¨w − h2
+4 w = −RH,
+w(0) = ˙w(0),
+therefore, taking into account (34), we have
+w(t)
+=
+1
+h2
+
+e− h2t
+2
+t
+�
+0
+R(τ)H(τ)e
+h2τ
+2 dτ − e
+h2t
+2
+t
+�
+0
+R(τ)H(τ)e− h2τ
+2 dτ
+
+ =
+1
+h2
+t
+�
+0
+R(τ)H(τ) sinh h2(τ − t)
+2
+dτ = O
+�1
+ν
+�
+,
+˙w(t) = O(1),
+ν → ∞.
+6. Now we show that for any fixed t > 0 as ν → ∞
+t
+�
+0
+H(τ)e− ν
+2 τR(τ)dτ = o
+
+
+t
+�
+0
+H(τ)e− ν
+2 τdτ
+
+ .
+(37)
+Indeed, (34) implies that there exists a constant R0 > 0 such that |R(t)| ≤
+R0e− ν−h2
+2
+t. Therefore
+��� 1
+R0
+t
+�
+0
+H(τ)e− ν
+2 τR(τ)dτ −
+t
+�
+0
+H(τ)e− ν
+2 τdτ
+��� ≤
+t
+�
+0
+���H(τ)
+���
+���R(τ)
+R0
+− 1
+��� e− ν
+2 τdτ ≤
+t
+�
+0
+| ¯H(τ) + w(τ)||e− ν−h2
+2
+τ − 1| e− ν
+2 τdτ =
+t
+�
+0
+��� ¯H(τ) + O
+�1
+ν
+� ��� e− ν
+2 τdτ · O
+� 1
+ν
+�
+=
+o
+�1
+ν
+�
+→ 0,
+ν → ∞,
+what implies (37).
+
+AXISYMMETRIC OSCILLATIONS OF A COLLISIONAL PLASMA
+13
+7. Thus, for a fixed t > 0 we have
+Ψ(H(t), ν) = ν
+2 (H(0) − ( ¯H(t) + w(t))e− ν
+2 t) + ˙H(0) − ( ˙¯H(t) + ˙w(t))e− ν
+2 t =
+Ψ( ¯H(t), ν) − ( ˙w(t) + ν
+2 w(t))e− ν
+2 t → 0,
+ν → ∞,
+due to (36). Together with (37) it implies (35).
+The asymptotic property (10) can be proved as in Theorem 1.
+6. Discussion
+We proved that for axisymmetric multidimensional oscillations of a cold plasma
+the constant linear dumping, which corresponds to a constant coefficient of the
+frequency of collisions between particles ν, serves as a mollifier. Moreover, Theorem
+3 tells us that for an arbitrary initial pulse we can choose such a large coefficient
+ν that the solution will remain smooth for all t > 0 and decay to the rest state.
+However, this scenario does not make physical sense, since we cannot control the
+collision rate, which is relatively small (ν ≪ 1) according to the measurements.
+The theoretical result of Theorem 1 is predictable. Physicists know that small
+axisymmetric smooth deviations of the rest state persist in collisional media, see
+[9], [7] for the cylindrical case and references therein. They would be interested in
+the more or less exact size of the neighborhood of the rest state corresponding to
+smooth solutions. The criterion of smoothness in the terms of initial data can be
+obtained analytically for d = 1, see [13]. Theorem 2 gives some information about
+the lifetime of a smooth solution for a fixed ν. However, this is not a criterion, but
+only sufficient conditions. The condition (7) is more precise, but it is difficult to
+use in practice, since we do not know the analytical solution (12). Condition (8)
+is more rough than (7), but more convenient, since we can check arbitrary initial
+data (4) and decide what the lifetime that we can guaranty for the solution of the
+Cauchy problem (3), (4).
+Note that the problem of blow-up or non-blow-up for specific initial data and a
+specific coefficient ν can still be solved numerically. Indeed, we solve system (12),
+(22) for each r and check the condition (23).
+Further, it should be noted that the constant collision frequency is only an
+assumption that simplifies the asymptotic analysis.
+Actually ν is a function of
+density n. It is shown in [14] that in the case d = 1 for ν = ν0nγ, γ > 1 each
+solution of the Cauchy problem is smooth for all initial data. A similar problem for
+the multidimensional case is completely open. It would be natural to expect that
+the form of ν(n) depends on d.
+Another important problem is to study how collisions between particles affect so-
+lutions without radial symmetry. The first approach to this difficult problem would
+be to study affine solutions for which (V, E) = (F(t)r, G(t)r), where F(t), G(t) are
+matrices (d × d). As shown in [15], under the assumption of radial symmetry, such
+solutions are globally smooth. Nevertheless, as was recently proved [16], an arbi-
+trarily small deviation from radial symmetry in the class of affine solution blows
+up, although the oscillation breaking mechanism is very subtle. The linearization
+shows that the constant damping prevents the blow-up of asymmetric affine so-
+lutions. However, it is interesting to investigate whether this property holds for
+arbitrary asymmetric oscillations.
+
+14
+OLGA S. ROZANOVA*, MARIA I. DELOVA
+Acknowledgments
+Supported by the Moscow Center for Fundamental and Applied Mathematics
+under the agreement 075-15-2019-1621.
+References
+[1] A.F. Alexandrov, L.S. Bogdankevich, and A.A. Rukhadze, “Principles of plasma electrody-
+namics,” Springer series in electronics and photonics, Springer, Berlin Heidelberg, 1984.
+[2] R. Bellman, “Stability Theory of Differential Equations,” Dover Books on Mathematics,
+Mineola, 1953.
+[3] D. Chae, and E. Tadmor, On the finite time blow-up of the Euler-Poisson equations in Rn,
+Commun. Math. Sci., Vol.6(3) (2008), 785–789.
+[4] Coddington E.A., and Levinson N. “Theory of Ordinary Differential Equations,” McGraw-
+Hill, New York, 1955.
+[5] R. C. Davidson, “Methods in nonlinear plasma theory,” Acad. Press, New York, 1972.
+[6] G. Freiling, A survey of nonsymmetric Riccati equations, Linear Algebra and its Applications,
+Vol.351-352 (2002), 243–270.
+[7] A.A. Frolov, and E.V. Chizhonkov, The effect of electron-ion collisions on breaking cylindrical
+plasma oscillations. Math Models. Comput. Simul. Vol.11 (2019), 438–450.
+[8] V. L. Ginzburg, “Propagation of electromagnetic waves in plasma,” Pergamon, New York,
+1970.
+[9] L.M. Gorbunov, A.A. Frolov, and E.V. Chizhonkov, N.E. Andreev, Breaking of nonlinear
+cylindrical plasma oscillations, Plasma Physics Reports, Vol. 36 (4) (2010), 345–356.
+[10] S. Engelberg, H. Liu, and E. Tadmor, Critical thresholds in Euler-Poisson equations, Indiana
+University Mathematics Journal, Vol.50 (2001), 109–157.
+[11] E. Esarey, C. B. Schroeder, and W. P. Leemans, Physics of laser-driven plasma-based electron
+accelerators, Rev. Mod. Phys., Vol. 81 (2009), 1229-1285.
+[12] W. T. Reid, “Riccati differential equations,” Academic Press, New York, 1972.
+[13] O. Rozanova, E. Chizhonkov, and M. Delova, Exact thresholds in the dynamics of cold plasma
+with electron-ion collisions, AIP Conference Proceedings, Vol.2302 (1) (2020), 060012.
+[14] O.S. Rozanova, Suppression of singularities of solutions of the Euler-Poisson system with
+density-dependent damping, Physica D: Nonlinear Phenomena Vol.429 (2022), 133077.
+[15] O.S. Rozanova, On the behavior of multidimensional radially symmetric solutions of the
+repulsive Euler-Poisson equations, Physica D: Nonlinear Phenomena Vol. 443 (2023), 133578.
+[16] O.S. Rozanova, and M.K. Turzinsky, On the properties of affine solutions of cold plasma
+equations, submitted, arXiv:2211.16894 (2022).
+[17] C. Tan, Eulerian dynamics in multidimensions with radial symmetry, SIAM Journal on
+Mathematical Analysis, Vol. 53 (3) (2021), 3040–3071.
+[18] D. Wei, E. Tadmor, and H. Bae, Critical thresholds in multi-dimensional Euler-Poisson
+equations with radial symmetry. Commun. Math. Sci., Vol. 10(1)(2012), 75–86.
+Mathematics and Mechanics Department, Lomonosov Moscow State University, Lenin-
+skie Gory, Moscow, 119991, Russian Federation, [email protected]
+

BdAyT4oBgHgl3EQf4Ppc/content/tmp_files/load_file.txt

@@ -0,0 +1,427 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf,len=426
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='00782v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='AP]  2 Jan 2023  ON MULTIDIMENSIONAL AXISYMMETRIC OSCILLATIONS OF  A COLLISIONAL COLD PLASMA  OLGA S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' ROZANOVA*, MARIA I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' DELOVA  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' We study the influence of the friction term on the radially sym-  metric solutions of the repulsive Euler-Poisson equations with a non-zero back-  ground, corresponding to cold plasma oscillations in many spatial dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  It is shown that for any arbitrarily small constant non-negative constant fric-  tion coefficient, there exists a neighborhood of the zero stationary solution in  the C1 norm such that the solution of the Cauchy problem with initial data  belonging to this neighborhood remains globally smooth in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' Moreover,  this solution stabilizes to the zero as t → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' This result contrasts with the  situation of zero friction, where any small deviation from the zero equilibrium  generally leads to a blow-up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' Our method allows to estimate the lifetime of  smooth solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' Further we prove that for any initial data one can find such  friction coefficient that the respective solution to the Cauchy problem keeps  smoothness for all t > 0 and stabilizes to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' Introduction  We study a frictional version of the repulsive Euler-Poisson equations  ∂n  ∂t + div (nV) = 0,  ∂V  ∂t + (V · ∇) V = k ∇Φ − ν V,  ∆Φ = n − n0,  (1)  where the the scalar functions n and Φ are the density and a repulsive (for k > 0)  force potential, respectively, the vector V is the velocity, they depend on the time  t and the point x ∈ Rd, d ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' Here n0 > 0 is the density background, ν > 0 is a  friction coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  If we denote ∇Φ = −E, and set n0 = 1, such that  n = 1 − div E,  (2)  we can remove n from (1) and rewrite it as  ∂V  ∂t + (V · ∇) V = −E − νV,  ∂E  ∂t + Vdiv E = V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  (3)  In this paper we study the Cauchy problem for (1) or (3) and our main concern  is to study initial data that guarantee a globally smooth solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  System (3) corresponds to the hydrodynamics of “cold” or electron plasma in the  non-relativistic approximation in dimensionless quantities (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=', [1], [5], [8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' In  this interpretation the friction coefficient characterizes the intensity of electron-ion  collisions during plasma oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' The cold plasma equations is now very popular  object of study, may be more popular than the Euler-Poisson equations themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  The reason is that the cold plasma in used in the accelerators of electrons in the  2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' Primary 35Q60;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' Secondary 35L60, 35L67, 34M10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' Euler-Poisson equations, quasilinear hyperbolic system, cold plasma,  blow up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  1  2  OLGA S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' ROZANOVA*, MARIA I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' DELOVA  wake wave of a powerful laser pulse [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' From this point of view the initial data  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' the initial laser pulse) that corresponds to a solution that cannot survive being  smooth are not applicable technically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  System (3) in 1D case was studied [13], however, in the multidimensional case  it is very difficult from both mathematical and physical points of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' Indeed, it  describes a non-hyperbolic superposition of different types of waves, each of them  have a tendency to break out in a finite time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' Therefore the theoretical results here  are very scarce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  The situation is more optimistic if we restrict ourselves to the class of axisym-  metric solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' Thus, we consider one-dimensional solutions in space, given on  the half-line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' In [10], [3], [18], [17] it was shown that for n0 = 0, k > 0 and n0 ≥ 0,  k < 0 in the non-frictional case there is a threshold in terms of the initial data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  Namely, one can specify exactly the class of initial data corresponding to a globally  smooth solution, and these data form a neighborhood of the stationary state in the  C1 -norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' As it has been recently shown [15], for n0 > 0, k > 0, ν = 0 the situation  is strikingly different: namely, for d ̸= 1 and d ̸= 4 an arbitrarily small pertur-  bation of the zero stationary state blows up in the general case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' The exception  is the initial data in the form of a simple wave, starting from which the solution  can remain globally smooth and tend to an affine solution as t → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' In any case,  the initial data corresponding to simple waves form a zero-measure manifold in the  neighborhood of the stationary state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  In this paper, we study the effect of constant friction on the blow-up process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  Namely, we establish that the presence of friction normalizes the situation with the  threshold for the initial data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' Namely, for an arbitrarily small ν > 0 and any d,  there exists a neighborhood of the zero stationary state in the C1-norm such that  the corresponding solution of the Cauchy problem preserves smoothness (Theorem  1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' For small ν, Theorem 2 gives sufficient conditions guaranteeing blow-up or non-  blow-up in terms of initial data, which can be applied to numerical tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' Besides,  we show that for any initial data, one can find ν such that the corresponding  solution of the Cauchy problem is globally smooth (Theorem 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' In other words,  this situation is absolutely analogous to d = 1, and the increase in spatial dimension  does not lead to any new phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  Thus, we consider axisymmetric solutions of (3)  V = F(t, r)r,  E = G(t, r)r,  where r = (x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=', xd) is the radius-vector, r =  �  x2  1 + x2  2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' + x2  d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  The initial data that correspond to these solutions are  (V, E)|t=0 = (V0(r), E0(r)) = (F0(r)r, G0(r)r),  (F0(r), G0(r)) ∈ C2(¯R+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  (4)  We assume that (V0(r), E0(r)) are bounded together with their derivatives uni-  formly on r ∈ ¯R+ and denote ∥f∥C1(R+) =  1�  i=0  sup  r∈R+  |f (i)(r)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  The physically natural condition n|t=0 > 0 dictates divE < 1, see (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  The main results of the paper are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' For arbitrary small ν > 0 there exists ε(ν) > 0, such that the  solution of the problem (3) - (4) satisfying  ∥V0(r), E0(r)∥C1(R+) < ε,  (5)  AXISYMMETRIC OSCILLATIONS OF A COLLISIONAL PLASMA  3  keeps C1 - smoothness for all t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' Moreover,  ∥V, E∥C1(R+) ≤ const e− ν  2 t → 0,  t → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  (6)  Let us denote  u0 = div V0 − dF0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  v0 = div E0 − dG0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  H0 = u0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  H1 =  �d − 2  2  F0 − ν  2  �  − v0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  φ = −d + 2  2  G + (d − 2)νF − (d − 2)(d − 4)  2  F 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  J+ = 1 − ν2  4 − d + 2  2  G�� + ν(d − 2)F+ + (1 − δ3d)(d − 2)(d − 4)  2  F 2  +,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  M± =  �1 − dG∓  1 − dG0  � d+2  2d  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  0 < M− < M+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  where (G,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' F) is the solution of the problem (12) subject to initial data (G0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' F0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  G− < 0 and G+ > 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' G+ < 1  d are the left and right roots of equation (15) or (14),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  F = 0 (they depend on (G0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' F0)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' F+ is given as (33) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' δij is the Kronecker symbol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  The next theorem gives more information about the size of the neighborhood of  the origin containing globally smooth solutions in the case of small ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' Let ν < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  a) A sufficient condition on initial data (4) that guaranties the smoothness of  the solution of the problem (3) - (4) for all t > 0 is the following:  inf  r∈R+)  F1(ν, V0(r), E0(r)) < 1,  (7)  F1(ν, V0(r), E0(r)) = 2  ν M+  �  H2  0 +  �  1 − ν2  4  �−1  H2  1  e  ∞  �  0  |φ(τ|dτ  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  b) If there exists T > 0 such that  inf  r∈R+)  F2(T, ν, V0(r), E0(r)) < 1,  (8)  F2(T, ν, V0(r), E0(r)) = 2  ν M+  �  H2  0 +  �  1 − ν2  4  �−1  H2  1  e  �  J+−1+ ν2  4  �  T ,  then the solution of the problem (3) - (4) preserves smoothness for t ∈ [0, T ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  c) If the initial data (4) are such that there exists a point r ∈ R+ for which  condition  F3(ν, V0(r), E0(r)) ≥ 1,  (9)  F3(ν, V0(r), E0(r)) = 2  ν M−  �  H2  0 + J−1  + H2  1,  H0 ≤ 0,  H1 < 0  holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' Then the solution of problem (3) - (4) blows up within t <  π  √  J+ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  4  OLGA S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' ROZANOVA*, MARIA I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' DELOVA  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' For arbitrary initial data (4) there exists such ν > 0 that the solution  of problem (3) - (4) keeps C1 - smoothness for all t > 0 and the asymptotic property  ∥V, E∥C1(R+) ≤ const e− ν−√  4−ν2  2  t → 0,  t → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  (10)  holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  Theorems 1, 2 and 3 can be reformulated in the terms of the Euler-Poisson  equations (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' The stationary stationary state in this case is  V = 0,  Φ = const,  n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  In this work we use the technique of linearization my means of the Radon lemma,  the same as in [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' It turn out to be very convenient for the analysis of the non-  strictly hyperbolic systems often arising when studying the reduced cold plasma  equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  The paper is organised as follows: Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='2 is devoted to auxiliary results on the  behavior of solution and its derivatives, Secs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='3, 4 and 5 contain the proofs of The-  orems 1, 2, and 3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='6 is devoted to a discussion on the importance  of the results for physics and the formulation of future problems in this area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' Behavior of solutions along characteristics  We use the fact that V = F(t, r)r, E = G(t, r)r and get  ∂F  ∂t + Fr∂F  ∂r = −F 2 − G − νF,  ∂G  ∂t + Fr∂G  ∂r = F − dFG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  (11)  (V0, E0) = (F(0, r)r, G(0, r)r) = (F0(r)r, G0(r)r),  (F0(r), G0(r)) ∈ C2(R+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' Physical constraints on solution components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' Let us fix r0 ∈ ¯R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' Along  the characteristics ˙r = ∂r  ∂t = Fr, r(0) = r0, of the system (11), the functions F  and G obey the system of equations  ˙F = −F 2 − G − νF,  ˙G = F − dFG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  (12)  Therefore  dG  F(1 − dG) = dr  Fr,  1 − dG = const · r−d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  (13)  Since r ≥ 0, the sign of the expression 1 − dG does not change, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' sign(1 − dG) =  sign(1−dG(0, r0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' Therefore, the motion on the phase plane (G, F) corresponding  to system (12) occurs either in the half-plane G < 1  d, or in the half-plane G > 1  d,  or on the line G = 1  d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  The equilibria of (12) are the following:  if ν <  2  √  d, then there exists the only point (F = 0, G = 0), a stable focus;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  if ν =  2  √  d, then there exist two points: (F = 0, G = 0), a stable focus, and  (F = − ν  2, G = 1  d), a saddle-node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  if ν >  2  √  d, then there exist three points: (F = 0, G = 0), a stable focus  (ν < 2) or a stable node, otherwise, and (F = −  ν±√  ν2− 4  d  2  , G =  1  d), a  saddle and an unstable node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  AXISYMMETRIC OSCILLATIONS OF A COLLISIONAL PLASMA  5  We see that there are no equilibria in the domain G >  1  d, hence there are no  bounded trajectories in this region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' If the motion on the plane (G, F) starts from  the point for which G(0, r(0)) >  1  d, then the phase trajectory rests in the half-  plane G >  1  d and G(t, r(t)) → +∞ for t → t∗ < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' Moreover, due to (13), we  have r(t) → 0 for t → t∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' In this case, we get a contradiction with the positivity of  density, since n = 1 − div E = 1 − Grr − dG > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' On the line G = 1  d the density is  zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  Thus, we study the problem only in the half-plane G < 1  d, F ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' Boundedness of the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' In the half-plane G < 1  d, F ∈ R system (12)  has one equilibrium (0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' It corresponds to the stationary state V = E = 0 and it  is stable for any values of the parameters d and ν > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' Namely, as a linear analysis  show,  if 0 < ν < 2 it is a stable focus;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  if ν = 2 it is a stable degenerate node;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  if ν > 2 it is a stable node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' There exists δ > 0 such that if the initial data (F0(r0), G0(r0)) belong  to the δ- neighborhood of the origin, r0 ∈ ¯R+, then any solution to (12) tends to  zero exponentially as t → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  The proof follows from the fact that (0, 0) is asymptotically stable for all choices  of parameters ν, d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' □  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' Let Φ(G, F) = Φ(G0, F0) be a closed phase curve corresponding to the  solution of system (12) for ν = 0 with initial data (F0, G0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' Then the phase curve  corresponding to the solution of system (12) ν > 0 with initial data (F0, G0) lies  strictly inside the curve Φ(G, F) = Φ(G0, F0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' Let us construct the phase curve of (12) at ν = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='e the solution of  dF  dG = −  F 2 + G  F(1 − dG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  It implies  dZ  dG = −  2  1 − dGZ −  2G  1 − dG,  Z(G) = F 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  The solution is  Φ(G, F) = (d − 2)F 2 − 2G + 1  (d − 2)(1 − dG)  2  d  = Φ(G0, F0) = Cd,  (14)  Cd = (d − 2)F 2  0 − 2G0 + 1  (d − 2)(1 − dG0)  2  d ,  for d ̸= 2  Φ(G, F) = 2F 2 + ln(1 − 2G)(1 − 2G) + 1  2(1 − 2G)  = Φ(G0, F0) = C2,  (15)  C2 = 2F 2  0 + ln(1 − 2G0)(1 − 2G0) + 1  2(1 − 2G0)  ,  for d = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' As it was shown in [15] The curves given as (15) and (14) are bounded,  they contain the origin and intersect the axis F = 0 in two points: (G−, 0), G− < 0,  and (G+, 0), G+ > 0, see the pictures in [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  6  OLGA S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' ROZANOVA*, MARIA I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' DELOVA  Let us consider V (t) = Φ(G, F) as a Lyapunov function in the half-plane G < 1  d,  F ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' The derivative of V (t) due to system (12) is  dV  dt = −  2νF 2  (1 − dG)  2  d ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  (16)  If we denote (G(t), F(t)) and ( ¯G(t), ¯F(t)) the point on the phase curve of (12) for  ν > 0 and ν = 0, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' and Thus, the distance |(G(t), F(t))| < |( ¯G(t), ¯F(t))|,  t > 0, since dV  dt = 0 if and only if F = 0 and F = 0 does not solve (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' □  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' System (12) has no limit cycles in the half-plane G < 1  d, F ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' We use the Lyapunov function from Lemma 2 to prove the absence of a  limit cycle by contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  Assume that a limit cycle (a closed trajectory Γ)  exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' Then it contains a stable equilibrium (0, 0) inside.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' We denote as d(Y1, Y2)  the distance between points Y1(t) = (G1(t), F1(t)), Y2(t) = (G2(t), F2(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  For  some initial point Y (t0) = (G∗, F∗) on Γ there exists a time t1 > t0 such that  Y (t1) = Y (t0) and, accordingly, d(Y (t0), Y (t1)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' The curve Γ contains (0, 0)  inside, so there are two points on this trajectory for which F = 0, they are (G+, 0)  and (G−, 0), 0 < G+ < 1  d, G− < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' At these points dV (G+,0)  dt  = dV (G−,0)  dt  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' At  other points of Γ we have dV  dt < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' Then the function V (t) does not increase along  Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' Moreover, dV  dt = 0 only at two points on Γ, so V (t) strictly decreases along the  trajectory, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=', V (t1) − V (t0) < 0 for t1 > t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' For the above points Y (t0), Y (t1),  therefore d(Y (t0), Y (t1)) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' Thus, we obtain a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' □  Thus, Lemmas 1, 2 and 3 imply the following property of the phase trajectories:  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' Let d ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' Then the phase curves of system (12) are bounded in the  half-plane G < 1  d, F ∈ R and tends to zero as t → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' Lemma 4 is not valid for d ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' Indeed, in this case the system (12)  coincides with the system (17) for the derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' As was shown in [13], for any  ν > 0 there exists a point on the phase plane such that the phase curve starting  from this point goes to infinity as t → t∗ < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' Study of the behavior of derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' This section closely follows [15], but  for the convenience of the reader we give a sketch of the reasonings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  Let us denote D = div V, λ = div E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' Equations (3) imply  ∂D  ∂t + (V · ∇D) = −D2 + 2(d − 1)FD − d(d − 1)F 2 − λ − νD,  ∂λ  ∂t + (V · ∇λ) = D(1 − λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  Along the characteristics given as ˙r = Fr the functions D, λ obey  ˙D = −D2 + 2(d − 1)FD − d(d − 1)F 2 − λ − νD,  ˙λ = D(1 − λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  (17)  We introduce new variables u = D − dF, v = λ − dG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' Systems (17) and (12) imply  ˙u = −u2 − 2uF − v − νu,  ˙v = −uv + (1 − dG)u − dFv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  (18)  System (18) can be linearized my means of the Radon lemma (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' [6], [12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  AXISYMMETRIC OSCILLATIONS OF A COLLISIONAL PLASMA  7  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' [The Radon lemma] A matrix Riccati equation  ˙W = M21(t) + M22(t)W − WM11(t) − WM12(t)W,  (19)  (W = W(t) is a matrix (n×m), M21 is a matrix (n×m), M22 is a matrix (m×m),  M11 is a matrix (n×n), M12 is a matrix (m×n)) is equivalent to the homogeneous  linear matrix equation  ˙Y = M(t)Y,  M =  �  M11  M12  M21  M22  �  ,  (20)  (Y = Y (t) is a matrix (n × (n + m)), M is a matrix ((n + m) × (n + m)) ) in the  following sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  Let on some interval J ∈ R the matrix-function Y (t) =  � Q(t)  P(t)  �  (Q is a  matrix (n × n), P is a matrix (n × m)) be a solution of (20) with the initial data  Y (0) =  �  I  W0  �  (I is the identity matrix (n × n), W0 is a constant matrix (n × m)) and det Q ̸= 0  on J .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' Then W(t) = P(t)Q−1(t) is the solution of (19) with W(0) = W0 on J .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  Let us (18) as (19) with  W =  �  u  v  �  ,  M11 =  �0�  ,  M12 =  �1  0�  ,  M21 =  �0  0  �  ,  M22 =  �−2 F − ν  −1  1 − d G  −d F  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  Then according Theorem 4 the solition of (18) is  W(t) = P(t)  Q(t),  where P(t) = (p1(t), p2(t))T and Q(t) solves the linear system  � Q  P  �·  = M  � Q  P  �  ,  M =  \uf8eb  \uf8ed  0  1  0  0  −2F − ν  −1  0  1 − dG  −dF  \uf8f6  \uf8f8  (21)  subject to the initial data  � Q  P  �  (0) =  �  1  W0  �  ,  W0 =  � u0  v0  �  =  � div V0 − dF0(r0)  div E0 − dG0(r0)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  Since the vector function P(t) and the function Q(t) are components of the solution  of a linear system of differential equations with continuous coefficients, these func-  tions do not go to infinity for any finite value of t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' Hence the functions u, v go to  infinity along the characteristic starting from the point r0 ∈ R if and only if there  exists a finite t∗ > 0 such that Q(t∗, r0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' Since u = D − dF, v = λ − dG and  the functions G, F are bounded, the derivatives of the solution of (3) are bounded  if and only if the functions u, v are bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' Thus, the conditions for the bound-  edness of derivatives coincide with the conditions under which the function Q(t)  does not vanish for any finite t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  System (21) implies  8  OLGA S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' ROZANOVA*, MARIA I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' DELOVA  Q(t) = 1 +  t  �  0  p1(τ)dτ,  ˙p1 = −(2F + ν)p1 − p2,  ˙p2 = (1 − dG)p1 − dFp2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  It follows  ¨p1 + ((d + 2)F + ν) ˙p1 + (2 ˙F + (1 − dG) + dF(2F + ν))p1 = 0,  and, taking into account ˙F = −F 2 − G − νF,  ¨p1 + ((d + 2)F + ν) ˙p1 + (2(d − 1)F 2 − (2 + d)G + (d − 2)νF + 1)p1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  We change  p1(t) = H(t)e− ν  2 te  − d+2  2  t�  0  F (τ)dτ  and obtain  ¨H + JH = 0,  (22)  with  J = 1 − 1  4ν2 − (d − 2)(d − 4)  4  F 2 + (d − 2)νF − (d + 2)  2  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  Thus,  Q(t) = 1 +  t  �  0  p1(τ)dτ = 1 +  t  �  0  H(τ)e− ν  2 τe  − d+2  2  τ�  0  F (ξ)dξ  dτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  Thus, for the boundedness of derivatives, it is necessary to require that for all  t > 0 condition  t  �  0  H(τ)e− ν  2 τe  − d+2  2  τ�  0  F (ξ)dξ  dτ > −1  (23)  holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  It is easy to check that  H(0) = H0 = u0,  ˙H(0) = H1 =  �d − 2  2  F0 − ν  2  �  u0 − v0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  (24)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' Proof of Theorem 1  First of all, we notice that condition (5) follows from  sup  r∈R+  |(rG0(r), rF0(r), u0(r), v0(r))| < ε,  (25)  where |(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' , xk)| =  �  x2  1 + · · · + x2  k, k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  Since we are interested in small values of ν, we restrict ourselves to the case of  ν < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' Let us fix r ∈ R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' The matrix of linearization of the system of four equations  (12), (18), consists of two blocks  �−ν  −1  1  0  �  , its complex conjugate eigenvalues  are λ1,2 = − ν±ih1  2  , where h1 =  √  4 − ν2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  therefore the equilibrium the origin  AXISYMMETRIC OSCILLATIONS OF A COLLISIONAL PLASMA  9  on the phase space G, F, u, v is asymptotically stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' This implies that for any  sufficiently small ε - neighborhood of the origin there exists δ(ε) < ε such that if  |(rG0, rF0, u0, v0)| < δ, then |(G, F, u, v)| < ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' Moreover,  |rG, rF, u, v| ≤ C1e− ν  2 t,  C1 = const,  (26)  the constant C1 depends on ν and ε, [4], Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='XIII, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='1, ε = ε(ν, r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  Thus, in condition (25) we take ε(ν) = inf  r∈R+  ε(ν, r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' The asymptotics (6) follows  immediately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' Proof of Theorem 2  There is an alternative method to proof Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' Namely, we can show that  for an arbitrary small ν > 0 there exists ε(ν) > 0 such that if (25) holds, then  ���  t  �  0  H(τ)e− ν  2 τe  − d+2  2  τ�  0  F (ξ)dξ  dτ  ��� < 1  (27)  for all t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' This condition evidently implies (23), therefore the derivatives of the  considered solution are bounded, and the solution of (3), (4) keeps smoothness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  However, detailed estimates of the functions under the sign of integral gives us  a possibility to obtain more or less practical sufficient condition that guarantees  smoothness of solutions in terms of initial data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' Let us denote S(t) = e  − d+2  2  t�  0  F (ξ)dξ  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' Then ˙S = − d+2  2 FS, and, as follows from  (12),  S =  ��� 1 − dG  1 − dG0  ���  d+2  2d ,  (28)  therefore, due to (16),  0 < M− < S < M+,  M+ =  ���1 − dG−  1 − dG0  ���  d+2  2d ,  M− =  ���1 − dG+  1 − dG0  ���  d+2  2d  (29)  where G− < 0 is the point, where the curve (14) (or (15)) intersects the axis F = 0,  see Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' To estimate H(t) we use the following result [2] (Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='2, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='2): all solution of  the equation  ¨z + (1 + ϕ(t))z = 0,  ∞  �  |ϕ(τ)|dτ < ∞  are bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' Moreover, z2 + ˙z2 ≤ (y2 + ˙y2) e  2  t�  0  |ϕ(τ)|dτ  , where y is a solution of  ¨y + y = 0 such that z(0) = y(0), ˙z(0) = ˙y(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' It implies  z2 ≤ (y2 + ˙y2) e  2  t�  0  |ϕ(τ)|dτ  = (y2(0) + ˙y2(0)) e  2  t�  0  |ϕ(τ)|dτ  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  (30)  In (22) we can change the time variable as t1 = h1t, to obtain ¨H + J1H = 0,  where J1 = 1 + ϕ1(t1),  ϕ1(t1) = 1  h1  �  −(d − 2)(d − 4)  4  F 2 + (d − 2)νF − (d + 2)  2  G  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  10  OLGA S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' ROZANOVA*, MARIA I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' DELOVA  From (26) we can conclude that |ϕ1(t1)| ≤ const e−  ν  2h1 t1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' Since  ∞  �  |ϕ1(τ)|dτ < ∞,  then |H(t1)| is bounded for all initial data H(0), ˙H(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' Now we go back to the time  variable t and use the notation of Theorem 2 for φ and J+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' Taking into account  (30) we get  |H(t)| ≤  �  H2  0 + 4H2  1  4 − ν2 e  ∞  �  0  |φ(τ)|dτ  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  (31)  and  |H(t)| ≤  �  H2  0 + 4 ˙H2  0  4 − ν2 e  t�  0  |φ(τ)|dτ  ≤  �  H2  0 + 4H2  1  4 − ν2 e  �  J+−1+ ν2  4  �  T ,  (32)  for every T > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' Note that due to Lemma 2, for all t > 0 the points (G, F) lie inside the  bounded curves (15) or (14), therefore the maximal (positive) G+ and minimal  (negative) G− values of G, as well as maximum of F 2, denoted as F 2  +, can be found  from the analytic expression for these curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' Therefore for every (G0, F0) we can  find J+ = const such that J ≤ J+, where J is given as (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  Estimates (29), (31), (27) imply condition (7), whereas (29), (32), (27) imply  condition (8), if we substitute (24) and use the notation of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  We can only notice that we do not need to know the value of F+, which appear  in J+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' Indeed, for d = 2 the expression for J+ does not contain F+, for d > 2 the  value of F+ can be found via G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  Let us prove the latter statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  At the maximum point of (14) we have  dF  dG = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' F 2 = −G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' Therefore, the value of G, at which the extremum is  reached, can be found from the equation  −G = 2G − 1  d − 2 + (1 − dG)  2  d Cd,  Cd = (d − 2)F 2  0 − 2G0 + 1  (d − 2)(1 − dG0)  2  d ,  which solution is G = 1  d  �  1 − ((d − 2)Cd)  d−2  d  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' Thus,  F+ = 1  d  �  ((d − 2)Cd)  d−2  d  − 1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  (33)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' Now we prove (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' Let us denote as H∗ < 0 the value of H(t) at the point t∗  of a negative minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' Assume that we know the estimate H∗ ≤ H∗  + < 0, then  t  �  0  H(τ)e− ν  2 τe  − d+2  2  τ�  0  F (ξ)dξ  dτ < 2H∗  +M+  ν  < −1,  is a sufficient condition for the blow-up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  Let H+(t) be the solution to the Cauchy problem  ¨H+ + J+H+ = 0,  H+(0) = H(0),  ˙H+(0) = ˙H(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  Indeed, it is easy to check that  d  dt  �  H2 +  ˙H2  J+  �  = 2(J+ − J)  J+  H ˙H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  AXISYMMETRIC OSCILLATIONS OF A COLLISIONAL PLASMA  11  Since H(0) ≤ 0, ˙H(0) < 0, then for t ∈ (0, t∗) we have H ˙H ≥ 0 and H2 +  ˙H2  J+ ≥  H2(0) +  ˙H2(0)  J+ , and in the point of minimum H2  ∗ ≥ H2(0) +  ˙H2(0)  J+  ≡ (H∗  +)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' Note  that H(t) obtains its minimum on the semi-period of H+, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' t∗ ≤  π  √  J+ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' Now it  rests to substitute (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  Thus, Theorem 2 is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' Proof of Theorem 3  Now we assume ν > 2 and fix r0 ∈ R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' The eigenvalues of the matrix of linearization of (12) are now real and nega-  tive: λ1,2 = − ν±h2  2  , where h2 =  √  ν2 − 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' Therefore ([4], Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='XIII, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='1)  |(G, F)| ≤ C2e− ν−h2  2  t,  C2 = const > 0,  (34)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' We change the time as t2 = h2  2 t, and rewrite (22) as ¨H − J2H = 0, where  J2 = 1 + ϕ2(t),  ϕ2(t) = − 4  h2  2  �  −(d − 2)(d − 4)  4  F 2 + (d − 2)νF − (d + 2)  2  G  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  The equation  u′′ − (1 + ϕ2(τ))u = 0,  ∞  �  |ϕ2(τ)|dτ < ∞,  has two solution such that u(t) ∼ eτ and u(t) ∼ e−τ as τ → ∞ [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' Moreover, for  |ϕ2| < 1 the solution is non-oscillating and has at most one root for t2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' Thus,  (22) has two non-oscillating solutions H(t) ∼ e  h2  2 t and H(t) ∼ e− h2  2 t as t → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' Due to (29), it is enough to prove that  ���  t  �  0  H(τ)e− ν  2 τdτ  ��� → 0,  ν → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  (35)  To this aim we perform twice the integration by parts to obtain  t  �  0  H(τ)e− ν  2 τdτ = Ψ(H(t), ν) +  t  �  0  H(τ)e− ν  2 τR(τ)dτ,  where  Ψ(H(t), ν) = ν  2 (H(0) − H(t)e− ν  2 t) + ˙H(0) − ˙H(t)e− ν  2 t,  R(t) = (d − 2)(d − 4)  4  F 2 + (d − 2)νF − (d + 2)  2  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  Taking into account (24),  Ψ(H(t), ν) = d − 2  2  F0u0 − v0 − ν  2 H(t)e− ν  2 t − ˙H(t)e− ν  2 t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' Let us denote as ¯H the solution of (22), (24) with R = 0, which formally  corresponds to F = G = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' This solution can be found explicitly as  ¯H(t) = H(0) cosh h2t  2 + 2 ˙H(0)  h2  sinh h2t  2 ,  12  OLGA S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' ROZANOVA*, MARIA I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' DELOVA  and  Ψ( ¯H(t), ν) = u0  h2  e− ν  2 t sinh h2t  2  +  �d − 2  2  F0u0 − v0  � �  1 − e− ν  2 t  �  cosh h2t  2 + ν  h2  sinh h2t  2  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  It can be readily checked that for any fixed t > 0 as ν → ∞ we have  1  h2  e− ν  2 t sinh 1  2h2t = 1  ν + O  � 1  ν2  �  ,  1 − e− ν  2 t  �  cosh h2t  2 + ν  h2  sinh h2t  2  �  = t  ν + O  � 1  ν2  �  ,  therefore Ψ( ¯H(t), ν) → 0 as ν → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' Further we are going to prove that  H(t) = ¯H(t) + O  �1  ν  �  ,  ˙H(t) = ˙¯H(t) + O(1),  ν → ∞,  t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  (36)  Indeed, w(t) = H(t) − ¯H(t) is the solution to the non-homogeneous problem  ¨w − h2  4 w = −RH,  w(0) = ˙w(0),  therefore, taking into account (34), we have  w(t)  =  1  h2  \uf8eb  \uf8ede− h2t  2  t  �  0  R(τ)H(τ)e  h2τ  2 dτ − e  h2t  2  t  �  0  R(τ)H(τ)e− h2τ  2 dτ  \uf8f6  \uf8f8 =  1  h2  t  �  0  R(τ)H(τ) sinh h2(τ − t)  2  dτ = O  �1  ν  �  ,  ˙w(t) = O(1),  ν → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' Now we show that for any fixed t > 0 as ν → ∞  t  �  0  H(τ)e− ν  2 τR(τ)dτ = o  \uf8eb  \uf8ed  t  �  0  H(τ)e− ν  2 τdτ  \uf8f6  \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  (37)  Indeed, (34) implies that there exists a constant R0 > 0 such that |R(t)| ≤  R0e− ν−h2  2  t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' Therefore  ��� 1  R0  t  �  0  H(τ)e− ν  2 τR(τ)dτ −  t  �  0  H(τ)e− ν  2 τdτ  ��� ≤  t  �  0  ���H(τ)  ���  ���R(τ)  R0  − 1  ��� e− ν  2 τdτ ≤  t  �  0  | ¯H(τ) + w(τ)||e− ν−h2  2  τ − 1| e− ν  2 τdτ =  t  �  0  ��� ¯H(τ) + O  �1  ν  � ��� e− ��  2 τdτ · O  � 1  ν  �  =  o  �1  ν  �  → 0,  ν → ∞,  what implies (37).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  AXISYMMETRIC OSCILLATIONS OF A COLLISIONAL PLASMA  13  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' Thus, for a fixed t > 0 we have  Ψ(H(t), ν) = ν  2 (H(0) − ( ¯H(t) + w(t))e− ν  2 t) + ˙H(0) − ( ˙¯H(t) + ˙w(t))e− ν  2 t =  Ψ( ¯H(t), ν) − ( ˙w(t) + ν  2 w(t))e− ν  2 t → 0,  ν → ∞,  due to (36).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' Together with (37) it implies (35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  The asymptotic property (10) can be proved as in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' Discussion  We proved that for axisymmetric multidimensional oscillations of a cold plasma  the constant linear dumping, which corresponds to a constant coefficient of the  frequency of collisions between particles ν, serves as a mollifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' Moreover, Theorem  3 tells us that for an arbitrary initial pulse we can choose such a large coefficient  ν that the solution will remain smooth for all t > 0 and decay to the rest state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  However, this scenario does not make physical sense, since we cannot control the  collision rate, which is relatively small (ν ≪ 1) according to the measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  The theoretical result of Theorem 1 is predictable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' Physicists know that small  axisymmetric smooth deviations of the rest state persist in collisional media, see  [9], [7] for the cylindrical case and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' They would be interested in  the more or less exact size of the neighborhood of the rest state corresponding to  smooth solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' The criterion of smoothness in the terms of initial data can be  obtained analytically for d = 1, see [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' Theorem 2 gives some information about  the lifetime of a smooth solution for a fixed ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' However, this is not a criterion, but  only sufficient conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' The condition (7) is more precise, but it is difficult to  use in practice, since we do not know the analytical solution (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' Condition (8)  is more rough than (7), but more convenient, since we can check arbitrary initial  data (4) and decide what the lifetime that we can guaranty for the solution of the  Cauchy problem (3), (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  Note that the problem of blow-up or non-blow-up for specific initial data and a  specific coefficient ν can still be solved numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' Indeed, we solve system (12),  (22) for each r and check the condition (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  Further, it should be noted that the constant collision frequency is only an  assumption that simplifies the asymptotic analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  Actually ν is a function of  density n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' It is shown in [14] that in the case d = 1 for ν = ν0nγ, γ > 1 each  solution of the Cauchy problem is smooth for all initial data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' A similar problem for  the multidimensional case is completely open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' It would be natural to expect that  the form of ν(n) depends on d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  Another important problem is to study how collisions between particles affect so-  lutions without radial symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' The first approach to this difficult problem would  be to study affine solutions for which (V, E) = (F(t)r, G(t)r), where F(t), G(t) are  matrices (d × d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' As shown in [15], under the assumption of radial symmetry, such  solutions are globally smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' Nevertheless, as was recently proved [16], an arbi-  trarily small deviation from radial symmetry in the class of affine solution blows  up, although the oscillation breaking mechanism is very subtle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' The linearization  shows that the constant damping prevents the blow-up of asymmetric affine so-  lutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content=' However, it is interesting to investigate whether this property holds for  arbitrary asymmetric oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
+page_content='  14  OLGA S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQf4Ppc/content/2301.00782v1.pdf'}
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BtFIT4oBgHgl3EQf_yzx/content/tmp_files/2301.11417v1.pdf.txt

@@ -0,0 +1,1007 @@
+Are Labels Needed for Incremental Instance Learning?
+Mert Kilickaya
+Eindhoven University of Technology
+[email protected]
+Joaquin Vanschoren
+Eindhoven University of Technology
+[email protected]
+Abstract
+In this paper, we learn to classify visual object instances,
+incrementally and via self-supervision (self-incremental).
+Our learner observes a single instance at a time, which
+is then discarded from the dataset. Incremental instance
+learning is challenging, since longer learning sessions ex-
+acerbate forgetfulness, and labeling instances is cumber-
+some. We overcome these challenges via three contribu-
+tions: i). We propose VINIL, a self-incremental learner that
+can learn object instances sequentially, ii). We equip VINIL
+with self-supervision to by-pass the need for instance la-
+belling, iii). We compare VINIL to label-supervised vari-
+ants on two large-scale benchmarks [6, 33], and show that
+VINIL significantly improves accuracy while reducing for-
+getfulness.
+1. Introduction
+This paper strives for incrementally learning to recognize
+visual object instances. Visual instance recognition aims to
+retrieve different views of an input object instance image.
+It can be seen as fine-grained object recognition, where the
+goal is to distinguish different instantiations of the same ob-
+ject, such as cup 1 from cup 2. Instance recognition finds
+applications in many domains, such as in visual search [40],
+tracking [5,48,49] and localization [60].
+Learning to distinguish across object instances is chal-
+lenging, as object instances differ from each other via only
+little nuances. To learn visual object instances, researchers
+generally resort to metric learning [52]. Two views of the
+same object, such as those, can be obtained via capturing
+the object from multiple angles, are fed to a deep convolu-
+tional network such as ResNet [22]. Deep net is then forced
+to pull representations of the same object together, while
+pushing the representations of all the other objects within a
+large batch.
+In doing so, researchers iterate over potentially million-
+scale datasets over and over to obtain a better metric space.
+Then, the deep net is used to query a large database of im-
+ages by comparing the feature representation of the query
+input image with the database representations. While work-
+ing well in practice, sifting through the whole dataset via
+multiple iterations may not be possible, due to privacy (a
+portion of the data may have to be deleted), or scale (i.e.
+scaling to a billion images).
+This paper builds upon incremental learning to miti-
+gate privacy and scale issues. In incremental learning, the
+learner observes images from a certain class for a num-
+ber of iterations. Then, the data of the previous class is
+discarded, and the learner receives examples from a novel
+category. Such approach is called class-incremental learn-
+ing, and receives an increasing amount of attention re-
+cently [27,36,37,57].
+Existing class-incremental learners are ill-suited for
+instance-incremental learning for two reasons. First, class-
+incremental learners rely on full label supervision. Collect-
+ing such annotation at the instance level is very expensive.
+Second, despite years of efforts, class-incremental learners
+are forgetful, since they lose performance on previously ob-
+served categories.
+This paper proposes Visual Self-Incremental Instance
+Learning, VINIL, to perform instance-incremental learn-
+ing, consider Figure 1. VINIL observes multiple views of
+a single instance at a time, which is then discarded from
+the dataset.
+Such examples can be easily captured via
+turntable cameras [6,18,29,38] or via hand-interactions [15,
+34, 50].
+Then, VINIL extracts its own supervision via
+self-supervision [56], therefore self-incremental.
+Self-
+incremental learning not only is label-efficient, it also con-
+sistently outperforms competitive label-supervised variants,
+as we will show. In summary, this paper makes three con-
+tributions:
+I. We propose VINIL, a realistic, scalable incremental in-
+stance learner,
+II. VINIL performs self-incremental learning, by-passing
+the need for heavy instance supervision,
+III. VINIL is trained without labels, and is consistently
+more accurate and less forgetful across benchmarks [6,
+33].
+arXiv:2301.11417v1  [cs.CV]  26 Jan 2023
+
+Label
+VINIL
+CNN
+Phone 1
+Phone 1
+…
+CNN
+Cup 50
+Phone 1
+Phone 2
+…
+Cup 50
+x
+y
+classifier
+t=0
+x
+y
+classifier
+t=1000
+VINIL
+SSL
+x
+x’
+t=0
+x
+x’
+t=1000
+VINIL
+SSL
+…
+VINIL
+SSL
+x
+x’
+t=1
+Figure 1. Top: Label-incremental learning requires instance-labels, and learns a new class weight per-instance. Therefore, it does not
+scale well with high number of visual instances, and is prone to forgetting previous instances. Bottom: In this paper we propose self-
+incremental instance learning: VINIL. VINIL solely focuses on learning a discriminative embedding. VINIL extracts its own supervision
+for incremental learning from different views of the same instance using Self-Supervised Learning (SSL). As a result, VINIL is not only
+label-free, but also more scalable and much less prone to forgetting.
+2. Related Work
+Visual Instance Recognition. Visual instance recognition
+aims to distinguish across different instances of an object
+category (i.e. bottle A from bottle B). Researchers re-frame
+many vision problem as visual instance search, to retrieve
+similar products [23, 32, 40, 52], to track target objects [5,
+48,49], or to geo-localize an image [31,46,51,53,59]. The
+dominant technique is to induce a discriminative embedding
+space, often with the help of metric learning [14,23]. These
+works demand access to the whole dataset at all times as
+well as fine-grained similarity labels. Instead, in this pa-
+per, we classify visual object instances, incrementally and
+without label supervision.
+Class-Incremental Learning. Class-incremental learning
+expands an existing deep classifier with novel objects [36].
+In doing so, the goal is to retain performance on the pre-
+vious categories (i.e. prevent forgetting). To prevent for-
+getting, two lines of research are popular: Regularization
+and Memory. Regularization prevents abrupt changes in
+network weights [28, 30, 43] whereas Memory techniques
+replay part of the previous data [4,24,45,47].
+We differ from conventional class-incremental learning
+in two major ways. First, class-incremental learning oper-
+ates on object-category level, whereas we operate on the in-
+stance level. The challenges of instance-incremental learn-
+ing goes far beyond that of class-incremental learning. Sec-
+ond, most of the class-incremental learners assume access
+to fully labeled datasets for learning, which is sub-optimal
+if not impossible in case of instance learning. To that end,
+we propose to utilize self-supervision, and adapt prominent
+techniques for evaluation.
+More specifically, we experiment with Elastic Weight
+Consolidation (EwC) as a regularization approach [28] and
+Replay as a memory approach [45] due to their ease of adap-
+tation in a label-free (i.e. self-supervised) setting.
+Self-Supervised Learning. Self-supervision designs pre-
+text tasks to learn deep representations without labels. Early
+approaches predict rotations [19] or patches [39], whereas
+recently contrastive learning dominates [8, 11, 12, 21]. In
+this work, we utilize self-supervision as a replacement of
+instance labels to extract learning signals. We experiment
+with BarlowTwins [56] for its high performance, and ease
+of integration to incremental learning setup.
+Incremental Self-Supervised Learning. Recently, there
+has been a surge of interest in use of self-supervision to re-
+place label supervision for incremental learning. We iden-
+tify three main directions.
+i) Pre-training:
+Researchers use self-supervised learn-
+ing either for pre-training prior to incremental learning
+stage [7, 17, 26] or as an auxiliary loss function to improve
+feature discrimination [58]. However, these papers still re-
+quire labels during the incremental learning stage.
+ii) Replay:
+Second line of techniques propose replay-
+based methods [10, 35, 42] to supplement self-supervised
+learners with stored data within the memory. However, they
+require large amounts of exemplars to be stored within the
+memory to work effectively.
+iii) Regularization: Third line of work proposes to reg-
+ularize self-learned representations [16,20,35].
+In this work, we focus on regularization-based self-
+incremental learning. More specifically, we closely follow
+UCL [35] and ask ourselves: What is the contribution of
+self-supervision for instance incremental learning? Instead
+of proposing a yet another model, we benchmark Barlow-
+Twins [56], and compare it to the strong baseline of label-
+supervised incremental learning.
+
+Method
+Supervision
+Input
+Memory
+Loss
+Fine-Tuning
+Label-supervised
+(x, y)
+
+CE(y, y′)
+Fine-Tuning
+Self-supervised
+(x)
+
+BT(x, x′)
+EwC
+Label-supervised
+(x, y)
+
+CE(y, y′) + Reg(Θ, y′)
+EwC
+Self-supervised
+(x)
+
+BT(x, x′) + Reg(Θ)
+Replay
+Label-supervised
+(x, y)
+(xm, ym)
+CE(y, y′) + CE(ym, ym′)
+Replay
+Self-supervised
+(x)
+(xm)
+BT(x, x′) + BT(xm, xm′)
+Table 1. VINIL performs incremental instance learning via self-supervision, and is compared with label-supervision. We use memory
+replay [45] and weight regularization [28] as well as simple fine-tuning. Fine-Tuning [44] relies on Cross-Entropy (CE) or BarlowTwins
+(BT) [56] to perform incremental learning. EwC [28] penalizes abrupt changes in network weights via regularization (Reg(·)). Replay [45]
+replays a part of previous data in the form of input-labels (label-supervised) or input-only (self-supervised).
+3. VINIL
+We present an overview of VINIL in Table 1. The goal
+of VINIL is to train an embedding network f(·)θt parame-
+terized by θt. The network maps an input image x to a D-
+dimensional discriminative embedding h = fθt(x) which
+will then be used to query the database to retrieve differ-
+ent views of the input query for instance recognition. Here,
+t denotes the incremental learning step, where the tasks
+are arriving sequentially: T = (T1, T2, ..., Tt). We train
+VINIL via minimizing the following objective:
+L = wc · Linst + (1 − wc) · Lincr
+(1)
+where wc controls the contribution of instance classification
+loss Linst and incremental learning loss Lincr. Incremen-
+tal learning loss either corresponds to memory replay [45]
+or weight regularization [28] whereas instance classifica-
+tion loss Linst is either cross-entropy with labels or a self-
+supervision objective.
+3.1. Incremental Learning
+Fine-Tuning.
+A vanilla way to perform incremental in-
+stance learning is to apply simple fine-tuning via SGD [44].
+In SGD, no incremental learning loss is applied (i.e. wc =
+1.0) and the sole objective is classification.
+In case of label-supervision, a task is defined by a dataset
+Dlabel
+t
+= {(xi,t, yi,t)kt
+i=1} where kt is the data size at time
+t. Then, SGD corresponds to instance discrimination via
+cross-entropy Linst = CE(yi,t, y′
+i,t). Here, instance cate-
+gory prediction for the instance i at time step t is obtained
+with a simple MLP classifier. Notice that this classifier will
+expand in size linearly with the number of instance cate-
+gories.
+In case of VINIL, a task is defined by a dataset Dself
+t
+=
+{(xi,t)nt
+i=1} (i.e. no labels).
+Then, SGD corresponds
+to minimizing the self-supervision objective Linst
+=
+BT(xi,t, x′
+i,t) where BT(·) is the BarlowTwins [56].
+EwC [28]. EwC penalizes big changes in network weights
+via comparing the weights in the current and the previous
+incremental learning step. Originally, EwC re-weights the
+contribution of each weight to the loss function as a function
+of instance classification logits (i.e. label-supervision). In
+VINIL, in the absence of labels, we omit this re-weighting
+and simply use identity matrix.
+Replay [45]. Replay replays a portion of the past data from
+previous incremental steps to mitigate forgetting. In case of
+label-supervision, this corresponds to replaying both the in-
+put data and their labels via cross-entropy: CE(ym
+i,t, ym′
+i,t )
+where ym′
+i,t is the instance categories for the memory in-
+stance i at time t. For VINIL, we simply replay the input
+memory data and its augmented view via self-supervsion of
+BarlowTwins as BT(xm
+i,t, xm′
+i,t ).
+3.2. Self-Supervised Learning
+In BarlowTwins, the features are extracted from the orig-
+inal and the augmented view of the input image with a
+siamese deep network, at time step t as: (zi,t, z′
+i,t) =
+(fθt(xi,t), fθt(x′
+i,t)) where x′
+i,t = aug(xi, t) is the aug-
+mented view of the input. BarlowTwins minimizes the re-
+dundancy across views while maximizing the representa-
+tional information. This is achieved via operating on the
+cross-covariance matrix via:
+BT =
+�
+i
+(1 − Cii)2 + wb ·
+�
+i
+�
+j̸=i
+(Cij)2
+(2)
+where:
+Cij =
+�
+β zβ,iz′
+β,j
+�
+β
+�
+z2
+β,i · �
+β
+�
+(z′
+β,j)2
+(3)
+is the cross-correlation matrix.
+Here,
+wb
+controls
+invariance-redundancy reduction trade-off, i and j corre-
+sponds to network’s output dimensions.
+
+4. Experimental Setup
+Implementation. All the networks are implemented in Py-
+Torch [41]. We use ResNet-18 [22] as the backbone f(·),
+and a single-layer MLP for the instance classifier. We train
+for 200 epochs for each incremental steps with a learning
+rate 0.001 decayed via cosine annealing. We use SGD op-
+timizer with momentum 0.9 and batch-size 256. We use
+random cropping and scaling for augmentation.
+We
+follow
+the
+original
+implementation
+of
+Bar-
+lowTwins [1]. 10% of the data is stored within the memory
+for replay [45]. We set scalars as: wc = 0.7, wb = 0.03
+Datasets. We evaluate VINIL on iLab-20M [6] and Core-
+50 [33], since they are large-scale, sufficiently different, and
+widely adopted in incremental learning.
+iLab-20M is a turntable dataset of vehicles. It consists
+of 10 objects (i.e. bus, car, plane) with varying ([25, 160])
+number of instances per category. Objects are captured by
+varying the background and the camera angle, leading to 14
+examples per-instance. We use the public splits provided
+in [3] with 125k training and 31k gallery images.
+Core-50 is a hand-held object dataset used in bench-
+marking incremental learning algorithms. The dataset in-
+cludes 10 objects (i.e. phones, adaptors, scissors) with 50
+instances per-category. Each instance is captured for 300
+frames, across 11 different backgrounds. We use 120k train-
+ing and 45k gallery images [2].
+Protocol. We first divide each dataset into 5 tasks, with 2
+object categories per-task. Then, each task is subdivided
+into N object instance tasks depending on the dataset. We
+discard the classifier of label-supervised variants after train-
+ing, and evaluate all models with instance retrieval perfor-
+mance via k-NN with k = 100 neighbors on the gallery set,
+as is the standard in SSL [8,11–13,21].
+We use the mean-pooled activations of LAYER4 of
+ResNet to represent images. All exemplars in the gallery
+set are used as query.
+Metrics. We rely on two well established metrics to eval-
+uate the performance of the models, namely accuracy and
+forgetting.
+i). Accuracy measures whether if we can retrieve differ-
+ent views of the same instance from the gallery set given a
+query. We measure accuracy for each incremental learning
+steps, which is then averaged across all sessions.
+ii).
+Forgetting measures the discrepancy of accuracy
+across different sessions. Concretely, it compares the max-
+imum accuracy across all sessions with the accuracy in the
+last step.
+5. Experiments
+Our experiments address the following research ques-
+tions: Q1: Can VINIL improve performance and reduce
+forgetting in comparison to label-supervision? Q2: Does
+VINIL learn incrementally generalizable representations
+across datasets? Q3: What makes VINIL effective against
+label-supervision?
+5.1.
+How
+Does
+VINIL
+Compare
+to
+Label-
+Supervision?
+First,
+we compare VINIL’s performance to label-
+supervision. The results are presented in Table 2.
+Method
+Supervision
+Core-50
+iLab-20M
+Accuracy (↑)
+Forgetting (↓)
+Accuracy (↑)
+Forgetting (↓)
+SGD
+Label
+71.450
+22.436
+89.340
+6.500
+SGD
+VINIL
+74.914
+4.802
+90.398
+0.000
+Replay
+Label
+88.180
+6.741
+84.464
+5.696
+Replay
+VINIL
+67.677
+10.095
+90.543
+0.000
+EwC
+Label
+75.117
+18.268
+87.690
+4.535
+EwC
+VINIL
+73.011
+2.167
+90.655
+0.000
+Table 2. Visual Incremental Instance Learning on Core-50 [33]
+and iLab-20M [6]. VINIL outperforms label-supervised variants
+for 4 out of 6 settings, while significantly reducing forgetfulness
+on both datasets.
+This indicates self-incremental learning is a
+strong, label-free alternative to label-supervision.
+VINIL Yields Competitive Accuracy. We first compare
+the accuracies obtained by VINIL vs.
+label-supervision.
+We observe that VINIL yields competitive accuracy against
+label-supervision: In 4 out of 6 setting, VINIL outperforms
+label-supervised variants.
+VINIL Mitigates Forgetting. Secondly, we compare the
+forget rates of VINIL vs. label-supervision (lower is better).
+We observe that VINIL consistently leads to much lower
+forget rates in comparison to label-supervision. On iLab-
+20M dataset, VINIL results in no forgetting. On the more
+challenging dataset of Core-50, the difference across forget
+rates are even more pronounced: Label-supervision suffers
+from 22% forget rate whereas VINIL only by 4%, a relative
+drop of 80% with SGD.
+Label-supervision Leverages Memory. Our last observa-
+tion is that memory improves the accuracy and reduces for-
+getfulness of label-supervision. In contrast, the use of mem-
+ory disrupts self-supervised representations. This indicates
+that replaying both inputs and labels ((xi, yi)) as opposed
+to input-only ((xi), as in self-supervision) may lead to im-
+balanced training due to limited memory size [9,25,54].
+In summary, we conclude that VINIL is an efficient,
+label-free alternative to label-supervised incremental in-
+stance learning.
+VINIL improves accuracy while reduc-
+ing forget rate.
+We also observe that label-supervision
+
+closes the gap when an additional memory of past data is
+present. This motivates further research for improving self-
+incremental instance learners with memory.
+5.2. Can VINIL Generalize Across Datasets?
+After confirming the efficacy of VINIL within the same
+dataset, we now move on to a more complicated setting:
+Cross-dataset generalization. In cross-dataset generaliza-
+tion, we first perform incremental training on Core-50, and
+then evaluate on iLab-20M. Then, we perform incremental
+training on iLab-20M and then evaluate on Core-50.
+Cross-dataset generalization between Core-50 and iLab-
+20M is challenging due to the following reasons: i). Cam-
+era: Core-50 is captured with a hand-held camera whereas
+iLab-20M is captured on a platform with a turntable cam-
+era, ii). Object Categories: Object categories are disjoint, as
+no common objects are present in each dataset, iii). Object
+Types: iLab-20M exhibits toy objects of vehicles whereas
+Core-50 exhibits hand-interacted daily-life objects.
+The results are presented in Table 3. We present train-
+and-test on the same dataset as well as the relative drop (∆)
+for reference.
+Train on=⇒
+Core-50
+iLab-20M
+iLab-20M
+Core-50
+Test on=⇒
+Core-50
+Core-50
+iLab-20M
+iLab-20M
+Method
+Supervision
+Accuracy
+Accuracy
+%∆(↓)
+Accuracy
+Accuracy
+%∆(↓)
+SGD
+Label
+71.450
+59.850
+16
+89.340
+67.249
+24
+SGD
+VINIL
+74.914
+66.704
+10
+90.398
+76.302
+15
+Replay
+Label
+88.180
+55.692
+36
+84.464
+69.412
+17
+Replay
+VINIL
+67.677
+61.857
+8
+90.543
+76.125
+15
+EwC
+Label
+75.117
+59.030
+21
+87.690
+70.087
+20
+EwC
+VINIL
+73.011
+70.648
+3
+90.655
+75.793
+16
+Table 3. Cross-Dataset Generalization on Core-50 and iLab-20M
+datasets. VINIL is consistently more robust in cross-dataset gen-
+eralization when compared with label-supervision. The results in-
+dicate that self-supervision improves the generality of visual rep-
+resentations, for instance-incremental setup.
+VINIL Yields Generalizable Representations. We first
+observe that VINIL consistently yields higher accuracy and
+lower drop rate across all 6 settings in both datasets. This
+indicates that self-supervision extracts more generalizable
+visual representations from the dataset.
+Label-supervision Overfits with Memory. Secondly, we
+observe that label-supervised variants with memory gener-
+alizes via overfitting on the training dataset. Replay with
+label-supervision leads to the biggest drop rate of 36% on
+Core-50, when trained with iLab-20M. This implies the use
+of the memory drastically reduces generality of visual rep-
+resentations. A potential explanation is that, since replay
+utilizes the same set of examples within the limited mem-
+ory repeatedly throughout learning, this forces the network
+to over-fit to those examples.
+T=0
+T=1
+T=2
+T=3
+T=4
+Incremental Time Steps
+0
+1
+2
+3
+4
+Accuracy per Task
+96.19
+85.25
+81.97
+86.22
+83.77
+74.19
+77.59
+70.08
+70.30
+69.10
+83.16
+83.71
+90.34
+83.47
+81.66
+93.58
+86.62
+84.92
+95.42
+88.08
+94.36
+84.13
+77.51
+89.85
+94.17
+70
+75
+80
+85
+90
+95
+Figure 2. Task-level performance of Label-supervision (SGD).
+Label-supervision is biased towards recent task.
+T=0
+T=1
+T=2
+T=3
+T=4
+Incremental Time Steps
+0
+1
+2
+3
+4
+Accuracy per Task
+75.18
+85.25
+91.50
+93.41
+95.27
+63.80
+68.95
+72.24
+73.02
+74.13
+66.55
+81.37
+87.78
+87.53
+88.06
+71.49
+82.92
+92.58
+95.81
+96.33
+72.01
+81.00
+93.54
+95.95
+98.20
+65
+70
+75
+80
+85
+90
+95
+Figure 3. Task-level performance of VINIL (SGD). VINIL im-
+proves its performance with incoming data, and is less biased to-
+wards recent task.
+We conclude that VINIL extracts generalizable visual
+representations from the training source to perform instance
+incremental training. We also conclude that the astound-
+ing performance of label-supervision equipped with mem-
+ory comes with the cost of overfit, leading to drastic drop in
+case of visual discrepancies across datasets.
+5.3. What Factors Affect VINIL’s Performance?
+VINIL Mitigates Bias Towards Recent Task. We present
+the heatmaps of the performance for all 5 main tasks, when
+each task is introduced sequentially, for label-supervision in
+Figure 2 and for VINIL in Figure 3 on iLab-20M [6]. Each
+row presents the accuracy for each task, as the tasks are in-
+troduced sequentially. For example, the entry (0, 2) denotes
+the performance on Task-0 when the Task-2 is introduced.
+Considering Figure 2 for label-supervision, observe how
+the tasks achieve their peak performance when they are
+being introduced to the model, hence the higher numbers
+
+within the diagonal. Then, the performance degrades dras-
+tically as more and more tasks are being introduced. This
+indicates label-supervision fails to leverage more data. We
+call such phenomenon ”recency bias”, as the model is bi-
+ased towards the most recently introduced task.
+In contrast, in Figure 3 for VINIL, the performance on
+each task improves sequentially with the incoming stream
+of new tasks. This indicates self-supervised representations
+are less biased towards the recent task, and can leverage
+data to improve performance. This renders them as a viable
+option when incremental learning for longer learning steps,
+such as in incremental instance learning.
+VINIL Focuses on the Object Instance. We present the
+activations of the last layer of ResNet, at different incre-
+mental time steps, in Figure 4.
+Label
+VINIL
+t=0
+t=1
+t=2
+t=3
+t=4
+Input
+Label
+Label
+Label
+Incremental Learning Time Steps
+VINIL
+VINIL
+VINIL
+Figure 4. Activations of the last layer of ResNet [22], throughout
+the incremental learning steps. We compare label-supervision with
+VINIL (SGD). Notice how the attention of the label-supervised
+variant is disrupted after a few learning tasks. Instead, VINIL
+learns to segment out the target object, successfully suppressing
+the background context, such as the hand or the background.
+Observe how VINIL learns to segment out the target ob-
+ject from the background. This allows the model to ac-
+curately distinguish across different instances of the same
+object sharing identical backgrounds.
+In contrast, label-
+supervised variant progressively confuses the object with
+the background. We call such a phenomenon ”attentional
+deficiency” of label-supervised representations.
+VINIL Stores Instance-level Information.
+We present
+nearest neighbors for three queries in Figure 5. We use
+the average-pooled activations of the last ResNet layer on
+Core-50 trained with SGD.
+Observe how VINIL retrieves the same instance in dif-
+ferent viewpoints, such as for the light bulb and can. In
+contrast, label-supervision is distracted by the background
+context, as it retrieves irrelevant objects with identical back-
+ground. This indicates self-supervision generalizes via stor-
+ing instance-level information. We present a failure case in
+the last row, as both models fail to represent an object with
+holes and un-familiar rotation.
+We conclude that VINIL can improve its performance
+with incoming stream of data, and generalizes via focusing
+on the target object and storing instance-level details to per-
+form instance-incremental learning.
+6. Discussion
+This paper presented VINIL, a self-incremental visual
+instance learner. VINIL sequentially learns visual object in-
+stances, with no label supervision, via only self-supervision
+of BarlowTwins [56]. Below, we summarize our main dis-
+cussion points:
+Self vs. Label-supervision? We demonstrate that self-
+supervision not only omits the need for labels, but it is also
+more accurate and less forgetful.
+W/ or W/o Memory? Our results show that the use
+of memory boosts label-supervised instance incremental
+learning, however the improvement comes with the cost of
+over-fitting on the training source.
+SGD [44] vs. Replay [45] vs. EwC [28]? We demon-
+strate that with the use of self-supervision, VINIL closes the
+gap between simple fine-tuning via SGD and more compli-
+cated, compute-intensive techniques like memory replay or
+regularization via EwC.
+What Makes VINIL Effective? VINIL retains repre-
+sentations across tasks, and is able to store and focus on
+instance-level information, which are crucial for instance-
+incremental learning.
+Limitation. VINIL is executed with regularization [28]
+and memory [45].
+One can also consider dynamic net-
+works [55] whose architectures are updated with incoming
+task data. VINIL is a scalable alternative to dynamic incre-
+mental network training due to abundant unlabeled data.
+
+Q0传
+00QDLabel
+Query
+Nearest Neighbors
+Label
+Descending
+Label
+VINIL
+VINIL
+VINIL
+Figure 5. Five nearest neighbors for three object instance queries on Core-50 [33] with SGD. Green is a success, red is a failure. Observe
+how VINIL retrieves object instances in different views. The last column showcases a failure case, where both models fail to represent an
+object with holes (scissor).
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+arXiv:2301.02187v1  [math.LO]  3 Jan 2023
+Growth of Log-Analytic Functions
+Tobias Kaiser
+Abstract.
+We show that unary log-analytic functions are
+polynomially bounded. In the higher dimensional case glob-
+ally a log-analytic function can have exponential growth. We
+show that a log-analytic function is polynomially bounded on
+a definable set which contains the germ of every ray at infinity.
+Introduction
+Log-analytic functions have been defined by Lion and Rolin in their seminal
+paper [5]. They are iterated compositions from either side of globally sub-
+analytic functions (see [2]) and the global logarithm.
+In [3] it was shown
+that from the point of view of differentiability log-analytic functions behave
+similarly to globally subanalytic functions. We have strong quasianalyticity
+and Tamm’s theorem hold. But with respect to growth properties log-analytic
+functions behave in a different way compared to globally subanalytic functions.
+Globally subanalytic functions are polynomially bounded. This holds also for
+log-analytic funtions of one variable. But in higher dimension surprisingly the
+situation changes. Although the global exponential function is not involved
+in the definition of log-analytic functions, a log-analytic function in at least
+two variables can have exponential growth. We construct an example where
+the function is not polynomially bounded on every dense definable set. But
+polynomially boundedness holds on a definable set which is ‘thick’ at infinity:
+We show that a log-analytic function is polynomially bounded on a definable
+set which contains the germ of every ray at infinity.
+Notations
+By N = {1, 2, . . .} we denote the set of natural numbers and by N0 = {0, 1, 2, . . .}
+the set of nonnegative integers.
+For t ∈ R we set R>t := {x ∈ R | x > t} and R≥t := {x ∈ R | x ≥ t}.
+Denoting by | | the euclidean norm on Rn we set Sn−1 := {x ∈ Rn | |x| = 1}.
+Given a subset A of Rn we denote by A its closure.
+By π : Rn × R → Rn, (x, y) �→ x, we denote the projection on all but the last
+coordinate. For a subset A of Rn × R and x ∈ Rn we set Ax := {y ∈ R |
+(x, y) ∈ A}.
+By expk respectively logk we denote the k-times iterated of the exponential
+function respectively the logarithm.
+2010 Mathematics Subject Classification: 03C64, 14P15, 26A09, 26A12, 32B20
+Keywords and phrases: log-analytic functions, polynomially bounded, exponential growth
+1
+
+The Results
+We assume basic knowledge of o-minimality (see for example van den Dries
+[1] and van den Dries and Miller [2]). By definable we mean definable in the
+o-minimal structure Ran,exp (with parameters).
+Setting and Preliminaries
+We recall the precise definition of a log-analytic function (see Lion and Rolin
+[5]) and state consequences of preparation results on special sets (compare with
+[3]).
+1. Definition
+Let X ⊂ Rn be definable and let f : X → R be a function.
+(a) Let k ∈ N0. By induction on k we define that f is log-analytic of order
+at most k.
+Base case: The function f is log-analytic of order at most 0 if f is
+piecewise the restriction of globally subanalytic functions; i.e. there is a
+finite decomposition Y of X into definable sets such that for Y ∈ Y there
+is a globally subanalytic function F : Rn → R such that f|Y = F|Y .
+Inductive step: The function f is log-analytic of order at most k if the
+following holds. There is a finite decomposition Y of X into definable sets
+such that for Y ∈ Y there are p, q ∈ N0, a globally subanalytic function
+F : Rp+q → R and log-analytic functions g1, ..., gp : Y → R, h1, . . . , hq :
+Y → R>0 of order at most k − 1 such that
+f|Y = F
+�
+g1, ..., gp, log(h1), ..., log(hq)
+�
+.
+(b) Let k ∈ N0. We call f log-analytic of order k if f is log-analytic of
+order at most k but not of order at most k − 1.
+(c) We call f log-analytic if it is log-analytic of order k for some k ∈ N0.
+2. Definition
+We call a definable cell Y ⊂ Rn+1 simple at infinity if for every x ∈ π(Y )
+we have Yx = R>dx for some dx ∈ R≥0.
+3. Remark
+Let Y be a definable cell decomposition of Rn × R>0. Then
+Rn =
+�
+{π(Y ) | Y ∈ Y simple at infinity}.
+2
+
+We set e0 := 0 and ek := exp(ek−1) for k ∈ N.
+4. Definition
+Let k ∈ N0. A cell Y ⊂ Rn+1 which is simple at infinity is called k-simple at
+infinity if inf Yx ≥ ek for all x ∈ π(Y ).
+5. Proposition
+Let f : Rn × R → R, (x, y) �→ f(x, y), be log-analytic of order k. Then there is
+a definable cell decomposition Y of Rn × R such that for every Y ∈ Y which is
+simple at infinity the cell Y is k-simple at infinity such that
+f|Y (x, y) = a(x)yq0 log(y)q1 · · · logk(y)qku(x, y)
+where
+(1) a : π(Y ) → R is log-analytic and continuous,
+(2) q0, . . . , qk ∈ Q,
+(3) u : Y → R is log-analytic and there is d ∈ R>0 such that 0 ≤ u(x, y) ≤ d
+for all (x, y) ∈ Y .
+Proof:
+This follows from [3, Theorem 2.30] using the substitution r �→ 1/r.
+■
+Statement and Proof of the Results
+6. Definition
+Let n ∈ N and let f : Rn → R be a function.
+(a) If n = 1 we say that f is polynomially bounded at infinity if there
+are constants t ∈ R>0 and N ∈ N such that |f(x)| ≤ xN for all x > t.
+(b) If n > 1 we say that f is polynomially bounded at infinity if there
+are constants t ∈ R>0 and N ∈ N such that |f(x)| ≤ |x|N for all |x| > t.
+Let f be as above and let A ⊂ Rn be unbounded. We say that f is polynomially
+bounded at infinity on A if
+1Af is polynomially bounded at infinity.
+We handle the unary case first.
+7. Proposition
+Let f : R → R be log-analytic. Then f is polynomially bounded.
+Proof:
+3
+
+By Proposition 5 we find k ∈ N0 and t ≥ ek such that
+f(x) = axq0 log(x)q1 · · · logk(x)qku(x)
+on R≥t where
+(1) a ∈ R,
+(2) q0, . . . , qk ∈ Q,
+(3) u : R>t → R is log-analytic and there is d ∈ R>0 such that 0 ≤ u(x) ≤ d
+for all x > t.
+This gives that f(x) behaves asymptotically as xq0 log(x)q1 · · · logk(x)qk at +∞
+(unless in the trivial case a = 0). By the growth properties of the logarithm
+we are done.
+■
+8. Definition
+A subset C of Rn is called a cone if x ∈ C implies rx ∈ C for all r ∈ R≥0.
+Given a cone C with C ⊋ {0} we denote by B(C) := C ∩ Sn−1 its base. Note
+that C = R≥0 · B(C).
+9. Proposition
+Let n ≥ 2 and let f : Rn → R be log-analytic. Then there is a cone C with
+nonempty interior such that f is polynomially bounded at infinity on C.
+Proof:
+We consider the polar coordinates ϕ : Sn−1 × R≥0 → Rn, (v, r) → rv. Let
+g : Sn−1 × R≥0 → R, (v, r) �→ f(ϕ(v, r)). By Remark 3 and Proposition 5
+we find k ∈ N0 and an open cell Y that is k-simple at infinity such that
+g|Y (x) = a(v)rq0 log(r)q1 · · · logk(r)qku(v, r) where
+(1) a : π(Y ) → R is log-analytic and continuous,
+(2) q0, . . . , qk ∈ Q,
+(3) u : Y → R is log-analytic and there is d ∈ R>0 such that 0 ≤ u(v, r) ≤ d
+for all (v, r) ∈ Y .
+Choose an open ball B in π(Y ) such that its closure is contained in π(Y ).
+Then by continuity a is bounded on B.
+By the growth properties of the
+iterated logarithms we get that g is polynomially bounded on Y ∩ (B × R).
+We consider the cone C := R≥0 ·B which has nonempty interior. By continuity
+there is R > 1 such that the function x �→ inf Yx on B is bounded from above
+by R. Therefore we find some N ∈ N such that |f(x)| ≤ |x|N for all x ∈ C with
+4
+
+|x| > R. By the very definition we obtain that f is polynomially bounded at
+infinity on C.
+■
+In the higher dimensional case global (polynomially) boundedness may fail
+simply if the pole locus is not bounded. Consider for example the function
+f : R2 → R, (x, y) �→
+
+
+
+1
+x−y
+x ̸= y,
+if
+0
+x = y.
+Then clearly sup|(x,y)|=r |f(x, y)| = ∞ for all r > 0.
+But even if one restricts to continuous functions a log-analytic function may
+be not be polynomially bounded if n ≥ 2.
+10. Proposition
+Let n ≥ 2. There is a continuous log-analytic function f : Rn → R which is
+not polynomially bounded at infinity.
+Proof:
+It suffices to deal with the case n = 2. Consider the function
+h : R>1 × R>0 → R, (x, y) �→ −y
+�
+(log y)2 − 2 log y + 2 − x
+�
+.
+Claim 1: The following holds:
+(1) The function h is log-analytic and continuous.
+(2) For every x > 1 there exists maxy>0 h(x, y) ∈ R.
+(3) The function α : R>1 → R, x �→ maxy>0 h(x, y), is given by α(x) =
+2exp(√x)(√x − 1).
+Proof of Claim 1:
+(1) being clear, we have to show (2) and (3). For x > 1 we have
+lim
+yր∞ h(x, y) = −∞, lim
+yց0 h(x, y) = 0
+and
+∂h
+∂y h(x, y) = −(log y)2 + x
+which vanishes exactly for y = exp(√x) and y = exp(−√x). We have
+h(x, exp(√x)) = 2exp(√x)(√x − 1), h(x, exp(−√x)) = −2exp(√x)(√x + 1).
+This implies that for x > 1 the function R>0 → R, y �→ h(x, y), attains its
+maximum at y = exp(√x) with this maximum being given by
+max
+y>0 h(x, y) = 2exp(√x)(√x − 1).
+5
+
+This shows (2) and (3).
+■Claim 1
+Let a ∈ R>1 be the (uniquely determined) value such that 2exp(√a)(√a−1) =
+1. Let
+g : R≥0×[0, 1] → R, (x, y) �→
+
+
+
+max
+�
+h(x, y/(1 − y)), 1
+�
+,
+(x, y) ∈ R>a× ]0, 1[,
+if
+1,
+(x, y) /∈ R>a× ]0, 1[.
+Claim 2: The following holds:
+(1) The function g is continuous and log-analytic.
+(2) The function β : R≥0 → R, x �→ max0≤y≤1 g(x, y), is given by β(x) = 1
+for x ≤ a and β(x) = α(x) for x > a.
+Proof of Claim 2:
+For (1) note that for b > a
+lim
+x→b,yր1 g(x, y) =
+lim
+x→b,yր∞ max
+�
+h(x, y), 1
+�
+= 1,
+lim
+x→b,yց0g(x, y) =
+lim
+x→b,yց0 max
+�
+h(x, y), 1
+�
+= 1,
+that for 0 < c < 1
+lim
+xցa,y→c g(x, y) =
+lim
+xցa,y→c max
+�
+h(x, y/(1 − y)), 1
+�
+= 1,
+and that
+lim
+xցa,yց0 g(x, y) =
+lim
+xցa,yց0 max
+�
+h(x, y/(1 − y)), 1
+�
+= 1,
+lim
+xցa,yր1 g(x, y) =
+lim
+xցa,yր1 max
+�
+h(x, y/(1 − y)), 1
+�
+= 1.
+For (2) note that for x > a
+max
+0≤y≤1 g(x, y) = max
+y>0 h(x, y) = 2exp(√x)(√x − 1).
+■Claim 2
+Let
+f : R2 → R, (x, y) �→
+
+
+
+g
+�
+|(x, y)|2, arg((x, y)/|(x, y)|)/2π
+�
+,
+(x, y) ̸= (0, 0),
+if
+1,
+(x, y) = (0, 0),
+where the argument function is given by arg : S1 → [0, 2π[ with arg((1, 0)) = 0
+and counterclockwise orientation. Then f is continuous and log-analytic. Let
+6
+
+γ : R≥0 → R≥0, r �→ max|(x,y)|=r |f(x, y)|. Then γ(r) = α(r2) for all r ≥ 0.
+Hence
+max
+|(x,y)|=r |f(x, y)| ≥ exp(r)
+for all sufficiently large r.
+■
+The question is how “big” we can choose a set where polynomially bounded-
+ness at infinity holds. In Proposition 9 we have shown that we can choose a
+nonempty open cone. By the continuity of the counterexample in Proposition
+10 we cannot hope for a dense definable set (or equivalently, a definable set
+with dimension of the complement being smaller than n):
+11. Corollary
+Let n ≥ 2. There is log-analytic function f : Rn → R such that f is not
+polynomially bounded on every dense definable subset.
+12. Remark
+Note that the above counterexample is globally given by composition of glob-
+ally subanalytic functions and the logarithm, not only piecewise.
+To formulate an optimal result we need to introduce some setting to speak
+about the ultimate size of a set at ∞. The first definition mimics the tangential
+cone at finite points (see for example Kurdyka and Raby [4]).
+We fix an unbounded definable subset A of Rn. We let dim∞ A to be dim(A ∩
+{x ∈ Rn | |x| > r}) for sufficiently large r (note that this stabilizes) and call
+it the dimension of A at infinity.
+13. Definition
+(a) We let B(A, ∞) to be the set of all v ∈ Sn−1 such that for every r, ε > 0
+there is x ∈ A with |x| > r and
+��x/|x| − v
+�� < ε. We call C(A, ∞) :=
+R≥0 · B(A, ∞) the tangent cone of A at infinity.
+(b) We let Bstr(A, ∞) to be the set of all v ∈ Sn−1 such that there is some
+t ∈ R≥0 with R≥t · v ⊂ A. We call Cstr(A, ∞) := R≥0 · Bstr(A, ∞) the
+strong tangent cone of A at infinity.
+14. Remark
+(1) We have Cstr(A, ∞) ⊂ C(A, ∞).
+(2) The tangent cone C(A, ∞) of A at infinity is closed and definable with
+dim C(A, ∞) ≤ dim∞ A.
+7
+
+(3) The strong tangent cone Cstr(A, ∞) of A at infinity is definable with
+dim Cstr(A, ∞) ≤ dim∞ A.
+(4) For r > 0 let B(A, r) := {x/r | x ∈ A and |x| = r}. Then B(A, ∞) is
+the Hausdorff limit of the family
+�
+B(A, r)
+�
+r∈R>0 (compare with Lion and
+Speissegger [6]) and
+Bstr(A, ∞) = lim sup
+r>0
+B(A, r) =
+�
+r>0
+�
+s>r
+B(A, s).
+The next concept will carry more information. A (closed) ray R in Rn is of
+the form R = a + R≥0 · v where a ∈ Rn and v ∈ Sn−1. We parametrize the
+set R of all rays by the bijection Rn × Sn−1 → R, (a, v) �→ a + R≥0 · v. For
+limit considerations it is natural to identify two rays R1 and R2 if R1 ⊂ R2 or
+R2 ⊂ R1. This is an equivalence relation ∼ on R. A canonical representative
+of the equivalence class of a ray R = a + R≥0 · v is given by o + R≥0 · v where
+o ∈ a + R · v with o ⊥ v (or, equivalently, o realizes the distance of the line
+a + R · v to the origin). A ray of this form is called a standardized ray. We
+identify the set R/ ∼ with the set of the standardized rays and parametrize it
+by the bijection S := {(o, v) ∈ Rn×Sn−1 | o ⊥ v} → R/ ∼, (o, v) �→ o+R≥0·v.
+15. Definition
+(a) We denote by RC(A, ∞) the union of all standardized rays R = o+R≥0·v
+such that for every r, ε > 0 there are x ∈ A and y ∈ R with |x| = |y| > r
+and |x − y| < ε and call it the the tangent ray cone of A at infinity.
+(b) We denote by RCstr(A, ∞) the union of all standardized rays R = o +
+R≥0 · v such that o + R≥t · v ⊂ A for some t ∈ R≥0 and call it the strong
+tangent ray cone of A at infinity.
+16. Remark
+(1) We have RCstr(A, ∞) ⊂ RC(A, ∞).
+(2) The tangent ray cone RCA,∞ of A at infinity is closed and definable with
+dim RCA,∞ ≤ dim∞ A.
+(3) The strong tangent ray cone RCstr
+A,∞ of A at infinity is definable with
+dim RCstr
+A,∞ ≤ dim∞ A.
+(4) We have C(A, ∞) ⊂ RC(A, ∞). In fact, the following stronger statement
+holds: A standardized ray o + R≥0 · v is contained in RC(A, ∞) if and
+only if R≥0 · v is contained in C(A, ∞).
+(5) We have Cstr(A, ∞) ⊂ RCstr(A, ∞).
+8
+
+17. Example
+Consider the half-strip
+S := {(x, y) ∈ R2 | x > 0, 0 < y < 1}.
+We have
+C(S, ∞) = R≥0 · (1, 0), Cstr(S, ∞) = ∅
+and
+RC(S, ∞) =
+�
+(0, t) + R≥0 · (1, 0)
+�� t ∈ R
+�
+,
+RCstr(S, ∞) =
+�
+(0, t) + R≥0 · (1, 0)
+�� 0 < t < 1
+�
+.
+18. Definition
+(a) We call A spherically dense at infinity if C(A, ∞) = Rn. We call A
+strongly spherically dense at infinity if Cstr(A, ∞) = Rn.
+(b) We call A ray dense at infinity if RC(A, ∞) contains every standard-
+ized ray. We call A strongly ray dense at infinity if RCstr(A, ∞)
+contains every standardized ray.
+19. Remark
+(1) A is spherically dense at infinity if and only if A is ray dense at infinity.
+(2) If A is strongly ray dense at infinity then A is strongly spherically dense
+at infinity. The converse does in general not hold.
+Proof:
+(1): The direction from right to the left being clear by definition we show the
+direction from left to the right. Let o+R≥0 ·v ∈ R/ ∼ where (o, v) ∈ S. Then
+R≥0 · v ∈ C(A, ∞) since A is spherically dense at infinity. By the definition of
+the tangent ray cone we obtain that o + R≥0 · v ⊂ RC(A, ∞).
+(2): The first statement ist clear. For the second one consider the complement
+of the above half-strip.
+■
+Hence the notion of ray density at infinity does not give anything new. We
+have included it for completeness and symmetry.
+Here is now the final optimal result.
+9
+
+20. Theorem
+Let n ≥ 2 and let f : Rn → R be log-analytic. Then there is a definable subset
+U of Rn which is strongly ray dense at infinity such that f is polynomially
+bounded at infinity on U.
+Proof:
+Consider the semialgebraic map Φ : S × R≥0 → Rn, (o, v, r) �→ o + rv, and
+the log-analytic function F := f ◦ Φ : S × R≥0 → R. Let F be log-analytic
+of order k ∈ N0. By Proposition 5 we find a definable cell decomposition Y of
+S × R≥0 such that for every Y ∈ Y which is simple at infinity the cell Y is
+k-simple at infinity such that
+F|Y (o, v, r) = a(o, y)rq0 log(r)q1 · · · logk(r)qku(o, v, r)
+where
+(1) a : π(Y ) → R is log-analytic and continuous,
+(2) q0, . . . , qk ∈ Q,
+(3) u : Y → R is log-analytic and there is d = dY ∈ R>0 such that 0 ≤
+u(o, v, r) ≤ d for all (o, v, r) ∈ Y .
+We fix Y ∈ Y simple at infinity. Let Z := π(Y ) and δ : Z → R≥0, (o, v) �→
+inf Y(o,v). We set frZS := (Z \ Z) ∩S. By passing to a finer cell decomposition
+of S we may assume that frZS ̸= ∅. For s ∈ R≥0 let
+Z(s) :=
+�
+(o, v) ∈ Z
+�� |(o, v)| ≤ s, dist((o, v), frSZ) ≥ s
+�
+.
+Then Z(s) is compact for every s ≥ 0. We set
+∆ : R≥0 → R≥0, s �→ max
+�
+|a(o, v)|
+�� (o, v) ∈ Z(s)
+�
+.
+Note that this is well-defined since a is continuous. Note that here by con-
+vention max ∅ = 0. The function ∆ is increasing and definable. Hence by van
+den Dries and Miller [2] it is bounded by an iterated exponential expl for some
+l ∈ N0. Choose N = NY ∈ N with N > |q0| + . . . + |qn|. We set
+WY :=
+�
+(o, v, r) ∈ S × R>0 | (o, v) ∈ Z(logl(r)), r > max{el, δ(o, v)}
+�
+.
+For (o, v, r) ∈ WY we have
+|F(o, v, r)| = |a(o, v)|rq0 log(r)q1 · · ·logk(r)qku(o, v, r) ≤ dY rrNY .
+We set VY := Φ(WY ). We obtain that |f(x)| ≤ dY |x|NY +1 on VY .
+10
+
+Let U be the union of all VY with Y ∈ Y simple at infinity. Then U is definable.
+We show that this U does the job. Let R = o + R≥0 · v be a standardized ray
+and let r > 0. By Remark 3 we find Y ∈ Y that is simple at infinity such
+that (o, v) ∈ Z. Note that we use the above notations. There is s ∈ R>0
+such that (o, v) ∈ Z(s). By the definition of WY we find t > 0 such that
+{(o, v)} × R≥t ⊂ WY . This gives o + R≥t · v ⊂ VY ⊂ U. So U is strongly ray
+dense. Let
+dU := max{dy | Y ∈ Y simple at infinity}
+and
+NU := max{Ny | Y ∈ Y simple at infinity}.
+Then |f(x)| ≤ dU|x|NU+1 for all x ∈ U. Hence f is polynomially bounded on
+U.
+■.
+21. Concluding Remarks
+In Corollary 11 we have found for n ≥ 2 a log-analytic function f : Rn → R
+and a definable open und unbounded set W such that r �→ infx∈W,|x|=r |f(x)| is
+of exponential growth. By Proposition 7 the set W cannot contain the image
+of an unbounded log-analytic curve. By the same methods as in the proof of
+Theorem 20 we can find an open and definable set U such that f is polyno-
+mially bounded at infinity on U and U contains the germ of every unbounded
+log-analytic curves up to a certain complexity (where the complexity is the
+complexity of terms in the language Lan(−1, ( n√...)n=2,3,..., log), compare with
+[3, Remark 1.2]). An open question is whether we can find such an U that
+contains the germ of every unbounded log-analytic curve.
+References
+(1) L. van den Dries: Tame Topology and O-minimal Structures. London Math. Soc.
+Lecture Notes Series 248, Cambridge University Press, 1998.
+(2) L. van den Dries and C. Miller: Geometric categories and o-minimal structures. Duke
+Math. J. 84 (1996), no. 2, 497-540.
+(3) T. Kaiser and Andre Opris: Differentiability Properties of Log-Analytic Functions.
+Rocky Mountain Journal of Mathematics 52 (2022) no. 4, 1423-1443.
+(4) K. Kurdyka, G. Raby: Densit´e des ensembles sous-analytiques. Ann. Inst. Fourier
+39 (1989), no. 3, 753-771.
+(5) J.-M. Lion, J.-P. Rolin: Th´eor`eme de pr´eparation pour les fonctions logarithmico-
+exponentielles. Ann. Inst. Fourier 47 (1997), no. 3, 859-884.
+(6) J.-M. Lion, P. Speissegger: A geometric proof of the definability of Hausdorff limits.
+Sel. Math., New Ser. 10 (2004), no. 3, 377-390.
+Tobias Kaiser, University of Passau, Faculty of Computer Science and Mathematics
+[email protected], D-94030 Germany
+11
+

CNE0T4oBgHgl3EQfQACa/content/tmp_files/load_file.txt

@@ -0,0 +1,294 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf,len=293
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='02187v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='LO]  3 Jan 2023  Growth of Log-Analytic Functions  Tobias Kaiser  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='  We show that unary log-analytic functions are  polynomially bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' In the higher dimensional case glob-  ally a log-analytic function can have exponential growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' We  show that a log-analytic function is polynomially bounded on  a definable set which contains the germ of every ray at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='  Introduction  Log-analytic functions have been defined by Lion and Rolin in their seminal  paper [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' They are iterated compositions from either side of globally sub-  analytic functions (see [2]) and the global logarithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='  In [3] it was shown  that from the point of view of differentiability log-analytic functions behave  similarly to globally subanalytic functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' We have strong quasianalyticity  and Tamm’s theorem hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' But with respect to growth properties log-analytic  functions behave in a different way compared to globally subanalytic functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='  Globally subanalytic functions are polynomially bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' This holds also for  log-analytic funtions of one variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' But in higher dimension surprisingly the  situation changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' Although the global exponential function is not involved  in the definition of log-analytic functions, a log-analytic function in at least  two variables can have exponential growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' We construct an example where  the function is not polynomially bounded on every dense definable set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' But  polynomially boundedness holds on a definable set which is ‘thick’ at infinity:  We show that a log-analytic function is polynomially bounded on a definable  set which contains the germ of every ray at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='  Notations  By N = {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='} we denote the set of natural numbers and by N0 = {0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='}  the set of nonnegative integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='  For t ∈ R we set R>t := {x ∈ R | x > t} and R≥t := {x ∈ R | x ≥ t}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='  Denoting by | | the euclidean norm on Rn we set Sn−1 := {x ∈ Rn | |x| = 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='  Given a subset A of Rn we denote by A its closure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='  By π : Rn × R → Rn, (x, y) �→ x, we denote the projection on all but the last  coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' For a subset A of Rn × R and x ∈ Rn we set Ax := {y ∈ R |  (x, y) ∈ A}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='  By expk respectively logk we denote the k-times iterated of the exponential  function respectively the logarithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='  2010 Mathematics Subject Classification: 03C64, 14P15, 26A09, 26A12, 32B20  Keywords and phrases: log-analytic functions, polynomially bounded, exponential growth  1  The Results  We assume basic knowledge of o-minimality (see for example van den Dries  [1] and van den Dries and Miller [2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' By definable we mean definable in the  o-minimal structure Ran,exp (with parameters).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='  Setting and Preliminaries  We recall the precise definition of a log-analytic function (see Lion and Rolin  [5]) and state consequences of preparation results on special sets (compare with  [3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' Definition  Let X ⊂ Rn be definable and let f : X → R be a function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='  (a) Let k ∈ N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' By induction on k we define that f is log-analytic of order  at most k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='  Base case: The function f is log-analytic of order at most 0 if f is  piecewise the restriction of globally subanalytic functions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' there is a  finite decomposition Y of X into definable sets such that for Y ∈ Y there  is a globally subanalytic function F : Rn → R such that f|Y = F|Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='  Inductive step: The function f is log-analytic of order at most k if the  following holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' There is a finite decomposition Y of X into definable sets  such that for Y ∈ Y there are p, q ∈ N0, a globally subanalytic function  F : Rp+q → R and log-analytic functions g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=', gp : Y → R, h1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' , hq :  Y → R>0 of order at most k − 1 such that  f|Y = F  �  g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=', gp, log(h1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=', log(hq)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='  (b) Let k ∈ N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' We call f log-analytic of order k if f is log-analytic of  order at most k but not of order at most k − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='  (c) We call f log-analytic if it is log-analytic of order k for some k ∈ N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' Definition  We call a definable cell Y ⊂ Rn+1 simple at infinity if for every x ∈ π(Y )  we have Yx = R>dx for some dx ∈ R≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' Remark  Let Y be a definable cell decomposition of Rn × R>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' Then  Rn =  �  {π(Y ) | Y ∈ Y simple at infinity}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='  2  We set e0 := 0 and ek := exp(ek−1) for k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' Definition  Let k ∈ N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' A cell Y ⊂ Rn+1 which is simple at infinity is called k-simple at  infinity if inf Yx ≥ ek for all x ∈ π(Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' Proposition  Let f : Rn × R → R, (x, y) �→ f(x, y), be log-analytic of order k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' Then there is  a definable cell decomposition Y of Rn × R such that for every Y ∈ Y which is  simple at infinity the cell Y is k-simple at infinity such that  f|Y (x, y) = a(x)yq0 log(y)q1 · · · logk(y)qku(x, y)  where  (1) a : π(Y ) → R is log-analytic and continuous,  (2) q0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' , qk ∈ Q,  (3) u : Y → R is log-analytic and there is d ∈ R>0 such that 0 ≤ u(x, y) ≤ d  for all (x, y) ∈ Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='  Proof:  This follows from [3, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='30] using the substitution r �→ 1/r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='  ■  Statement and Proof of the Results  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' Definition  Let n ∈ N and let f : Rn → R be a function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='  (a) If n = 1 we say that f is polynomially bounded at infinity if there  are constants t ∈ R>0 and N ∈ N such that |f(x)| ≤ xN for all x > t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='  (b) If n > 1 we say that f is polynomially bounded at infinity if there  are constants t ∈ R>0 and N ∈ N such that |f(x)| ≤ |x|N for all |x| > t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='  Let f be as above and let A ⊂ Rn be unbounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' We say that f is polynomially  bounded at infinity on A if  1Af is polynomially bounded at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='  We handle the unary case first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' Proposition  Let f : R → R be log-analytic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' Then f is polynomially bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='  Proof:  3  By Proposition 5 we find k ∈ N0 and t ≥ ek such that  f(x) = axq0 log(x)q1 · · · logk(x)qku(x)  on R≥t where  (1) a ∈ R,  (2) q0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' , qk ∈ Q,  (3) u : R>t → R is log-analytic and there is d ∈ R>0 such that 0 ≤ u(x) ≤ d  for all x > t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='  This gives that f(x) behaves asymptotically as xq0 log(x)q1 · · · logk(x)qk at +∞  (unless in the trivial case a = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' By the growth properties of the logarithm  we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='  ■  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' Definition  A subset C of Rn is called a cone if x ∈ C implies rx ∈ C for all r ∈ R≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='  Given a cone C with C ⊋ {0} we denote by B(C) := C ∩ Sn−1 its base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' Note  that C = R≥0 · B(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' Proposition  Let n ≥ 2 and let f : Rn → R be log-analytic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' Then there is a cone C with  nonempty interior such that f is polynomially bounded at infinity on C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='  Proof:  We consider the polar coordinates ϕ : Sn−1 × R≥0 → Rn, (v, r) → rv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' Let  g : Sn−1 × R≥0 → R, (v, r) �→ f(ϕ(v, r)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' By Remark 3 and Proposition 5  we find k ∈ N0 and an open cell Y that is k-simple at infinity such that  g|Y (x) = a(v)rq0 log(r)q1 · · · logk(r)qku(v, r) where  (1) a : π(Y ) → R is log-analytic and continuous,  (2) q0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' , qk ∈ Q,  (3) u : Y → R is log-analytic and there is d ∈ R>0 such that 0 ≤ u(v, r) ≤ d  for all (v, r) ∈ Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='  Choose an open ball B in π(Y ) such that its closure is contained in π(Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='  Then by continuity a is bounded on B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='  By the growth properties of the  iterated logarithms we get that g is polynomially bounded on Y ∩ (B × R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='  We consider the cone C := R≥0 ·B which has nonempty interior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' By continuity  there is R > 1 such that the function x �→ inf Yx on B is bounded from above  by R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' Therefore we find some N ∈ N such that |f(x)| ≤ |x|N for all x ∈ C with  4  |x| > R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' By the very definition we obtain that f is polynomially bounded at  infinity on C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='  ■  In the higher dimensional case global (polynomially) boundedness may fail  simply if the pole locus is not bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' Consider for example the function  f : R2 → R, (x, y) �→  \uf8f1  \uf8f2  \uf8f3  1  x−y  x ̸= y,  if  0  x = y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='  Then clearly sup|(x,y)|=r |f(x, y)| = ∞ for all r > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='  But even if one restricts to continuous functions a log-analytic function may  be not be polynomially bounded if n ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' Proposition  Let n ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' There is a continuous log-analytic function f : Rn → R which is  not polynomially bounded at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='  Proof:  It suffices to deal with the case n = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' Consider the function  h : R>1 × R>0 → R, (x, y) �→ −y  �  (log y)2 − 2 log y + 2 − x  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='  Claim 1: The following holds:  (1) The function h is log-analytic and continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='  (2) For every x > 1 there exists maxy>0 h(x, y) ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='  (3) The function α : R>1 → R, x �→ maxy>0 h(x, y), is given by α(x) =  2exp(√x)(√x − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='  Proof of Claim 1:  (1) being clear, we have to show (2) and (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' For x > 1 we have  lim  yր∞ h(x, y) = −∞, lim  yց0 h(x, y) = 0  and  ∂h  ∂y h(x, y) = −(log y)2 + x  which vanishes exactly for y = exp(√x) and y = exp(−√x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' We have  h(x, exp(√x)) = 2exp(√x)(√x − 1), h(x, exp(−√x)) = −2exp(√x)(√x + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='  This implies that for x > 1 the function R>0 → R, y �→ h(x, y), attains its  maximum at y = exp(√x) with this maximum being given by  max  y>0 h(x, y) = 2exp(√x)(√x − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='  5  This shows (2) and (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='  ■Claim 1  Let a ∈ R>1 be the (uniquely determined) value such that 2exp(√a)(√a−1) =  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' Let  g : R≥0×[0, 1] → R, (x, y) �→  \uf8f1  \uf8f2  \uf8f3  max  �  h(x, y/(1 − y)), 1  �  ,  (x, y) ∈ R>a× ]0, 1[,  if  1,  (x, y) /∈ R>a× ]0, 1[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='  Claim 2: The following holds:  (1) The function g is continuous and log-analytic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='  (2) The function β : R≥0 → R, x �→ max0≤y≤1 g(x, y), is given by β(x) = 1  for x ≤ a and β(x) = α(x) for x > a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='  Proof of Claim 2:  For (1) note that for b > a  lim  x→b,yր1 g(x, y) =  lim  x→b,yր∞ max  �  h(x, y), 1  �  = 1,  lim  x→b,yց0g(x, y) =  lim  x→b,yց0 max  �  h(x, y), 1  �  = 1,  that for 0 < c < 1  lim  xցa,y→c g(x, y) =  lim  xցa,y→c max  �  h(x, y/(1 − y)), 1  �  = 1,  and that  lim  xցa,yց0 g(x, y) =  lim  xցa,yց0 max  �  h(x, y/(1 − y)), 1  �  = 1,  lim  xցa,yր1 g(x, y) =  lim  xցa,yր1 max  �  h(x, y/(1 − y)), 1  �  = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='  For (2) note that for x > a  max  0≤y≤1 g(x, y) = max  y>0 h(x, y) = 2exp(√x)(√x − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='  ■Claim 2  Let  f : R2 → R, (x, y) �→  \uf8f1  \uf8f2  \uf8f3  g  �  |(x, y)|2, arg((x, y)/|(x, y)|)/2π  �  ,  (x, y) ̸= (0, 0),  if  1,  (x, y) = (0, 0),  where the argument function is given by arg : S1 → [0, 2π[ with arg((1, 0)) = 0  and counterclockwise orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' Then f is continuous and log-analytic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' Let  6  γ : R≥0 → R≥0, r �→ max|(x,y)|=r |f(x, y)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' Then γ(r) = α(r2) for all r ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='  Hence  max  |(x,y)|=r |f(x, y)| ≥ exp(r)  for all sufficiently large r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='  ■  The question is how “big” we can choose a set where polynomially bounded-  ness at infinity holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' In Proposition 9 we have shown that we can choose a  nonempty open cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' By the continuity of the counterexample in Proposition  10 we cannot hope for a dense definable set (or equivalently, a definable set  with dimension of the complement being smaller than n):  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' Corollary  Let n ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' There is log-analytic function f : Rn → R such that f is not  polynomially bounded on every dense definable subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' Remark  Note that the above counterexample is globally given by composition of glob-  ally subanalytic functions and the logarithm, not only piecewise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='  To formulate an optimal result we need to introduce some setting to speak  about the ultimate size of a set at ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' The first definition mimics the tangential  cone at finite points (see for example Kurdyka and Raby [4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='  We fix an unbounded definable subset A of Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' We let dim∞ A to be dim(A ∩  {x ∈ Rn | |x| > r}) for sufficiently large r (note that this stabilizes) and call  it the dimension of A at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' Definition  (a) We let B(A, ∞) to be the set of all v ∈ Sn−1 such that for every r, ε > 0  there is x ∈ A with |x| > r and  ��x/|x| − v  �� < ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' We call C(A, ∞) :=  R≥0 · B(A, ∞) the tangent cone of A at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='  (b) We let Bstr(A, ∞) to be the set of all v ∈ Sn−1 such that there is some  t ∈ R≥0 with R≥t · v ⊂ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' We call Cstr(A, ∞) := R≥0 · Bstr(A, ∞) the  strong tangent cone of A at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' Remark  (1) We have Cstr(A, ∞) ⊂ C(A, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='  (2) The tangent cone C(A, ∞) of A at infinity is closed and definable with  dim C(A, ∞) ≤ dim∞ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='  7  (3) The strong tangent cone Cstr(A, ∞) of A at infinity is definable with  dim Cstr(A, ∞) ≤ dim∞ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='  (4) For r > 0 let B(A, r) := {x/r | x ∈ A and |x| = r}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' Then B(A, ∞) is  the Hausdorff limit of the family  �  B(A, r)  �  r∈R>0 (compare with Lion and  Speissegger [6]) and  Bstr(A, ∞) = lim sup  r>0  B(A, r) =  �  r>0  �  s>r  B(A, s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='  The next concept will carry more information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' A (closed) ray R in Rn is of  the form R = a + R≥0 · v where a ∈ Rn and v ∈ Sn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' We parametrize the  set R of all rays by the bijection Rn × Sn−1 → R, (a, v) �→ a + R≥0 · v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' For  limit considerations it is natural to identify two rays R1 and R2 if R1 ⊂ R2 or  R2 ⊂ R1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' This is an equivalence relation ∼ on R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' A canonical representative  of the equivalence class of a ray R = a + R≥0 · v is given by o + R≥0 · v where  o ∈ a + R · v with o ⊥ v (or, equivalently, o realizes the distance of the line  a + R · v to the origin).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' A ray of this form is called a standardized ray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' We  identify the set R/ ∼ with the set of the standardized rays and parametrize it  by the bijection S := {(o, v) ∈ Rn×Sn−1 | o ⊥ v} → R/ ∼, (o, v) �→ o+R≥0·v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' Definition  (a) We denote by RC(A, ∞) the union of all standardized rays R = o+R≥0·v  such that for every r, ε > 0 there are x ∈ A and y ∈ R with |x| = |y| > r  and |x − y| < ε and call it the the tangent ray cone of A at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='  (b) We denote by RCstr(A, ∞) the union of all standardized rays R = o +  R≥0 · v such that o + R≥t · v ⊂ A for some t ∈ R≥0 and call it the strong  tangent ray cone of A at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' Remark  (1) We have RCstr(A, ∞) ⊂ RC(A, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='  (2) The tangent ray cone RCA,∞ of A at infinity is closed and definable with  dim RCA,∞ ≤ dim∞ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='  (3) The strong tangent ray cone RCstr  A,∞ of A at infinity is definable with  dim RCstr  A,∞ ≤ dim∞ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='  (4) We have C(A, ∞) ⊂ RC(A, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' In fact, the following stronger statement  holds: A standardized ray o + R≥0 · v is contained in RC(A, ∞) if and  only if R≥0 · v is contained in C(A, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='  (5) We have Cstr(A, ∞) ⊂ RCstr(A, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='  8  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' Example  Consider the half-strip  S := {(x, y) ∈ R2 | x > 0, 0 < y < 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='  We have  C(S, ∞) = R≥0 · (1, 0), Cstr(S, ∞) = ∅  and  RC(S, ∞) =  �  (0, t) + R≥0 · (1, 0)  �� t ∈ R  �  ,  RCstr(S, ∞) =  �  (0, t) + R≥0 · (1, 0)  �� 0 < t < 1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='  18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' Definition  (a) We call A spherically dense at infinity if C(A, ∞) = Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' We call A  strongly spherically dense at infinity if Cstr(A, ∞) = Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='  (b) We call A ray dense at infinity if RC(A, ∞) contains every standard-  ized ray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' We call A strongly ray dense at infinity if RCstr(A, ∞)  contains every standardized ray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' Remark  (1) A is spherically dense at infinity if and only if A is ray dense at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='  (2) If A is strongly ray dense at infinity then A is strongly spherically dense  at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' The converse does in general not hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='  Proof:  (1): The direction from right to the left being clear by definition we show the  direction from left to the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' Let o+R≥0 ·v ∈ R/ ∼ where (o, v) ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' Then  R≥0 · v ∈ C(A, ∞) since A is spherically dense at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' By the definition of  the tangent ray cone we obtain that o + R≥0 · v ⊂ RC(A, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='  (2): The first statement ist clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' For the second one consider the complement  of the above half-strip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='  ■  Hence the notion of ray density at infinity does not give anything new.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' We  have included it for completeness and symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='  Here is now the final optimal result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='  9  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' Theorem  Let n ≥ 2 and let f : Rn → R be log-analytic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' Then there is a definable subset  U of Rn which is strongly ray dense at infinity such that f is polynomially  bounded at infinity on U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='  Proof:  Consider the semialgebraic map Φ : S × R≥0 → Rn, (o, v, r) �→ o + rv, and  the log-analytic function F := f ◦ Φ : S × R≥0 → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' Let F be log-analytic  of order k ∈ N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' By Proposition 5 we find a definable cell decomposition Y of  S × R≥0 such that for every Y ∈ Y which is simple at infinity the cell Y is  k-simple at infinity such that  F|Y (o, v, r) = a(o, y)rq0 log(r)q1 · · · logk(r)qku(o, v, r)  where  (1) a : π(Y ) → R is log-analytic and continuous,  (2) q0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' , qk ∈ Q,  (3) u : Y → R is log-analytic and there is d = dY ∈ R>0 such that 0 ≤  u(o, v, r) ≤ d for all (o, v, r) ∈ Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='  We fix Y ∈ Y simple at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' Let Z := π(Y ) and δ : Z → R≥0, (o, v) �→  inf Y(o,v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' We set frZS := (Z \\ Z) ∩S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' By passing to a finer cell decomposition  of S we may assume that frZS ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' For s ∈ R≥0 let  Z(s) :=  �  (o, v) ∈ Z  �� |(o, v)| ≤ s, dist((o, v), frSZ) ≥ s  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='  Then Z(s) is compact for every s ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' We set  ∆ : R≥0 → R≥0, s �→ max  �  |a(o, v)|  �� (o, v) ∈ Z(s)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='  Note that this is well-defined since a is continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' Note that here by con-  vention max ∅ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' The function ∆ is increasing and definable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' Hence by van  den Dries and Miller [2] it is bounded by an iterated exponential expl for some  l ∈ N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' Choose N = NY ∈ N with N > |q0| + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' + |qn|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' We set  WY :=  �  (o, v, r) ∈ S × R>0 | (o, v) ∈ Z(logl(r)), r > max{el, δ(o, v)}  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='  For (o, v, r) ∈ WY we have  |F(o, v, r)| = |a(o, v)|rq0 log(r)q1 · · ·logk(r)qku(o, v, r) ≤ dY rrNY .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='  We set VY := Φ(WY ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' We obtain that |f(x)| ≤ dY |x|NY +1 on VY .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='  10  Let U be the union of all VY with Y ∈ Y simple at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' Then U is definable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='  We show that this U does the job.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' Let R = o + R≥0 · v be a standardized ray  and let r > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' By Remark 3 we find Y ∈ Y that is simple at infinity such  that (o, v) ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' Note that we use the above notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' There is s ∈ R>0  such that (o, v) ∈ Z(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' By the definition of WY we find t > 0 such that  {(o, v)} × R≥t ⊂ WY .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' This gives o + R≥t · v ⊂ VY ⊂ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' So U is strongly ray  dense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' Let  dU := max{dy | Y ∈ Y simple at infinity}  and  NU := max{Ny | Y ∈ Y simple at infinity}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='  Then |f(x)| ≤ dU|x|NU+1 for all x ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' Hence f is polynomially bounded on  U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='  ■.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='  21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' Concluding Remarks  In Corollary 11 we have found for n ≥ 2 a log-analytic function f : Rn → R  and a definable open und unbounded set W such that r �→ infx∈W,|x|=r |f(x)| is  of exponential growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' By Proposition 7 the set W cannot contain the image  of an unbounded log-analytic curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' By the same methods as in the proof of  Theorem 20 we can find an open and definable set U such that f is polyno-  mially bounded at infinity on U and U contains the germ of every unbounded  log-analytic curves up to a certain complexity (where the complexity is the  complexity of terms in the language Lan(−1, ( n√.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=')n=2,3,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=', log), compare with  [3, Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' An open question is whether we can find such an U that  contains the germ of every unbounded log-analytic curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='  References  (1) L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' van den Dries: Tame Topology and O-minimal Structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' London Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='  Lecture Notes Series 248, Cambridge University Press, 1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='  (2) L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' van den Dries and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' Miller: Geometric categories and o-minimal structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' Duke  Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' 84 (1996), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' 2, 497-540.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='  (3) T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' Kaiser and Andre Opris: Differentiability Properties of Log-Analytic Functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='  Rocky Mountain Journal of Mathematics 52 (2022) no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' 4, 1423-1443.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='  (4) K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' Kurdyka, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' Raby: Densit´e des ensembles sous-analytiques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' Fourier  39 (1989), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' 3, 753-771.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='  (5) J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' Lion, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' Rolin: Th´eor`eme de pr´eparation pour les fonctions logarithmico-  exponentielles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' Fourier 47 (1997), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' 3, 859-884.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='  (6) J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' Lion, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' Speissegger: A geometric proof of the definability of Hausdorff limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='  Sel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=', New Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' 10 (2004), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content=' 3, 377-390.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='  Tobias Kaiser, University of Passau, Faculty of Computer Science and Mathematics  tobias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='kaiser@uni-passau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}
+page_content='de, D-94030 Germany  11' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfQACa/content/2301.02187v1.pdf'}

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